University of Birmingham School of Physics and Astronomy Asteroseismology & Finding Planets Group Studies Investigating the Relationship Between the Characteristics of an Extrasolar System and the Properties of its Host Star Final Report Jeremy Galiszewski (Project Leader) (JG), Alexander Willett (Editor-in-Chief) (AW), Mark Gilbert (Sub-Group Leader) (MG), Peter Steele (Sub-Group Leader) (PS), Matthew Harrison (Sub-Group Editor) (MH), Louis Pollock (Sub-Group Editor) (LP), Fergus Cowie (FC), Asher Ezekiel (AE), Oliver Hall (OH), Morgan Lamb (ML), Chester Lewis (CL), Gareth Miller (GM), Edward Murton (EM), Laura Scott (LS) March 26, 2015 Abstract (JG) We have carried out an investigation into how the characteristics of an extrasolar system correspond with those of its host star. This has been done using luminosity-time and frequency-power data of 39 stars observed by NASA’s Kepler mission. From this data, along with additional values of temperature and metallicity, we have calculated the mass, radius, age and core hydrogen fraction of each of the stars in question using methods of direct power spectrum analysis and asteroseismic diagrams. In the data set provided to us, the calculated stellar mass range was 0.91 ± 0.17 M� to 2.32 ± 1.05 M�, the stellar radii ranged from 0.83 ± 0.10 R� to 4.53± 0.55 R� and the age range was 0.52± 0.13 Gyrs to 8.06± 2.03 Gyrs. In addition to the results from asteroseismology, 30 exoplanets have been detected in the luminosity-time graphs, from which the orbital parameters for each planet could be calculated. This was achieved via the creation of a transit detection code with accuracy determined by a separate phase folding code. A transit fitting code was used to fit a model to the data and output more accurate values, taking into account stellar limb darkening. Comparisons of this model to an independent numerical model revealed that limb-darkening coefficients produced by the fit were inaccurate, decreasing the accuracy on the planetary radii as a result. With this in mind, the calculated range of planetary radii was 1.407 ± 0.180R⊕ to 1.364 ± 0.162RJup and the calculated range for orbital periods was 2.204735299± 0.000000139days to 179.4304524± 0.00213173days. The computational methods used in obtaining these period values justify the precision in the errors. For stars with more accurate asteroseismic results, additional planetary characteristics such as mass, orbital eccentricity, composition and equilibrium temperature were estimated or constrained. Contents 1 Introduction 1 1.1 Introduction to the Report (AW & MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Aims of the Report (JG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Introduction to Asteroseismology (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2.1 Discovery and Development (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 MOST, CoRoT, and Kepler (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.3 The Current State and Future of Asteroseismology (MH) . . . . . . . . . . . . . . . . . . 2 1.3 Introduction to Planet-Detection (LP & AW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3.1 Transit Method (LP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.2 Astrometric Wobble and Doppler Wobble Method (LP) . . . . . . . . . . . . . . . . . . . 3 1.3.3 Transit Timing Variation (TTV) Method (AE) . . . . . . . . . . . . . . . . . . . . . . . . 3 2 The Kepler and PLATO Missions 4 2.1 The Kepler Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Mission Design (PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Spacecraft Design (PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The PLATO Mission (EM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Asteroseismology Theory 6 3.1 Asteroseismic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.1 The Description of Oscillations (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.2 The Origin of Oscillations (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Analysis of Asteroseismic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.1 Frequency-Power Spectra (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.2 Data Visualisation (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.3 Spectrum Noise (MG & LS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2.4 Scaling Relations (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2.5 Asteroseismic Diagrams (MH, EM & MG) . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Results from Asteroseismic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3.1 Stellar Properties (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3.2 Stellar Evolutionary Theory (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 Planet-Finding Theory 16 4.1 Planet Formation (AE & EM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Orbital Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2.1 Kepler’s Laws (AE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2.2 Two- and Three-Body Problem (LP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Planet-Finding Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3.1 Transit Method (LP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3.2 Transit Timing Variations (TTV) (AE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3.3 Other Planet Detection Methods (CL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.4 Stellar Limb Darkening (CL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.5 Mass Constraining Methods (PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.6 Auxiliary Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.6.1 Stellar Variability (ML) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.6.2 The Effect of Starspots on Transits (CL) . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.6.3 Hill Spheres (EM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.6.4 Roche Limit (EM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.6.5 Three-Day Pileup (EM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 i 4.6.6 Exomoon Detection (AE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.6.7 Circumstellar Habitable Zone (EM & MG) . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5 Asteroseismology Method 38 5.1 Initial Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.1.1 Plotting Raw Data (JG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.1.2 Filtering and Removing Noise (GM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.1.3 Isolating Regions of Interest (GM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.2 Manually Measuring Stellar Oscillation Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2.1 Measuring Large and Small Frequency Spacing (JG & ML) . . . . . . . . . . . . . . . . . 39 5.3 Computationally Measuring Stellar Oscillation Properties . . . . . . . . . . . . . . . . . . . . . . 39 5.3.1 Measuring Large and Small Frequency Spacing (LS) . . . . . . . . . . . . . . . . . . . . . 39 5.3.2 Measuring Frequency of Maximum Power (LS & GM) . . . . . . . . . . . . . . . . . . . . 40 5.4 Obtaining Asteroseismic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.4.1 Using Scaling Relations (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.4.2 Using Asteroseismic Diagrams (GM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6 Planet-Finding Method 44 6.1 Manual Methods (LP, CL & AE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.2.1 Introduction (FC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.2.2 Transit Detection Code (FC & AW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.2.3 Phase Folding Code (FC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.2.4 Transit Fitting Code (OH & FC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.2.5 Modelling for Stellar Limb Darkening (CL & OH) . . . . . . . . . . . . . . . . . . . . . . 49 6.3 Processing of Results (PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7 Results and Discussion 55 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.1.1 Key to Results Tables (MH & PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.1.2 Nomenclature (PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.2 Full Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.2.1 Stellar, Orbital, and Planetary Results (PS, AW, JG & ML) . . . . . . . . . . . . . . . . 55 7.2.2 Omissions (JG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.2.3 Results from Asteroseismic Diagrams (GM) . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.2.4 Comparison Between Scaling Relation and Asteroseismic Diagram Values (GM) . . . . . . 80 7.3 General Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.3.1 Relationship Between Stellar Mass and System Age (GM) . . . . . . . . . . . . . . . . . . 82 7.3.2 Relationship Between Stellar Mass and Luminosity (GM) . . . . . . . . . . . . . . . . . . 82 7.3.3 Relationship Between Stellar Mass and Radius (MH) . . . . . . . . . . . . . . . . . . . . . 83 7.3.4 Relationship Between Stellar Metallicity and Planet Composition (MH) . . . . . . . . . . 83 7.3.5 Relationship Between Stellar Radius and Planet Radius (JG) . . . . . . . . . . . . . . . . 83 7.3.6 Frequency of Orbital Period for Discovered Exoplanets (PS) . . . . . . . . . . . . . . . . . 84 7.3.7 The Mass-Radius Relation for Discovered Exoplanets (PS) . . . . . . . . . . . . . . . . . 84 7.3.8 Evidence for Tidal Circularisation (PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.4 Comparison of Stellar Limb-Darkening Coefficients (CL & OH) . . . . . . . . . . . . . . . . . . . 85 8 Conclusions 90 8.1 General Conclusions (JG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8.2 Stellar Properties (MG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8.3 Exoplanet Detection Methods and Properties (LP, FC & PS) . . . . . . . . . . . . . . . . . . . . 91 8.4 Relationships Between Characteristics of the Extrasolar Systems and the Properties of their Host Star (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 ii Chapter 1 Introduction 1.1 Introduction to the Report (AW & MH) Asteroseismology - the study of oscillations in stars - and the hunt for extrasolar planets are both rela- tively recent fields, having their beginnings as serious research areas within the last half century. They are highly interconnected, with similar technical require- ments and using similar data, which has resulted in combined space-borne missions such as CoRoT and Ke- pler. A key area of interest with respect to exoplanets is whether or not they would be able to support life as we know it. This means special attention is given to finding Earth-like planets orbiting Sun-like stars, in an orbital region known as the ‘Habitable Zone’, where water can exist in liquid state. However, the study of exoplanets goes beyond the simple detection of their existence, as it involves the analysis of the physical and dynamical processes of the systems and can give an insight into their formation. This requires detailed knowledge of the physical characteristics of both the planets and their host star. It is here that both aster- oseismology and the different planet-detection meth- ods come into play. Asteroseismology provides de- tails about the host star, while various planet-detection methods can provide information about the planets rel- ative to their host star. Together, these two fields of study can build a detailed picture of the properties of the star-planet systems. With enough systems doc- umented, apparent trends and relationships between stellar properties and planetary properties can be iden- tified, which can then increase our understanding of all planetary systems, including our own, and their forma- tion. In this project, we research and investigate a selec- tion of the methods available to astrophysicists in these areas, and then go on to apply some of these methods to Kepler data that we were given on 39 stars (exclud- ing GSIC 28 which was found to be a duplicate of GSIC 1, and GSIC 37, a duplicate of GSIC 11) with a range of mass, temperature, metallicity, age, and number of detectable planets, in an attempt to derive as much information about those stars and planets as possible from the data we have. We then characterise the stars and planets, and attempt to discern trends between properties of each of the host stars and their planet or planets. 1.1.1 Aims of the Report (JG) This report first of all aims to provide a sufficient background to provide the reader with a context into which the rest of the work can be placed. This comes in the form of a short history of asteroseismology and planet-finding, an overview of the general methods used in both fields, and a discussion of the work carried out by the Kepler mission and the planned strategy of the PLATO mission. The next aim is to provide a thorough description of all the theory used by the members of the project in compiling this report. This is required by the reader to understand the rest of the report as all sections will refer back to the theory chapters. The theory chapters will contain all equations used later on in the report and explain their origin, culminating in a useful com- pilation of information given in a wide range of astero- seismology and planet-finding papers published by the astrophysics community in recent years. Once the theory has been discussed the report will take the reader step-by-step through the methods used in obtaining useful results from the data given. There were both manual and computational methods of data analysis used in this project and members of each team will explain their own methods and how they compare to those used by their counterparts. The final aim of the report is to outline the results in a thorough and informative manner. An overall de- scription of each star and detectable planet is given, aiming to be consistent with the accepted practices used in announcing newly discovered planets. Results from a variety of methods will also be displayed. This section will then discuss in more detail the initial re- sults by analysing trends between the stellar and plan- etary properties. 1.2 Introduction to Asteroseismology (MH) Asteroseismology (from the Greek aster, seismos, logia meaning star-earthquake-study) as a field has its beginnings in the study of our own star, the Sun, known as helioseismology. This field of research, pioneered in Birmingham, involves measuring the complex oscilla- tions that manifest on the surface of the Sun due to standing waves set up by various internal mechanisms, and then analysing the oscillation spectra to obtain 1 information on the internal structure. Asteroseismol- ogy applies these methods to other observable stars, with the principal goals of advancing theories of stellar structure and evolution. Clearly, given the proximity of the Sun, it can be observed in much greater detail than other stars, but the principles remain the same in many types of stars. We can treat the Sun as any other star for comparison by viewing it in integrated light as we do the others, rather than with the detailed imaging possible. 1.2.1 Discovery and Development (MH) In the 1960s solar oscillation modes were detected for the first time. Leighton, Noyes and Simon (1962) used a Doppler shift in spectral lines technique to ob- serve radial velocity variations, in time and with po- sition on the disk, to identify oscillations with a five minute period [1]. In the early 1970s these oscilla- tions were explained [2] and in the following years and decades the theory was developed and eventually ex- tended to other stars in the 1990s - probably the first certain detection of individual modes in another star was by Kjeldsen et al. in 1995 [3]. Initially performed with ground-based telescopes, asteroseismic observa- tions suffered from the usual problems of ground-based astronomy, such as rotation of the Earth, atmosphere- related scintillation, and inclement weather. The launch of the first dedicated space-based as- teroseismology instrument, the Canadian-run MOST (Microvariability and Oscillations of STars or Micro- variabilité et Oscillations STellaires) marked the be- ginning of much better asteroseismic data. Following on from MOST, the French-led CoRoT (Convection, Rotation, and planetary Transits or Convection, Rota- tion et Transits planétaires) and NASA’s Kepler space- based missions permitted an explosion in the amount of data and the number of stars being investigated astero- seismically. Kepler in particular, with its dual, comple- mentary asteroseismology and exoplanet-hunting mis- sions, made a huge contribution: asteroseismic obser- vations often provide the best way to constrain the global properties of exoplanets by constraining the stel- lar properties relative to which exoplanet properties are measured using transit methods. 1.2.2 MOST, CoRoT, and Kepler (MH) Launched in mid-2003, MOST was capable of ob- serving one star at a time for up to 60 days [4]. Tar- geting solar-like and also metal-poor subdwarf stars, MOST was able to obtain impressive data relative to its size (approximately 50kg and the size of a suitcase), thanks to its attitude control system and set of reac- tion wheels, on stars of apparent magnitude V = 6 and brighter. As of June 2007, MOST had conducted 64 campaigns, collecting data for more than 850 stars [5]. Beginning observations at the end of 2006, the Eu- ropean Space Agency’s CoRoT mission was extended twice and eventually lasted 2137 days [6], almost six years. CoRoT was able to observe thousands of stars simultaneously, for up to 150 days without interrup- tion. CoRoT obtained asteroseismic data on several hundred stars before the failure of several components in late 2012 [7]. NASA’s Kepler telescope launched in 2009 and is capable of observing around 150,000 stars simultane- ously in long cadence (29.4 minutes of exposure and readout time) and just over 500 stars simultaneously in short cadence (58.9 seconds of exposure and readout time) [8] - for explanations of the cadences, see Chapter 2. Kepler is responsible for the detection and confirma- tion of more than a thousand exoplanets, which can be characterised thanks to combining transit data with asteroseismically-derived stellar properties. The Ke- pler Asteroseismic Science Consortium (KASC), with more than 500 members around the world, is dedicated to maintaining the Kepler Asteroseismic Science Oper- ations Centre (KASOC) database which holds all the photometry data collected by the telescope [9]. 1.2.3 The Current State and Future of Asteroseismology (MH) The CoRoT and Kepler missions in particular have generated a wealth of data on hundreds of thousands of stars, much of which remain to be exploited. Aside from the simple global stellar properties that can be de- rived from asteroseismic data, novel methods are con- stantly being devised to interpret the data in order to, for example, detect binary systems that cannot be re- solved with photometry [10], measure stellar differen- tial rotation [11], or investigate the excitation of modes by gravitational waves [12], to name but a few. Not only do we still have a lot to gain from analysing the data produced by those missions, but with the launch of NASA’s TESS (Transiting Exo- planet Survey Satellite) expected in 2017, and ESA’s PLATO planned for 2024, there will be no shortage of new and improved data. This promises both im- proved accuracy for currently known stellar properties - and therefore exoplanet properties - and a large in- crease in the number of stars which we can analyse asteroseismically for exoplanet purposes or to better our understanding of stellar evolution. 1.3 Introduction to Planet Detection (LP & AW) The term extrasolar planet, or exoplanet, refers to a planet orbiting a star other than our own Sun. The existence of these has been theorised for centuries, beginning with Giordano Bruno, an Italian Catholic monk and supporter of heliocentricity, controversially asserting the existence of “countless suns and count- less Earths all rotating around their suns” in 1584 [13]. However, it hasn’t been until the last 23 years that the field has yielded any results, although there had pre- viously been a couple of unconfirmed detections. For example, in 1988 a suspected planet was detected out- side of the Solar System [14], however due to the low precision of measurements, this could not be confirmed 2 until 2003 [15]. A more significant development was made in 1992 when Dr. Aleksander Wolszczan pub- lished his findings of the first confirmed exoplanets, or- biting a pulsar, PSR B1257+12 [16]. The planets were discovered by studying the variations in the pulsations of the pulsar, brought about by the gravitational ef- fects of the mass of each planet. It wasn’t until 1995 that the first exoplanet orbiting a main-sequence (MS) star, 51 Pegasi, was discovered by Michael Mayor and Didier Queloz of the University of Geneva [17]. It was this discovery that has now led to the discovery of over 1800 confirmed exoplanets within the last twenty years, with projects such as CoRoT and Kepler working on this task [18]. The detection of extrasolar planets is very chal- lenging due to the low apparent size and brightness of distant stars. With exoplanets only a fraction of the size of these stars and with a negligible flux, very high precision telescopes and spectroscopes are required to locate these planets. It is generally more straight- forward to detect large, massive planets than small, Earth-like planets, as such planets have greater gravi- tational influences on their host stars. It is for this rea- son that the majority of exoplanets located to date ex- hibit Jupiter-like properties. There are many methods that can be used to detect exoplanets, the most com- mon of which are the transit method and the Doppler wobble method. The Doppler method was generally used to discover the first extrasolar planets, however in recent years the most successful means of discovery for new planets has been via the transit method. The Kepler mission has been instrumental in this, with the vast majority of these transit detections having been achieved using data from the Kepler spacecraft. The Doppler wobble method and transit method are de- scribed below, along with the less practical astrometric method. 1.3.1 Transit Method (LP) The transit method takes into account any peri- odic changes in the flux of a distant star that may indicate the presence of a transiting planet [19]. When an exoplanet moves between its host star and an ob- server on Earth, a small fraction of the flux from the star is blocked by the planet. Dips in flux from a star are common however, and many considerations need to be made before confirming the presence of a planet; these are discussed in Chapter 4. The transit method provides information on the period of the orbit of the planet as well as the ratio of the planetary radius to the stellar radius. Phenomena such as stellar limb darken- ing and varying impact parameters add complications to calculations that determine planetary characteris- tics; these are also discussed later in the report. 1.3.2 Astrometric Wobble and Doppler Wobble Method (LP) The astrometric wobble method considers the rel- ative position of the stars in the night sky, with the positions of stars changing due to the gravitational at- traction of orbiting bodies. When a planet orbits a star, the gravitational pull of the planet can move the star to such a degree that its motion can be detected. The stellar motion is largest and therefore most easily detectable with a massive planet in orbit with a large semi-major axis. The mass of the star is also an im- portant factor in the motion, with less massive stars wobbling to a larger degree. To detect planets using this method, data regarding the positions of stars with time is required. No extrasolar planets have yet been confirmed using this method, as movements of distant stars due to orbiting planets are too small to detect from Earth. The Doppler method, on the other hand, consid- ers the changing wavelength of light detected from a distant star that indicates the presence of a compan- ion planet. A distant star moving towards and away from us will give a sinusoidally varying spectrum of shifted wavelengths, from which the period of orbit of its companion planet can be extrapolated. Over 300 extrasolar planets have been found to date using this method with the vast majority lying within 3 AU of their host star [20]. On top of this, the majority of the located planets possess masses close to that of Jupiter, with the masses increasing as the distance from the star increases. The method only works if the planet and star are moving towards and away from us. The radial velocity can be found using this method (the velocity towards or away from the observer). From this a min- imum mass for the planet can be found - inclination of the system will reduce the radial velocity component, so we would see an effect on the star as though from a less massive planet. If the spectral data of the planet are known, this can be used to find the planet’s actual mass. The Doppler wobble of distant stars has been detected by such facilities as PRIMA [21] - an instru- ment used in the VLT (Very Large Telescope) which detects Jupiter-like planets. 1.3.3 Transit Timing Variation (TTV) Method (AE) Transit Timing Variation (TTV) is a technique that looks for changes in the timings of transits for a planet detected by the transit method to infer the presence of another planet. When using the transit method to detect planets, the transits should be strictly periodic, that is the time between each transit should be exactly the same. If the time of each transit is measured accurately and a variation in transit timing can be seen, this suggests the presence of another body, usually a planet, grav- itationally perturbing the orbit of the planet around the star. By measuring the variation in transit timing and observing it as a function of time, it is possible to determine properties of the perturbing body such as mass and orbital radius [22]. Natural satellites orbit- ing exoplanets (or exomoons) can also cause TTVs and this is also discussed. 3 Chapter 2 The Kepler and PLATO Missions 2.1 The Kepler Mission 2.1.1 Mission Design (PS) Launched in March 2009, the Kepler spacecraft was designed to observe one specific region of The Milky Way in optical wavelengths, with the aim of detecting extrasolar planets via the transit method. The data from the initial data collection period has been used to confirm the existence of over 1000 extrasolar planets to date, with the possibility of many more detections to come [23]. The spacecraft orbits the Sun in an ‘Earth-trailing’ orbit of period 372.5 days, appearing to slowly recede from the planet. In order to prevent light from the Sun saturating the telescope, the spacecraft executes a 90◦ roll four times each year in order to orient the Sunshade correctly. The Kepler spacecraft was planned to be stabilised by three reaction wheels (with a fourth wheel in re- serve). These wheels rotated the spacecraft in three dimensions to both ensure that the telescope aper- ture continued observing the intended area, and also to maximise the area of the solar panels in direct sunlight. Unfortunately, by May 2013 two of these wheels had failed, jeopardising future data collection [24]. In re- sponse to this, a second mission plan, K2, was devised. The spacecraft will be oriented along the ecliptic plane of the solar system, with stellar radiation pressure then used to balance the spacecraft in lieu of a third reac- tion wheel, so that several regions on the plane of the ecliptic can be studied [25]. 2.1.2 Spacecraft Design (PS) The Kepler spacecraft comprises a large photome- ter, along with non-measurement-oriented systems to transmit collected data to Earth, control spacecraft at- titude and regulate spacecraft power and temperature. The photometer is the instrument from which all the data used in this report ultimately stems, and thus it is appropriate to discuss its design in greater detail here. Figure 2.1.1 – Basic structure of the Kepler spacecraft, showing several important systems [26]. Photometer The telescope for the Kepler spacecraft is a Schmidt camera of aperture diameter 0.95m, which provides a 101deg2 field of view. To prevent saturation of pix- els by contaminant light from the Sun, a large shade protects the aperture of the photometer. The photometer itself comprises a total of 42 50x25mm CCDs, each with 2200x1024 pixels. The CCDs are mounted on a slightly curved base, as the primary mirror produces a curved focal plane. As the telescope’s field of vision is fixed, each star in Kepler ’s view corresponds to a certain pixel or group of pixels, the data from which are read out at a constant interval 4 of 6 seconds in order to prevent pixel saturation. Only pixels corresponding to stars of interest in the study are used. The data from these pixels is then integrated into a cadence, which comprises 270 readings, mean- ing that each cadence records a total amount of light absorbed from each star over a period of 0.49 hours, known as a ‘long cadence’. For stars of particular in- terest, ‘short cadence’ data can also be recorded, with a length of 1 minute [27]. In addition to the 42 ‘science’ CCDs, 4 further CCDs in the corners of the array are used to monitor 40 ‘guide’ stars, in order to ensure that the instrument is stable and the field of view remains the same. Data from these CCDs has shown the spacecraft to be drift- ing only a few millipixels from its intended position over each 3 month period of observation [28]. Figure 2.1.2 – Cross-section of the Kepler photometer, showing focusing of input light via the primary mirror onto the CCD array [26]. 2.2 The PLATO Mission (EM) PLAnetary Transits and Oscillations of stars is a mission planned by ESA that will be the spiritual successor to the NASA Kepler mission. PLATO is planned for launch in 2024 when it will be launched up to the L2 Lagrange point [29], an orbit that con- stantly shields the spacecraft from much of the sun- light. The mission plans for 6 years of observations from the shadow of the Earth where the 34 indepen- dent telescopes and cameras will cover ∼ 2250 square degrees of the sky per pointing. Two of the cam- eras are short 2.5 second cadence cameras looking at the brighter stars with the remaining thirty-two long 25 second cadence cameras recording data from stars that have visual magnitudes greater than eight. [30] The mission will be providing vital data to continue progress made by the now limited capabilities of the Kepler mission and is planned to last for 6 years and observe over 1 million stars [29] Photometer Each camera on the PLATO spacecraft is made up of an array of four 4510x4510 pixels each of side 18μm. Different regions within each pointing are monitored by different numbers of cameras. The centre is ob- served by 32 cameras, 24 are used in the observation of another region with 16 and 8 for a further two. Figure 2.2.1 – Field of view of the PLATO mission divided into sections corresponding to number of telescopes moni- toring each region [31] For every two cameras there is one Data Process- ing Unit which completes the necessary calculations and modifications on the data before it is sent to the Instrument Control Unit, which sends the data to the ground in order to be analysed. 5 Chapter 3 Asteroseismology Theory 3.1 Asteroseismic Oscillations 3.1.1 The Description of Oscillations (MH) Before discussing the mechanics of the oscillations, it is useful to introduce the notation used to describe them. The oscillations within a star cause the surface to expand and contract periodically, thus a star that has only radial oscillations would simply expand and contract uniformly and symmetrically. However, the introduction of non-radial oscillations causes spherical symmetry to be broken. Parts of the surface expand while other areas contract, with node lines between these areas where there is neither expansion nor con- traction. This results in a huge variety of different sur- face patterns of expansion and contraction dependent on the combinations of radial and non-radial oscillation modes. Figure 3.1.1 – Depiction of the oscillations for several values of l and m. The dark grey regions are expanding, the light grey regions contracting, and the white lines represent node lines. The equator and pole are marked [32]. The radial components are described by the radial overtone number n (sometimes written in the literature with k), the number of nodes along the radius. The spherical components (corresponding to θ in spheri- cal polar coordinates) are described by the angular or spherical degree l , the number of node lines parallel to the equator between the two poles, and finally the azimuthal components are described by the azimuthal order, m (corresponding to φ), the number of node lines that pass through the pole. The angular degree l and azimuthal order m also define the spherical harmonic functions that describe the small-amplitude displacements [19]: Yml (θ,φ) = (–1) mcl,mP m l cos(θ)exp(imφ) (3.1) where θ and φ are the latitude and longitude angles respectively, cl ,m are normalisation constants and P m l are Legendre polynomials. Figure 3.1.1 shows a visuali- sation of the manifestation of these spherical harmonics for some low-degree modes. An oscillation with angular degree l = 0 is a radial one which passes along a diameter through the centre of the star. These oscillations therefore have the longest surface wavelength and the lowest frequency for any particular radial overtone n. 3.1.2 The Origin of Oscillations (MH) Oscillations in a star can be generated by several mechanisms, among them the kappa-mechanism which is the mechanism for strongly varying classical pul- sators like Cepheids. This mechanism results from lay- ers of high opacity (due to ionised helium) acting like a valve and blocking radiation from escaping. This causes a build up of pressure which eventually results in a rapid expansion of the star as the energy is re- leased. The star can then contract and heat up again, and the process repeats. These types of pulsation can display visual amplitudes of up to two magnitudes, and periods of days to weeks [33]. The important class of oscillations for this project is solar-like oscillators, so named because this is how our Sun oscillates. Solar-like oscillations are typically on the order of several parts per million in amplitude, with frequencies of minutes to hours. There exist two main types of oscillations: p- mode acoustic oscillations for which pressure gradients provide the restoring force, and g-mode buoyancy oscil- lations which are damped by gravity. The frequency re- gions of the two types can be defined with reference to two characteristic frequencies: the Brunt-Väisälä and the Lamb frequency, NBV and Ll respectively [32]: N2BV = g ( 1 Γ1 dln(P) dr – dln(ρ) dr ) (3.2) L2l = l(l + 1)c2 r2 (3.3) 6 where g is the gravitational field strength, Γ1 is the first adiabatic exponent, P is the pressure, ρ is the density, l is the angular degree, c is the local speed of sound, and r is the radius. Oscillations with frequency higher than both the Lamb and the Brunt-Väisälä fre- quency are p-modes; oscillations with frequency less than both the characteristic frequencies are g-modes. As a star evolves, the g-mode frequencies increase and the p-mode frequencies decrease. This can result in interactions between the two modes, producing oscilla- tions that combine both g- and p-behaviour, known as mixed-modes. P-mode Oscillations In stars which have near-surface convection zones (CZs), i.e. generally those stars with mass less than 1.1M� and temperature less than 7500K [19], turbu- lence in the outermost part of the convective regions randomly excites acoustic oscillations. These sound waves can travel through the rest of the star, and pen- etrate to different depths depending on the angle at which they are propagated from the CZ. Due to the increase of temperature going inwards towards the cen- tre of the star, wavefronts are refracted away from the centre back towards the surface, and because, above the surface, the solar atmosphere is much less dense, waves cannot be sustained and are reflected back in, thus oscillations are trapped within the star and travel around beneath the surface. There is therefore a fun- damental maximum frequency of the oscillations, the acoustic cut-off frequency, νac . The frequency is de- fined by the speed of sound in that medium and the scale height, H [3]: νac ∝ c 2H ∝ gT–1/2eff (3.4) If the frequency of a wave is higher than this, i.e. its period is short enough, it can be sustained in the lower density and will escape from the stellar interior into the atmosphere. Therefore only oscillations of frequencies lower than νac reflect back in and become trapped and generate the standing waves which cause the surface expansion and contraction. Figure 3.1.2 shows how a wave that propagates deeper into the star before turn- ing back up towards the surface will reach the surface fewer times on its trip around the star, so will exhibit a longer wavelength on the surface and will have a lower frequency (for the same radial order), denoted by a lower angular degree, l . P-mode oscillations are observed to follow the asymptotic relation, established by Tassoul (1980) [34], which is valid for high values of the radial overtone [35]: νn,l ' ( n + l 2 + ε ) Δν0 – l(l + 1)D0 (3.5) where ε is related to physics of the surface layers and D0 depends on the sound speed gradient near the core of the star. The capabilities of the telescopes used to detect the oscillations in other stars by their changes in bright- ness limit the number of degrees we can observe. Since the amplitude of the oscillations decreases with increas- ing l , the highest degree oscillation measured in other stars is l = 3 , for stars with particularly good obser- vational data and high signal to noise ratios (SNR). This is less problematic than it may seem, as these low-degree modes are those that penetrate most deeply into the star and hence are most useful for probing stel- lar structure, as well as being used to determine global stellar properties. The radial overtones of the frequen- cies measured are typically in the range n = 10 – 30 . Figure 3.1.2 – Oscillations of varying angular degree, l . The lower the angular degree, the deeper into the star the oscil- lation penetrates, down to l = 0 , the radial mode passing through the centre [36]. G-mode Oscillations In the central regions of solar-like stars, which are stable against convection, the displacements of pock- ets of gas upwards and their subsequent sinking due to gravity result in oscillatory motions around their po- sition of equilibrium. This motion in turn regularly displaces adjacent material perpendicular to the ra- dial direction, and hence g-mode oscillations are al- ways non-radial. They are also restricted to the ra- diative central regions of the star; they cannot cross the convective boundary as in the convectively unsta- ble zones the oscillatory motion of pockets of gas does not occur, and they therefore manifest only with very small amplitudes at the surface of the star, making them usually very difficult to detect. This is unfor- tunate, as g-mode oscillations, being very sensitive to conditions in the core of the star, are very useful for investigating stellar evolution. G-modes are much eas- ier to detect when their frequencies increase and they undergo avoided crossing interactions with p-modes, i.e. in evolved stars. In these cases, mixed-mode os- cillations allow g-mode properties to be brought to the surface via their interactions with p-modes. Thus g- modes are more often detected in stars evolved off the main sequence (MS). 7 F-mode Oscillations F-mode, or fundamental mode oscillations, are the oscillations with radial order n = 0 . These oscillations behave similarly to p-mode oscillations, but are tech- nically distinct. At high angular degrees l > 20 , where they penetrate only very slightly into the star from the surface, they can be approximated as surface gravity waves - like ripples on the surface of a large body of water - with frequency given by [36]: ω = √ g0kh (3.6) where g0 is the equilibrium value of the surface gravity and kh is the horizontal wavenumber given by [19]: kh = √ l(l + 1) R (3.7) where l is the angular degree and R is the stellar radius. Mixed-mode Oscillations After some evolution, typically into the subgiant and red-giant phases, when the core has contracted and the envelope expanded, g-mode frequencies are in- creased and p-mode frequencies are decreased due to the decrease in mean density, eventually to the extent that the frequency regimes of the two types of oscilla- tion overlap. When a g-mode and p-mode of similar frequency and the same angular degree (except l = 0 for which there are no g-modes) approach, they can interact and undergo an ‘avoided crossing’, resulting in a change in the frequencies called mode-bumping which causes the oscillations to no longer follow the asymptotic relation. This phenomenon can make the frequency-power spectrum harder to interpret as the mode peaks become more irregular, but the interac- tion can be witnessed in an Échelle diagram as the lower end of the l = 1 ridge fragments due to the ef- fect on the large spacing (see Section 3.2.2). Avoided crossings of l = 2 modes are much weaker. 3.2 Analysis of Asteroseismic Data 3.2.1 Frequency-Power Spectra (MH) The observational data obtained from telescopes such as CoRoT and Kepler are used to generate plots of the frequencies detected. The Doppler shifts or inten- sity changes in emitted light are measured for the inte- grated light from the disk, and the variation with time is analysed by a Fourier Transform [32]. The result- ing frequency-power spectrum (FPS), shown for one of the stars from our dataset (GSIC 0) in Figure 3.2.1, when smoothed and filtered to account for noise (see Section 3.2.3), shows the relative power of the oscilla- tions as a function of frequency and from these plots can be extracted several useful quantities. The region of interest, where the amplitudes of the oscillations on the surface are large enough to be measured above the noise (either intrinsic to the star or instrumental), lies within a region bounded by two frequency limits. The maximum frequency is the acoustic cut-off frequency mentioned above. The lower limit is the frequency of the fundamental oscillation, where n = l = 0 . Figure 3.2.1 – Part of the raw frequency-power spectrum for GSIC 0 from our set of stellar data. Peaks of l = 0 , 1 , 2 can be discerned (and are labelled) even in the raw spectrum. First, the visible frequencies can be fitted with a Gaussian envelope in order to find the frequency for which the oscillation power is greatest (not just the mode with the strongest amplitude), νmax . This fre- quency is observed to scale with the acoustic cut-off frequency [3]: νmax ∝ νac ∝ gT –1/2 eff (3.8) such that for solar-like stars the ratio νmax/νac is the same. Next, the ‘frequency separations (or spacings)’ can be measured. The large spacing, Δν, is the difference between the frequencies of two oscillation modes of the same angular degree and consecutive radial overtone: Δνn,l = νn,l – νn–1,l (3.9) The large spacing corresponds to the inverse of the time taken for sound to travel the diameter of the star. The small separation, δν, is given by the difference in frequency of oscillation modes of consecutive radial overtone and odd or even consecutive angular degree: δνn,l = νn,l – νn–1,l+2 (3.10) The large spacing can be combined with the fre- quency for maximum power and the effective tempera- ture of the star (measured independently by photome- try) to estimate, to a reasonable degree of accuracy, global properties like mass and radius, from which mean density and surface gravity can be obtained (see Section 3.2.4). The large and small spacings can be used together to determine stellar properties such as mass, hydrogen concentration, and age; the methods are described in Section 3.2.5. As implied above, one can also use the large spacing to calculate the acoustic radius of the star, that is the time taken for sound waves to travel the radius of the 8 star, given by [37]: T0 = ∫ R 0 1 cs dr ≈ 1 2〈Δνn,l〉 (3.11) where T0 is the acoustic radius and cs is the speed of sound. This quantity can be useful as it can be used to estimate the depth of the base of the sub-surface convection zone. At the base of the convective zone there is a sharp change in structure. If the region of the change, known as a glitch, lies within the cavity probed by an oscillation mode, a periodic variation is introduced into the frequency, a component of which is the acoustic depth of the glitch from the surface. The sinusoidal variation can be fitted to find the acoustic depth - which is proportional to the period of the vari- ation, which can be compared to the acoustic radius of the star in order to find an absolute value for the radius of the base of the convection zone [37], if this is the feature causing the glitch. In Solar-type stars, the base of the convection zone lies at an acoustic depth of approximately T0/2 or deeper [37]. Also from the FPS, by examining the oscillation peaks in fine detail, it is possible to find evidence of a star’s rotation, from which can be determined the angle of inclination of the rotation axis [38]. This is useful information, providing a likely value of the incli- nation of the plane of any planets in the system - spin- orbit misalignment is possible but uncommon. When a star has significant, but not too fast, rotation, and has an inclination, i , which means its rotation axis is not parallel to the line of sight (which would be i = 0 ◦), oscillation frequencies are shifted: higher when the os- cillation moves with the rotation and lower when it moves against. Thus peaks of degree l are split into either triplet or doublet peaks of degree and order l , |m| ≤ l . The relative power of these peaks can be used to calculate the angle of inclination [38]. The relative power of the central peak to the split peaks, εl ,m , is re- lated to the inclination, as shown in Figure 3.2.2, and depends on the angular degree [38]: ε1,0(i) = cos 2(i) (3.12) ε1,±1(i) = 1 2 sin2(i) (3.13) ε2,0(i) = 1 4 (3cos2(i) – 1)2 (3.14) ε2,±1(i) = 3 8 sin2(2i) (3.15) ε2,±2(i) = 3 8 sin4(i) (3.16) where ∑ m εl,m(i) = 1 (3.17) There are some limitations on the usability of this technique. This method works best for angles of in- clination less than 30◦, approximately 15% of stars. At least a useful estimate value can be obtained from spectra with an SNR down to 20. If the rotation axis is parallel to the line of sight, no splitting is seen, and it is not possible to distinguish the cases of no incli- nation and no rotation without other information. For the splitting to be resolvable, the angular velocity, Ω, must be greater than twice the line width of the peak. Thus, as the line-width increases with frequency [38], lower frequency modes offer the best chance for this analysis. Ω can be obtained approximately from [38]: νn,l,m = νn,l,0 + mΩ (3.18) Figure 3.2.2 – Rotational splitting of frequency peaks for two different inclinations. The greater inclination has split peaks with a higher relative power [38]. 3.2.2 Data Visualisation (MH) Figure 3.2.3 – The frequency-power spectrum for GSIC 40. This is an evolved star, and with the p-mode frequencies spikes labelled, g-modes can be seen between them. 9 Figure 3.2.4 – Some examples of Échelle diagrams. The upper panel (MS star 16 Cyg A) shows very clean ridges of each angular degree, even faintly l = 3 ; the lower panel (star KIC6442183, a subgiant) shows evidence of an avoided crossing [37]. The Échelle diagram provides a useful method of looking at a frequency-power spectrum’s information. The diagram is created by dividing the FPS into slices of width Δν and essentially superposing them on top of each other. On the horizontal axis is frequency mod- ulo the large spacing, ν%Δν, and on the vertical axis is frequency. The frequencies of the oscillation modes in the slices is plotted, and thus vertical lines are visible, each corresponding to a value of l , because the large spacing is approximately the same for each l . In reality the ridges are almost always curved, concave towards increasing ν%Δν. This is an artifact of the inaccu- racy of the simplified asymptotic relation which doesn’t take into account the change of the large separation with frequency: the large frequency is usually mea- sured around νmax , so for higher frequencies the sepa- ration is greater and for lower frequencies it is smaller. An Échelle diagram can be used to identify avoided crossings and mixed-mode oscillations, as shown in the lower panel in Figure 3.2.4: the fragmented lower end of the l = 1 ridge is evidence of the bumped frequencies and subsequently affected large spacing. This could be employed in helping to identify modes in irregular spec- tra. For example, without the labels on Figure 3.2.3 which comes from an evolved star, the spectrum would look very irregular due to the presence of the g-modes. An Échelle diagram could be used to find the verti- Figure 3.2.5 – Granulation as it appears on the surface of the Sun. Hot, rising plasma will appear brighter than the cooler, sinking intergranular lanes between cells [39] cal ridges corresponding to p-modes and thus identify those peaks. 3.2.3 Spectrum Noise (MG & LS) Granulation (MG) Stellar granulation is a highly significant contribu- tor of noise in the FPS. Granules are cells of convection that appear on surface of stars, such as the Sun, with effective temperatures lower than about 7500K that have an outer convective zone. The typical diameter of these cells is 1000-1500km. They arise from convec- tion currents which transport hot plasma from below the surface upward, with the intergranular lanes in be- tween cells providing the return flow of cooler, hence less luminous, plasma. Importantly, the lifetimes of the granules are similar to the periods of the p-mode oscillations so attempting to model and reduce contri- bution of granulation to the noise in the FPS would greatly benefit analysis. The effect of granulation in the FPS can be seen predominately at the lower frequencies (≤1 mHz) where the noise can be seen to increase as the fre- quency decreases. Mesogranulation and supergranula- tion – groups of convective cells of increasing diameter and lifetime – along with granulation create the layer of noise as described above. On the stellar surface the vertical granule velocity (ranging up to the order of kms–1) far exceeds that of the oscillations we wish to study and, in the FPS of evolved stars, the frequency of p-modes can have been reduced low enough to over- lap with the granulation noise, making it difficult to analyse these oscillations (especially at low signal to noise ratios). One distinguishing property of granula- tion that can be exploited is its coherence as discussed 10 by Elsworth and Thompson (2004) [40]. Oscillations can rise above the granulation background provided that both the variations are observed for long enough and the mode lifetimes are long enough. Modelling the signature of this phenomenon should be possible and, if achieved, would greatly improve the ability to analyse stars where the granulation noise in the FPS significantly affects detection of the oscillations. One obstacle faced when trying to build such a model is the unpredictable nature of granulation from one star to another. A successful attempt to model solar granula- tion was made by Samadi et al. (2013) [41] with the aim of extending their methods to stars being observed by CoRoT and Kepler. They produced a simple, 1D theoretical model which reproduced the observed solar granulation spectrum, as well as a more complex 3D ra- diative hydrodynamical model which saw granulation in disk-integrated intensity (as is seen in our FPS) rep- resentative of the surface layers of F-type dwarfs and red giant stars. These models, however, are non-trivial and beyond the scope of this investigation, as much of the focus is on analysis of the FPS and creation of asteroseismic diagrams detailed later in Section 3.2.5. Reducing the Noise (LS) If a star’s modes of oscillation are to be found mea- sures must be taken to account for the noise in the FPS. One way of doing this would be to smooth the raw data using a boxcar filter, as was done in this project. This can be done by convolving the data with an ar- ray whose size is known as the window size and whose elements are all one, or equivalently by calculating a moving average. The effect of the boxcar filter is to smooth the data, with a larger window size producing more severe smoothing. The boxcar is most effective at dealing with white noise as the filter has a constant amplitude. However, the total noise in the FPS is not just white. The low-frequency areas of the FPS have more powerful noise than the high-frequency areas due to granulation. Applying the boxcar filter to an FPS will not remove the increase in power seen in the low- frequency areas. Although the stars studied in this project did not require anything more than the boxcar filter (see Section 5.1.2), it may not be adequate for other stars. In more evolved stars for example, where the modes appear at lower frequencies, the signal may be well within the region of granulation noise. Modelling the noise and then subtracting the model away from the data is one way to deal with the frequency-dependance of the noise. In order to do this the noise due to each component of stellar activity must be summed. The power of an individual component of the stellar activity noise can be written as [42]: P(ν) = η(ν)2 4ζσ2τ 1 + (2πντ)c (3.19) where ν is the frequency, η is a damping factor arising from the discrete nature of the data, ζ is a normalisa- tion constant that depends on the stellar activity pro- cess, σ is the granulation RMS velocity, τ is the char- acteristic timescale of the noise-causing process and c is an exponent equal to 4. The form of equation 3.19 is based on the model proposed by Harvey (1985) [43] that estimates the ef- fects of granulation as being a pulse in power that sud- denly appears and then exponentially decays. In Har- vey’s model the value of c is 2 (making the noise power a Lorentzian function). Corrections made by Kallinger et al. (2014) [42] when better data became available modified the value of c to be 4. This is consistent with a power pulse which symmetrically and exponen- tially rises and decays with a characteristic timescale, τ. The rise and decay of the power suggested by this model is representative of the fact that the granules on a star’s surface appear and disappear with those same timescales. Granulation, mesogranulation and super- granulation all have different values of τ. 3.2.4 Scaling Relations (MH) Once the FPS has been filtered, smoothed, and the useful quantities extracted, we can use those quanti- ties to obtain estimates of the global properties of the star. For solar-like stars, i.e. those that have near- surface convection zones, if we assume that for the main-sequence (MS) and red-giant (RG) stages the os- cillation properties scale well against a solar reference, we can derive relations to estimate properties of these stars just from the FPS and an independently mea- sured temperature. Kjeldsen and Bedding (1995) de- rive relations for the large spacing and the frequency of maximum power [3]. The large spacing, Δν, is approx- imately equal to the inverse of the time for sound to cross the star, which is related to the temperature and the density and hence the mass and radius. The fre- quency for maximum oscillation power, νmax , can also be scaled via the cut-off frequency, νac , which defines a dynamical timescale and hence is related to the surface gravity and temperature. From these relations, scaling laws can be derived for these two quantities [3]: νmax νmax,� = ( M M� )( R R� )–2( Teff Teff,� )–1/2 (3.20) Δν0 Δν0,� = ( M M� )1/2( R R� )–3/2 (3.21) These relations can be easily rearranged to give equivalent reverse scaling relations to find estimates of the mass and radius, and hence density and surface gravity of any star for which the large spacing, fre- quency for maximum oscillations power, and effective temperature are known [37]: R R� ' ( νmax νmax,� )( 〈Δνn,l〉 〈Δνn,l〉� )–2( Teff Teff,� )1/2 (3.22) M M� ' ( νmax νmax,� )3( 〈Δνn,l〉 〈Δνn,l〉� )–4( Teff Teff,� )3/2 (3.23) 11 These scaling relations will give errors on the order of one to ten percent [37], of which the error on mass will be larger. If the goal is to constrain the mass and radius of extrasolar planets around a star analysed in this way, these large errors render these values of limited use. Other methods must therefore be used to retrieve accurate deductions of the value of these properties. 3.2.5 Asteroseismic Diagrams (MH, EM & MG) Attempting to use a Hertzsprung-Russell diagram to match an observed star to an evolutionary track is difficult due to the degeneracy of the HR diagram: cer- tain regions of the diagram have many tracks, of widely varying properties such as mass and age, very close to- gether such that only a small error on the temperature and luminosity of the star can result in a large num- ber of possibilities. Hence, so-called ‘asteroseismic HR diagrams’ (AHRDs) are used instead. These are plots of asteroseismic quantities which result in much more distinct evolutionary tracks and a much lower degree of degeneracy, particularly for solar-like stars on the MS. Stellar modelling is used to generate a large number of model stars with a range of masses, radii, metallicities, and other initial parameters, including relevant stel- lar physics, whose evolution is simulated. From these model stars, simulated oscillations are generated using the asymptotic relation in Equation (3.5), in which D0 is related to the small frequency spacing by: δνn,l = (4l + 6)D0 (3.24) The spectra are analysed and the asteroseismic quantities are calculated and plotted against each other on an AHRD. The small and large spacing are used particularly because they are sensitive to the evolution and mass of stars. The large spacing is proportional to the square root of the mean density of the star, which increases with stellar mass and decreases slightly with evolution. D0 is sensitive to the sound speed gradient. With evolution, the sound speed decreases towards the centre as the proportion of helium and therefore the mean molecular weight, μ, rises, and so D0 decreases with evolution. The small spacing δν therefore also decreases. Thus on this grid of stellar models, evolu- tionary tracks as lines of constant mass, hydrogen con- tent (isopleths), and age (isochrones) can be drawn, and a real star whose FPS has been analysed can be located on this plot to give a very good estimate of its global properties. Figure 3.2.6 shows an asteroseismic diagram with the small separation plotted against the large separation, to give a grid with lines of constant mass and hydrogen content. Figure 3.2.6 – An asteroseismic HR diagram (or JCD or C-D diagram) of the small separation plotted against the large separation for MS stars. The solid lines are lines of constant mass, and the dotted lines are lines of constant hydrogen content [44]. This is the most common form of AHRD, as the hydrogen fraction is a well-defined quantity, whereas age is more subject to other parameters. The age can be approximately obtained, assuming correct models, from the hydrogen fraction [19]: T = 10 – 14.3X M/M� x109years (3.25) where T is the age of the star, Xc is the hydrogen fraction and M is the mass of the star. It can also be useful to plot a ratio of the small and large separations, rl ,l+2 : r0,2(n) = δν0,2(n) Δν1(n) (3.26) or r1,3(n) = δν1,3(n) Δν0(n + 1) (3.27) against the large spacing, as this quantity is not sen- sitive to surface effects or the structure of the outer layers of a star, which can vary strongly between mod- els [45]. It should be noted that Chaplin et al. (2005) found evidence that for the Sun as a star, the frequency ratio changes with the activity level [46], and so varies sinusoidally on a timescale of decades. This timescale is much longer than the timebase of our data, so should be insignificant, assuming our stars have similar activ- ity periods, if they have activity. White et al. (2011) also noted that for the l = 1 modes there is departure from the asymptotic relation for more evolved stars, making Δν0 more reliable, and that when there are no avoided crossings Δν0 ' Δν1, so there is little impact on the diagram from changing the definition of the ra- tio [44]. Hence the most useful form is: r = δν0,2 Δν0 (3.28) Another important building block in AHRDs is metallicity. Metallicity defines the fraction of heavy 12 elements, or metals, in a star. In this context a metal is any element heavier than helium. Stellar hydrogen, helium and metal fractions – X, Y and Z, respectively (where X + Y + Z = 1) – are used to describe the ma- terial content inside stars and play a significant role in how both actual stars, and the stellar models used to build the AHRDs, evolve. Metallicity values are often presented as the logarithm of the ratio between concentrations of iron (Fe) and hydrogen (H) observed near the stellar surface, calibrated with the Sun (Z� = 0.017).[ Fe H ] = log10 ( NFe NH ) ∗ – log10 ( NFe NH ) � (3.29) where NFe and NH are the respective number densities of iron and hydrogen near the surface of the stars. With reference to HR diagrams, a relatively low metallicity, solar-like, Zero Age Main Sequence (ZAMS) star will appear bluer, i.e. of higher temperature, compared to a star of higher metallicity, due to decreased opacity and line blanketing which increases its effective tempera- ture. The luminosity will vary very little with metal- licity, therefore it can be inferred that the radius of a low metallicity star will be lower than that of a high metallicity star of the same mass to maintain similar luminosities (see Equation 4.84). Having a reduced ra- dius will increase the overall density of a low metallicity star so the value of the large spacing would be expected to increase, and vice versa. A small change in the metallicity value of the stars used to model the grid in the AHRDs will have a great impact on position of the grid itself, as can be seen in Figures 3.3.1 and 3.3.2 in Section 3.3.2. It is an apparent fact for any AHRD, such as the those illustrated, that metal-poor stars have their tracks moved up and to the left – with increased small spacing and decreased large spacing – with the oppo- site for metal-rich stars. This effect (at least for the large spacing) is contrary to what was previously ex- pected. It is, however, very important to state how the parameters on which the large spacing is dependent, previously outlined in the text, are only approxima- tions and it is likely that there are many other subtle dependencies which will affect it. When stars evolve off from the MS the evolutionary tracks of stars of dif- ferent masses appear to converge as both the small and large spacings decrease in value over time. It is often the case that the effect of metallicity on the position of the tracks becomes less important as the difference becomes similar to the error on the small spacing (See Figure 3.3.1). 3.3 Results from Asteroseismic Data 3.3.1 Stellar Properties (MH) The accuracy of the scaling relations quoted above is, as previously mentioned, not very good. They can, however, be improved by a second-order asymptotic treatment that takes into account more precise varia- tion in the frequencies. For example, the large spacing increases with frequency, hence the positive curvature of the ridges on Échelle diagrams calculated using a constant Δν. The second order asymptotic relation is [35]: νn,l = ( n + l 2 + ε ) Δν – [l(l + 1)d0 + d1] Δν2 νn,l (3.30) where d0 is related to the gradient of the speed of sound and d1 is a correction related to surface boundary con- ditions. Following this relation, we can obtain an equa- tion to modify the observed large spacing, Δνobs , to get an asymptotic large spacing, Δνas : Δνas = (1 + ζ)Δνobs (3.31) where ζ = 0.57/nmax for MS stars, nmax = νmax/Δνobs , and ζ = 0.038 for RG stars. ζ is a parameter calculated as a measure of the offset between the observed and asymptotically-derived values. The scaling relations can then be rewritten: R R� ' ( νmax νref )( 〈Δνn,l〉 〈Δνn,l〉ref )–2( Teff Teff,� )1/2 (3.32) M M� ' ( νmax νref )3( 〈Δνn,l〉 〈Δνn,l〉ref )–4( Teff Teff,� )3/2 (3.33) where νref and 〈Δνn,l 〉ref are reference values differ- ent to the Solar values, derived from a set of model stars. Mosser et al. (2013) give the values 3104μHz and 138.8μHz respectively [35]. These forms of the scaling relations have uncertainties of 4% and 8% for radius and mass respectively. Mosser et al. suggest that these relations are appropriate for stars of mass less than 1.3M� and temperatures in the range 5000- 6500K. Additionally, estimates of the radius and mass of stars that were calculated using first-order asymp- totic relation-derived scaling relations could be altered using: Ras ' (1 – 2ζ)Robs (3.34) Mas ' (1 – 4ζ)Mobs (3.35) In addition, White et al. (2011) find that the rela- tion between mean stellar density and the large spacing has a temperature dependence, by observing deviation from the scaling relations. They deduce that an im- proved form of the relation should be [44]: ρ̄ ρ̄� = ( Δν Δν� )2 · (f (Teff))–2 (3.36) where f (Teff) = –4.29 ( Teff 104K )2 + 4.84 ( Teff 104K ) – 0.35 (3.37) 13 for stars with temperature between 4700K and 6700K and mass above ∼ 1.2M�. They found that this ad- justment gives an error on the density scaling relation of approximately 1%. Asteroseismic diagrams, on the other hand, should offer a much more accurate value for the age and the mass of a star, provided appropriate models are used. However, because the large spacing is less sensitive to mass for stars above 1.4M� [47], the evolutionary tracks become more degenerate above this value, re- ducing the accuracy of this method for mass. Similarly, as stars evolve off the MS, there is a higher degeneracy in the lines of constant hydrogen content as the small spacing changes less with evolution up the red-giant branch. Asteroseismic diagram grids are particularly strongly dependent on metallicity. For example, Lebre- ton and Montalbán found that halving the metallicity from Z = 0.02 to 0.01 caused a 15 to 30% change in the age value obtained [48]. In addition, they can also be affected significantly by the inclusion or alteration of the overshooting parameter, αov [48], which mea- sures the carrying over of material from the convective zone into the convectively-stable zone beneath it due to inertia while sinking, and by changes to the initial hydrogen content [47]. Convective core overshooting can also result in the intake of more nuclear fuel to the core, affecting the evolution of the star. A change from 0.0 to 0.2 of this parameter can cause the lines of constant hydrogen content to stretch apart by up to 50% their original spacing [48]. Figure 3.3.2 displays these effects. Changing the initial hydrogen and he- lium content has a smaller but still noticeable effect on the mass result. In general this method, assuming the correct input physics and parameters to the models used, should be able to give stellar ages with an uncertainty of less than 10% and mass with an uncertainty of several percent. 3.3.2 Stellar Evolutionary Theory (MH) So far we have a reasonably poor understanding of the physics of the surface layer of stars, with models, as previously noted, often differing strongly on this part of the star. We have a much better understanding of the deeper layers and cores of stars. As discussed in Section 3.2.5 above, the small separation δν is very sensitive to the structure of the stellar core and hence to the evolutionary state of the star. This makes the small separation a valuable diagnostic tool for obtaining the hydrogen fraction and therefore estimating the age, as featured in Figure 3.2.6 and with Equation (3.25). The study of stellar surface layers using asteroseis- mology could be greatly augmented with improved ob- servational equipment and techniques. Modes of higher angular degree penetrate less deeply into the star and hence would be more useful for probing the surface structure. The ability to resolve the surface of stars and measure smaller amplitudes with less noise would permit the measurement of modes of higher angular degree, which with current techniques is not possible due to their small amplitudes and the self-cancellation in low-resolution integrated-light observations. We can also use asteroseismic results to test cur- rent stellar models. We can measure some stars’ global properties independently, for example in eclipsing bi- nary systems whose mass can be measured with pho- tometry and spectroscopy [49], or bright stars whose radius can be measured by ground-based interferome- try or parallax techniques which require the distance and magnitude to be known as well as the tempera- ture [50]. Then, asteroseismic results that differ from the known values can reveal discrepancies in the model used. Equivalently, asteroseismic results that agree well with independently established properties will give confidence to the scaling relations or AHRD models used. 14 Figure 3.3.1 – Small against large spacing AHRD with evolutionary tracks for stars that are metal-poor (Z0 = 0.011, blue) and metal-rich (Z0 = 0.028, red). Isochrones are not shown in this example[44] Figure 3.3.2 – Asteroseismic diagrams showing the effect of changing the metallicity or overshooting parameters in the models used [48]. On the left, metallicity changes from Z = 0.02 (black lines) to 0.01 (light grey lines). On the right, the overshooting parameter is changed from 0.0 (black lines) to 0.2 (grey lines). 15 Chapter 4 Planet-Finding Theory In our study, we were provided with long and short cadence Kepler data showing the flux changes from 40 different stars over time. Such data can only be anal- ysed using the transit method and so 40 long and short cadence light curves were produced using a Python code. The Planetary group was tasked with detect- ing possible transits in these data. After detecting the transits and analysing their shape, relevant data about the various star-planet systems could be produced in concert with the Asteroseismology group. 4.1 Planet Formation (AE & EM) Much of what is known about the formation of plan- ets is based on the planets in our solar system. These planets can be grouped into two categories. The ter- restrial planets, Mercury, Venus, Earth and Mars, are predominantly rocky and orbit close to the sun, all with a semi-major axis of 1.52AU or less [51]. The giant planets, Jupiter, Saturn, Uranus and Neptune, have small rocky cores surrounded by huge amounts of ice and gas, and are more massive than terrestrial planets with larger orbits. Giant planets can be further divided into gas giants such as Jupiter and Saturn which con- tain mostly hydrogen and helium, and icy giants such as Uranus and Neptune which contain heavier, volatile elements such as oxygen, carbon, and sulphur. The formation of stars and planets occurs in dense collections of gas and dust particles called Giant Molec- ular Clouds (GMCs). Approximately 10 Gyrs after a GMC is formed, sections of the cloud collapse due to gravitational instabilities, which results in the forma- tion of a protostar. This attracts further matter into a disk surrounding the protostar. Disc dissipation fol- lows and the material in the disk surrounding the pro- tostar is transported inwardly towards it [52]. This process continues until the protostar reaches an age of 1 Myr, at this point the protostar finishes this accu- mulation and becomes a T Tauri star [53]. As a result of this process, planetary formation can begin. Terrestrial planets form via the Planetesimal Ac- cretion Model which states that rocky planets form from dust grains that collide and combine into plan- etesimals which in turn grow in size to eventually be- come a planet [54]. As these planets form in the inner solar nebula, temperatures are so high that the only dust particles available for accretion are those with a high melting point such as iron and aluminium. As these molecules are relatively rare, there is only a small amount of material available to the planets, therefore they have a small size. The currently favoured model for the formation of giant planets is the Core Accretion Model. As with terrestrial planets, the core of the planet forms from solid particles. However, it is thought that giant plan- ets form further out from the protostar, past a point known as the ice line. The ice line is the region around a protostar where the energy transferred from light and pressure in the nebula enable volatile molecules to freeze. Since boiling point and volatility are dependent on the molecular type, the ice line varies depending on the molecule in question. Ices, with the exception of water, are more dense than their previous gaseous state and can therefore exert a stronger gravitational force and form planetesimals with the already present dust grains [55]. Since the ices are abundant, the core is larger than the case for terrestrial planets and so has a greater gravitational attraction. This larger core can attract the huge amounts of gas that are present in giant planets. From observations of other stellar systems, planets with a rocky core surrounded by an envelope of gasses such as hydrogen, helium and volatiles exist. These are given the name gas dwarfs as they contain gasses like the giants but have a lower mass. Buchhave et al. (2014) constrained the radii of each type of planet. They constrained terrestrial-like planets to less than 1.7R⊕, gas dwarf planets to between 1.7R⊕ and 3.9R⊕ and giant planets to larger than 3.9R⊕ [56]. The first planet found orbiting a Sun-like star, 51 Pegasi b, was found to have a mass around half that of Jupiter, suggesting that it was a gas giant. How- ever given a semi-major axis of 0.052AU and a higher surface temperature than Jupiter [17], 51 Pegasi b be- came the prototype of a class of planets labelled ‘hot Jupiters’. Since then, many more planets with sim- ilar properties have been discovered. These planets call into question the planetary formation theories that were developed by observing our solar system, in which there are no hot Jupiters. According to these theories, it is not possible for a planet the size of Jupiter to have formed this close to the star. Giant planets are only able to grow so large due to the fact that their core contains volatile elements that have frozen. This freez- 16 ing only occurs when the the planets orbit lies beyond the ice line. For this reason, these planets must have formed further out and somehow moved to a closer or- bit [57]. The most developed model to explain this phe- nomenon is migration. This describes how when plan- ets form, tidal interactions between the planet and the surrounding protoplanetary disk result in the exchange of angular momentum and so the orbit of the planet can change [58]. There are two main types of migra- tion regarding planets. Type I describes the relatively fast migration of planets with masses less than 10M⊕ whereas type II describes the slower migration of plan- ets with masses between 10M⊕ and 30M⊕ [59]. As hot Jupiters have a high mass, they fall under the cate- gory of type II migration. In this migration, due to the high mass of the planet, the planet and disk inter- act so strongly that a gap in the disk actually opens up where the planet orbits, forming a barrier. Some gas does leak into the path of planetary orbit and is either accreted by the planet or passes straight through. As the particles in the protoplanetary disk are generally moving inwards, the planet, which is obstructing the viscous evolution of the disk, moves in the same direc- tion to a smaller orbit. This migration slows down as the disk dissipates since there is less force exerted on the planet enabling it to settle into a smaller orbit [60]. Another way to explain the presence of hot Jupiters is to examine the situation where a giant planet inter- acts with a smaller body in the solar system, reducing the energy of its own orbit. This model is examined in more detail in Subsection 4.6.5 which discusses the three-day pileup of hot Jupiters. It is worth mentioning that hot Jupiters present other surprising properties. An example is that obser- vations show that some hot Jupiters orbit their host stars in the opposite direction to the direction of ro- tation of the star. This poses a problem as all plan- etary formation theories predict that planets formed from the same disk of gas and dust as their parent star should orbit in the same direction. Li et al. (2014) propose a mechanism to explain this seemingly unnat- ural observation. They describe how the orbit of the inner hot Jupiter is greatly perturbed to an extremely elliptical orbit by an outer orbiting planet until the or- bit of the inner planet suddenly flips over. The outer planet steadily removes angular momentum from the inner planet until, when the inner planet is essentially on a collision course with the star and has very little angular momentum, a small gravitational kick applied to the planet can flip it over and reverse the spin and orbital direction [61]. 4.2 Orbital Mechanics 4.2.1 Kepler’s Laws (AE) Celestial mechanics is a branch of astronomy that deals with the motion of celestial objects such as plan- ets and moons under gravity. In the early 17th century, Johannes Kepler developed three laws that describe the motion of planets around their respective stars in the sky. A summary of the laws are: · All planets move in elliptical orbits around their star which is at a focus. · The area swept out by a planet in a given time is always the same. · The period of the orbit of a planet squared is proportional to the semi-major axis of the orbit cubed. Since the planets are in elliptical orbits, the distance between the planet and star will vary depending on the point in orbit of the planet, unlike a circular orbit, as shown in Figure 4.2.1[62]. Figure 4.2.1 – Kepler’s first law describing the elliptical orbit of a planet around the Sun [62]. As Figure 4.2.2shows, Kepler’s second law means that when the planet is at the perihelion of the orbit, it must be moving faster than at the aphelion so that the areas swept out by both positions are the same. Figure 4.2.2 – Kepler’s second law showing the area swept out by a planet as it moves around the star [62]. 17 The mathematical representation of Kepler’s third law is: P2 = 4π2 GM∗ a3 (4.1) where P is the orbital period of a planet around a star of mass M∗, G is the gravitational constant and a is the semi-major axis of the orbit. The semi-major axis, defined as the mean of the perihelion and aphe- lion, must be used in this law instead of the radius of orbit since the planet follows an elliptical orbit and so the distance between the planet and star will vary from a minimum at the perihelion to a maximum at the aphelion. When using the transit method to detect a planet, it is possible to constrain the eccentricity above a certain minimum by comparing the semi-major axis calculated using Kepler’s third law to the distance be- tween the planet and star at the time of transit. A detailed explanation of this can be found in Section 6.3. 4.2.2 Two- and Three-Body Problem (LP) The two and three body problems are used to deter- mine the motion of bodies that have noticeable gravita- tional influence over each other, such as orbiting plan- ets or stars. In the two body problem, the two objects are given set masses, velocities and different starting positions. From this information, the change in motion of the two bodies can be determined as a function of time. The three body problem is much more complex than the two body problem as a third object is added to the system with its own velocity, mass and position. The two body problem yields clear and understand- able results via methods of integration, the three body problem on the other hand is too complex to solve an- alytically and is yet to be completely understood. Two-Body Problem (LP) The most basic two body problem scenario is to consider two masses m1 and m2 with a common barycentre. The vectors from each respective object to the centre of mass are defined as r1 and r2. With this, the following equation can be written [63]: m1r1 + m2r2 = 0 (4.2) The total mass of the system (m1 plus m2 ) is equal to M and the vector displacement between the two bodies is r. The distance between the two objects is labelled d , which is equal to the sum of r1 and r2 . From this and the above equation, further identities can be derived: r1 = m2 m1 r2 = m2 M – m2 (d – r1) (4.3) r1(1 + m2 M – m2 ) = r1( M M – m2 ) = d( m2 M – m2 ) (4.4) These equations can then be used to derive the sim- plified equations shown below: r1 = d( m2 M ) (4.5) r2 = ( m1 m2 )r1 = d( m1 M ) (4.6) The above equations can be written in vector form with vectors r1 and r2 displayed in terms of r, the vector displacement: r1 = –( m2 M )r (4.7) r2 = ( m1 M )r (4.8) Differential equations of motion can be written, de- scribing the force that each body exerts on one another: m1r̈1 = ( Gm1m2 r2 )r̂ (4.9) m2r̈2 = –( Gm1m2 r2 )r̂ (4.10) Using substitutions between equations (4.7) and (4.8), and equations (4.9) and (4.10), the following equation of motion can be derived: r̈ = –( GM r2 )r̂ (4.11) Here the unit vector r̂ is included which is equal to one. These two body considerations are made with the assumption that the two bodies move in a plane with respect to each other, the centre of mass frame. It is also possible to derive equations that consider the total energy of a two body system: E = 1 2 (m1ṙ 2 1 + m2ṙ 2 2) – Gm1m2 r (4.12) Similarly, the total angular momentum of a two body system can be written: L = m1r 2 1θ̇+ m2r 2 2θ̇ (4.13) Three-Body Problem (LP) Although the general three body problem has no analytical solution, an idealised, restricted three body problem can be considered. This three body problem involves two bodies with masses m1 and m2 and a third body, m3 with a much lower mass that has a negligible gravitational effect on the other two bodies. The two more massive bodies are in orbit and both lie along the x-axis of a Cartesian co-ordinate system (x , y , z ) at a time t = 0 . The origin of the system can be considered as the centre of mass C and the distance between m1 and m2 remains a constant distance R. The distance between m1 and C is constantly r1 whilst the distance between m2 and C is constantly r2 . An equation for the orbital angular velocity of the system 18 can be written using M as the total mass of the system [64]: ω2 = GM R3 (4.14) We can also state the following relationship: m1 m2 = r1 r2 (4.15) For simplicity the values of R and GM are made to be equal to 1. Using Equation (4.14) it can be clearly seen that this also results in ω being equal to 1. We take the following identities: μ1 = Gm1 (4.16) μ2 = Gm2 (4.17) Hence: r1 = μ2 (4.18) r2 = 1 – r1 = μ1 (4.19) From these equations, position vectors can be as- signed to m1 and m2 . The position vector for m1 is r1 = (x1, y1, 0) with the position vector for m2 as r2 = (x2, y2, 0), where: r1 = μ2(–cosωt, –sinωt, 0) (4.20) r2 = μ1(cosωt, sinωt, 0) (4.21) At this point, the third mass can finally be consid- ered with a position vector of r = (x, y, z). Cartesian equations of motion of the particle can therefore be found using this consideration. These are displayed below: ẍ = –μ1 (x – x1) ρ31 – μ2 (x – x2) ρ32 (4.22) ÿ = –μ1 (y – y1) ρ31 – μ2 (y – y2) ρ32 (4.23) z̈ = –μ1 z ρ31 – μ2 z ρ32 (4.24) where: ρ21 = (x – x1) 2 + (y – y1) 2 + z2 (4.25) ρ22 = (x – x2) 2 + (y – y2) 2 + z2 (4.26) The light particle m3 is considered as a moving par- ticle in the rotating frame, forces on stationary bodies in this frame can be represented by using a Roche po- tential. When two bodies orbit one another circularly, their gravitational fields interact with one another and merge to form one field. Contoured images can be pro- duced that display areas of equal and varying gravita- tional potential. This is known as the Roche potential named after Édouard Roche [65]. Figure 4.2.3 – The Contours of a Roche Potential [66] This diagram not only shows the relative positions of the bodies and the gravitational field, but it also shows the five Lagrangian points in the system. At these points, a stationary body would feel no overall force. These positions are often ideal for satellites. Mathematically, the Roche potential is defined as [67]: ΦR(r) = – Gm1 |r – r1| – Gm2 |r – r2| – ω2r2 2 (4.27) In the equation, the third term accounts for the centrifugal force. The five Lagrangian points appear when ΔΦR = 0. An understanding of the Roche po- tential can be used to model the motion of a third, less massive body through systems of this type. 4.3 Planet-Finding Methods 4.3.1 Transit Method (LP) When an object passes between a distant star and an observer on Earth, there is a drop in the amount of flux reaching the observer as a fraction of the light from the distant star has been blocked. The drop in flux corresponds to the area of the star blocked by the passing object. Often, such occurrences are as a result of a planet orbiting the star. The presence of a planet can be confirmed by detecting such flux changes pe- riodically, with the change in flux being the same for every transit. Since planets have a very small radius compared to that of stars, the change in detected flux during the transit of an exoplanet on a distant star is very small. For this reason, the instruments that de- tect changes in flux of distant stars must have a high precision. 19 Figure 4.3.1 – Transit Example [68]. Figure 4.3.2 – Impact Parameter Diagram with Basic Light Curve [69]. When recording data on a distant star, a light curve is often produced, displaying the flux of the star against time. The radius of a transiting planet has an effect on the transit depth of the light curve. A light curve is shown in Figure 4.3.1 with one transit displayed. The depth of the transit δ is displayed as a fraction of the total flux of the host star and it can be related to the ratio of the planetary radius and stellar radius [70]: δ = ( Rp R∗ )2 (4.28) So long as the star remains much greater in size than the planet, the above equation can be applied. These transits reveal more than just the ratio between planetary and stellar radii; they also reveal the pe- riod of transit of the planet and the time that it takes for a transit to occur. The period can be inferred by the time between transits with the transit length mea- sured by considering the time it takes from the start of a transit to its end. Detection of many transits is required to ensure that these values remain constant over a substantial period of time. With these values known, further information regarding exoplanets can be inferred. The transit time and period of orbit can be related by [70]: τ = PR∗ πa (4.29) Here, τ corresponds to the transit time, R∗ to the stellar radius and a to the semi major axis of the planet. This equation is somewhat simplified, as it assumes that the planet moves in a straight line across the star and does not account for any varied planetary motion. When a planet transits a distant star, it is rare for it to take the longest path across the star, along the stars’ radius. For this reason an impact parameter is introduced to account for the variety of possible paths a transiting planet can take in front of a star. The impact parameter is represented by the letter b, and is a measure of how off-centre a transit is. The geometry is depicted in Figure 4.3.2. The impact parameter of a planet affects the transit duration due to the circular appearance of a star. This can be related to the transit depth, transit time and time of partial transit t . The partial transit time is the sum of the ingress and egress time where a fraction of the planet blocks the star, but not the whole of the planet. The ingress marks the beginning of a transit, the egress marks the end of a transit [71]: b = 1 – τ √ δ t (4.30) The impact parameter can also be found by consid- ering the inclination of the system, as follows: b = a cos(i) R∗ (4.31) This equation can then be combined with Equation (4.29) to get a more accurate transit time: τ = Psin–1 (√ (Rp+R∗)2–b2 a ) π (4.32) Many stars host exoplanets that cannot be detected by this method as the planets do not transit the star relative to our position on Earth. The probability of detecting the planet for a system with a star and one exoplanet, using the transit method, can be quantified using: p = R∗ a (4.33) The probability of detecting an exoplanet when only analysing one star is very low, however when hun- dreds of stars are analysed, the probability of finding exoplanets soon becomes very high. Equation (4.33) only takes into consideration planets with a circular orbit; for an elliptical orbit, Equation (4.34) must be applied [72]: p = R∗ a(1 – e2) (4.34) As well as detecting primary transits, where a planet passes in front of a star relative to an observer, 20 secondary transits can also be detected. A secondary transit occurs when an exoplanet is ‘hidden’ from an observer on Earth by its host star. Although planets do not emit light, they are able to reflect star light which can be detected on Earth. When an exoplanet is obstructed from view by its host star, the reflected light is no longer able to reach us and a dip in the flux from the system is noticed. The depth of a secondary transit is significantly smaller than a primary transit and therefore these transits are much more difficult to detect. The collection of secondary transit data can be used to determine the albedo of the exoplanet. The albedo is the reflection coefficient for a planet which if known, can be used as an indication of the nature of the atmosphere or surface of a planet. The detection of a transit does not necessarily mean that a planet is present, as there are often many ‘false transits’ detected. The majority of false transits are as a result of binary stars, where the smaller of the two stars moves in front of the larger star. This re- sults in a dip in flux of the system which usually has a much greater transit depth than that of a typical planet-star transit. For this reason, the majority of bi- nary star transits can be identified easily. If the smaller of the two stars only eclipses the larger star partially, then a dip comparable to that of a planet can be de- tected. In this situation, the spectroscopic make up of the system must be analysed. Since stars emit light of different wavelengths, any dips in the intensity of particular wavelengths of light can be monitored to in- dicate whether the passing body is a star or planet. As planets emit no light of their own, when a planet tran- sits a star, all emitted wavelengths from the system will drop in intensity. The luminosity of a star changes due to internal oscillating standing waves that alter the size of the star. The luminosity changes periodically and there- fore this must be considered when analysing transit data. A change in flux due to fluctuating luminosity occurs more gradually than the dip in flux seen when a body transits the star, however such changes do make it more difficult to detect transits in the first place. The luminosity of a star can change by a factor of 10–5. Although this seems small, this value is comparable to the change in flux of a Sun-like star by the transit of an Earth-like planet (8.4x10–5W). Observing stars from Earth produces a limit on detecting changes of luminosity smaller than 8x10–4W [73]. The presence of sunspots on stars can result in a dip in the flux de- tected as the stars rotate. This can be mistaken for a transit, however further analysis of such stars will reveal that as time goes on, the numbers of sunspots will change and so there will be no periodicity to the changes in flux. When a planet transits a star, some of the light from the star can pass through the planetary atmosphere be- fore it reaches Earth. Spectroscopic examination of the light could reveal the contents of the atmosphere with a detection of either O2 or O3 indicating a potential for the presence of life [74]. This information is practi- cally impossible to obtain using light reflected from an orbiting planet that is not in transit. If an exoplanet has an approximately circular orbit, and the radius of its host star is known, the planetary radius, semi-major axis, orbital period and inclination can all be calculated. The transit method is most useful when finding planets with large radii and small semi- major axes as these are the most likely types of planets to be detected. The semi-major axis is important as it allows us to determine whether a planet is in the hab- itable zone. This describes the distance from the star at which water can collect on the surface of a planet without boiling or freezing. This, together with spec- troscopic observations, can determine the make up of the atmosphere of the planet, which is of paramount importance when searching for habitable planets. 4.3.2 Transit Timing Variations (TTV) (AE) In a multiple planet system, the motions of the star and its orbiting planets will not follow Kepler’s Laws, but slightly irregular orbits due to the gravitational influence of the other bodies in the system. This pro- duces a slight variation in the length of the planet’s orbital period, which can be detected by the variation in time between transits. From this, the presence of a second planet can be inferred. As of February 2015, 15 planets have been discovered using TTVs [18], 14 of which were detected using the Kepler spacecraft. The first planet detected using this technique was Kepler- 19c in 2011 [75]. Take the case looking at j transits, where 0 ≤ j ≤ N and N is the total number of transits ob- served. The time of any given transit is given by: t(j) = jP + t(0) (4.35) where P is the period of transits and t(0 ) the time at which the the first transit begins. If there is a pe- riodic change in this time, there is a transit timing variation given by: δt(j) = t(j) – jP (4.36) Transit timing variations can be categorised into four situations [76]. 1. Non-Interacting Inner Planet Take the case of observing transits of a planet far from a star. Timings of the transits will vary if the star has formed a binary system with another planet close to the star. As the star orbits the centre of mass of the binary system and changes its relative position in the sky, the line of sight of the transit will be shifted peri- odically so the time between transits will change. This case is represented in Figure 4.3.3 which shows how the point in a planet’s orbit during transit is dependent on the position of the star as it wobbles. 21 Figure 4.3.3 – TTVs caused by an inner planet forming a binary system with the star. The yellow sphere represents the star and the blue and green spheres represent the inner and transiting planets respectively [76]. If the star is to the right of the centre of mass of the binary system, as is the case for the left side part of Figure 4.3.3, the planet has to move further round in order to transit the star. For this reason there will be a larger time between this transit and the previous one resulting in a positive δt(j). The opposite happens when the star is to the left of the centre. This results in a sinusoidal TTV signal and it is clear to see that the larger the mass of the inner planet, the larger the wobble of the star, and so the TTV signal is greater. This case assumes that the two planets are far enough away to not interact with each other. The TTV for a given transit can be given by [77]: δt(j) ≈ – Pt 2π ac at mc M∗ sin ( 2π(jPt – t(o)) Pc ) (4.37) where Pc , Pt , ac and at are the periods and radii of the planet close to the star causing the perturbation and the transiting planet respectively and mc and M∗ are the masses of the inner planet and the star. A useful measure of the time variation is the root mean square given by [77]: 〈δt〉RMS = 1√ 2 Pt 2π ac at mc M∗ (4.38) as this just leaves two unknown variables, ac and mc , assuming the period and radius of the transiting planet are known. 2. Interaction from Exterior Planet in Large Orbit In this situation, an inner circular orbiting system consisting of a star and transiting planet is perturbed by an exterior planet on a large orbit with high ec- centricity producing TTVs in the transiting planet’s period, as is shown in the Figure 4.3.4. Figure 4.3.4 – TTVs caused by an outer planet with a high eccentricity perturbing a binary system consisting of the star and an inner transiting planet. The spheres represent the same bodies as for case one [76]. Agol et al. (2005) use Jacobian coordinates to de- rive a first order Legendre series approximation for the perturbing effective force on the inner binary system. This approximates the TTV signal as [77]: δt = β(1 – e20) –3/2[f0 – n0(t – τ0) + e0 sin(f0)] (4.39) where e0 is the eccentricity of the outer planet, n0 (t – t0 ) is the mean motion of the planet, f0 is the true anomaly of the planet, which is defined as the angular distance between the perihelion of the planet and the planet itself observed from the star it orbits. β is given by: β = m0 2π(M∗ + mt) P2t P0 (4.40) The RMS of the TTV is [77]: 〈δt〉RMS = 3βe0√ 2(1 – e20) 3/2 [ 1 – 3e20 16 – 47e40 1296 – 413e60 27648 ]1/2 (4.41) This model is particularly useful for finding exterior planets perturbing a transiting hot Jupiter as the TTV signal is highest when mt > m0 . 3. Interactions Between Two Planets Consider a case where two planets with orbits that are close to one another, but not in resonance, perturb each other. When the two planets are at conjunction (the orbits of the planets are at their closest positions), the planets interact most strongly [78]. This results in the orbit of the transiting planet being perturbed from circular to eccentric as Figure 4.3.5 shows. 22 Figure 4.3.5 – TTVs caused by two planets with initially circular orbits close to one another. The orbit of the tran- siting planet, shown in green, is perturbed from circular to eccentric so a TTV signal is produced [76]. Since the planets are not in resonance, the position of conjunction will not be constant and the perturba- tions will cancel out when the angular position moves by π radians. This means that the eccentricity will grow over half a period of circulation of the position of conjunction. The closer the planets are to resonance, the longer it takes for the position of conjunction to complete a full cycle, therefore the eccentricity and hence the period of orbit are affected greatly. There are two factors which contribute to the TTV: fluctuations in the mean motion of the planets and changes due to the eccentricity not remaining constant. For the equations below, it is assumed we have a case where the planets are near but not at resonance so there is a transit ratio j : j + 1 with a high value of j . When eccentricity dominates, the TTV is given by [77]: δt ≈ μpertε–1P (4.42) where μpert is the ratio of the mass of the perturbing planet and the star and ε is the fractional distance from resonance: ε = |1 – (1 + j–1)P1 P2 |. (4.43) When fluctuations in mean motion dominate, the resulting TTV for the lighter planet is [77]: δtlight ≈ μ2ε–3P (4.44) where μ is the mass ratio of the heavy planet and the star. Due to conservation of energy, the TTV for the heavier planet is reduced to [77]: δtheavy ≈ mlight mheavy μ2ε–3P (4.45) As the equations show, when ε > μ1/2 the eccentric- ity term dominates the mean motion fluctuation term and the opposite is true when ε < μ1/2 until ε < j1/3μ2/3 at which point the planets are in mean motion reso- nance. If both planets in the system transit the star (which is not improbable as planets tend to orbit on similar planes), it will be possible to derive values for both planets individually. This data would then be used to test our model and show whether the observed TTV is caused by the planets observed or by other, non- observed planets. 4. Interactions of Two Planets in Mean-Motion Resonance Take the case of two planets with initially circu- lar orbits with a first order resonance. Initially, con- junctions occur at exactly the same position in space relative to the star. Since the interaction between the planets is strong at the conjunctions, perturbations are caused in the orbit of the lighter planet making it ec- centric, and so the semi-major axis and period change. These changes result in movements of the position of conjunction and, like the previous case, when the po- sition shifts by approximately π radians, the pertur- bations start to cancel out causing the eccentricity to decrease. So over a cycle of 2π radians, the eccentrici- ties and change in period increase to a maximum before returning to their original values. Assuming a large j so that j ≈ j + 1 and P1 ≈ P2 as before, the maximum transit timing variation for the lighter planet is [77]: δtmax,light ∼ P j (4.46) and for the heavier planet is [77]: δtmax,heavy ∼ mlight mheavy P j (4.47) The time it takes for the conjunction to return to its original position is called the libration period and is [77]: Plib ∼ 0.5j–1ε–1P ∼ 0.5j–4/3μ–2/3P. (4.48) In the context of this project, the accuracy with which we are able to measure the period of transits is too low to be able to detect a TTV and hence infer the presence of extra planets in the system with any certainty. How- ever, these equations are important as they are neces- sary in calculating properties of the planets detected by TTVs, if more accurate periods can be obtained. 4.3.3 Other Planet Detection Methods (CL) With the Kepler data provided being the only data available for study, exoplanets could only be detected using above listed methods. However, there are many other methods by which planets can be detected. Some of these methods, which were not used in this study, have been outlined in this section. It is common for more than one method to be used to determine the properties of a exoplanet and confirm its existence. 23 Doppler Wobble (Radial Velocity) (CL) In the presence of an orbiting planet, a star moves around the centre of mass, or barycentre, of the two- body system (See Section 4.2.2). This motion of the star can be seen in the periodic shift of the spectral lines from the light emitted by the star. The wave- length of the light is alternately blue and red shifted as the star moves towards and away from the observer, respectively. For high precision measurements of the wavelength shifts, the stellar spectra observed need to contain many absorption lines. Stars with a low num- ber of features in their spectra have a low precision. The wavelength shifts in the observed light are used to determine the radial velocity at which the star is mov- ing with respect to the observer, allowing properties of the orbiting planet to be calculated [79]. The observed radial velocity of the star, Vdop, is given by [80]: Vdop = V∗ sin (i) (4.49) where V∗ is the radial velocity of the star caused by the orbiting body and i is the angle of inclination of the system. Equation (4.49) shows that the change in radial velocity seen by the observer is dependent on the value of the inclination and the magnitude of the star’s movements. This means that this method is most sensitive to planets that are orbiting with an inclination of 90◦. This equation also shows that the Doppler method is more sensitive to massive planets, as these bodies have a greater gravitational influence over their host star. This in turn maximises the move- ment and changes in radial velocity of the star. On the contrary, a planet orbiting a star with an inclination angle of 0◦, will not have an effect on the radial veloc- ity of the star with respect to the observer. Therefore, any planet orbiting a star in such an orientation would not be detected [81]. Simple orbital mechanics can be used to find the velocity of the planet, by equating the forces acting upon the planet in circular motion as it orbits. The result is given by [82]: Vp = √ GM∗ a (4.50) where Vp is the radial velocity of the planet, G is the Gravitational constant, M∗ is the mass of the star and a is the semi-major axis of the planetary orbit. The value Vp can be calculated if the mass of the star is known, this can be derived from asteroseismic data. The value for a is determined by using Kepler’s third law, inputting the period of oscillation of the star, P∗ [62]: a3 = GM∗ 4π2 · P2∗ (4.51) The mass of the planet, Mp is then easily deduced from Equation (4.52), where the conservation of mo- mentum in the star-planet system is used: Mp = M∗V∗ Vp (4.52) A final equation for the mass of the planet can then be found, by combining all of the previous relations [83]: Mp = Vdop sin (i) ( P∗M2∗ 2πG )1/3 (4.53) This is the equation for the mass of a planet in a cir- cular orbit. If a planet is orbiting in an eccentric orbit, Equation 4.53 becomes [83]: Mp = Vdop sin (i) ( P∗M2∗ 2πG )1/3√ 1 – e2 (4.54) where e represents the value of the eccentricity of the orbit. Unfortunately, the Kepler spacecraft does not have the capacity to carry out stellar spectroscopy, therefore it can not produce radial velocity measure- ments. Kepler has had great success in detecting the possible presence of exoplanets from light curves and the transit method. However, star systems where Ke- pler is able to detect transits are usually then sub- ject to further ground-based observations and radial velocity measurements. The transit and radial veloc- ity methods work well together because of their similar dependence on inclination. A planet that can be ob- served by the transit method must be closely aligned to a 90◦angle of inclination, meaning that the likelihood of also being able to detect the planet by the radial velocity method is significantly increased. The follow- up measurements are used to eliminate false detections that are produced from the Kepler light curves. The large majority of such false positives are found, and eliminated, by using statistical tests on the Kepler data itself. These false-positives are caused by the common occurrence of astrophysical signals that mimic the sig- nals received by transiting planets [84]. If the possible planet passes these statistical tests, then an observa- tion using the radial velocity method is used to rule out another common source of false-positives; stellar binary systems. An eclipsing binary can be very dif- ficult to distinguish from a planet in the transit data. However, due to the relatively large masses of stars compared to planets, the radial velocity method can distinguish these two objects. In the presence of an eclipsing binary, the radial velocity variations are much larger, ≈ 1km s–1 [84]. The follow-up radial velocity observations can not only confirm the presence of an exoplanet in a system, they also give values for key planetary properties that can not be obtained by Kepler alone, like the mass of the planet. The combination of the planetary radius values obtained by the transit method and the mass values obtained by the radial velocity methods means that the density of the planet can be calculated. This allows conclusions to be drawn on the composition of the planet [85]. Astrometric Wobble (CL) Astrometric wobbles are observed by Astrometry, a branch of astronomy which takes precise measure- ments of the relative positions and movements of stars. 24 An astrometric wobble describes the apparent motion of a star on the celestial sphere which is caused by the gravitational effects of orbiting planets about it. Astrometry is the oldest theorised method of detect- ing extrasolar planets. In the 18th century, William Herschel was researching the small, visible angular dis- placement of faint stars and from this it was concluded that these stars had an unseen companion affecting the position of the observed star [86]. This proves that as- trometric wobbles can even be seen by eye. However, in Herschel’s case, the effects were due to a system of binary stars, not the motion of a star as a result of an orbiting planet. In order to detect exoplanets by us- ing astrometric methods, more accurate methods using photographic and telescopic techniques must be used. The size of the wobble is dependent on the mass of the star and the planet, the radius of the orbit of the planet about the star and the inclination of the orbital plane, relative to the observer. If the angle of inclination is i = 90 ◦, so that the orbital plane is in line with the line of sight of the observer, the mo- tion of a star due to the presence of an orbiting planet would appear to move from side to side. If the orbital plane were aligned however so that it is perpendicu- lar to the observers line of sight, i = 0 ◦, then the star would display a small circular motion as the planets gravitational force pulls the star out of position [87]. In a star-planet system, the distance from the star to the system centre of mass, r1 , is given by the equa- tion [71]: r1 = a m M + m = a 1 + Mm (4.55) where M is the mass of the star, m is the mass of the planet and a represents the semi-major axis. From this equation it can be stated that the motion of the star is greatest when the barycentre of the system is further away from the star. For this reason, astrometric wob- bles are larger in systems with small stars, large planets and large orbital radii. This is because the barycentre lies further from the centre of the star meaning that the star must move a greater distance about this point. The angular displacement caused by the stars orbital motion, at distance d away from the star is given by [87]: Δθ ≈ r1 d ≈ (m M )(a d ) (4.56) Here, the small angle approximation is applied and it is assumed that M � m . This confirms what was previously stated; as the centre of mass moves further away from the star (increased r1 ) then the angular dis- placement gets larger. Equation (4.56) also illustrates that the motion of a star is most obvious as the dis- tance to the star decreases, relative to an observer. This is because the angular displacement gets larger as the observation distance decreases. The astrometric method is most obvious in two-body systems where a planet orbits a star at a large orbital radius. This is in contrast to the transit method, which is most sen- sitive when detecting planets that orbit close to the star. For the case where the astrometric wobble effect is caused by the orbit of one planet, the mechanics of the situation are governed by the two-body problem (See Section 4.2.2). The problem in the advancement of Astrometry as a detection method is the precision to which telescopes can resolve an angular change in position. The Earth’s atmosphere is a barrier to improving the precision of ground-based telescopes. The atmosphere degrades the quality of astronomical images collected and noise is produced by distortions in the telescope’s structure. These distortions are caused by the weight of the struc- ture and also the thermal response of the surrounding building and the telescope itself [88]. Ground-based telescopes also have fixed locations, resulting in a lim- ited coverage of the sky. In order to increase this cover- age, data is combined from different observatories from around the world. However, this process also produces uncertainties. For these reasons, ground-based tele- scopes have not been able to surpass the barrier of position measurements with a precision greater than one hundredth of an arc second [88]. For this rea- son, moving telescopes to space seemed the logical so- lution. The European Space Agency’s Gaia mission (Global Astrometric Interferometer for Astrophysics), which launched in 2013, has planned to find thousands of planets using the astrometric wobble method. With an astrometric precision of up to 0.00001 arc seconds, Gaia will determine positions of stars out to 30,000 light-years away [89]. Gaia will be able to measure a star’s position and motion 200 times more accurately than previous astrometric detection attempts. A good analogy to show the precision of Gaia is that it is able to the measure the angle that corresponds to the length of an astronauts thumbnail, who is standing on the moon, from Earth [90]. This traditional method of pos- sible planet detection has been reinvigorated with the advance of instrumentation in astrometric techniques. On Gaia itself, astrometric, photometric (transit) and spectrospcopic (radial velocity) techniques will be com- bined to produce data from which it is easy to identify, characterise and confirm planets that orbit a star [89]. 4.4 Stellar Limb Darkening (CL) Stellar limb darkening is a phenomenon that is ob- served with varying effect in all stars, including our own sun. It is given its name simply due to the appearance of the edges, or limbs, of the star being darker than its centre. The consequence of this is that the specific intensity, Iλ, at a particular point on the star is de- pendent on the distance that the point is away from the centre of the star [91]. This is illustrated in Figure 4.4.1. 25 Figure 4.4.1 – Image of the Sun showing the decreased in- tensity of light emitted at the edges [92]. The centre of the Sun has a much greater inten- sity of light compared to that of the limbs of the Sun. This gives the sun its spherical, three-dimensional ap- pearance even though it is observed in a two dimen- sions. This is partially due to the effect of the effec- tive temperature decreasing as the radial distance from the centre of the star increases. The intensity of radia- tion produced is largely dependent on the temperature, which can be very simply demonstrated by the Stefan- Boltzmann Law, where the intensity is proportional to temperature to the power of four [93]. However, the temperature dependence of the radial distance alone does not explain the limb-darkening effects observed. It is also reliant upon the concept known as optical depth, which provides a dimensionless measure of the depth we can observe into a partially transparent gas [19]. At the centre of the star disk, the observed light rays are penetrating radially outwards from relatively deep with in the stars photosphere. At this point the temperature is relatively high and the intensity is at a maximum. Light emitted from the limbs of the star originate in the upper, cooler regions of the photo- sphere - compared to the lower depths at which the optical depth allows at the centre. Therefore, the in- tensity of light observed at the same optical depth fur- ther from the centre is reduced due to the decrease in temperature [94]. The modelling of the effects of stellar limb dark- ening is extremely important in obtaining an accurate estimate of planetary characteristics from their transit data [95]. The stars in our data that have been ob- served by Kepler show the limb-darkening effect. This effect causes issues with the transit method; the pres- ence of limb darkening causes a change in the inten- sity. Transit detection relies upon intensity changes produced only by the planet blocking out the star’s light. Figure 4.4.2 shows the transits of four planets of different sizes (scaled to the radius of the Earth) or- biting with the same period. These modelled transits take place around a star of a constant size with a uni- form intensity across the whole disc, therefore the only significant source of change in intensity comes from the light blocked out by the transiting planets. Figure 4.4.2 – Simulated light curves for planets of varying radii transiting a star of uniform intensity, with a radius of 0.8R� [96]. The transit shapes in Figure 4.4.2 are sharply de- fined in a box shape, with an almost instantaneous drop in intensity as the planet moves in front of the star. The larger the planet the more light that it blocks out and the larger the drop in intensity. The points plotted by the model that lie between the normal in- tensity of the star and the dropped transit intensity are due to partial coverage of the planets disc as they begin to move in front and away from the star’s disc. This is an ideal model and in practice the light curves received from stars with transiting planets present do not look as perfect as these models. This is mainly due to limb darkening causing a variation in the intensity of the light emitted and blocked out by the transiting planets. Applying the effects of stellar limb darken- ing to the model shown in Figure 4.4.2 produces the transits shown in Figure 4.4.3 Figure 4.4.3 – Simulated light curves for planets of varying radii transiting a star of radius 0.8R�, with limb darkening present [96]. The shapes of the transits in Figure 4.4.3 are signif- icantly different to the shapes of the transits in Figure 26 4.4.2, where limb darkening is not accounted for. The difference is that as the planet moves over the limbs of star, a smaller proportion of total light is absorbed compared to when the planet transits over the centre of the star. This is because the star is brightest at the centre. When the planet passes over the centre of the star the maximum drop in intensity occurs. All of these transits are assumed to have an impact parame- ter equal to zero, where it transits along the radius of the star. To calculate the intensity due to limb darkening at a specific point upon a star, a method to mathemati- cally mark a specific point on the star, relative to the centre, is first needed. This is done by defining the quantity, μ, as the cosine of the angle between the line of sight of the observer and the normal to the surface of the star at that point. The geometry of this is il- lustrated in Figure 4.4.4 and shown mathematically by the following equation: μ = cos(θ) = √ R2∗ – r2 R2∗ (4.57) As Equation (4.57) shows, μ can also be represented in terms of the radial distance from the centre r and the stellar radius R∗. Figure 4.4.4 – Geometry of observer and normal vectors representing the quantity μ [97]. This angular distance, μ, can be used with the knowledge of how temperature changes with radius, and how opacity and density change with radius, to predict the change in surface brightness as a function of angular distance [94]. This is often expressed as the ratio of the intensity at a point on the star over the intensity at the centre (μ = 1). The intensity ratio re- lations have only been fully determined by observations for a few stars, including the Sun. This means that in order to calculate the effects of limb darkening for stars with transiting planets, the relations and coefficients need to be produced by theoretical models of the stel- lar atmospheres [98]. These relations and approxima- tions have evolved over many years, as more advanced models have been produced to analyse limb darkening in stellar atmospheres, unlike that of the Sun. How- ever, initially the most used intensity relation was the linear limb-darkening law, which was based upon the limb-darkening features of the solar atmosphere [99]. The temperature and physical properties of the Sun are comparable to that of that of a grey atmosphere, where limb-darkening is well approximated to be linear [98]: Iλ(μ) Iλ(1) = 1 – u (1 – μ) (4.58) The value u in Equation (4.58) is known as a limb- darkening coefficient (LDC) and varies depending on the wavelengths of light observed and ultimately the temperature of the star. The study of other stellar atmospheres has changed our understanding of the in- tensity laws. Other, non-linear, laws have been found to describe the limb-darkening effects to a greater accu- racy than the linear law when analysing a star outside the temperature range of the Sun. The quadratic law, shown in Equation (4.59), can be used for this purpose and gives a more general intensity law for stars with different properties to the Sun [98]: Iλ(μ) Iλ(1) = 1 – a (1 – μ) – b (1 – μ)2 (4.59) The quadratic law has two limb-darkening coeffi- cients, a and b, which need to be determined in order to produce values for the intensity ratio. In 1970, the logarithmic intensity law was proposed. The results from data taken showed a close fit between the ob- served relations and this theoretical approximation of the intensity variation in stars with the effective tem- peratures in the range 10000K < Teff < 40000K [100]: Iλ(μ) Iλ(1) = 1 – c (1 – μ) – fμ ln (μ) (4.60) At this temperature range, Equation (4.60) only really applies to the stars in the spectral classes O and B, as these are the more massive and hotter stars. These types of stars are harder to produce follow up measurements for as radial velocity measurements are at their most precise when the stellar spectra contain large numbers of absorption lines. O and B group stars do not produce many features compared to MS stars. Limb darkening can also have an effect on the shape of the transit depending on the value of the impact pa- rameter, b. Figure 4.4.5 shows a light curve for mod- elled transits at varying impact parameters, where an impact parameter of 0.9 revealing a transit of an object with an almost negligible effect on the intensity of light received. 27 Figure 4.4.5 – Simulated light curves for a planet of 10 Earth radii, orbiting a star with uniform intensity and a radius of 0.8×Solar Radius, at various values for the impact parameter [96]. As the impact parameter value increases, the tran- sit time decreases due to the distance travelled across the stellar disk shortening. This is caused because the orbital plane, with respect to the observer, moves fur- ther away from the equator of the star as b increases. The transit where b = 0.9 has a slightly lower depth than the other transits. In this case, this will most probably be caused by part of the planet transiting outside of the observers view of the stellar disk, there- fore the intensity drop is reduced, as less of the stellar disks intensity is blocked. Figure 4.4.6 shows the same transits as in Figure 4.4.5, however, the effects of limb darkening are included and the stellar disk is no longer of uniform intensity. Figure 4.4.6 – Simulated light curves for a planet of 10 Earth radii, orbiting a star with limb darkening present and a radius of 0.8R�, at various values for the impact parameter [96]. In the presence of limb darkening, the shortening effect of the transit times is still present for increasing values of b. However, the depth of the transits also sig- nificantly decreases as the value for b increases. This is caused by the decrease in intensity of the limbs of the star. As earlier explained, an increase in the im- pact parameter results in a transit across the star at a greater height relative to its centre. This means that the planetary disk blocks out a smaller proportion of light whilst transiting near the limbs and the change in intensity of the transit is less than that of a transit across the equatorial plane. Introducing the limb darkening phenomenon has an effect on the transit observed, which in turn affects the measurement of the planetary radius and the impact parameter; it is therefore crucial to correctly model for limb darkening in order to obtain accurate values of these quantities. 4.5 Mass Constraining Methods (PS) In general, the mass of transiting exoplanets can be determined by the use of a second detection method, such as the Doppler Method. However, by assuming certain parameters concerning the composition of the planet in question based on known observables, it is possible to constrain the mass of the planet detected via the Transit Method to within a mass range without recourse to a second detection. By considering theoretical planet compositions, mass-radius relations can be determined which will give rough values for exoplanet masses (Figure 4.5.1). From top to bottom the plotted lines on this graph show planets composed of: · hydrogen (cyan solid line) · a hydrogen-helium mixture with 25% helium by mass (cyan dotted line) · water ice (blue solid line) · water planets with 75% water ice, a 22% silicate shell, and a 3% iron core (blue dashed line) · water planets with 45% water ice, a 48.5% sili- cate shell, and a 6.5% iron core (blue dot-dashed line) · water planets with 25% water ice, a 52.5% sil- icate shell, and a 22.5% iron core(blue dotted line) · silicate (MgSiO3 perovskite) (red solid line) · Earth-like silicate planets with 32.5% by mass iron cores and 67.5% silicate mantles (red dashed line) · Mercury-like silicate planets with 70% by mass iron core and 30% silicate mantles (red dotted line) · solid iron planets (green solid line) Several planets of known mass and radius are also shown for illustrative purposes. 28 Figure 4.5.1 – Theoretical mass-radius relationship for solid planets of various compositions. Solar system planets ap- pear as blue triangles, while extrasolar planets (with signif- icant uncertainties on each value) appear in pink. [101]. By considering orbital and planetary characteris- tics, unlikely planetary compositions can be ruled out, further narrowing the mass range. For example, the equilibrium temperature of a transiting planet can be determined from its radius and semi major axis, if its albedo and the effective temperature of the star are also known. If this temperature is sufficiently high that liq- uid water cannot exist on the surface of the planet, these compositions can be discarded, greatly reducing the possible mass range. Note that in Figure 4.5.1, radius values tail off and actually decrease at very high masses. This is due to electron degeneracy pressure, as the Pauli Exclu- sion Principle forbids two electrons occupying the same quantum state. This adds a significant uncertainty to calculation of the mass ranges of planets with particu- larly large radii. Furthermore, for gaseous planets, the radius is de- fined not just by the mass of material present but ad- ditionally by its temperature, as the volume of a gas expands with temperature. Thus, for close orbiting gas giants (hot Jupiters) the lower mass limit must be re- duced to account for the ‘puffiness’ caused by increased equilibrium temperature. There is also a discrepancy in the radius calculated by the transit method for a gas planet and the ra- dius given by the mean distance between the centre of a planet and its ‘surface’, as the latter is defined as the point at which the atmospheric pressure is 1 bar [102]. The ‘true’ radius of a gas giant will therefore be slightly lower than that calculated by the transit method, as above the defined surface there still exists gas which absorbs light and thus reduces the measured stellar intensity. For close orbiting gas giants (hot Jupiters), it can also be useful to consider the Hill Sphere and Roche limit (see Sections 4.6.3 and 4.6.4) in order to constrain the mass of the planet. In order to transit multiple times, the Hill sphere of the planet at closest approach to the star must be at least as large as the volume of the planet. This can provide a lower mass limit for particularly large planets, although most planets do not orbit close enough to their host stars for this to be a productive result. This method is particularly useful in categorising small planets of around the radius of the Earth, as for this radius a gas planet is extremely unlikely, especially considering the bias towards detection of close-orbiting planets inherent in the use of the transit method. The planet can therefore safely be assumed to be rocky, and the mass can thus be constrained to within relatively narrow bounds. 4.6 Auxiliary Theory 4.6.1 Stellar Variability (ML) Stellar variability is the fluctuation of the apparent magnitude of a star. This can be caused by intrinsic or extrinsic factors. Intrinsic factors (when a star’s lu- minosity actually changes) include pulsation, eruption and novae (including supernovae). Extrinsic factors (changes in brightness due to external forces) include binary eclipses, planetary eclipses and rotation. Pulsation is where the star expands and contracts. MS stars typically have a constant luminosity, but gi- ants’ and supergiants’ radii may vary as they evolve, leading to a variation in their apparent magnitude [103]. Stellar eruptions include flares and coronal mass ejections. Flares are violent surface eruptions, result- ing in the release of energetic particles and electromag- netic waves, whereas coronal mass ejections consist of a slow release of large spheres of gas over the course of numerous hours. Both of these contribute to the apparent magnitude of a star when they occur [104]. Novae are nuclear explosions caused when a cer- tain amount of hydrogen has accreted from a red giant onto the surface of a white dwarf, where the two stars comprise a binary system. The white dwarf will settle down after the explosion and will begin to accrete hy- drogen again [105]. In contrast, supernovae occur when a massive star’s core collapses, releasing huge amounts of gravitational potential energy in a cataclysmic ex- plosion [106]. Binary eclipses result in a lower apparent magni- tude when either star is being eclipsed, and the high- est possible apparent magnitude when the stars are entirely uneclipsed [107]. Planetary eclipses occur when a planet orbiting the star moves between the observer and the star, prevent- ing a fraction of the star’s light reaching the observer. This leads to a reduction in apparent magnitude, which results in periodic dips sometimes seen in the light curves of certain stars. 29 Starspots are caused by the differential rotation of a star. This is when the different latitudes of the star rotate at different velocities. The magnetic field associ- ated with these latitudes gets wound around the star as the equator rotates quicker than the poles. Some mag- netic field lines exit the star one side of the equator and enter the other hemisphere. These exit and entry points are cooler than the surrounding environment, so appear darker [108] [109]. 4.6.2 The Effect of Starspots on Transits (CL) Starspots are a common feature of stars. They are active regions upon a star’s surface where the magnetic field penetrates to the surface. These spots appear darker due to these areas dispersing less energy and oc- cupying lower temperature ranges compared to the sur- rounding stellar surface [110]. Starspots are not con- tinuous, they can have lifetimes of a few hours to a few months, as the magnetic fields that cause them fade, so therefore the number and position of the starspots vary heavily with time. Figure 4.6.1 shows how these starspots appeared on the Sun on the 27th September 2001. Figure 4.6.1 – NASA’s image of the Sun showing the pres- ence of Sunspots on the 27th September 2001 [111] By studying the positions of these starspots it is possible to determine the stellar rotational period of the star, this is achieved by a measuring the move- ment of a spot with respect to time. The sun and its spots have been studied for centuries and around 400 years ago the first measurement for the rotational pe- riod of our Sun was achieved by directly tracking the movement of these areas of much lower intensity [112]. However, for stars further away from the Sun, it is not possible to resolve the starspots by direct imaging, so therefore other techniques are used in order to deter- mine the stellar rotation from the starspots. Today, there are three main methods in which to determine the rotational period of the star. The first is using spectroscopy to analyse the broadening of the spectral lines caused by the rotation of the star [112]. How- ever, this method does introduce a factor of sin (i) due to the observed inclination of the Doppler signal. The second method uses the principle that the stellar flux changes as the starspots move from the front to the back of the stars. This intensity pattern in a star’s light curve will repeat periodically, therefore giving a value of the period of rotation [113]. This method is superior when measuring stars with long rotational pe- riods, as the broadening of spectral lines cannot be re- solved at low rotational velocities. This method also has the advantage that the uncertainty factor of sin (i) is not incorporated into calculations [112]. The final method determines the rotational period using information of the starspots position from a tran- sit in the light curve. When a transiting planet passes over the starspot, there is a rise in intensity at that point causing a small positive bump in the light curve and misshaping the symmetric bottom of the transit [113]. This effect is shown in Figure 4.6.2. 30 Figure 4.6.2 – Top panel - Planetary Transit in light curve of the Sun recorded on the 26th April 2000 (Black) and the 29th April 2000 (Grey). Bottom panel - subtraction of the two light curves in the top panel. The arrows and dotted lines indicate the flux variations caused by the Sunspots. Data taken by the Big Bear Solar Observatory (BBSO).[112] The abnormality and rise in intensity is due to the starspot not being as bright as the surrounding photo- sphere, meaning that the average intensity of the star increases when the planet is covering this darker area [113]. In order, to determine a value for the rotational period of the star, the starspot needs to be detected in multiple transits. The ability to detect the same star spot in multiple transits depends on two main factors: the obliquity of the system (angle between the orbital axis and stellar rotational axis) and the rate of transit compared to the rate of stellar rotations. If the axis of the orbit of the transiting planet is aligned with the axis of rotation of the star then subsequent transits will be detected, as long as there is a second transit before the starspot has rotated around to the unob- servable side of the star. Misalignment of theses axes could cause a starspot to appear in the first transit but not in subsequent transits, even if the rotation of the star was sufficiently slow (See later in this section for further discussion) [113]. The period of rotation can then be easily calculated by considering the change in longitude of the starspot on the star in the time be- tween the transits, Δt. This is depicted in Equations 4.61 and 4.62: Ps = 2π Δt θ1 – θ2 (4.61) where θ1 and θ2 are the longitude values of the starspot at each point of measurement in each transit [112]. These values can be calculated using the equation: θi = arcsin ([ a Rs cos (lat) ] sin [2π (f i – 0.5)] ) (4.62) where the first transit corresponds to i = 1 and the sec- ond to i = 2. ‘lat’ is the latitude of the starspot, which can be deduced from the inclination and the phase f i is found from the phases of the positive and negative peaks in the subtracted light curve (See the bottom panel of Figure 4.6.2). The calculated values of Ps carry an uncertainty, this uncertainty can be reduced by repeating the measurements over many transits and taking an average value for Ps [112]. In reality, there are gaps in light curve data from space telescopes, therefore it is not possible to simply subtract one light curve from the other - as it was done for the Sun in Figure 4.6.2. Instead, a transit is mod- elled by using the predetermined parameters of the star and its planet. This produces a model transit without starspots that is then subtracted from the transit data with the spots present, to obtain a subtracted light curve highlighting the starspots. In order for the spot- less transit to be accurate enough to produce useful results, accurate values of the planet radius, stellar ra- dius, inclination and orbital radius need to be input into a model that accounts for limb darkening [112] (See Section 6.2.5 for more detail on the Stellar Limb Darkening Model). As mentioned previously, the presence of starspots in transits can also give information on the obliquity of the star-planet system. The Rossiter-Mclaughlin (RM) technique is the most commonly used technique to measure the obliquity of exoplanet systems. However, this spectroscopic technique proves to be challenging when analysing faint host stars that are typical of the exoplanet systems that Kepler has discovered. Many of these stars are also slow rotators, which is a disad- vantage to the RM method but crucial in the transit starspot technique [114]. Therefore, in order to ob- tain obliquities for these systems, the transit starspot method must be used. The route that a transiting planet moves across the planet is defined as the transit chord. If this transit chord lies over a starspot then the received intensity will be temporarily brighter and a bump will be pro- duced in the light curve, as explained before. A slowly rotating star in a perfectly aligned system will see the 31 presence of the same starspot in many recurring tran- sits. This is due to the transiting planet tracing out exactly the same transit chord with every full orbit, which is parallel and aligned with the starspots motion due to rotation. The starspots position will appear to move along the transit chord in subsequent transits as the star rotates. In this case, the starspot appears in all transits until the starspot is rotated onto the hidden side of the star [114]. In a misaligned system, the transit chord is not par- allel with the direction of the rotating starspots. This means that if a starspot is covered in one transit, the spot would have moved away from the transit chord when it transits for the second time and won’t be de- tected [114] [115]. Therefore, the probability of the rotational period of the star and the transit period be- ing related such that the starspot interacts with the transit chord again, with in the next several transits, is extremely low [113]. Figure 4.6.3 illustrates the two alignments discussed. Figure 4.6.3 – A simulation of the effect of a single starspot on four consecutive transits, in an aligned system (Top panel) and a misaligned system (Bottom panel). The ro- tation of the star is ten times slower that the orbit of the planet [114] These alignments can then be translated into esti- mates on the obliquity of the system, where the per- fectly aligned system has an obliquity of 0◦. This method is being used to determine the reasons be- hind the high spin-orbit angles measured between hot Jupiters and their host stars. In some cases, the di- rection of rotation of the star can be in reverse to the direction of the orbit of the planet [116]. It is cur- rently believed that the high obliquities in hot Jupiter systems are caused by one of two theories. The first is that the dynamical interactions, such as planet-planet scattering, tilt the orbits of the planets with respect to the stars rotation axis. The second is that the spin axis of the star can get tilted from its original position, in line with the protoplanetary disk, by chaotic ac- cretion, torques from neighbouring stars or magnetic interactions [114]. In principle, the latter of the two theories would be applicable to all of the exoplanet systems [114]. By investigating the obliquities using the starspot transit method for the many exoplanet systems that Kepler has discovered, the validity of the latter theory can be deduced [115]. So far, obliquities of these systems have been found to be low, suggest- ing that the high spin-orbit angles in the hot Jupiter systems are caused by dynamic interactions. However, this is ongoing research, more systems need to be ob- served in order for this conclusion to hold [114][115]. 4.6.3 Hill Spheres (EM) The Hill Sphere is the region surrounding a massive body in which the gravity of the body is dominant, within this region objects will orbit the larger body. The radius of the Hill Sphere is given by [67]: RH = ( m 3M )1/3 a (4.63) where M is the mass of the larger body if M >> m, m is the mass of the smaller body, and a is the separation of the two bodies. 4.6.4 Roche Limit (EM) The Roche limit is the boundary of the region around any massive body inside which, if a smaller body were to orbit, tidal interactions would cause the smaller mass to break apart. An estimate of the equa- tion for the Roche Limit is [67]: aR = 1.44 (M m )1/3 R (4.64) where M is the mass of the larger body if M >> m, m is the mass of the smaller body, and R is the radius of the orbiting body. This is derived from the equa- tion of the Hill Sphere by balancing the gravitational attraction for a particle on the surface of the smaller body against ‘self-gravity’ of the particle. This orbital radius is only an estimate as it is derived under the as- sumption that the small body undergoes no distortion; it simply calculates at which point the Hill Sphere is equal to the radius of the object. The more rigorously derived Roche Limit [67]: aR = 2.456 (M m )1/3 R (4.65) has a larger factor - 2.456 vs 1.44. The larger factor of 2.456 arises from tidal forces acting on the planet that 32 cause it to deform. This decreases the gravitational attraction between the object and its outer layers as they are pulled away from the centre of mass. 4.6.5 Three-Day Pileup (EM) When studying the distribution of periods of gas giants around a host star there is a peak found at a period of ∼ 3 days with very low eccentricity orbits. Planets found in this region are known as hot Jupiters and current understanding suggests that these planets formed past the ice line (see Section 4.1) and migrated to their current orbital radius either by interacting with the disk from which they formed, [57] or by ejecting smaller bodies from the solar system. The total energy of an orbit is given by: EOrbit = – GMm 2a (4.66) Ejecting matter reduces the orbital energy of the larger body as some is transformed into the kinetic energy of the smaller body. If enough mass is ejected in a short time, for instance a small planet being expelled, the result is a highly eccentric orbit. The subsequent planet-star separation corresponding to the periastron is important in the circularisation of the orbit. If the periastron is close enough to the star, the planet will undergo tidal interactions with the star. This process hinders the free rotation of the planet and acts to dis- sipate the orbital energy making it more negative and hence a more circular orbit. If the periastron is too close to the star it will enter the Roche Lobe and be torn apart, if it is too far from the star the tidal inter- actions will not dissipate enough energy to have any tangible effect on the orbit. The angular momentum of an orbit is described mathematically as: L = m(GM) 1/2 √ a(1 – e2) (4.67) where m is the mass of the planet, M is the mass of the star, a is the semi-major axis, and e is the eccentricity. If we assume that the eccentricity is approximately 1 then the initial orbital angular momentum is: Lini = m √ 2GMa(1 – e) (4.68) and equating this to the orbital angular momentum of the circular orbit gives: acirc = 2rper (4.69) As these planets are found with orbital periods of ∼ 3 days, a periastron value of ∼ 1.5 days appears to dis- sipate enough energy to circularise the orbit without pulling the planet apart. This phenomenon is only valid for Sun-like stars as much more massive stars have a larger Roche Lobe and can even have radii large enough that an 3 day orbital period would be inside the star. 4.6.6 Exomoon Detection (AE) Exomoons are natural satellites that orbit extraso- lar planets. Like the planets themselves, they are ex- tremely difficult to see as they have such a low apparent size and brightness. Although no exomoons have cur- rently been detected, there are theoretical methods of detection and the chances of discovering an exomoon are higher than before due to the increasing number of exoplanets being discovered [117]. The two main methods of detection are observing the light curve for a transiting planet and looking for unexpected dips, and by looking for TTVs in the transiting planet that could be caused by the presence of an exomoon. Detection by Light Curve Observation If an exomoon is present around a transiting planet, it can be detected by looking for distortions in the light curve of the flux of the star when the planet transits. This is because the moon also transits the star, as is shown in Figure 4.6.4, and so will also reduce the de- tected brightness of the star. Without the exomoon, the light curve should have a smooth, symmetric dip when the planet transits, as shown in Figure 4.4.6 in Section 4.4. Distortions appear when an exomoon is present be- cause what is actually being observed is a superposition of the transits of the planet and the moon. Figure 4.6.5 shows a case where the satellite is ahead of the planet (in terms of their orbit around the star) and so there is an initial small dip caused by the moon followed by a larger dip caused by the transit of both of the bodies before the dip slightly reduces in depth when the satel- lite is no longer transiting until finally the depth goes back to zero when the entire transit of the planet-moon system has completely finished. Figure 4.6.4 – The two transits of the planet and moon. 33 Figure 4.6.5 – Superposition of the two transits caused by the planet and exomoon [118]. As the relative position of the moon around the planet will vary from transit to transit, the distortion of the planetary dip will change. Detection by TTV Subsection 4.3.2, describes how TTV signals can be used to detect additional planets orbiting a star. In a similar way, this method can theoretically be used to detect the presence of satellites orbiting a transiting planet, i.e. exomoons. The following derivations as- sume the case where there are no other planets in the star system and only one satellite is orbiting the planet. Additionally, it can be assumed that the orbital planes of the star-planet and planet-moon systems are aligned with an inclination very close to 90 degrees. The planet orbits the barycentre (the centre of mass of the planet-moon system), with a semi-major axis aw . Since there will normally be a difference between the times at which the planet is at the mid-transit point (which is the point at which TTVs are measured) and at which the barycentre is at the mid-transit point we will expect to see transit timing variations. For a cir- cularly orbiting satellite, the TTV signal observed is simply given by the equation [119]: δtcir = ( asMsPp 2πapMp ) cos(fm) (4.70) where ap and as are the semi-major axis of the planet around the star and the exomoon around the barycentre of the planet-moon system respectively, Mp and Ms are their masses, Pp is the orbital period of the planet, and fm is the true anomaly of the moon which is defined as the angular distance between the perihelion of the moon (the point at which the moon is closest to the planet) and the actual position of the moon as seen from the planet. The RMS of the TTV signal is then [119]: δtcir,RMS = asMsPp apMp √ 2π (4.71) For an eccentric orbit the TTV signal has a more complicated form. There are now additional variables: the eccentricity of the planet and satellite denoted by ep and es respectively, and the pericentre positions de- noted by ω̄p and ω̄s. The RMS value of the TTV signal is now given by the equation [120]: δtell,RMS = 1√ 2 a 1/2 p asMsM –1 sp [G(M∗ + Msp)] 1/2 ζT(es, ω̄s) Υ(ep, ω̄p) (4.72) where Msp is the total mass of the planet-moon system, and ζT and Υ are given by the equations [120]: ζT = (1 – e2s ) 1/4 es [(e2s +cos(2ω̄s))(2(1–e 2 s ) 3/2–2+3e2s )] 1/2 (4.73) and Υ = cos [ tan–1 ( – ep cos(ω̄p) 1 + ep sin(ω̄p) )] · ( 2(1 + ep sin(ω̄p)) (1 – e2p) – 1 )1/2 (4.74) A range of as can be obtained by noting that the semi-major axis of a satellite around a planet must lie between the Roche limit, denoted by aR, and the Hill radius, denoted by RH . The Roche limit, given by Equation (4.65) in Sec- tion 4.6.4 is the minimum radius where particles can coalesce into a moon, or for an existing moon to be sta- ble and not be torn apart by the gravity of the planet it orbits [121]. The Hill radius is the maximum distance at which the gravity of a planet dominates that of the star so that the satellite orbits the planet, and is given by Equation (4.63) in Section 4.6.3. An estimate of as can be obtained by expressing it as a fraction of RH: as = χRH (4.75) where χ is a number between aR and 1. In fact, χ can be further constrained to have a maximum value of approximately 1/3 [122] because the Hill sphere does not take into account factors such as radiation pressure that would move the orbit of the satellite outside the Hill sphere. The period of the exomoon can theoretically be cal- culated by taking the ratio of the period of the planet around the star and the moon around the planet. The period of the moon Ps is given by Kepler’s third law: Ps = ( 4π2a3s GMsp )1/2 (4.76) and the period of the planet, Pp , is: Pp = ( 4π2a3p G(Msp + M∗) )1/2 ' ( 4π2a3p GM∗ )1/2 (4.77) Since as is a fraction of the Hill radius, it can be expressed as: as = χap ( Msp – Ms 3M∗ )1/3 ' χap ( Msp 3M∗ )1/3 (4.78) 34 and substituting this relation into Equation (4.76) gives Ps in the form: Ps = ( 4π2χ3a3p 3GM∗ )1/3 (4.79) and so the ratio of Ps and Pp is: Ps Pp ' √ χ3 3 . (4.80) The issue in the method of detecting exomoons us- ing TTVs is that there are two variables that cannot be observed: the mass of the exomoon and the distance between the moon and planet. The ratio of periods equation seems to reduce this problem since a finite range of values for the period of the moon and hence the size of its orbit can be calculated as shown above. Knowing as means that the only unknown in the equa- tion for the RMS of a TTV due to an exomoon on an eccentric orbit is the mass of the exomoon so this can be calculated. In a similar way to the TTV method, observing the duration of transit over many transits and looking for small variations can allow the inference of the presence of an exomoon. Under the same assumptions as the TTV method, a transit duration variation (TDV) must be due to the changing velocity of the planet due to its ‘wobble’ around the barycentre of the planet-moon system. The amplitude of the TDV signal is given by [120]: δTDV = (ap as )1/2( M2s Msp(Msp + M∗) )1/2 τ̄√ 2 ζD(es, ω̄s) Υ(ep, ω̄p) (4.81) where τ̄ is the average transit duration over many transits and ζD(es, ω̄s) is given by [120]: ζD(es, ω̄s) = ( 1 + e2s – e 2 s cos(2ω̄s) 1 – e2s )2 (4.82) It is interesting to note that the TDV signal in- creases with increasing exomoon eccentricity whereas the TTV signal decreases. By taking the ratio of the TDV and TTV signals, it is possible to remove Ms and hence be able to solve the equation to find Ps , and hence as . Using the as- sumption of a circular orbit for the satellite (es = 0 ), so that ζT = ζD = 1, and cancelling equal terms, the ratio of signals, denoted by η, is [120]: η = δTDV δTTV ' 2πτ̄ Pp √ 3 χ3/2 = 2πτ̄ Ps (4.83) Ps is the only unknown variable so it can be calcu- lated. as can then be calculated simply using Kepler’s third law. Once as is known, its value can be input back into Equation (4.76) to determine a value of Ms . So, although using either the TTV or TDV method in isolation to determine the distance of orbit and mass of an exomoon cannot give an accurate value, using both signals together does enable an accurate value to be determined. However, in the scope of this project, the accuracy of the TTV or TDV signals that can be detected are too low to be able to make any measure- ments about an exomoon or even infer its existence, just like when using TTVs to detect planets. 4.6.7 Circumstellar Habitable Zone (EM & MG) The Circumstellar Habitable Zone (CHZ) is a re- gion around a star where water can exist as a liquid. The CHZ is of special interest, particularly to astro- biology, as it gives a limit on where life, to our un- derstanding, can exist. If the planet can have liquid water on its surface then the conditions such as tem- perature and pressure would at least be similar to that of the Earth, providing the possibility of life. Deter- mining the CHZ for stars proves to be an incredibly challenging process as many factors have to be con- sidered and extensively modelled before an accurate result can be obtained. These factors include (but are not limited to) atmospheric composition, density, dis- tribution, tidal locking, stellar spectral type [123]. Habitable Zone Limits (EM & MG) There are several different methods for estimating limits on the inner and outer habitable zones. In in- creasing radial distance from star, these include: Re- cent Venus, Runaway Greenhouse, and Moist Green- house for the inner habitable zone (IHZ), and Maxi- mum Greenhouse and Early Mars for the outer habit- able zone (OHZ). The Moist Greenhouse limit is the point at which a planet’s stratosphere becomes dom- inated by water vapour [124] when surface tempera- tures are slightly higher than that of the Earth. An H2O-dominated atmosphere becomes opaque to infra- red radiation preventing the planet from re-emitting the radiation into space and preventing the cooling of the surface. A planet capable of retaining a moderate atmosphere which orbits close to its host star, within an orbit similar to Venus for an Earth-like planet and a sun-like star, is subject to the Runaway Greenhouse gas effect. The increased surface temperatures convert more liquid water into vapour - an effective greenhouse gas. The greater content of water vapour will, in turn, increase the atmosphere’s ability to contain the infra- red radiation. This cycle repeats creating a runaway effect, raising the surface temperature to levels which would be unlikely to support life, or at least anything other than extremophiles. An estimate of the inner limit of the CHZ is determined by considering at what point this runaway effect occurs [123]. The final model for the inner limit is the Recent Venus estimate. It con- siders that there has been no surface water on Venus for approximately 1 billion years and what solar lu- minosity was compared to current values, essentially an empirical method of the Runaway Greenhouse for values in the solar system. 35 The Maximum Greenhouse relies on a CO2 atmo- sphere providing the greatest heating possible for a planet, however if the concentration of CO2 is too high the albedo of the planet will be high and much of the incident radiation will merely be reflected into space. It requires a balance between the saturation of CO2 and the albedo as a result of this atmosphere. Finally the Early Mars limit recognises that the early Mars maintained liquid surface water 3.8 billion years ago, much like the Recent Venus the solar luminosity can be calculated at that time and compared to current solar flux at Mars to provide an OHZ limit. Stellar Spectral Type (EM) The spectral type of the star has a significant ef- fect on the CHZ. O- and B-type stars have high mass, which means the star formed more rapidly than a star such as the Sun. This restricts the planets to form for a shorter period reducing their accretion time, lessening the maximum atmospheric density. They also have in- credibly high effective temperatures, due to their high mass, and emit intensely in the UV. The simple as- sumption for the requirement of liquid water falls short in this instance as UV radiation is damaging to cells, the energy of the light being sufficient for ionisation to occur. The planet would need shielding against such radiation in order to be habitable; the planet needs a thick atmosphere capable of absorbing the bulk of the incident UV radiation. There is a limiting factor on a sufficient atmosphere, enough time has to pass during gaseous accretion during the planetary forming phase of the proto-stellar nebula. This requires that the star does not eject substantial matter from the system, i.e enter the T Tauri phase [125]. Combining this fact with the necessity for a solid surface upon which liquid water can lie, high mass stars present several momen- tous obstacles that must be overcome. Furthermore, the planet has to be detected before it is analysed. The large semi-major axis required to be in the CHZ means the planet is orbiting farther from the star, making a transit is exceedingly unlikely see Equation (4.34). Even with a transit present in the light curve data, de- tecting it will be incredibly difficult as the fractional change in intensity will be minute as the ratio of the radii will be very small see Equation (4.28). K-type stars have a different problem with habit- able zones. For a planet to have the correct equilib- rium temperature around a K-type star it must have an orbit that is significantly closer than that of the Earth’s. In principle detecting planets around K-type stars is easier and they are far more likely to be hab- itable than planets around giants. The habitable zone is located in the inner region of the system, therefore, a smaller perturbation of the planet’s orbit is required to cause it to become sufficiently close to the host star that it becomes tidally locked. A planet that under- goes tidal locking will have one face that remains il- luminated whilst the other is in perpetual darkness, thus an intense temperature gradient is engendered. This environment would hold the water primarily in two forms, vapour and solid on the respective sides of the planet, and prevent there from being substantial amounts of liquid water present on the surface. Habitable Zone Limit Calculation (EM) In order to calculate the habitable zones around another star, the luminosity of the star is required: L = 4πσSR 2T4eff (4.84) where σS is the Stefan-Boltzmann constant, R is the radius of the star, and Teff is the effective temperature of the star. Once the luminosity of the star has been calculated, the spectral flux must be computed: Seff = Seff� + aT∗ + bT 2 ∗ + cT 3 ∗ + dT 4 ∗ (4.85) where the coefficients correspond to those presented by Kopparapu et al. (2013) and are dependent on the limit used, T∗ = Teff –5780. The stellar luminosity and the spectral flux are combined with the solar luminosity in the following equations: ri = √ Lstar/L� S�,i (4.86) ro = √ Lstar/L� S�,o (4.87) where S�,i and S�,o are the solar fluxes calculated at the inner habitable zone and outer habitable zone respectively. The results of Equation (4.85) are tab- ulated in Section 7.2. These equations produce re- sults that apply to Earth-like planets in terms of mass, 1M⊕ – 10M⊕. Every planet discovered has had a semi-major axis calculated that was used in conjunction with the limits on the CHZ to categorise the planets as either orbiting within the CHZ of their host star or not. Unfortunately of all the planets that have been found and their orbital parameters computed, none have orbits that are per- manently place them inside the CHZ regardless of the limiting assumption used. No quantitative errors have been calculated for the limits of the CHZ. The limits themselves are based on a number of assumptions that do not account for every variable that is involved in a planet’s atmosphere such as cloud reflections regulat- ing the temperature. Each value is only an estimate of the limits and obtaining a numerical error would be infeasible. Mass Dependence (MG) The mass of a planet has also been shown to ef- fect where the boundaries of its CHZ lie, especially for the IHZ. The density of a terrestrial planet’s at- mosphere will increase with its mass resulting in the atmospheric column depth (scale height) of H2O being smaller than that of a less massive planet. This means that higher temperatures are required before enough water vapour is present in the atmosphere to dominate the outgoing longwave radiation, hence the Runaway 36 Greenhouse limit moves inward for high-mass terres- trial planets. The OHZ is not significantly affected by changing planetary mass as the increased albedo com- petes with the Maximum Greenhouse limit. This mass dependence of HZs around MS stars was explored by Kopparapu et al. (2014) [126] where planets of between 0.1M⊕ and 5M⊕ were modelled using a 1D, radiative- convective, cloud-free climate model using atmospheres of H2O with N2 as a background gas. The OHZ used CO2 instead of H2O as the main constituent of the at- mosphere. With the inclusion of 3D global atmosphere models from a variety of different studies, the limits of different HZs could be outlined for stars with an effec- tive temperature of 2600 K – 7200 K. From the results of this study, shown in Figure 4.6.6, it is clear that the IHZ limit is extended inward when the mass of a ter- restrial planet is increased, and vice versa, whereas the OHZ is not greatly affected. Figure 4.6.6 – HZ limits labelled for different planetary masses. The x-axis is scaled using the solar constant. Rapidly rotating and tidally locked planets (assuming a 4.5 Gyr tidal locking timescale) are divided with a horizontally sloped, black-dashed line. The coloured squares represent tidally locked inner-edge limits around cool stars from a different study. [126] The stellar fluxes used in Figure 4.6.6 are calculated using Equation 4.85. As discussed in Section 4.6.7, the IHZ of planets around cool stars is considerably closer than for Sun-like stars. Overall, it has been shown that the inner limit of the HZ moves outward for low-mass terrestrial planets – where the flux apparent is ∼10% lower than at Earth for a planetary mass of 0.1M⊕ – and moves inward for high-mass planets – where the flux apparent is ∼7% higher than at Earth. The impli- cations of this mass dependence prove to be positive for the search for life-harbouring, habitable planets using the planet-finding methods outlined in this chapter. A more massive planet with a similar density to Earth will have a greater radius. A greater radius means the planet will have a greater probability of being detected by the transit method. The probability of a transit oc- curring increases as the semi-major axis of the planet decreases. This all points toward the fact that a ter- restrial planet is more likely to be detected if it has a large mass – hence a large radius – and orbits close to its host star, therefore making it significant having the IHZ limit move inward with increasing planetary mass. Exomoons and Habitable Zones (EM) As discussed in 4.6.6, exomoons are currently purely theoretical but regardless are quite likely to ex- ist. Exomoons can orbit planets that are far beyond the limits of the habitable zone, and can still contain liquid water. It has recently been suggested and is quite likely that Ganymede along with Europa, moons of Jupiter, have a liquid ocean beneath their crusts [127] [128]. The orbit of Jupiter at 5AU is beyond the OHZ (1.7AU) [123] requiring a different source of energy heating the interiors of the moons. The tidal forces acting on a planet by their host star that cause orbits to circularise as discussed in 4.6.5 are the same that heat up the moon. Flexing the whole body away from its equilibrium generates heat through friction as the body distorts and reforms, the energy transferred in this way can be increased by the presence of other bodies. This is the case in the Jovian system where multiple moons interact at different times due to vary- ing orbital radii. The presence of multiple moons or- biting planets in our own solar system makes it likely that there will be similar structures in extrasolar sys- tems. Tidal flexing is related to habitability as it can result in warm vents on the ocean floor which are a place life can thrive, demonstrated in our very own oceans. Moons with liquid oceans may have analogues to warm vents and providing the necessary molecules are present, life is a distinct possibility. This means that despite none of the planets discovered orbiting within the stellar CHZ, there may be moons orbiting within a moon spe- cific HZ around their host planet. When it becomes possible to observe and analyse exomoons determining whether or not there are sufficient tidal interactions be- tween the satellite and the planet for an ocean to exist will be truly exciting. Delving even deeper into theoret- ical astrobiology opens the possibility for life that exists in the most extreme conditions. An example being Ti- tan, the largest moon of Saturn, undergoing precipita- tion cycles akin to that of the Earth but with methane as the molecule found in different states. Whilst com- pletely hypothetical, including moons as possible hab- itable bodies would completely change what is consid- ered the “Habitable Zone”. 37 Chapter 5 Asteroseismology Method 5.1 Initial Data Analysis 5.1.1 Plotting Raw Data (JG) The Asteroseismology Sub-Group was provided with frequency-power spectra for 40 stars. These came in the form of text files which were then input to MAT- LAB to obtain the spectra visually. Understandably, there was a wide range in the quality of spectra given, with the region of interest overwhelmed by noise in approximately half of the individual data sets. The spectra with a low signal to noise ratio required filter- ing, but those with a high signal to noise ratio could be used immediately by the team aiming to manually measure the large and small frequency spacing. 5.1.2 Filtering and Removing Noise (GM) In order to better identify the regions of interest on the frequency-power spectra and isolate the oscil- lation peaks, the data needed to be filtered to remove as much noise as possible. Two filtering methods were developed in order to achieve this. The first, a MATLAB function called sm3 , mainly used the inbuilt MATLAB function filter . The filter function was a one-dimensional digital filter, that used a rational transfer function as a moving-average filter. The moving-average filter was represented by the dif- ference equation y(n) = 1 W (x(n) + x(n – 1) + ... + x(n – (W – 1))) (5.1) where W refers to the window size of the filter. While this produced a moving average filter for the majority of the data, it did cause significant problems towards the two ends of the data set. This was due to the first value being calculated as y(1) = x(1)/W, the second as y(2) = (x(1)+x(2))/W and so on, so until there were the same number of x values as the window size, the averaged y value was always much smaller. With a window size of a few hundred points this meant that the ends of the filtered spectra were significantly smaller than expected and the problem was only accentuated with multiple iterations of the function on the same data set. To alleviate this problem, a second MATLAB func- tion was written which did not rely on the inbuilt filter function. This function, smooth, used a boxcar func- tion to average the data of a set size T with a window size W. Firstly the function calculated c = 1/W: the scaling amount for each value to be averaged. It then summed the first W values and multiplied them by c to calculate the average of that window and assigned it to the first value of the filtered set. The function then iterated (M-W+1) times through the entirety of data set, calculating the subsequent filtered values using the equation y(n) = y(n – 1) + (x(n + W – 1) – x(n – 1)) ∗ c (5.2) By using equation 5.2 rather than calculating the average for each subsequent window size, only three calculations were needed, rather than having to calcu- late the sum of W terms for every value to be filtered. For large window sizes this greatly reduced the num- ber of calculations that needed to be completed and therefore the time it took to complete each filtering process. This function could also very easily be looped to act as a double- or even triple-boxcar filter and fur- ther smooth the data. The only disadvantage to this function was that, due to the average of the window being assigned to the first value in the window, at the end of the data sets a set of (W-1) points was dis- carded. Looping the function a significant number of times could cause the vast majority of the data to be discarded. For a double-boxcar filter however, used to initially filter all of the frequency-power spectra, this was not a problem and smooth was used with great preference over sm3. 5.1.3 Isolating Regions of Interest (GM) After filtering, it was beneficial to isolate the re- gions of interest to aid in calculating Δν and δν. With several hundred thousand data points in each frequency-power spectra and the regions of interest generally only occupying around 20% of the spectrum, isolating these areas and discarding the unnecessary data would save processing power, and therefore time, and would potentially allow easier identification of smaller details in the spectra. Isolating the relevant regions of the spectra was done manually from MAT- LAB figures of the filtered data. This was done by highlighting the area of interest on the figure, which 38 MATLAB would automatically convert into a new set of data. This was done for all but three of the spectra, for which no region of interest could be found. 5.2 Manually Measuring Stellar Oscillation Properties 5.2.1 Measuring Large and Small Frequency Spacing (JG & ML) Initially, the power spectra with a high signal to noise ratio were plotted using MATLAB. Identifying and labelling the l = 0 peaks was an important part of the process to gain an accurate value for the large frequency spacing. On the spectra with a high signal to noise ratio, the l = 2 peaks were prominent in close proximity to the left of the l = 0 peaks, allowing us to distinguish between these and the otherwise very similar looking l = 1 peaks. For the power spectra with a slightly lower signal to noise ratio, the l = 2 peaks were identifiable af- ter a small amount of filtering, but this was not al- ways the case. For the power spectra with no identifi- able l = 2 peaks, the slightly irregular spacing between l = 1 peaks had to be used to distinguish them from the l = 0 peaks. If the irregular spacing was not appar- ent then it made less of an issue which peaks were used to identify the large frequency spacing, as this method would not be used to yield final results. To measure the large frequency spacing, the power spectra were first smoothed by the computational anal- ysis team, as mentioned above. This made the centre of each peak much easier to identify. MATLAB’s data cursor tool was then used to accurately determine the frequencies of the centre of the peaks. The large fre- quency spacing was measured for several pairs of peaks and an average taken to create a more accurate value. This process was repeated for the small frequency spac- ing where possible. This process provided the manual measurements team with familiarity with each power spectrum. As the computational analysis team were using a built in function to identify peaks, incorrect spacing values were obtained for a small number of spectra. Once all spacings had been measured computationally, re- sults were compared with the manual measurements and incorrect results were identified. The computa- tional measurement process was then repeated with the aid of the manual analysis team to correct these few discrepancies. 5.3 Computationally Measuring Stellar Oscillation Properties 5.3.1 Measuring Large and Small Frequency Spacing (LS) The computational method for finding the fre- quency spacings involved identifying peak positions and from there calculating the spacings between rele- vant peaks. The MATLAB iPeak function, written by Thomas O’Haver, was used to find peaks in the data and flag them depending on threshold values for peak parameters [129], namely the peak amplitude, gradi- ent, sharpness and width. These thresholds could be altered interactively whilst simultaneously viewing a plot of the signal, allowing the important peaks to be identified for each star. The function also allowed the frequency position of the peaks to be exported with high precision. The iPeak function worked by using the derivative of the data looking for zero-crossings, which indicated turning points. To help to eliminate the effects of noise, it first smoothed the derivative. The results could be further narrowed down by imposing a minimum value on both the slope of the derivative at the turning point and the amplitude of the peak. The position of the peak was then identified by fitting a portion of the unsmoothed signal at the turning point, as smooth- ing distorts the shape of the peak. The width of the smoothing filter, the amplitude, the slope of the deriva- tive and the amount of the original peak fitted to de- termine the position were all parameters which could be altered as required. However, as it was simple to pick out the relevant peak positions, only the ampli- tude threshold was altered to reduce particularly large numbers of false mode detections due to noise. After exporting the data for each power spectrum from MATLAB into a spreadsheet, a number of values for Δν were calculated around the mode which approx- imately corresponded to νmax . Four values of Δν were calculated where there were sufficient numbers of l = 0 peaks and the mean of these values was calculated to give a single value for the frequency-power spectrum. The same process was then used to calculate δν where the l = 2 could be identified. A more simplistic approach to peak identification, which was discarded in favour of iPeak, was to apply a power spectral density amplitude threshold to the data and to use data points above that threshold to find peaks. The major failing of the threshold method was its inflexibility. For instance, l = 2 modes often merged into the l = 0 peaks in filtered data. It was very difficult for such l = 2 modes to be detected us- ing the threshold method as a peak was considered to begin when the data were above the threshold and end when they went under. The peak position was given by the position of the maximum power spectral den- sity in that frequency window. Therefore, for an l = 2 mode which only manifested as a small bump on the 39 side of an l = 0 mode, the threshold value needed to be in a narrow range of values between the bottom and the top of the l = 2 mode in order for both peaks to be detected. This was impractical to implement as the correct threshold for one mode often failed for another. Another alternative method that was considered for extracting the frequency spacings was to use the auto- correlation of the signal. The autocorrelation is found by multiplying the signal by itself with a certain ap- plied lag in frequency and then integrating over all values of frequency in the signal. It can be used to find periodicity as it peaks when the signal and its lagged version are similar to each other. The auto- correlation function of a star’s power spectral density plotted against frequency lag will therefore show peaks for the frequency spacings. The peak corresponding to the large frequency spacing would be the second main peak from the centre of the function, with the peak first from the centre being the average frequency spac- ing between l = 0 and l = 1 modes. In signals where the l = 2 modes are visible, plotting the autocorrela- tion function would also show a smaller peak beside the large frequency spacing peak. The distance between this smaller peak and the large frequency spacing peak would give the small frequency spacing. The autocorrelation method was simple to imple- ment, but noise in the signal made peaks in the func- tion much less obvious. This could be circumvented somewhat by setting points which were below the noise threshold to zero before autocorrelating. However, the small frequency spacing often became obscured as it was not unusual for some l = 2 modes to be less pow- erful than the noise. This was another reason that the iPeak function was the most suitable method for extracting the frequency spacings. It did not matter if the data was moderately noisy, because iPeak still identified small l = 2 modes even if it also picked up spikes in the noise. For our stars, it was not difficult to pick out only the relevant peaks in the exported data so that the noise could be ignored. 5.3.2 Measuring Frequency of Maximum Power (LS & GM) To find νmax , the frequency of maximum power, a Gaussian curve was fitted to the signal using the MATLAB function gaussfit [130]. This function used an iterative least-mean-squares algorithm to optimise the position and width of a Gaussian peak that best matched the data. The function used the position of the maximum power as an initial guess for the position of the Gaussian peak. After one hundred iterations, the final value of the centre of the Gaussian was taken to be the value of νmax . Previously, it had been attempted to find νmax by repeatedly smoothing the data until the signal was re- duced to a broad peak. However, it was difficult to select an appropriate window size as both small and large windows had separate disadvantages. Small win- dows needed many repeated runs for any significant smoothing to be seen. Due to the powerful noise in the low-frequency areas of the power spectrum, large win- dow sizes caused issues with skewing the signal to be more powerful at the low-frequency end. In addition, the smoothing method was computationally expensive when the whole dataset was smoothed. In an effort to correct for this, the signal was cut out of the data before smoothing. This was not suitable either as edge effects, either at the beginning or the end of the data depending on the smoothing code, caused the loss of significant amounts of the signal. Experiments also showed that fitting a Gaussian curve to the smoothed data using gaussfit produced almost exactly the same value for νmax than using gaussfit on the unsmoothed data, making it an almost redundant effort. 5.4 Obtaining Asteroseismic Results 5.4.1 Using Scaling Relations (MH) Initial results from the scaling relations were avail- able as soon as a value for νmax and Δν0 were obtained manually for each star. The simplicity of the scaling relation method made it easy to create a spreadsheet that used the relations to calculate the mass and ra- dius as soon as the manually measured νmax and large spacing were entered, along with the effective tempera- ture for each star. The uncertainties for the measured quantities were also entered in order to calculate an error for the mass and radius. The errors were calcu- lated by adding the uncertainties on each component of the scaling relation in quadrature, with an extra error that accounts for the intrinsic inaccuracy of the scaling relations due to the assumptions that they require. Once the computational methods to obtain more accurate values for νmax and Δν0 were developed, these values were similarly added to the spreadsheet and the scaling relations were used to calculate a sec- ond set of values for the mass and radius of each star. The errors on the values measured computationally were significantly lower, and this was reflected in the calculated errors for the mass and radius. These more accurate results were then passed to the Planet-Finding Sub-Group for use in producing absolute, as opposed to relative, values for quantities including planet radius. 5.4.2 Using Asteroseismic Diagrams (GM) The Asteroseismology Sub-Group was provided with the oscillation information for approximately 70,000 stellar models computed using the Modules for Experiments in Stellar Astrophysics (MESA) stellar evolution code. The models ranged from a mass of 0.8 to 1.5 solar masses with metallicities of Z = 0.005 to Z = 0.08. Each combination of mass and metal- licity had approximately 70 models throughout their evolution, generally starting with a central hydrogen mass fraction, Xc of 0.7 and ending at around 0.05 at 40 the end of the MS. The frequency data were detailed for modes with angular degree ranging from l = 0 – 2 and l = 0 generally had radial orders from n = 0 – 30 . The number of radial orders for l = 1 and l = 2 varied significantly for each model. Extracting Frequency Data A C++ program was written to automatically ex- tract and calculate the relevant values from the model data. The program worked by reading the frequency data file path and model parameters - namely stellar mass, radius, metallicity and hydrogen mass fraction - and opening the frequency data file. The program calculated νmax for the model using the scaling rela- tion in Equation (3.8) and the model parameters, then iterated through each of the l = 0 radial modes to find the closest mode to νmax . While the scaling relation for νmax could lead to slightly inaccurate results for the less solar-like stars, as it was only used to find the ap- proximate area to calculate Δν and δν this would not have much, if any, effect on the final values so was an acceptable method to use. The program first counted the total number of radial orders for l = 0 and l = 2 angular degrees, before iterating through the frequen- cies of all of the l = 0 modes to find the radial order closest to νmax . The program then checked that this radial order was not greater than the highest available radial order for the l = 2 angular degree in order to en- sure that δν could be calculated. If the selected l = 0 radial mode was greater, the program decreased the radial order incrementally until an acceptable radial mode was found or the program ran out of radial or- ders for l = 0 , at which point the model was discarded. Following this, the program selected the three ra- dial modes above and below νmax and calculated five values for Δν from the difference between adjacent l = 0 modes. To calculate δν, the program calculated the difference between each selected l = 0 radial mode and the previous l = 2 mode, to give six values. The mean of these values was then calculated to give singu- lar values for Δν and δν and the program input these values and model parameters into a .txt file for later use. If the program could not compute Δν and δν for some reason - due to lack of radial orders for l = 0 or no corresponding l = 2 modes, or the calculation pro- duced an unexpected or negative result - the model was discarded. Erroneous values were mainly calculated for models nearing the end of the MS due to the rapidly changing properties of these stars. Plotting the Asteroseismic Diagrams All calculated values for Δν and δν were grouped in files corresponding to the model mass and metallicity and these were loaded into MATLAB as arrays. The model data were then plotted using a MATLAB func- tion, which mainly used the inbuilt MATLAB function scatter. Using a scatter graph for the mass lines rather than a line graph meant that the data did not need to first be ordered chronologically and meant any anoma- lous results were easily identified and could be ignored. The function iterated through the various masses for each metallicity and plotted the small frequency spac- ing against the large frequency spacing. Only inter- vals of 0.1 solar masses were plotted so that the graph was still legible - plotting all available masses for each metallicity meant that it was almost impossible to see howΔν and δν varied as the star evolved for each mass. For masses greater than 1.2M�, the mass lines tended to be in much closer proximity to each other and in some cases were almost overlapping. Adding ten ex- tra mass lines between these intervals would have been unfeasible. In order to extract data about the stellar ages, the hydrogen mass fraction of the star needed to be deter- mined from hydrogen mass fraction isopleths. Unfor- tunately the data for various masses on the same graph did not necessarily have the same exact hydrogen mass fraction values and the closest value to a desired hy- drogen mass fraction often varied by 0.01 or greater for different masses of the same metallicity. In order to get true isopleths the values for regular isopleth intervals needed to be known. For the ZAMS age, this was less important, however it was decided to use intervals of 0.1 with values for ZAMS (around 0.7), 0.6, 0.5, 0.4, 0.3, 0.2, 0.1 and 0.05. The final value was used only on mass lines less than one solar mass, or in some cases 1.1 solar mass, as these were the masses for which val- ues of Δν and δν tended to be less volatile as the star reached the end of the MS. While the isopleth separation was not a linear re- lationship, for small intervals in the hydrogen mass fraction of around 0.01 it could be approximated as one. In order to calculate values for Δν and δν for the isopleths, the known values of the isopleths immedi- ately above and below the desired isopleth value were weighted in order to give the closest match to the iso- pleth value. For example, for the Xc = 0.4 isopleth for the 0.4M�, Z = 0.017 model set, the known values of Δν and δν were for Xc values of 0.410321 and 0.394821. The values for Δν and δν were thus weighted 33% and 67% respectively in order to calculate the correspond- ing values for an Xc value of 0.4. By calculating the isopleths using this method it meant that the isopleth line was the best representative for the respective hy- drogen mass fraction and improved the accuracy when adding the project’s stellar data. It also allowed any outliers on the mass lines to be easily identified as, if an anomalous value was used to calculate the isopleth, it would not follow the expected trend. This method allowed these values to easily be discarded and the next closest known isopleth value to be used and weighted differently. In total, 14 asteroseismic diagrams with complete hydrogen mass fraction isopleths were com- pleted to be used in the next stage of the project and an example of one can be seen in 5.4.1. While the sub-group had been given metallicity values for the stars, these were in the form of [Fe/H] metallicities. Asteroseismic diagrams generally use the heavy mass fraction, or Z value, so these values needed to be converted to the correct form before the stars 41 Figure 5.4.1 – The completed asteroseismic diagram for metallicity Z=0.09 could be plotted on the correct asteroseismic diagram. The total metallic abundance of the star, [M/H], is re- lated to the metallicity by [131]:[ M H ] = log10 ( Z/X Z�/X� ) (5.3) The total metal abundance [M/H] is a more general expression than the iron abundance [Fe/H] and the two are related through a constant, A, by [131]:[ M H ] = A ∗ [ Fe H ] (5.4) where A generally takes values between 0.9 and 1. By equating Equations (5.3) and (5.4) the relation be- tween Z and [Fe/H] can be written as: log10 ( Z/X Z�/X� ) = A ∗ [ M H ] (5.5) Rather than spend time writing a program to com- pute the Z value from the [Fe/H] value, it was found that calculators to do this already existed online [132]. By using this calculator, the corresponding Z value for each star was computed. As the Z values for the stel- lar models did not correspond exactly to the Z values of our stars, any small inaccuracy caused by using a non-specific calculator was rendered insignificant when plotting the stars on a non-perfect asteroseismic dia- gram. Since the asteroseismic diagrams were plotted asΔν against δν, only stars where both of these values had been calculated could be added. This requirement im- mediately reduced the number of stars that were avail- able to plot, as for some of the frequency-power spectra the signal to noise ratio had not been great enough to calculate δν. It was decided to also only use the com- putationally calculated values to plot the stars, which reduced the total number of stars to be plotted to 22 stars. These stars had a metallicity range of Z = 0.09 to 0.35 which corresponds to roughly 0.5 to 1.5 solar metallicity. The stars were grouped by the closest as- teroseismic diagram metallicity, with all stars having no greater than a 0.003 difference to its asteroseismic diagram metallicity, which would not make any signif- icant difference when the stellar values were extracted. Only six asteroseismic diagrams would be needed to plot all available stars, with metallicities of Z = 0.009, 0.011, 0.014, 0.017, 0.023, 0.029. To add the stars to the diagrams, a MATLAB func- tion was written where the user could open the relevant diagram and input the values for Δν, δν and the star number. The star would then be plotted in the correct place on the diagram and would be labelled automati- cally. The function also plotted the stars as a different colour and marker than the mass lines and hydrogen mass fraction isopleths so they were easily identifiable on the graph. Error bars were initially added to the star plots to make constraining the mass and age of the star easier when it came to extracting stellar proper- ties, however these could not be identified easily enough due to the size of the asteroseismic diagram so were left out of the main plots. Extracting Stellar Properties Due to the design of the asteroseismic diagrams, ex- tracting the mass and hydrogen mass fraction was very simple, and was done by calculating its position relative to the closest mass lines and isopleths. This was done manually, and the variation in mass and hydrogen mass 42 fraction was approximated as a linear relationship be- tween adjacent values. Using this method all but two of the values were found. One of these values was only slightly beyond the scope of the asteroseismic diagram to which it was added so could be extracted, albeit to a lower accuracy, however no values could be extracted for the other due to its position on the asteroseismic diagram. In order to further constrain the masses and ages for each star an attempt was made to produce smaller- scale asteroseismic diagrams for each star - showing mass lines with separations of 0.01M� and hydrogen mass fraction isopleths with separations of 0.001. On these asteroseismic diagrams the errors for Δν and δν for each star were more significant and were also plot- ted with the star. However, it was found after complet- ing this for GSIC 3 that the error for the mass and hy- drogen mass fraction value was not significantly differ- ent from the values obtained by the previous method. Calculating each isopleth for the smaller separations took significantly longer than for the greater sepa- rations and it was decided that the amount of time needed to do this for each star was too significant for the small reduction in the constraint boundaries it would produce. GSIC 3 remains the only star this further method was completed on as an example of its results. The stellar ages were related to the hydrogen mass fraction of the star - the lower the hydrogen mass frac- tion, the older the star would be. This was, however, only true for stars on the same mass line due to stars with different masses ageing at vastly different rates. The stellar ages, radii, luminosity and effective temper- ature of each model were supplied alongside the mass, metallicity and hydrogen mass fraction of each, so com- puting these values for the project’s stars was a case of matching the star to the closest available model. This was an optimisation problem and while a search pro- gram could have been written to do this automatically, it was easy enough to go through the model data man- ually to obtain this values. Due to the sheer number of models supplied, for the vast majority of the stars an exact match to the extracted hydrogen mass fraction could be found and for the others it was never more than a 0.01 difference. Due to the age, radius, luminosity and effective temperature values being dependent on two variables, the mass and the hydrogen mass fraction, an error in either of these values would mean a different stel- lar model would have to be used. Therefore, to con- strain them, the values for each from the models at the maximum and minimum mass and hydrogen mass fraction were calculated and the greatest difference to the already-calculated stellar properties were used as the error for that value. Amalgamating the Data In order to compare the masses, ages, radii, lu- minosities and effective temperatures of the stars and compare the two methods used to compute the values, it was decided to display all of the data for the stars on several graphs. These were created using MATLAB’s inbuilt errorbar function, which plotted a line graph of the data with the various errors for each value. By removing the line using MATLAB’s figure editor and adding point markers to the various points, it was very easy to compare the various data. Several graphs were plotted in this way and are discussed later in this report in Section 7.2.3. 43 Chapter 6 Planet-Finding Method 6.1 Manual Methods (LP, CL & AE) A Python code was used to plot light curves for the short and long cadence Kepler data separately. The plots produced by this light curve reader code could be analysed manually to find any dips in flux that may cor- respond to a transiting planet. The code also counted how many data points were present in each data set (long and short cadence) to enable the user to get a feel for the size of the data set, useful for running fur- ther codes outlined in Section 6.2. Initially, a total of forty light curves were analysed using manual methods by a team within the Planet-Finding Sub-Group. Dur- ing the analysis, any periodic dips in flux were noted with the periodicity, transit depths and transit dura- tions analysed. The curves produced were varied, some displayed clear transits with little noise, therefore en- abling results to be collected with low error values. Other curves however were more difficult to analyse. With greater noise levels, values could not be collected with as much certainty and error values were greater. The short cadence data contained more data points and therefore was used for analysis in favour of the long cadence data, as values collected from the curves would be more accurate. On top of this, the transits were much easier to spot on the short cadence curves than the long cadence curves. Apparent recurring transits were searched for along each light curve. Once the shape and periodicity of the transits were considered constant, analysis of the transits could begin. Transits were analysed in groups of ten, with data for consecutive transits being ideal as variations in period could more easily be noted. On top of this, three of the light curves revealed two different transit patterns, which were analysed separately. This indicated that multiple planets were in orbit around these particular stars, rather than just one. Fluctuations in flux were apparent throughout each light curve even with transit dips disregarded. In some systems, the flux readings consistently fluctuated, whilst in others, the changes were more gradual. One possible reason for such flux changes are the oscillations of the host stars. Changes in stellar luminosity occur frequently when stars expand and contract. The ap- pearance of starspots on stars also results in changing stellar luminosity. Gaps also occur when looking at the data, preventing us from seeing any transits or fluctu- ations in flux, during particular time windows. These gaps are generally as a result of scheduled movements or rolls of the spacecraft and down-linking events [133]. These factors resulted in some slight difficulties when searching for consecutive transits manually. The extraction of numerical data from the tran- sits was a relatively simple, yet laborious process, with all values, (including errors) checked manually. The transit depth was measured by recording the flux just before the start of a transit, and simply subtracting this value from the lowest flux value at the bottom of the same transit. Any uncertainty in these values due to noise was recorded and used to find errors in each value for transit depth. The values for the error were collected by considering the upper and lower limits of flux just outside the transit and at the bottom of each transit. The transit duration was recorded using a similar method, with the times at which the transits begin and end recorded. The beginning of a transit was taken to be the point at which the flux begins to fall, with the end taken to be the point at which the flux levels off after rising back up from the transit well. Again, un- certainty is present for these times as the beginning and end of a transit is unclear at times. This is particularly true when limb-darkening and varying impact param- eters are considered, which give varying gradients of slopes at which the ingress and egress occur. Values for the error were collected by considering the upper and lower limits for the time at which the beginning or end of a transit could be considered. Period values were found in a slightly more com- plex way. The times for the beginning and end of each transit were added together and then divided by two to find a central point in time for each transit. The times recorded between each of these average values were then considered as period values for the transits. Error values for the period were found by using pre- viously obtained error values for the transit duration, with the manipulation of these values used to validate them for the central points of each transit. After ten sets of data had been collected for each ap- parent planet, average values for period, transit depth and transit duration were taken with errors included in the calculations. The values were then used to aid the computational team, giving a preliminary indication of 44 the properties of the transits of each planet. 6.2 Computational Methods 6.2.1 Introduction (FC) Once all the transit property values had been calculated manually, the next step was to refine and build upon them using computational methods. Using a computer program to determine these quantities enabled higher accuracy and greater precision, which was necessary in order to obtain meaningful results for planetary and stellar properties. The hand-calculated values were useful as first estimates for the parameters input into the Python codes created, but could not be used for final results as the uncertainty on these values would be far too high. Several codes were built in Python 3.4, which all worked in col- laboration to achieve the high accuracy values for planetary and stellar properties found in the results in Chapter 7. The most important of these codes are described in detail in this section, explaining how they function and why they were necessary for the obtention of results. The codes to be discussed include: · Transit detection code · Phase folding code · Transit fitting model (uniform source and limb- darkening versions) · Limb-darkening numerical model Each of these codes worked in tandem with one an- other in a step by step process of obtaining increas- ingly accurate planetary property values from light curves. In particular, the orbital period was first esti- mated manually using a simple light curve reader code, then improved upon using the transit detection code, then further improved using the uniform source tran- sit fitting model, and finally the most accurate and precise value possible was determined using the limb- darkening transit-fitting model. The highly accurate period value could then also be used to help constrain other parameters in the limb-darkening fitting model such as planet radius and the semi-major axis. 6.2.2 Transit Detection Code (FC & AW) This computer code was built with the primary pur- pose of detecting transits and acquiring more accurate planetary orbital period values. A more accurate pe- riod would not only clarify information about the exo- planetary system, but is essential for successful phase folded transits, and for use in the transit fitting models. The process for detecting transits involved binning the light curve into bins of equal width, and taking the mean of each bin. The code initially plotted the light curve for the input star, as mentioned previously for the manual methods. This image, along with the hand-calculated values, allowed the user to gain an idea of the most efficient bin size to use, as the bins had to be small enough for a drop in the mean intensity to be noticeable, but not so small that the code took too long a time to process this simple step. The code then calculated the difference between the mean values of each consecutive bin, and flagged up any significant drop in intensity. The user also had to define what drop in intensity is considered ‘significant’ based upon the hand-calculated values of the transit depth. These flags are shown as the red points in Figure 6.2.1. This also gave a rough estimate of the number of transits detected within the data set looked at by the code. However, this was dependent on the quality of the ini- tial user-input values, and so this step often required repeating several times until there was approximately one flag per transit. However, this step did not need to be completely accurate, as the code was able to account for discrepancies such as missed transits, double detec- tions of a single transit and also spikes in the signal noise. The next step was to find the central point for each transit. To do this, a small range of data was taken either side of each flagged point, and further binned using very small bins (ranging from 3 to 5 data points). The range either side of the flagged coordinate to be inspected was input as the hand-calculated value for the transit width plus some additional leeway in or- der to encompass the entire transit. Subsequently the mean of each bin was once again calculated, after which the minimum mean value in the inspected region was taken, representing the centre of the transit. These points were then plotted onto the light curve, an ex- ample of which is shown in Figure 6.2.2, demonstrating the effectiveness of the code to find the centre of each transit. Any of these points which may not precisely be located at the centre of the transits, due to irregularly shaped transits or particularly large spikes in noise, will be averaged out for the most part by inspecting a vast number of transits for each star. The average difference between consecutive transit centre times was then taken in order to determine the orbital period. The differences between every other and every third transit were also calculated, and divided by 2 and 3 respectively. This was to increase the number of values used to calculated the mean period, with the hope of reducing the standard deviation on the result. In order to account for false detections, such as noise detections and double transits detections, as well as missed transits, any differences between transit centre times that were found to be too dissimilar from the hand-calculated period were discounted. This also en- sured that the period was only calculated for a single orbiting planet at any given time, and that the cal- culated period was not influenced by the transits of a second planet. Once all outlying values had been discounted, the mean orbital period with standard de- viation could be taken and output. Due to the constraint built into the code for the period value, if this value were changed to reflect the approximate period value of a second orbiting planet, then the transit detection code was able to detect the second planet and determine the orbital period. How- 45 Figure 6.2.1 – Flagged transits for GSIC 1 short cadence data. Figure 6.2.2 – A single transit for GSIC 1 short cadence data. ever, the code was only able to detect one transiting planet at a time, a limitation requiring the light curves to be inspected manually initially in order to deter- mine whether or not it was necessary to look for a sec- ond planet. Although, from the output period values, one could analyse this list manually and see if a signifi- cant number of similar values were present for another value of the period (which is not an integer multiple of the primary transiting planet) which would imply a second transiting planet. This ‘quasi-computational’ approach was actually utilised successfully for a couple of very noisy light curves such as GSIC 8 (which was also found to have two transiting planets) and GSIC 24, where it was easier and faster to distinguish pat- terns in the output list of period values manually, than it was to change the code to reflect these issues. On the other hand, for the cleaner light curves where the noise did not obstruct the detection of tran- sits, the transit detection code could calculate orbital periods in days to a reliable precision of approximately three to four decimal places, which was found to be superior to those calculated manually. Although some- times struggling with more noisy light curves, if sup- plied with enough transits to calculate over, the code could still output the periods to a higher accuracy than the hand-calculated values. Another example, in addi- tion to GSIC 8 and GSIC 24, where the transit detec- tion code was particularly useful was for determining the period of the planet from the GSIC 22 light curve. This data was also very difficult to analyse due to lots of noise and relatively small dips in intensity during the transits. Furthermore, the period value calculated from the manual methods approach was not accurate enough to achieve a successful phase folded light curve. However, using the transit detection code, the period was found accurately enough and hence could be used in the transit fitting models to further improve on the accuracy, as well as helping to determining other pa- rameters, discussed in Section 6.2.4. 6.2.3 Phase Folding Code (FC) Following on from the transit detection code, a transit phase folding code was created. This was re- quired as the light curves needed to be phase folded in order to determine whether or not the period value 46 calculated was accurate enough. For this process to be successful, it was essential that the period was very accurate, as when hundreds of transit curves are over- laid on top of one another, it only takes a very small difference in the period for the transit to no longer be visible in the phase fold. Figure 6.2.3 shows an exam- ple of a successful phase folded light curve (of GSIC 34), output by the phase folding code. The code requests the required GSIC number from the user, followed by the period value in days. This pe- riod value could either be automatically taken from the text file output by the transit detection code, or could be input manually. The actual phase fold is achieved by dividing all the time coordinates by the input pe- riod value and taking the remainder. This remainder is then plotted against the flux, resulting in a graph similar to that shown in Figure 6.2.3. The estimated phase value of the transit was also obtained from this plot for each planet, which was used as one of the input parameters for the transit fitting code. Figure 6.2.3 – Phase folded light curve for GSIC 34. The tight distribution of data points means an accurate period value was used. 6.2.4 Transit Fitting Code (OH & FC) To obtain more accurate results and more precise errors, as stated above, it was essential that a more sophisticated method was used to extract results from the light curves, besides studying the data manually. For this, an analytical model was fitted to the long ca- dence light curves to produce parameters for a best fit. The model used was that set forth by Mandel & Agol (2002) [134], and transcribed into Python by Crossfield [135]. This model was fitted to the data using the em- cee Python implementation of Goodman & Weare’s Affine Invariant Markov Chain Monte Carlo Ensemble Sampler [136] [137]. Two versions of this code were produced: one fit- ting a uniform source model to the data, and one fitting a quadratic stellar limb darkened model to the data. An example of the resulting shape of these models, as fitted to phase folded light curves, can be seen in Fig- ures 6.2.4 and 6.2.5 respectively. The final version of the uniform source model code required input of sev- eral parameters: the orbital period of the planet, the semi-major axis in units of stellar radius, the planetary radius in units of the stellar radius (Rp/R∗) and either an estimation of the time of the first visible transit or the phase of a visible phase fold, of which the latter was used as input in almost all instances. Besides this, the limb darkened model also required first guesses of the inclination of the system, and two quadratic limb- darkening LDCs. The limb-darkening equation utilised in this model, obtained from Mandel & Agol’s work on analytical transit curves, uses a small planet assump- tion: the ratio of the planet radius to stellar radius is 0.1 or less. This drastically reduces the complex- ity of the model and allows for more rapid computing of transit light curves whilst maintaining a reasonable model accuracy [134]. A comparison to an analytical limb-darkening model and an analysis thereof can be found in Sections 6.2.5 and 7.4 respectively. Figure 6.2.4 – A fit of the uniform source model to a phase fold of GSIC 1. As can be seen, the period is especially well constrained. 47 Figure 6.2.5 – A fit of the limb-darkened model to a phase fold of GSIC 1. As can be seen, the depth is especially well constrained. As this figure was produced on a low amount of iterations, the fit is imperfect, as can be seen by the width of the fit, which is too small. After setting reasonable ranges in which to look for the best fit, emcee then performed a random walk within this range attempting to match the parameters to one another and to the data by working to find a better value of χ2, or the likelihood, which related to the error on the flux values and the difference between the model and the data. It did so using a set amount of ‘walkers’ each ‘stepping’ a set number of iterations. If, after a random step, a walker was found to wander further from the best fit, it reversed its ‘direction’ so that the next step would tend more towards an ideal fit. As such, if run for long enough using a large enough number of walkers and iterations, the program returns a visualisation of the most probable results found to be a best fit. This visualisation can be seen for the limb-darkened fit of GSIC 35b in Figure 6.2.6. As can be seen, there are seven 1D histograms, one for each parameter, which hold the results with the high- est probability of a perfect fit. The mean value of these histograms is then taken to be the final result, with the error in this result being the standard deviation on these values as calculated by Python. The plots found at an intersection between two different parame- ters represent the 2D probability density of correlation between the two parameters, which in an ideal case is circular in shape, as can be seen in the majority of plots in Figure 6.2.6. These serve as a clear indica- tion of how successful the code has been at finding a high probability of similar results (as can be seen in this figure). This particular figure contains plots for seven parameters, whilst the plots output by the uni- form source transit fitting model would only contain five parameters, as this simpler model does not require the input of values of the inclination or LDCs. As such, one can imagine that the limb darkened fitting code would take significantly longer to run, as was in- deed the case. It is also important to note that the case of GSIC 35b returned very successful results, and that not all resulting histograms were Gaussian in shape nor the probability densities circular. While an increased number of iterations or walkers for the emcee func- tion naturally increases the accuracy of the results and the visualisation thereof, it also significantly increases the time required to run the code. As such, the final results are limited, to a degree, by the time and com- puting power available during this project. The final version of both programs output a low resolution plot of the phase fold to gauge the success of the program, a plot of the 1D and 2D probability distributions as de- scribed above, and the resulting values and standard deviations in ASCII format. As close initial estimates for the exact parameters, most importantly the period, were essential to the ac- curacy of the final results and the limiting of the code’s run time, the faster uniform source model was run us- ing the estimated period values determined manually, after which the results from this model were used as the input parameters for the limb darkened model. Ideally, the uniform source model would have used the period values determined by the transit detection code for all the stars, instead of the manual estimates for most, as this method would have constrained the region that emcee had to analyse further and thus sped up the fitting process. A better initial estimate would result in a more accurate result from the uniform source fit, which in turn would ensure a decreased run time of the limb darkened model. This in turn would have enabled more iterations and walkers to be used to increase the accuracy of the final parameter values. However, the transit detection code itself took a long time to com- pute the period, especially using short cadence data (which was preferable in order to achieve a better value for the period from the larger number of data points), so the transit detection code was only utilised for de- tection of the period of planetary orbits found in some of the more noisy light curves, including planets GSIC 8b, 22b and 24b where the manual estimates were not accurate enough to achieve a clear phase folded transit, as mentioned previously. On the other hand, the uniform source model was run on all the stars with confirmed planets and avail- able parameter values with emcee using 50 walkers and 500 iterations. In addition, the limb-darkened model was subsequently run using the output from the uniform source model, producing clear phase folds and accurate values, including errors. This was initially performed by members of the research group with em- cee using 100 walkers and 1000 iterations. This proved a taxing and time consuming process, due to the limi- tations set forth by the computers available. As such, the data were run simultaneously on a more sophisti- cated system with emcee using 200 walkers and 400 iterations. Values were compared between both the latter data set and those that were run beforehand on the group’s computers, of which the most accurate re- sults with corresponding errors were selected. In the case of GSIC 8b and GSIC 40b, the results from both 48 Figure 6.2.6 – A collection of the histogram and probability density plots of values for parameters entered into emcee for the limb-darkened model, for GSIC 35b, after running on 200 walkers and 400 iterations. As can be seen, the histograms are mostly Gaussian in shape, and the 2D probability density plots appear to be mostly spherical in shape. data sets appeared to be unsatisfactory, therefore the code was run again for these planets with emcee using 200 walkers and 2000 iterations. Unfortunately, due to time constraints, this level of accuracy was unobtain- able for all discovered planets. 6.2.5 Modelling for Stellar Limb Darkening (CL & OH) To further study the effects of stellar limb dark- ening on the transit method, and to ensure the accu- racy of the limb-darkened transit fitting model used, a numerical limb-darkening model was constructed in Python, allowing use of both linear and quadratic limb- darkening models as a function of the effective temper- ature. Finding Limb-Darkening Coefficients (CL) As mentioned previously and as is clear from the use of limb darkening in the transit fitting, the modelling of the effects from the phenomenon of limb darkening is extremely important in obtaining an accurate estimate of the planets characteristics from its transit data [95]. Limb-darkening coefficients (LDC) have only been di- rectly deduced for relatively very few stars apart from the Sun. Due to this, when producing light curve syn- thesis programs, the coefficients are usually interpo- lated from tables of theoretical values calculated from atmosphere models [98]. 49 In order to compute the specific intensity relations discussed in 4.4 for a modelled star, the LDCs must first be deduced for each varying equation. To make the transit model comparable to the data available in this study, the LDCs need to be computed as a function of effective temperature. The effective temperature, Teff , is obtained from the Kepler data and can be easily input for each star. Table 6.1 – Kepler Stellar limb-darkening Coefficients [138]. Teff Linear Quadratic Quadratic u a b 4000 0.6888 0.5079 0.2239 4250 0.7215 0.6408 0.0999 4500 0.7163 0.6483 0.0842 4750 0.6977 0.6036 0.1164 5000 0.6779 0.5528 0.1548 5250 0.6550 0.4984 0.1939 5500 0.6307 0.4451 0.2297 5750 0.6074 0.3985 0.2586 6000 0.5842 0.3539 0.2851 6250 0.5640 0.3198 0.3023 6500 0.5459 0.2901 0.3167 6750 0.5312 0.2672 0.3267 7000 0.5191 0.2478 0.3358 7250 0.5085 0.2308 0.3437 7500 0.5003 0.2165 0.3512 Table 6.1 was formulated from results from D.K. Sing’s research on the LDCs for data taken from Ke- pler and CoRot. The values of LDCs: u, a and b were obtained as these apply to the linear and quadratic in- tensity laws used in this study. Values for c and f were not included in this table as none of the stars in this study have effective temperatures greater than 6500K; the logarithmic law applies to stars with effective tem- peratures between 10,000K and 40,000K. The values for each LDC were plotted on a graph as a function of the effective temperature, to deduce a relationship for each. This is shown in Figure 6.2.7. Figure 6.2.7 – Fitted relations for limb-darkening coeffi- cients for Kepler as a function of effective temperature, Teff . Values are from Table 6.1, fit with Equation (6.1). A Weibull function was deduced to be the best fit for the data. This equation was of the form: H = k ( m – 1 m ) 1–m m ( Teff – x0 l + ( m – 1 m ) 1 m )m–1 · ( Teff – x0 l + ( m – 1 m ) 1 m )m–1 · exp [ Teff – x0 l + ( m – 1 m ) 1 m ]m + m – 1 m + y0 (6.1) where H is the values of the LDC. The fit to each set of data was performed by the dynamic fit wizard in SigmaPlot, by varying 5 parameters: y0, x0, k, l and m. The non-linear regression statistical measure R- squared was used to analysis the fit to the data. This produced R-squared values of 0.9997±0.0015, 0.9999± 0.0022 and 0.9993 ± 0.0030 to the fits of u, a and b respectively. As the best fit value of R-squared is 1 this shows that the data fits very well to the fitted relation (Equation 6.1). The values for the parameters that form the LDC-Teff relations for u, a and b from Equation 6.1 are included in Tables 6.2, 6.3 and 6.4 respectively. Table 6.2 – Fit parameters for fit to coefficient u from Fig- ure 6.2.7. Fit Parameter Value k 0.2656± 0.0063 l 1749.7597± 45.6889 m 1.2544± 0.0264 x0 4327.5242± 10.6594 y0 0.4555± 0.0059 Table 6.3 – Fit parameters for fit to coefficient a from Fig- ure 6.2.7. Fit Parameter Value k 0.4706± 0.0043 l 1393.0336± 16.8337 m 1.2766± 0.0116 x0 4365.3571± 6.9795 y0 0.1800± 0.0040 Table 6.4 – Fit parameters for fit to coefficient b from Figure 6.2.7. Fit Parameter Value k –0.2679± 0.0033 l 1153.4187± 19.2744 m 1.3530± 0.0213 x0 4400.0749± 12.5456 y0 0.3548± 0.0029 These tables, along with the Weibull function in Equation 6.1, form the three equations that relate the 50 LDCs to the effective temperature of the star. This completes the linear and quadratic intensity relations, so they can be used to model the limb-darkening effects on any of the stars included in the data. Stellar Limb Darkening Model (OH) Using these LDCs and Equation (6.1), a model could be constructed which depended closely on the effective temperature. Using Python, a code was writ- ten to take various input parameters and output a light curve displaying a transit as calculated using the model. In part inspired by a similar approach taken by Addison, Durrance and Schwieterman (2010) [139], both the star and the planet were visualised as arrays of pixels, or ‘boxes’. To ensure that the ratio between the planetary and stellar radii was maintained, an ar- ray of zeroes of 10 by 10 pixels was created to repre- sent the planet. Using the input planetary radius, the size which a box represents in reality could be found, allowing the array representing the star to be scaled accordingly. To allow the model to incorporate the ef- fects of egress and ingress, the total array was enlarged by 20 boxes in both width and height. The very centre of the stellar array was taken to have a radius value of zero. Simply using Pythagoras’ theorem, the radius value outward from the centre of each box in the array was calculated. Using Equations (4.58) and (4.59) de- scribed in Section 4.4, the intensity of each box rel- ative to the intensity at the centre of the star could be calculated as a function of the radius assigned to each box through the variable μ as given in Equation (4.57). To ensure the star did not overrun its boundaries, any box with a radius value that exceeded the stellar ra- dius was set to zero. For simplicity, the intensity at the centre of the star was chosen to be 1, ensuring that the individual intensities of boxes were between the val- ues of 0 and 1. The total intensity of the star-planet system was then taken to be the sum of the intensity values of all the boxes in the stellar array. A clarifying visualisation of the model can be seen in Figure 6.2.8. Figure 6.2.8 – A contour plot of the stellar array of in- tensities, including a schematic representation of a planet transiting the star (not to scale) and the pixel grid. Note how the effects of limb darkening are clearly visible. On the Figure, b represents the impact parameter and p represents the distance to the planet, both relative to the centre of the star. The planet was then shifted along the array one box at a time, whilst its position was recorded rel- ative to the centre of the star. For every step, the boxes which were found to be ‘covered’ by the plane- tary array were set to be zero, thus removing a num- ber of intensity values from the total during transit. The sum of the intensity values was then taken and recorded. The stellar array was then reset for the next step, and this process repeated. As is to be expected, the planetary array would ‘cover’ boxes with a higher relative intensity the closer it got to the centre of the star, thus recreating the effects of limb darkening that appear clearly on transit light curves.The recorded in- tensity values were then plotted against the position of the centre of the planetary array on the stellar disk as a function of the stellar radius. An example of a resulting set of light curves can be seen in Figure 6.2.9. As it depends on the individual intensity values of the boxes on the stellar array, the total intensity differs depending on whether the first or second order equa- tion is used, despite all other input parameters being the same. As such, the intensity values in Figure 6.2.9 have been normalised using the differences between the mean intensity of each light curve. 51 Figure 6.2.9 – A comparison between light curves output by the stellar limb-darkening model, using the parameters of star GSIC 22 and planet GSIC 22b. The difference between the shapes of the first (in blue) and second (in red) order models is clear. As can be seen from Figure 6.2.9, the shape output by the model appears to be very smooth, and remi- niscent of the shape of an actual transit (such as that shown in Figure 6.2.2 for GSIC 1), which is encourag- ing for the accuracy of this numerical model. However, as the array sizes are still fixed to a relatively low, finite value, the resulting model will still include an intrin- sic error on its accuracy. It is worth noting that it is likely that the second order model provides a better fit in this case, as GSIC 22 has an effective temperature significantly higher than the Sun, making the second order model more applicable [100]. Furthermore, this difference between the models as seen in Figure 6.2.9 has been observed to change depending on the input parameters, indicating that the effect described by the different models indeed varies significantly. The final version of this model allowed for user input of the pe- riod, semi-major axis, effective temperature, planetary radius, stellar radius, inclination of the system, and which stellar limb-darkening model (first or second or- der) to be used. As such, the model could be very accu- rately characterised for different stars. To ensure that it was as accurate as possible, the code also allowed the reading in of results received from the limb-darkened transit fitting code to act as the relevant parameters. To test the accuracy of the model used in the transit fitting code, both the first and second order models were plotted over a phase fold of the quadratic limb- darkening model as given by Mandel & Agol [134] for a given star, using the same parameters. To ensure that all three models overlapped, the x-axis value for the transit fitting model was divided by the period to range between values of 0 and 1 for the phase. This was done by ensuring that the phase fold produced a transit at exactly a phase of 0.5, by defining the data set to start at the centre of a secondary transit. The x-axes of the numerical models were then accordingly transcribed from the impact parameter to the centre of the stellar array to the phase, as if(b ≤ 0) : φ = arccos ( b a ) · 0.25(π 2 ) + 0.25 if(b > 0) : φ = arcsin ( b a ) · 0.25(π 2 ) + 0.5 (6.2) where a is the semi-major axis, b is the impact pa- rameter, φ is the phase and P is the period. Besides this, the LDCs produced by the transit fitting code and those for the second order model produced by Equa- tion (6.1) were compared, as they represent constants for the same limb-darkening equation (namely Equa- tion (4.59)).The results and a discussion of these com- parisons can be found in Section 7.4. 6.3 Processing of Results (PS) Using these various computational methods, accu- rate values for the period of a planet’s orbit (P), the ratio of the planet’s radius to the star (Rp/R∗) and the inclination (i) were obtained. These results were then used in conjunction with data collected by the Aster- oseismology group in order to characterise planets as fully as possible. As this study only used the transit method to de- tect planets, values such as the mass of a planet and the eccentricity of its orbit could not be calculated an- alytically, but novel methods were devised in order to obtain as much information about the detected plan- ets as possible. The methods used to calculate each parameter will now be explained. Planetary Radius This value is easily obtained using the ratio of radii provided by the transit fitting limb-darkened model code, and the radius of the host star, R∗. As there is a systematic uncertainty in the method used to cal- culate R∗ (see Section 3.2.4) this results in a rather large error on this result of around 12%, almost all of which is due to the uncertainty on the calculation of stellar radius. The accuracy of this value is also depen- dent on the quality of the limb darkening coefficients used by the program, which affect the measurement of transit depth. Semi-Major Axis of Orbit Again, this value is easy calculable using Kepler’s third law (Equation (4.51)), using the period P of the planet (provided by the transit fitting limb-darkening model) and the mass of the host star M∗, which can be obtained with the use of asteroseismic scaling relations. Although the systematic uncertainty on M∗ often gives an error of 20%. As this value is raised to the 1/3 power this effect is reduced, resulting in an uncertainty on a of around 6%. As planetary transits occur with high regularity, the uncertainty on the value of P provided by the code is very low, often being known to one part in a million or better. 52 Transit Duration This value, τ, was measured manually from a phase folded light curve, which was produced using the peri- odicity P provided by the code. As noted, this value is often exceptionally precise, resulting in almost all cases in a clearly phase folded curve comprised usu- ally of several hundred transits, in which the transit was clearly defined. The width of the transit was then measured from the start of ingress to the end of egress, and suitable errors assigned based on the clarity of the transit phase fold. For some planets, the value of P was not quite sufficient for a clear phase fold, and so for these planets τ was not measured. In some other cases the depth of transit was too small to be measur- able manually, and thus measurement of τ was again not possible. As the clarity of phase fold varied signifi- cantly, the uncertainty of τ also varied between 2% and 26%, but the vast majority of measurements produced values with uncertainties of around 6%. Star-Planet Separation at Point of Transit This value, s, is obtained from a modified version of Equation (4.29), which provides a value for a in the case of a circular orbit. When eccentricities other than 0 are considered, the equation is more properly rendered as: τ = PR∗ πs (6.3) In the above equation, s is the distance between the star and the planet at the moment of transit, rather than the semi-major axis of the orbit. As this value is dependent on R∗, which as noted had a large sys- tematic uncertainty, we find a similar uncertainty on s of around 15%, which is also a product of the manual method used to measure τ. Impact Parameter This value, b, can be obtained by similarly rear- ranging Equation (4.31). Once again, however, it must be remembered that this equation is valid only for a circular orbit, and so substituting a for s is therefore necessary to obtain a more accurate value. This gives a final equation of: b = s cos(i) R∗ (6.4) Although the value of i is obtained computationally to a precision of a few tenths of a degree, this equation is also dependent on s and R∗, which as previously discussed have large systematic errors associated with them. The uncertainty on b is thus very large, and in several cases renders the value statistically meaning- less. Eccentricity The true eccentricity of a planet’s orbit is not an- alytically obtainable by the use of the transit method, and is generally found by Doppler wobble methods. However, as we had calculated both the semi-major axis of the planet’s orbit a, and the separation between the planet and the star at transit s, by comparing these values we were able to constrain the eccentricity some- what in most cases. In the case that s was greater than a, s was treated as the minimum value for the apoastron of the orbit, and thus eccentricity was constrained to: emin = s – a a (6.5) and in the case where s was less than a, s was treated as the maximum value for the periastron, giving an eccentricity constraint of: emin = a – s a (6.6) In reality, since it is unlikely that the transit occurs at the periastron or apoastron, the eccentricity may well be significantly larger than this value, but con- sidering the lack of data which forbids an analytically calculated value this constraint will have to suffice for this study. Due to the relatively high error on the mea- surement of s, an uncertainty on the minimum eccen- tricity was found which often exceeded the magnitude of the result, but nonetheless this method was usable to detect non-circular orbits in a few cases. Equilibrium Temperature For this result, each planet was treated as a black body, heated only by radiation from its host star. Other sources of heat, such as radiation in the planet, and other effects such as the insulating effect of atmo- spheres, were not considered. The planets were all treated as having an emissivity of 1, and as reflecting a fraction of incident light given by A, the albedo. Based on the radius of the planet, among other considerations, an estimated albedo of 0.3 was assigned to rocky planets, and 0.5 for gas giants. For Gas dwarfs, a range of values from 0.3-0.5 were used. These values were chosen after considering the planetary albedos of solar system planets [140] [102]. The equilibrium temperature was then calculated as the point at which the incident radiation energy ab- sorbed by the planet is equal to that radiated from its surface, according to the equation: Teq = Teff(1 – A) 1/4 √ R∗ 2a (6.7) In the above equation a circular orbit of semi-major axis a around a star of radius R∗ and effective temper- ature Teff is assumed. As this equation provides only a very rough value for the equilibrium temperature, and this only at the semi major axis, the value it provides is quoted here as a range including what we consider to be a sensible range of temperatures, which to some extent allows for variation in albedos and atmospheric insulation. This is in part to emphasise that this value is given merely as a ‘ballpark figure’, in order that composition and the habitable zone may be meaningfully discussed with respect to each planet. 53 Planetary Mass and Composition The mass of a planet is not directly obtainable by analytical methods from the transit method. Neverthe- less, we are able to at least constrain the mass of the planets we detected by reference to the other results from our data. By using analytical models for planet composition, a relation between the mass and the radius can be de- rived (see Section 4.5), and thus from our relatively precise results for planet radius, a reasonable estimate for the mass can be obtained. The smaller the planet, the more tightly the mass can be constrained, and by using other results, like the equilibrium tempera- ture, we can further discount certain possibilities. The ranges obtained vary significantly, but we are confident that we have devised a reasonable method of determin- ing mass ranges which can allow us to characterise the planet by its likely composition. 54 Chapter 7 Results and Discussion 7.1 Introduction In this section we will briefly discuss each star and all detected planets. As the previous sections show, many different methods were adopted to calculate a wide range of information. The tables display every parameter obtained by this project. 7.1.1 Key to Results Tables (MH & PS) Stellar Properties In the tables of stellar properties, the parameters are as follows: · Δν0, the large spacing, computational value · νmax , the maximum power frequency, computa- tional value · δν0,2, the small spacing, computational value unless specified as manual · Teff , the effective temperature, provided · [Fe/H], the metallicity, provided · Xc , the hydrogen mass fraction, asteroseismic diagram value · M∗, the stellar mass, computational scaling re- lation value using Equation (3.23) · R∗, the stellar radius, computational scaling re- lation value using Equation (3.22) · the age of the star, stellar model comparison value The inner and outer Habitable Zone limits, IHZ and OHZ respectively, are calculated with the following methods (see Section 4.6.7): · RG - Runaway Greenhouse · MaG - Maximum Greenhouse Planet Orbital Properties In the tables of planet orbital properties, the pa- rameters are as follows: · P , the period of the orbit, computational value · a, the semi-major axis of the orbit, calculated using Equation (4.51) · τ, the duration of the transit, measured from a transit phase fold · s, the separation of the star and planet at the point of transit, calculated using Equation (6.3) · e, the eccentricity of the orbit, constrained using Equations (6.5) and (6.6) · i , the inclination of the orbit, computational value · b, the impact parameter of the orbit, calculated using Equation (6.4) Planet Properties In the tables of planet characteristics, the parame- ters are as follows: · R, the radius of the planet, calculated from the radius ratio (computational) and R∗ · the rough composition of the planet, deduced by reference to other properties · A, the rough albedo of the planet, suggested based on composition · Teq , the equilibrium temperature of the planet, calculated using Equation (6.7) · M , the mass of the planet, constrained by ref- erence to other properties 7.1.2 Nomenclature (PS) In keeping with conventional exoplanet nomencla- ture, our detected planets have been named after the stars around which they orbit, which have been identi- fied to us with GSIC numbers. Thus, a planet around star GSIC 3 will be identified as GSIC 3b (GSIC 3a technically refers to the star, but for simplicity stars will not be assigned letters). Planets in multiple planet systems will be assigned letters continuing through the alphabet in order of increasing period. In actuality these are usually assigned by date of discovery, but as all have been detected as a result of the same study they must be sorted by another method. 7.2 Full Results It should be noted that the results for stars GSIC 2, GSIC 11, GSIC 28 and GSIC 37 have been omitted for various reasons. These are explained in Section 7.2.2 located at the end of the main results. 7.2.1 Stellar, Orbital and Planetary Results (PS, AW, JG & ML) GSIC 0 The power spectrum from GSIC 0 was very easy to read. The signal to noise ratio was very high so only a 55 small amount of filtering was required, resulting in low errors on the calculated mass and radius of the star. The high signal to noise ratio also resulted in the visi- bility of the l = 2 peaks, meaning the small frequency spacing, and hence the age of the star, could be esti- mated more accurately. The maximum frequency for oscillations was calculated by the computational team without any trouble. This can be assumed for every star in this report unless otherwise stated. Table 7.1 – Stellar Parameters for GSIC 0 Parameter Value Error Δν0 (μHz) 116.819 0.058 νmax (mHz) 2.41467 0.00145 δν0,2 (μHz) 6.35 0.31 Teff (K) 5766 60 [Fe/H] 0.052 0.021 Xc 0.13 0.02 M∗ (M�) 0.937 0.165 R∗ (R�) 1.098 0.127 Age (Gyrs) 4.77 1.29 RG IHZ (AU) ∼ 1.07 - MaG OHZ (AU) ∼ 1.85 - From the mass, radius and effective temperature values given in Table 7.1, GSIC 0 can be deduced to be very similar to the Sun. This gives a habitable zone close to 1AU, but unfortunately there were no planets detected orbiting this particular star. This star has an age of 4.77Gyrs compared to the Sun’s age of 4.6Gyrs. The only distinguishing factors between this star and the Sun are its metallicity and hydrogen fraction, which for the Sun are 0 and 0.74 respectively [141]. GSIC 1 The power spectrum from GSIC 1 was more dif- ficult to read due to the lower signal to noise ratio. Once filtered, however, the l = 0 and l = 1 peaks were easily identifiable, making the large frequency spacing calculations more simple. The small frequency spacing could also be calculated after filtering. Table 7.2 – Stellar Parameters for GSIC 1 Parameter Value Error Δν0 (μHz) 59.469 0.229 νmax (mHz) 1.08099 0.00106 δν0,2 (μHz) (BE) 3.52 0.7 Teff (K) 6350 50 [Fe/H] 0.26 0.08 Xc 0.14 0.02 M∗ (M�) 1.446 0.260 R∗ (R�) 1.990 0.236 Age (Gyrs) 1.82 0.226 RG IHZ (AU) ∼ 2.28 - MaG OHZ (AU) ∼ 3.88 - It can be seen from Table 7.2 that this star is sig- nificantly larger and hotter than the Sun, creating a habitable zone centred roughly 3AU from the star. As this star is more massive than the Sun it will progress through the MS faster. This is shown by a low hydro- gen fraction, suggesting that GSIC 1 is near the end of the MS, and the age is approximately half that of the Sun. There was a single planet detected orbiting GSIC 1. GSIC 1b Table 7.3 – Orbital Parameters for GSIC 1b Parameter Value Error P (days) 2.204735299 1.39E – 07 a (AU) 0.037489 0.000245 τ (days) 0.1427763 5.70E – 06 s (AU) 0.045505 0.000396 e > 0.214 0.161 i (◦) 89.587 0.215 b 0.4005 0.0318 Table 7.4 – Planetary Characteristics for GSIC 1b Parameter Value R (R⊕) 15.380± 1.826 Composition Gas giant A 0.5 Teq (K) 1800-2000 M (M⊕) 350-1000 This is an archetypal example of a hot Jupiter planet, orbiting very close to its host star. It is likely that this planet formed much further from its star and then migrated inwards through the ejection of other planets. This would likely result in an eccentric or- bit, which may be under the process of circularisation by tidal forces, which will act particularly strongly on the planet at the periastron, the distance of closest ap- proach to its host star. As the radius of this planet is so high, it is partic- ularly difficult to constrain the mass, as factors such as electron degeneracy become relevant. However, this high radius value also clearly indicates a gaseous com- position (other compositions could not produce plan- ets of this radius) which in turn leads us to an albedo comparable to that of Jupiter. The equilibrium tem- perature of this planet is very high due to proximity to the star, which is expected to artificially inflate the at- mosphere in a ‘puffy’ effect, which also influenced our choice of mass constraints. In comparison to the properties of Jupiter, this planet has a radius of 1.364 ± 0.162RJ and a mass of 1.101-3.145MJ. It is worth noting that as the transits for this planet are particularly deep, and as the star undergoes al- most negligible background variations in flux over the measuring time, the errors obtained on these data are among the most precise in our study. 56 GSIC 3 For GSIC 3, the power spectrum required filtering, but only by a small amount. After filtering there were very faint l = 2 peaks present, allowing the identifica- tion of the l = 0 peaks and the calculation of the small frequency spacing. Table 7.5 – Stellar Parameters for GSIC 3 Parameter Value Error Δν0 (μHz) 145.534 0.039 νmax (mHz) 3.31667 0.00167 δν0,2 (μHz) 8.89 0.2 Teff (K) 5669 75 [Fe/H] –0.18 0.1 Xc 0.28 0.02 M∗ (M�) 0.982 0.177 R∗ (R�) 0.963 0.112 Age (Gyrs) 6.26 1.64 RG IHZ (AU) ∼ 0.92 - MaG OHZ (AU) ∼ 1.58 - The mass, radius and temperature values in Ta- ble 7.5 suggest another star very similar to the Sun. It’s worth noting the slightly higher than average value of the maximum frequency for oscillations, suggesting that GSIC 3 is a star currently towards the start of its MS lifetime. This is in contrast to the star’s age and hydrogen fraction calculated from the asteroseismic di- agrams, which would put a star of this mass well into its MS lifetime. GSIC 3b Table 7.6 – Orbital Parameters for GSIC 3b Parameter Value Error P (days) 4.72674034 1.10E – 06 a (AU) 0.05479 0.00329 τ (days) 0.1188 0.01 s (AU) 0.05676 0.00817 e > 0.036 0.161 i (◦) 89.301 0.419 b 0.1545 0.0970 Table 7.7 – Planetary Characteristics for GSIC 3b Parameter Value R (R⊕) 1.427± 0.167 Composition Rocky A 0.3 Teq (K) 900-1100 M (M⊕) 2-4 This planet’s small radius and extremely close orbit indicate a rocky ‘Super-Earth’ planet. This small ra- dius value, coupled with the high equilibrium tempera- ture which forbids surface water, allows us to discount this as a possible composition and thus gives us a very small mass range . It is to be noted here that the cal- culated minimum eccentricity is quite imprecise, and given the low period it is very likely that this planet follows an almost circular orbit. GSIC 4 The power spectrum for GSIC 4 was another spec- trum that was very easy to read, with a high signal to noise ratio and a visible small frequency spacing be- fore filtering. This made the l = 0 peaks, and hence the large frequency spacing, easily detectable. Table 7.8 – Stellar Parameters for GSIC 4 Parameter Value Error Δν0 (μHz) 60.677 0.537 νmax (mHz) 1.07660 0.00116 δν0,2 (μHz) 3.98 1.12 Teff (K) 6305 50 [Fe/H] –0.03 0.1 Xc 0.19 0.02 M∗ (M�) 1.304 0.258 R∗ (R�) 1.897 0.243 Age (Gyrs) 1.54 0.160 RG IHZ (AU) ∼ 2.15 - MaG OHZ (AU) ∼ 3.66 - Table 7.8 shows that GSIC 4 is visibly larger, hot- ter, more massive and younger than the Sun. The star’s age in comparison to its mass places it towards the end of the MS, an assumption that agrees with the low hydrogen fraction. There were no planets detected in orbit around this star. GSIC 5 GSIC 5 had a power spectrum with a large amount of noise, enough for the l = 2 peaks to be undetectable by computational analysis. Therefore there was no age calculation from the asteroseismic diagrams. The rest of the table can, however, be used to generate an age estimate. Table 7.9 – Stellar Parameters for GSIC 5 Parameter Value Error Δν0 (μHz) 128.525 0.683 νmax (mHz) 2.91945 0.00174 δν0,2 (μHz) - - Teff (K) 6134 91 [Fe/H] –0.24 0.1 Xc - - M∗ (M�) 1.240 0.237 R∗ (R�) 1.131 0.139 Age (Gyrs) - - RG IHZ (AU) ∼ 1.22 - MaG OHZ (AU) ∼ 2.09 - 57 The slightly higher than solar mass and radius val- ues given in Table 7.9 correspond to the increased ef- fective temperature. The value for the maximum fre- quency of oscillations is similar to that of the Sun, indi- cating that GSIC 5 is at a similar point in its evolution. Putting this with the mass value suggests that GSIC 5 is younger than the Sun by a small amount. GSIC 5b Table 7.10 – Orbital Parameters for GSIC 5b Parameter Value Error P (days) 11.87291881 5.79E – 06 a (AU) 0.10941 0.00696 τ (days) 0.09959 0.01 s (AU) 0.1996 0.0317 e > 0.825 0.312 i (◦) 88.722 0.019 b 0.846 0.170 Table 7.11 – Planetary Characteristics for GSIC 5b Parameter Value R (R⊕) 2.055± 0.253 Composition Gas dwarf/ rocky A 0.3-0.5 Teq (K) 750-900 M (M⊕) 3-15 The radius of this planet is above, but still close to the theoretical boundary between solid rocky planets and those which have collected a significant envelope of gas. As it is impossible from our data to determine the extent of the planet’s atmosphere, both possibili- ties have been accounted for here, and this is reflected in the large mass range, the upper reaches of which describe a rocky planet, and the lower end of which would characterise a composition and density similar to that of Neptune. Also notable is the planet’s high eccentricity. Al- though our calculation is imprecise, this still indicates a planet which is surprisingly eccentric, especially con- sidering its close proximity to the host star. This will also give rise to greater variation in temperature (and possibly radius, depending on composition) than may be implied by our data. GSIC 6 The power spectrum of GSIC 6 had a slightly higher signal to noise ratio than that of GSIC 5, meaning the small frequency spacing could be measured and an age can be discussed. Table 7.12 – Stellar Parameters for GSIC 6 Parameter Value Error Δν0 (μHz) 97.876 0.499 νmax (mHz) 2.14031 0.00152 δν0,2 (μHz) 8.64 0.58 Teff (K) 6270 79 [Fe/H] –0.04 0.1 Xc 0.48 0.02 M∗ (M�) 1.501 0.282 R∗ (R�) 1.445 0.176 Age (Gyrs) 1.13 0.172 RG IHZ (AU) ∼ 1.62 - MaG OHZ (AU) ∼ 2.76 - All values in the above table correspond to a star in the very middle of the MS. The mass of GSIC 6 is greater than solar mass and it is a younger star. Just under half of the mass of GSIC 6 is composed of hydrogen. This is the first of three stars in the data set to have multiple planets detected in its orbit. GSIC 6b Table 7.13 – Orbital Parameters for GSIC 6b Parameter Value Error P (days) 6.23853699 1.44E – 06 a (AU) 0.07593 0.00476 τ (days) 0.146 0.01 s (AU) 0.0915 0.0128 e > 0.204 0.185 i (◦) 89.384 0.370 b 0.1462 0.0918 Table 7.14 – Planetary Characteristics for GSIC 6b Parameter Value R (R⊕) 2.764± 0.338 Composition Gas dwarf A 0.3-0.5 Teq (K) 1100-1300 M (M⊕) 2-9 This planet is likely to be a small gas planet, having a relatively large rocky core surrounded by a thick gas envelope. As we are unable to tell with certainty the composition of the envelope, an albedo range has been given. Being relatively small for a gas planet, the core must take up a sizable portion of the volume, and thus a higher density has been assumed than any gas planet in our solar system, which leads us to the mass range stated. The lower range of this mass range shows a Neptune-like composition, whereas the higher reaches describe a planet with a large rocky core surrounded by a gas envelope. The lower end of the range has also been lowered slightly to account for the ‘puffy’ effect 58 which may inflate the atmosphere for close orbiting planets such as this one. The period of the orbit seems slightly inconsistent with the eccentricity calculated. It is likely that the true eccentricity is lower than this value, as tidal forces will act to circularise such low period orbits. GSIC 6c Table 7.15 – Orbital Parameters for GSIC 6c Parameter Value Error P (days) 12.7203787 1.87E – 05 a (AU) 0.12210 0.00765 τ (days) - - s (AU) - - e - - i (◦) 89.491 0.037 b - - Table 7.16 – Planetary Characteristics for GSIC 6c Parameter Value R (R⊕) 4.330± 0.530 Composition Gas giant A 0.5 Teq (K) 800-1000 M (M⊕) 10-30 This planet is larger than the other in this system, and likely to be of similar composition. This planet is likely to be less dense than its companion, as it is larger and thus more of its volume will be taken up by a gaseous envelope. This planet was therefore assigned a density similar to that of Neptune, and thus a relatively similar mass range was suggested, taking the inflating effect of temperature into account. Unfortunately, due to a slightly imprecise value of P , a successful phase fold was not possible for this planet, and thus the transit time was not measurable. This in turn meant that we were unable to calculate the separation at transit or the eccentricity of the system, although as this is a close orbiting planet with another in the system, it is likely that both planets orbit at near 0 eccentricity. It is also notable that the periods of the two planets are very nearly in the resonance ratio 1:2. This is the clearest example of orbital resonance seen in the data. In comparison to the properties of Jupiter, this planet has a radius of 0.3841±0.0470RJ and a mass of 0.0315-0.0994MJ. GSIC 7 GSIC 7 has a slightly lower signal to noise ratio in its power spectrum than GSIC 6. Unfortunately this difference is great enough to make any possible l = 2 peaks unreliable. Table 7.17 – Stellar Parameters for GSIC 7 Parameter Value Error Δν0 (μHz) 82.0.32 0.609 νmax (mHz) 1.61821 0.00125 δν0,2 (μHz) - - Teff (K) 5845 88 [Fe/H] 0.07 0.11 Xc - - M∗ (M�) 1.182 0.233 R∗ (R�) 1.501 0.190 Age (Gyrs) - - RG IHZ (AU) ∼ 1.50 - MaG OHZ (AU) ∼ 2.58 - It is a shame that no small frequency spacing can be calculated for GSIC 7, as this is one of the more interesting extrasolar systems in the data set and an age calculation would lead to greater discussion. How- ever, despite the lack of small frequency spacing, an age can still be discussed through comparison with similar stars. The value for the maximum frequency of oscilla- tions in Table 7.17 puts the star firmly on the MS and is fairly low compared to the value for more solar-like stars, implying it is more evolved. The mass of GSIC 7 is similar to solar mass, suggesting a greater age than many of its neighbours but likely lower than the Sun. GSIC 7 is the second of three stars with multiple plan- ets detected in orbit. GSIC 7b Table 7.18 – Orbital Parameters for GSIC 7b Parameter Value Error P (days) 15.9653315 1.04E – 05 a (AU) 0.13118 0.00863 τ (days) 0.1854 0.01 s (AU) 0.1914 0.0263 e > 0.459 0.223 i (◦) 88.619 0.167 b 0.660 0.147 Table 7.19 – Planetary Characteristics for GSIC 7b Parameter Value R (R⊕) 3.230± 0.412 Composition Gas dwarf A 0.3-0.5 Teq (K) 800-900 M (M⊕) 7-15 This planet’s radius indicates a gas dwarf, and the mass range and albedo have been chosen to reflect this. The close orbit is also a factor once more in the mass range, and the lower mass bound has again been re- duced to account for thermal gas expansion. This 59 planet also has a relatively eccentric orbit, which is slightly surprising given its low period, but as the other planet in this system has such a large period the orbits are unlikely to affect each other. It is quite possible that these two planets did inter- act at some point in the past, as this planet is unlikely to have formed this close to the star, and the eccen- tricity value may be an indication that this happened relatively recently as tidal forces would normally be expected to circularise this orbit. GSIC 7c Table 7.20 – Orbital Parameters for GSIC 7c Parameter Value Error P (days) 179.43045 2.13E – 03 a (AU) 0.6582 0.0433 τ (days) - - s (AU) - - e - - i (◦) 89.627 0.256 b - - Table 7.21 – Planetary Characteristics for GSIC 7c Parameter Value R (R⊕) 1.407± 0.180 Composition Rocky A 0.3 Teq (K) 300-400 M (M⊕) 0.8-4 It is quite surprising to detect a planet with this high a period by planetary transit, and that this planet should prove to be rocky makes this result particularly exciting. Even better, the equilibrium temperature of this planet places it within the range possible for liquid water to exist on its surface - one of the few planets in this study for which this is the case. This also affects our mass constraints, as ‘water planet’ compositions must be included. Due to the high period only a few transits were recorded for this planet, and thus the value obtained for the period was of insufficient precision for a usable phase fold, especially considering that this planet had a low signal to noise ratio. The transit time, separation, impact parameter and eccentricity of this planet can therefore not be calculated. This is unfortunate, as the latter may have been interesting as it could well have interacted with the closer orbiting planet in this system. GSIC 8 Similarly to the previous star, GSIC 8 had a power spectrum with just enough noise to prevent the detec- tion of l = 2 peaks. Table 7.22 – Stellar Parameters for GSIC 8 Parameter Value Error Δν0 (μHz) 90.69 2.29 νmax (mHz) 1.80987 0.00141 δν0,2 (μHz) - - Teff (K) 5952 75 [Fe/H] –0.08 0.1 Xc - - M∗ (M�) 1.139 0.299 R∗ (R�) 1.387 0.223 Age (Gyrs) - - RG IHZ (AU) ∼ 1.43 - MaG OHZ (AU) ∼ 2.45 - The values for mass, radius and effective tempera- ture given in Table 7.22 are very similar to the values for GSIC 7, but with a slightly higher ratio of temper- ature to radius. This means similar assumptions can be made about its age. This is the final star in the set with multiple planets detected in orbit, so it is a shame that a calculation of the age of the extrasolar system was not possible. GSIC 8b Table 7.23 – Orbital Parameters for GSIC 8b Parameter Value Error P (days) 6.48164 9.66E – 03 a (AU) 0.07104 0.00622 τ (days) 0.15177 0.04 s (AU) 0.0877 0.0271 e > 0.235 0.396 i (◦) 89.093 0.957 b 0.215 0.239 Table 7.24 – Planetary Characteristics for GSIC 8b Parameter Value R (R⊕) 2.251± 0.566 Composition Gas dwarf A 0.3-0.5 Teq (K) 1000-1200 M (M⊕) 2-6 This planet appears to be a close-orbiting gas dwarf, and thus again is unlikely to have formed at such a close orbit. It is likely that it has migrated inwards by ejecting other planets, and given the low period of both planets in this system they are likely to have interacted at some point. The eccentricity value is likely to be near 0, as tidal forces would be relatively strong at this distance and would therefore circularise an eccentric orbit quite quickly. 60 GSIC 8c Table 7.25 – Orbital Parameters for GSIC 8c Parameter Value Error P (days) 21.2227181 4.38E – 05 a (AU) 0.1566 0.0137 τ (days) 0.2696 0.01 s (AU) 0.1617 0.0267 e > 0.032 0.193 i (◦) 89.907 0.078 b 0.0406 0.0352 Table 7.26 – Planetary Characteristics for GSIC 8c Parameter Value R (R⊕) 2.284± 0.367 Composition Gas dwarf A 0.3-0.5 Teq (K) 700-800 M (M⊕) 2-6 This planet is very similar in nature to its compan- ion, and is very likely to contain a similar composition. This leads us to very similar values for each planet - the only significant difference being the equilibrium tem- perature, which is significantly lower here - although still not nearly low enough to approach habitability. GSIC 9 The GSIC 9 power spectrum had a high signal to noise ratio and detectable l = 2 peaks. Table 7.27 – Stellar Parameters for GSIC 9 Parameter Value Error Δν0 (μHz) 90.079 0.256 νmax (mHz) 1.90682 0.00122 δν0,2 (μHz) 8.35 0.6 Teff (K) 6169 50 [Fe/H] 0.09 0.08 Xc 0.52 0.02 M∗ (M�) 1.444 0.255 R∗ (R�) 1.508 0.176 Age (Gyrs) 0.792 0.150 RG IHZ (AU) ∼ 1.65 - MaG OHZ (AU) ∼ 2.81 - Table 7.27 shows that GSIC 9 is one of the youngest stars in the sample. It has a mass greater than solar mass, therefore will evolve faster. The hydrogen frac- tion puts this star towards the middle of the MS. A single planet has been detected orbiting GSIC 9, mak- ing this one of only two extrasolar systems in the data set with a calculated age of less than one billion years. GSIC 9b Table 7.28 – Orbital Parameters for GSIC 9b Parameter Value Error P (days) 5.85993065 1.77E – 06 a (AU) 0.07189 0.00424 τ (days) 0.1637 0.01 s (AU) 0.07993 0.0106 e > 0.112 0.161 i (◦) 87.318 0.021 b 0.533 0.094 Table 7.29 – Planetary Characteristics for GSIC 9b Parameter Value R (R⊕) 2.518± 0.295 Composition Gas dwarf A 0.3-0.5 Teq (K) 1100-1300 M (M⊕) 3-8 In this system another close-orbiting gas giant is seen, with a period low enough such that tidal forces are likely to have circularised the orbit. Although no other planets were detected in this system, it is likely that they are or were present at some point, as this planet is likely to have migrated inwards. The mass range takes into consideration the ‘puffy’ effect of high temperature on the radius of the planet. GSIC 10 The power spectrum from GSIC 10 had a very low signal to noise ratio and as a result was nearly impossi- ble to read without lots of filtering. This resulted in a high uncertainty on the calculated mass and radius of the star. The signal to noise ratio was too low for the l = 2 peaks to be found, so the age of the star could not be determined. Table 7.30 – Stellar Parameters for GSIC 10 Parameter Value Error Δν0 (μHz) 67.328 0.228 νmax (mHz) 1.27508 0.00228 δν0,2 (μHz) - - Teff (K) 5871 94 [Fe/H] 0.17 0.11 Xc - - M∗ (M�) 1.284 0.242 R∗ (R�) 1.761 0.212 Age (Gyrs) - - RG IHZ (AU) ∼ 1.77 - MaG OHZ (AU) ∼ 3.05 - From the mass, radius and effective temperature values given in Table 7.30 it can be deduced that GSIC 10 is larger than the Sun. Due to its large mass and 61 radius, an argument could be made for this star be- ing classed as a subgiant. One orbiting planet was detected. GSIC 10b Table 7.31 – Orbital Parameters for GSIC 10b Parameter Value Error P (days) 13.74884265 8.78E – 06 a (AU) 0.12208 0.00766 τ (days) 0.2495 0.01 s (AU) 0.1437 0.0183 e > 0.177 0.167 i (◦) 85.263 0.900 b 1.448 0.374 Table 7.32 – Planetary Characteristics for GSIC 10b Parameter Value R (R⊕) 3.991± 0.506 Composition Gas dwarf/gas giant A 0.3-0.5 Teq (K) 900-1000 M (M⊕) 10-30 This planet occurs almost perfectly on the bound- ary between gas dwarf and gas giant classification, and thus has been estimated as having a Neptune-like mass and composition. The inclination of the orbit is quite low, resulting in a transit which may very well only ‘graze’ the star - this is likely the source of the unphys- ical value obtained for the impact parameter, which is likely to be close to 1. GSIC 12 The power spectrum from GSIC 12 was fairly sim- ple to read. The signal to noise ratio was moderately high, so not much filtering was needed. The adequate signal to noise ratio allowed the l = 2 peaks to be seen which resulted in values for the small frequency spacing along with the age of the star. Table 7.33 – Stellar Parameters for GSIC 12 Parameter Value Error Δν0 (μHz) 77.057 0.110 νmax (mHz) 1.47346 0.00117 δν0,2 (μHz) 5.29 0.21 Teff (K) 5825 75 [Fe/H] 0.02 0.1 Xc 0.28 0.02 M∗ (M�) 1.142 0.206 R∗ (R�) 1.547 0.181 Age (Gyrs) 1.85 0.417 RG IHZ (AU) ∼ 1.54 - MaG OHZ (AU) ∼ 2.65 - From the mass, radius and effective temperature values given in Table 7.33 it can be deduced that GSIC 12 is larger than the Sun, and as a result is has used up over two-thirds of its core hydrogen despite being less than half as old. One planet was detected orbiting GSIC 12. GSIC 12b Table 7.34 – Orbital Parameters for GSIC 12b Parameter Value Error P (days) 12.8158875 1.02E – 05 a (AU) 0.11201 0.00673 τ (days) 0.2655 0.01 s (AU) 0.1106 0.0136 e > 0.012 0.135 i (◦) 89.468 0.377 b 0.143 0.104 Table 7.35 – Planetary Characteristics for GSIC 12b Parameter Value R (R⊕) 2.112± 0.248 Composition Gas dwarf A 0.3-0.5 Teq (K) 800-1000 M (M⊕) 2.5-6 This planet is likely to be a gas dwarf, with a den- sity greater than that of Neptune due to a larger rel- ative core size. The mass radius has again been sug- gested considering the ‘puffy’ effect of high tempera- ture. Once again it is likely that this planet has mi- grated inwards and then undergone tidal circularisa- tion. GSIC 13 The power spectrum from GSIC 13 was also quite simple to read. The signal to noise ratio was moder- ately high, so again, not much filtering was needed to find the large frequency spacing. However, the l = 2 peaks were hard to discern without further filtering. These values were eventually obtained and the small frequency spacing could then be found. 62 Table 7.36 – Stellar Parameters for GSIC 13 Parameter Value Error Δν0 (μHz) 58.287 0.157 νmax (mHz) 0.61878 0.00074 δν0,2 (μHz) 3.78 0.34 Teff (K) 5896 75 [Fe/H] –0.17 0.1 Xc - - M∗ (M�) 0.2630 0.0479 R∗ (R�) 1.143 0.135 Age (Gyrs) - - RG IHZ (AU) ∼ 1.16 - MaG OHZ (AU) ∼ 1.99 - Due to an over-estimation of large frequency spac- ing, the mass value should be higher than shown in Table 7.36. This also had repercussions on our abil- ity to plot this star on the asteroseismic diagrams. As the mass was too low, the asteroseismic diagrams were unable to be used to find the star’s age or Hydrogen fraction. Unfortunately, this error was found close to the deadline, leaving too little time to correct it. How- ever, one planet was detected orbiting GSIC 13. GSIC 13b Table 7.37 – Orbital Parameters for GSIC 13b Parameter Value Error P (days) 105.880655 1.10E – 04 a (AU) 0.2806 0.0170 τ (days) - - s (AU) 0 0 e - - i (◦) 89.143 0.002 b - - Table 7.38 – Planetary Characteristics for GSIC 13b Parameter Value R (R⊕) 3.722± 0.441 Composition Gas dwarf/water planet A 0.3-0.5 Teq (K) 400-600 M (M⊕) 10-60 The equilibrium temperature of this planet indi- cates that there is a chance that it could support liq- uid water at its surface. This in turn means that this planet cannot simply be classified as a gas dwarf, and thus the rather large mass range has been suggested to include masses of a very large rocky planet covered by water (upper reaches of the range) in addition to the more probable gas dwarf composition. Unfortunately the period of this transit was not suf- ficiently precise for a clear phase fold, and thus several values were not calculable. As the period of this planet is particularly large for a transiting planet, a large ec- centricity range is theoretically possible, and if indeed the eccentricity is large the temperature at periastron would likely have been sufficiently high to rule out the possibility of a water-covered planet. Due to significant uncertainty on stellar values for this system, the characteristics of GSIC 13b are likely to be significantly less accurate than our errors imply. GSIC 14 GSIC 14 had a power spectrum with a very low signal to noise ratio, so a lot of filtering was required to be able to clearly separate the l = 2 peaks from the l = 0 peaks. This was successful, and as a result, the mass, radius and age of the star were obtained. Table 7.39 – Stellar Parameters for GSIC 14 Parameter Value Error Δν0 (μHz) 88.425 0.429 νmax (mHz) 1.83660 0.00153 δν0,2 (μHz) 8.8 0.14 Teff (K) 6463 110 [Fe/H] 0.09 0.11 Xc 0.56 0.02 M∗ (M�) 1.490 0.286 R∗ (R�) 1.543 0.190 Age (Gyrs) 0.515 0.134 RG IHZ (AU) ∼ 1.82 - MaG OHZ (AU) ∼ 3.09 - From the mass, radius and effective temperature values given in Table 7.39 it can be deduced that GSIC 14 is larger and hotter than our Sun, and is likely to be a new MS star. The hydrogen mass fraction, however, puts this star in the middle of its MS lifetime. It is also the youngest star to have a planet orbiting it. GSIC 14b Table 7.40 – Orbital Parameters for GSIC 14b Parameter Value Error P (days) 18.0115861 2.66E – 05 a (AU) 0.15358 0.00984 τ (days) 0.2754 0.04 s (AU) 0.1495 0.0285 e > 0.027 0.196 i (◦) 89.569 0.335 b 0.156 0.127 63 Table 7.41 – Planetary Characteristics for GSIC 14b Parameter Value R (R⊕) 1.552± 0.192 Composition Rocky A 0.3 Teq (K) 850-950 M (M⊕) 2.5-5 This system shows a close-orbiting rocky planet, which is likely to be in a low-eccentricity orbit due to the low value of a. It has been possible to constrain the mass considerably, as gaseous and liquid-covered planet compositions have been safely discarded. GSIC 15 GSIC 15 also had a power spectrum with a low signal to noise ratio, so a lot of filtering was needed to be able to identify the l = 2 peaks separately from the l = 0 peaks. This was successful, and the mass, radius and age of the star were obtained as a result. Table 7.42 – Stellar Parameters for GSIC 15 Parameter Value Error Δν0 (μHz) 68.591 0.813 νmax (mHz) 1.2821 0.00128 δν0,2 (μHz) 5.98 0.79 Teff (K) 6072 75 [Fe/H] –0.09 0.1 Xc 0.41 0.02 M∗ (M�) 1.275 0.269 R∗ (R�) 1.735 0.233 Age (Gyrs) 1.03 0.0984 RG IHZ (AU) ∼ 1.85 - MaG OHZ (AU) ∼ 3.16 - From the mass, radius and effective temperature values given in Table 7.42, it can be deduced that GSIC 15 is larger and hotter than our Sun and is on the main sequence. The age and hydrogen fraction are typical for a star of this mass in the middle of its MS lifetime. No planets were detected orbiting GSIC 15. GSIC 16 The power spectrum from GSIC 16 was initially hard to read due to its low signal to noise ratio. After filtering, the l = 0 and the l = 2 peaks were located with minimum effort, and the stellar properties easily calculated. Table 7.43 – Stellar Parameters for GSIC 16 Parameter Value Error Δν0 (μHz) 93.253 0.369 νmax (mHz) 1.93452 0.00144 δν0,2 (μHz) 7.9 1.04 Teff (K) 6239 94 [Fe/H] –0.14 0.1 Xc 0.42 0.02 M∗ (M�) 1.335 0.251 R∗ (R�) 1.436 0.174 Age (Gyrs) 1.42 0.140 RG IHZ (AU) ∼ 1.60 - MaG OHZ (AU) ∼ 2.72 - Looking at the mass, radius and effective tempera- ture values in Table 7.43, it can be deduced that GSIC 16 is a young MS star and is larger and hotter than the Sun. Its age and hydrogen core mass fraction put the star in the middle of the MS. A single planet was detected orbiting the star. GSIC 16b Table 7.44 – Orbital Parameters for GSIC 16b Parameter Value Error P (days) 100.2829887 6.93E – 05 a (AU) 0.4651 0.0291 τ (days) 0.4824 0.01 s (AU) 0.4419 0.0542 e > 0.050 0.131 i (◦) 89.939 0.049 b 0.0708 0.0579 Table 7.45 – Planetary Characteristics for GSIC 16b Parameter Value R (R⊕) 2.592± 0.314 Composition Gas dwarf/water planet A 0.3-0.5 Teq (K) 400-500 M (M⊕) 3.5-20 This planet orbits at a relatively large distance, and it is quite surprising that it was detectable at all, es- pecially with an almost equatorial transit. The planet is also potentially cool enough for liquid water to exist on the surface, which widens the mass range consider- ably, as multiple compositions must be accounted for. Were the planet hotter, the mass could be constrained to below around 8 M⊕. GSIC 17 The GSIC 17 power spectrum had a low signal to noise ratio, and was therefore hard to read. When filtered, the l = 0 peaks were clear to see; however the 64 l = 2 peaks could not be seen, so only the mass and radius have been calculated. Table 7.46 – Stellar Parameters for GSIC 17 Parameter Value Error Δν0 (μHz) 83.188 0.030 νmax (mHz) 1.60261 0.00133 δν0,2 (μHz) - - Teff (K) 5699 74 [Fe/H] 0.3 0.1 Xc - - M∗ (M�) 1.047 0.188 R∗ (R�) 1.428 0.167 Age (Gyrs) - - RG IHZ (AU) ∼ 1.37 - MaG OHZ (AU) ∼ 2.36 - From the mass, radius and effective temperature values given in Table 7.46, GSIC 17 was deduced to be larger than the Sun, but with approximately the same mass and effective temperature. A single planet was detected orbiting GSIC 17. GSIC 17b Table 7.47 – Orbital Parameters for GSIC 17b Parameter Value Error P (days) 11.52309135 4.85E – 06 a (AU) 0.10136 0.00607 τ (days) - - s (AU) - - e - - i (◦) 87.849 0.281 b - - Table 7.48 – Planetary Characteristics for GSIC 17b Parameter Value R (R⊕) 3.350± 0.393 Composition Gas dwarf A 0.3-0.5 Teq (K) 850-950 M (M⊕) 8-20 This planet is likely to be in a low-eccentricity orbit due to tidal forces, although due to an imprecise pe- riod value it was not possible to obtain a satisfactory phase fold for this planet, and thus τ and those values dependent on it could not be found. GSIC 18 GSIC 18’s power spectrum had a low signal to noise ratio, so it needed thorough filtering to identify the l = 0 and l = 2 peaks. The large and small frequency spacings were then found, and the mass, radius and age of the star were calculated. Table 7.49 – Stellar Parameters for GSIC 18 Parameter Value Error Δν0 (μHz) 178.664 0.230 νmax (mHz) 4.40160 0.00165 δν0,2 (μHz) 13.39 1.02 Teff (K) 5417 75 [Fe/H] –0.32 0.07 Xc 0.42 0.02 M∗ (M�) 0.944 0.171 R∗ (R�) 0.829 0.097 Age (Gyrs) 6.60 1.39 RG IHZ (AU) ∼ 0.73 - MaG OHZ (AU) ∼ 1.27 - The results in Table 7.49 show that GSIC 18 is slightly smaller and cooler than our Sun, as a result it is nearly one and a half times older but remains in the middle of its MS. GSIC 18 was found to have one planet orbiting it. GSIC 18b Table 7.50 – Orbital Parameters for GSIC 18b Parameter Value Error P (days) 39.79215303 3.32E – 06 a (AU) 0.2238 0.0135 τ (days) 0.1856 0.01 s (AU) 0.2632 0.0340 e > 0.176 0.168 i (◦) 89.981 0.020 b 0.0226 0.0240 Table 7.51 – Planetary Characteristics for GSIC 18b Parameter Value R (R⊕) 1.885± 0.221 Composition Gas dwarf/ Rocky A 0.3-0.5 Teq (K) 400-500 M (M⊕) 2-12 This planet is potentially small enough to be solid, but is likely to have at least some atmospheric enve- lope. The planet also orbits far enough from its host star to put its equilibrium temperature potentially on the inside edge of the habitable zone, meaning water compositions must also be considered. This leads us to a larger mass range than would have been the case for a larger planet, due to the uncertainty of composition. GSIC 19 The power spectrum from GSIC 19 was initially very hard to read due to its low signal to noise ratio. After filtering, the l = 0 peaks could be found com- putationally, although the l = 2 peaks could not be 65 located at all, therefore no result for the star’s age has been logged. Table 7.52 – Stellar Parameters for GSIC 19 Parameter Value Error Δν0 (μHz) 61.937 1.559 νmax (mHz) 1.15814 0.00123 δν0,2 (μHz) - - Teff (K) 6144 106 [Fe/H] 0.13 0.1 Xc - - M∗ (M�) 1.439 0.380 R∗ (R�) 1.933 0.311 Age (Gyrs) - - RG IHZ (AU) ∼ 2.10 - MaG OHZ (AU) ∼ 3.58 - The results in Table 7.52 show that GSIC 19 is about eight times as voluminous as the Sun and has one and a half times the mass. One planet was de- tected orbiting the star. GSIC 19b Table 7.53 – Orbital Parameters for GSIC 19b Parameter Value Error P (days) 42.8822 1.63E – 02 a (AU) 0.2707 0.0238 τ (days) 0.4002 0.01 s (AU) 0.3068 0.0500 e > 0.133 0.210 i (◦) 89.64 2.83 b 0.216 1.69 Table 7.54 – Planetary Characteristics for GSIC 19b Parameter Value R (R⊕) 1.890± 0.335 Composition Gas dwarf/ Rocky A 0.3-0.5 Teq (K) 600-800 M (M⊕) 2-12 This planet is again potentially rocky or gaseous, however in this case the presence of liquid water can be discounted due to high temperature. Due to a poor fit to the inclination of the system, the value for the impact parameter of the system has been rendered sta- tistically useless. Once again, a large mass range has been suggested due to uncertainty of composition. GSIC 20 The power spectrum for GSIC 20 required signif- icant filtering to identify the region of interest, along with a significant amount of analysis to identify the l = 0 peaks. Fortunately the spacings between l = 1 and l = 0 peaks were almost identical, so any errors made had minimal effect on the large frequency spac- ing. Eventually the small frequency spacing was also calculated. Table 7.55 – Stellar Parameters for GSIC 20 Parameter Value Error Δν0 (μHz) 61.893 0.414 νmax (mHz) 1.13110 0.00097 δν0,2 (μHz) 5.02 0.96 Teff (K) 5945 60 [Fe/H] 0.17 0.05 Xc - - M∗ (M�) 1.279 0.244 R∗ (R�) 1.860 0.231 Age (Gyrs) - - RG IHZ (AU) ∼ 1.91 - MaG OHZ (AU) ∼ 3.28 - Although the small frequency spacing could be cal- culated for GSIC 20, the asteroseismic diagrams failed to obtain any reliable results for the age or hydrogen fraction. This is due to the mass calculated by the as- teroseismic diagrams, 1.53M�, being slightly too large for any valid age calculations. This mass lies just over one error above the mass calculated from the scaling re- lations. As the mass was at the top end of those used in the asteroseismic diagrams, we could approximate the age being just below that of GSIC 4 (1.54Gyrs), as this is the most similar star according to the asteroseismic diagrams. GSIC 20b Table 7.56 – Orbital Parameters for GSIC 20b Parameter Value Error P (days) 4.7768795 2.18E – 05 a (AU) 0.06025 0.00383 τ (days) - - s (AU) - - e - - i (◦) 89.840 0.228 b - - Table 7.57 – Planetary Characteristics for GSIC 20b Parameter Value R (R⊕) 1.693± 0.211 Composition Rocky A 0.3 Teq (K) 1300-1500 M (M⊕) 3.5-8 This planet is below the radius threshold for a gas dwarf, but may still have some atmosphere. The tem- perature is very high due to an extremely small period 66 value, and thus the possibility of surface water can be safely discounted. In this case, the transit depth was so small that in the phase fold obtained it was not distinguishable from background noise manually, and thus τ and dependent values were not found. However, a low eccentricity orbit can be expected, due to tidal circulation. GSIC 21 The signal to noise ratio on the power spectrum of GSIC 21 was low, so no small frequency spacing could be calculated. Table 7.58 – Stellar Parameters for GSIC 21 Parameter Value Error Δν0 (μHz) 49.91 1.93 νmax (mHz) 0.95116 0.00180 δν0,2 (μHz) - - Teff (K) 5882 87 [Fe/H] 0.16 0.1 Xc - - M∗ (M�) 1.771 0.560 R∗ (R�) 2.393 0.449 Age (Gyrs) - - RG IHZ (AU) ∼ 2.42 - MaG OHZ (AU) ∼ 4.15 - This is one of the more interesting stars in terms of evolutionary stage. Table 7.58 shows a higher than av- erage radius and mass compared to other stars in this data set. The high mass means that GSIC 21 will move off the MS sooner than the majority of other stars. The lower than average value of maximum frequency for oscillations also agrees with this assumption. Un- fortunately the age couldn’t be calculated, but due to its high mass and proximity to the MS, GSIC 21 is likely younger than the Sun. There was a single planet detected in orbit around this star. GSIC 21b Table 7.59 – Orbital Parameters for GSIC 21b Parameter Value Error P (days) 46.1511576 5.69E – 05 a (AU) 0.3046 0.0321 τ (days) - - s (AU) - - e - - i (◦) 88.945 0.008 b - - Table 7.60 – Planetary Characteristics for GSIC 21b Parameter Value R (R⊕) 8.139± 1.527 Composition Gas giant A 0.5 Teq (K) 600-800 M (M⊕) 40-150 This gas giant is considered a hot Jupiter despite orbiting comparatively far from the star by the stan- dards of this study. Unfortunately the period value obtained for this orbit was not sufficiently precise to allow a readable phase fold, and thus values dependent on τ were not found. Of these, the eccentricity had the potential to be interesting, as this planet is far enough from the star that tidal circularisation may not have occurred, and considering that this planet has likely migrated from further out in the system the orbital eccentricity may well be high. In comparison to the properties of Jupiter, this planet has a radius of 0.722 ± 0.135RJ and a mass of 0.126-0.472MJ. GSIC 22 The signal to noise ratio for GSIC 22 was high enough for the small frequency spacing to be calcu- lated after filtering. Table 7.61 – Stellar Parameters for GSIC 22 Parameter Value Error Δν0 (μHz) 94.025 0.688 νmax (mHz) 1.96181 0.00147 δν0,2 (μHz) 12.39 0.38 Teff (K) 6325 75 [Fe/H] 0.01 0.1 Xc - - M∗ (M�) 1.375 0.267 R∗ (R�) 1.442 0.182 Age (Gyrs) - - RG IHZ (AU) ∼ 1.64 - MaG OHZ (AU) ∼ 2.79 - Although the small frequency spacing was visible for this star, there were issues with plotting it on the asteroseismic diagrams. GSIC 22 did not fall onto any of the mass lines, so no accurate age or hydrogen frac- tion could be calculated. This has very recently been put down to a single error in the peak finding code used by the computational analysis team. An age estimate can be made, however, looking at the mass value in Table 7.61. The mass of GSIC 22 is significantly greater than solar mass and the maximum frequency for oscillations places this star somewhere in the middle of its MS lifetime. Therefore its age will be far lower than the age of the Sun, likely by a factor of two. There was one planet detected in orbit around GSIC 22. 67 GSIC 22b Table 7.62 – Orbital Parameters for GSIC 22b Parameter Value Error P (days) 17.83367414 3.08E – 06 a (AU) 0.14855 0.00962 τ (days) 0.2006 0.01 s (AU) 0.1898 0.0257 e > 0.278 0.192 i (◦) 89.335 0.051 b 0.3285 0.0658 Table 7.63 – Planetary Characteristics for GSIC 22b Parameter Value R (R⊕) 2.755± 0.345 Composition Gas dwarf A 0.3-0.5 Teq (K) 750-900 M (M⊕) 3-8 In this system there is, once more, a hot close- orbiting gas dwarf. The mass range has once again been suggested considering thermal atmospheric infla- tion. This planet has a higher eccentricity than might generally be expected, and may be undergoing tidal circularisation. GSIC 23 GSIC 23 has a power spectrum with a very high signal to noise ratio, so all desired parameters could be calculated. Table 7.64 – Stellar Parameters for GSIC 23 Parameter Value Error Δν0 (μHz) 59.666 0.384 νmax (mHz) 1.08708 0.00137 δν0,2 (μHz) 4.63 0.54 Teff (K) 6253 85 [Fe/H] –0.13 0.1 Xc 0.30 0.02 M∗ (M�) 1.418 0.274 R∗ (R�) 1.973 0.246 Age (Gyrs) 1.28 0.132 RG IHZ (AU) ∼ 2.20 - MaG OHZ (AU) ∼ 3.75 - Table 7.64 shows that this is another one of the larger stars in the data set. The high mass shows that GSIC 23 will progress off the MS sooner than our sun, and a hydrogen fraction value of 0.3 agrees with this suggestion. However, the age given from the asteroseis- mic diagrams is typical for a star of this mass around the middle of the MS. There was a single planet de- tected in orbit around GSIC 23. GSIC 23b Table 7.65 – Orbital Parameters for GSIC 23b Parameter Value Error P (days) 53.50521447 8.12E – 05 a (AU) 0.3122 0.0201 τ (days) 0.4362 0.01 s (AU) 0.3583 0.0454 e > 0.148 0.163 i (◦) 89.932 0.060 b 0.0464 0.0419 Table 7.66 – Planetary Characteristics for GSIC 23b Parameter Value R (R⊕) 2.268± 0.283 Composition Gas dwarf A 0.3-0.5 Teq (K) 600-700 M (M⊕) 3-7 This gas dwarf has an orbital period allowing the potential for significant eccentricity, as tidal forces will be weak at this separation. As it is quite small for a gas dwarf, it is likely to have a dense composition. GSIC 24 The region of interest on the power spectrum of GSIC 24 was visible without filtering, but filtering was required to find the small frequency spacing. Table 7.67 – Stellar Parameters for GSIC 24 Parameter Value Error Δν0 (μHz) 53.067 0.334 νmax (mHz) 0.99350 0.00126 δν0,2 (μHz) 5.11 1.07 Teff (K) 6174 92 [Fe/H] 0.22 0.1 Xc - - M∗ (M�) 1.698 0.329 R∗ (R�) 2.265 0.282 Age (Gyrs) - - RG IHZ (AU) ∼ 2.48 - MaG OHZ (AU) ∼ 4.23 - Although the small frequency spacing was visible, the asteroseismic diagrams weren’t able to find an age or hydrogen fraction for this star. Similarly to GSIC 20, this is due to the mass being greater than 1.5M� and the model on the asteroseismic diagrams not ac- counting for this. The large mass means that this star will evolve off of the main sequence sooner than more Sun-like stars. The value for the maximum frequency for oscillations is less than 1mHz, which implies that this star is reaching the end of its MS lifetime, though 68 its age will likely be far less than that of the Sun. There was one planet detected in orbit around GSIC 24. GSIC 24b Table 7.68 – Orbital Parameters for GSIC 24b Parameter Value Error P (days) 10.27783263 5.34E – 06 a (AU) 0.11034 0.00713 τ (days) - - s (AU) - - e - - i (◦) 88.899 0.421 b - - Table 7.69 – Planetary Characteristics for GSIC 24b Parameter Value R (R⊕) 3.140± 0.392 Composition Gas dwarf A 0.3-0.5 Teq (K) 1100-1300 M (M⊕) 7-15 This is a fairly large gas dwarf, with a high equi- librium temperature which will inflate its atmosphere. Due to an imprecise period value, the phase fold for this planet was not usable, resulting in no measurement for τ and dependent values. This planet is likely to orbit with low eccentricity due to tidal forces however. GSIC 25 GSIC 25 had a very standard power spectrum that yielded the small frequency spacing after a small amount of filtering. Table 7.70 – Stellar Parameters for GSIC 25 Parameter Value Error Δν0 (μHz) 153.232 0.337 νmax (mHz) 3.52729 0.00183 δν0,2 (μHz) 8.94 0.61 Teff (K) 5460 75 [Fe/H] 0.08 0.1 Xc 0.29 0.02 M∗ (M�) 0.909 0.166 R∗ (R�) 0.907 0.107 Age (Gyrs) 8.06 2.03 RG IHZ (AU) ∼ 0.81 - MaG OHZ (AU) ∼ 1.41 - Table 7.70 shows a mass and radius each within an error of solar values, with a lower temperature. This star is also the oldest in the data set for which the age could be calculated. It is a little under twice the age of the Sun with a significantly lower hydrogen fraction. A hydrogen fraction of 0.29 does, however, suggest that GSIC 25 has a reasonable amount of time left on the MS. There were no planets detected in orbit around GSIC 25. GSIC 26 GSIC 26 was another star with a low signal to noise ration on its power spectrum, but as the small spacing is visible after filtering this is not a problem. Table 7.71 – Stellar Parameters for GSIC 26 Parameter Value Error Δν0 (μHz) 69.940 0.493 νmax (mHz) 1.27947 0.00135 δν0,2 (μHz) 6.1 0.49 Teff (K) 5784 98 [Fe/H] –0.11 0.11 Xc 0.42 0.02 M∗ (M�) 1.090 0.216 R∗ (R�) 1.625 0.206 Age (Gyrs) 1.00 0.0940 RG IHZ (AU) ∼ 1.60 - MaG OHZ (AU) ∼ 2.75 - Table 7.71 shows another mass within one error of solar mass, but with a greater radius. Unfortunately, the initial mass value obtained through computational analysis does not agree with the value obtained in the asteroseismic diagrams, which was 1.5M�. The AHRD-derived mass would explain why GSIC 26 has a hydrogen fraction of less than half at such a young age. There were no planets detected in orbit around this star. GSIC 27 The power spectrum for GSIC 27 had a low signal to noise ratio, hence no l = 2 peaks were detected. Table 7.72 – Stellar Parameters for GSIC 27 Parameter Value Error Δν0 (μHz) 69.790 0.939 νmax (mHz) 1.36544 0.00249 δν0,2 (μHz) - - Teff (K) 5770 75 [Fe/H] 0.29 0.1 Xc - - M∗ (M�) 1.331 0.290 R∗ (R�) 1.740 0.240 Age (Gyrs) - - RG IHZ (AU) ∼ 1.70 - MaG OHZ (AU) ∼ 2.93 - The temperature displayed in Table 7.72 is around the temperature of the Sun, which is unexpected for a star of greater mass and radius. From the value of the maximum frequency for oscillations, it appears that 69 GSIC 27 isn’t too close to evolving off of the MS and with a large mass it can be deduced that this is a fairly young star, likely around 1Gyr. There were no planets detected in orbit around GSIC 27. GSIC 29 The power spectrum for GSIC 29 had a high sig- nal to noise ratio so that the l = 2 peaks were visible without filtering. Although, like for all other stars in the data, the power spectra underwent smoothing to find accurate values for the small and large frequency spacings. Table 7.73 – Stellar Parameters for GSIC 29 Parameter Value Error Δν0 (μHz) 102.915 0.276 νmax (mHz) 2.11675 0.00168 δν0,2 (μHz) 6.76 0.49 Teff (K) 6104 74 [Fe/H] –0.2 0.1 Xc 0.15 0.02 M∗ (M�) 1.141 0.207 R∗ (R�) 1.276 0.151 Age (Gyrs) 3.56 0.423 RG IHZ (AU) ∼ 1.37 - MaG OHZ (AU) ∼ 2.34 - It can be seen from Table 7.73 that the mass and radius of GSIC 29 are slightly greater than that of the Sun. This star is also slightly younger than the Sun, but due to its greater mass has a much lower hydrogen fraction, as it will progress off of the MS sooner. There were no planets detected in orbit around GSIC 29. GSIC 30 The power spectrum from GSIC 30 was hard to read due to the low signal to noise ratio. The spectrum was filtered and the l = 0 peaks were found, although the l = 2 peaks could not be located. This led to the calculation of GSIC 30’s mass and radius. Table 7.74 – Stellar Parameters for GSIC 30 Parameter Value Error Δν0 (μHz) 107.752 0.368 νmax (mHz) 2.2482 0.00124 δν0,2 (μHz) - - Teff (K) 5982 82 [Fe/H] –0.02 0.1 Xc - - M∗ (M�) 1.104 0.204 R∗ (R�) 1.224 0.146 Age (Gyrs) - - RG IHZ (AU) ∼ 1.27 - MaG OHZ (AU) ∼ 2.18 - From the mass, radius and effective temperature values in Table 7.74, GSIC 30 was deduced to be a little larger and hotter than the Sun, with one orbiting planet detected. GSIC 30b Table 7.75 – Orbital Parameters for GSIC 30b Parameter Value Error P (days) 41.74594751 2.97E – 05 a (AU) 0.2434 0.0150 τ (days) 0.1918 0.01 s (AU) 0.3944 0.0515 e > 0.621 0.234 i (◦) 89.384 0.017 b 0.133 0.179 Table 7.76 – Planetary Characteristics for GSIC 30b Parameter Value R (R⊕) 2.406± 0.289 Composition Gas dwarf A 0.3-0.5 Teq (K) 500-600 M (M⊕) 3.5-7 This planet is another gas dwarf, with a particularly high value for the eccentricity. It may very well be the case that this planet has ejected other material to now reside in its current orbit, which may be too far from the host star for tidal circularisation to have taken place since. GSIC 31 The power spectrum from GSIC 31 was very simple to read due to the high signal to noise ratio. No filtering was needed, as both the l = 0 and l = 2 peaks were clearly visible. This led to the calculation of GSIC 31’s mass, radius and age. Table 7.77 – Stellar Parameters for GSIC 31 Parameter Value Error Δν0 (μHz) 101.434 0.053 νmax (mHz) 2.05453 0.00152 δν0,2 (μHz) 5.51 0.21 Teff (K) 5793 74 [Fe/H] 0.12 0.07 Xc 0.11 0.02 M∗ (M�) 1.022 0.183 R∗ (R�) 1.242 0.145 Age (Gyrs) 4.47 0.0543 RG IHZ (AU) ∼ 1.22 - MaG OHZ (AU) ∼ 2.11 - From the values given in Table 7.77, it can be de- duced that GSIC 31 is similar to the Sun in terms of age, effective temperature and mass, though it has a 70 radius about 20% larger than that of the Sun. It has almost finished core hydrogen-burning as it only has 11% left. One planet was detected orbiting GSIC 31. GSIC 31b Table 7.78 – Orbital Parameters for GSIC 31b Parameter Value Error P (days) 5.398754123 8.34E – 07 a (AU) 0.060668 0.00363 τ (days) - - s (AU) - - e - - i (◦) 89.558 0.337 b - - Table 7.79 – Planetary Characteristics for GSIC 31b Parameter Value R (R⊕) 2.142± 0.250 Composition Gas dwarf A 0.3-0.5 Teq (K) 1000-1200 M (M⊕) 2.5-6 This gas dwarf orbits extremely close to its star, resulting in a high equilibrium temperature which will inflate it somewhat. Although a phase fold was not obtained for this planet, tidal circularisation means a low eccentricity orbit is to be expected. GSIC 32 The GSIC 32 power spectrum was also moderately difficult to read. The signal to noise ratio was low, so the spectra needed a filtering to obtain l = 0 and l = 2 peaks. This led to the large and small frequency spacing being found, and the age, radius and mass of the star were then acquired. Table 7.80 – Stellar Parameters for GSIC 32 Parameter Value Error Δν0 (μHz) 68.115 0.358 νmax (mHz) 1.24896 0.00114 δν0,2 (μHz) 5.41 0.22 Teff (K) 5911 66 [Fe/H] –0.2 0.06 Xc 0.33 0.02 M∗ (M�) 1.164 0.218 R∗ (R�) 1.691 0.206 Age (Gyrs) 1.15 1.54 RG IHZ (AU) ∼ 1.72 - MaG OHZ (AU) ∼ 2.96 - The values in Table 7.80 show that GSIC 32 is a young, Sun-like star with a radius about 70% bigger than the Sun’s. Its low core hydrogen content and age also reveal that GSIC 32 is large, as more massive stars burn hydrogen and therefore evolve quicker. One planet was detected orbiting the star. GSIC 32b Table 7.81 – Orbital Parameters for GSIC 32b Parameter Value Error P (days) 16.2318437 4.12E – 05 a (AU) 0.13197 0.00823 τ (days) - - s (AU) - - e - - i (◦) 89.912 0.057 b - - Table 7.82 – Planetary Characteristics for GSIC 32b Parameter Value R (R⊕) 2.463± 0.300 Composition Gas dwarf A 0.3-0.5 Teq (K) 800-1000 M (M⊕) 3.5-7 This gas dwarf’s period was not calculated to suffi- cient precision for a clear phase fold, but a low eccen- tricity can be expected due to tidal forces in this close orbit. The mass range was chosen assuming a density exceeding that of Neptune due to a larger fraction of the radius taken up by the core. GSIC 33 In the GSIC 33 power spectrum, the peaks were simple to find, but the spectrum was filtered to obtain more prominent peaks, as the signal to noise ratio was preventing the l = 2 peaks from being seen. The age, radius and mass of the star were then acquired by using the large and small frequency spacing. Table 7.83 – Stellar Parameters for GSIC 33 Parameter Value Error Δν0 (μHz) 75.888 0.733 νmax (mHz) 1.48114 0.00141 δν0,2 (μHz) 7.07 0.05 Teff (K) 6225 75 [Fe/H] 0 0.08 Xc 0.48 0.02 M∗ (M�) 1.362 0.276 R∗ (R�) 1.658 0.216 Age (Gyrs) 0.798 0.0911 RG IHZ (AU) ∼ 1.84 - MaG OHZ (AU) ∼ 3.13 - 71 From the values in Table 7.83, GSIC 33 appears to be a hot star a little larger than the Sun. It’s relatively young, at less than a billion years old and appears to be in the middle of its MS lifetime from its core hydrogen fraction. No planets were detected orbiting GSIC 33. GSIC 34 The power spectrum from GSIC 34 had a very low signal to noise ratio, therefore it was difficult to find the peaks. After filtering, the l = 0 peaks could be located, but the l = 2 peaks remained hidden. This meant that the mass and radius values could be determined, but not the age of the star. Table 7.84 – Stellar Parameters for GSIC 34 Parameter Value Error Δν0 (μHz) 74.705 0.167 νmax (mHz) 1.42627 0.00108 δν0,2 (μHz) - - Teff (K) 5781 76 [Fe/H] 0.09 0.1 Xc - - M∗ (M�) 1.159 0.211 R∗ (R�) 1.587 0.187 Age (Gyrs) - - RG IHZ (AU) ∼ 1.56 - MaG OHZ (AU) ∼ 2.68 - Due to the values of radius, mass and effective tem- perature in Table 7.84, it can be deduced that GSIC 34 is a little larger than the Sun and is approximately the same temperature. One planet was detected orbiting the star. GSIC 34b Table 7.85 – Orbital Parameters for GSIC 34b Parameter Value Error P (days) 3.213668950 8.69E – 07 a (AU) 0.0448 0.00271 τ (days) 0.162 0.01 s (AU) 0.04664 0.00621 e > 0.042 0.152 i (◦) 88.820 0.903 b 0.130 0.102 Table 7.86 – Planetary Characteristics for GSIC 34b Parameter Value R (R⊕) 3.998± 0.472 Composition Gas dwarf/ gas giant A 0.3-0.5 Teq (K) 1350-1550 M (M⊕) 8-25 This planet is approaching the size where it could be described as a hot Jupiter, with a period of around 3 days, which is around the minimum possible. As expected, the eccentricity is found to likely be low, al- though this was almost definitely not the case in the past, as the planet must have migrated inwards. It is likely that the planet is tidally locked to the star, with the same face constantly on the day side. The atmo- sphere of this planet will be significantly inflated by temperature, and so a relatively low mass range has been suggested for this planet. GSIC 35 The power spectrum from GSIC 35 was hard to read due to the low signal to noise ratio. Filtering the spectra resulted in the appearance of the second degree mode peaks, which in turn resulted in the values of the age, mass and radius of GSIC 35. Table 7.87 – Stellar Parameters for GSIC 35 Parameter Value Error Δν0 (μHz) 117.502 0.087 νmax (mHz) 2.43778 0.00128 δν0,2 (μHz) 5 0.54 Teff (K) 5647 74 [Fe/H] –0.15 0.1 Xc 0.06 0.02 M∗ (M�) 0.913 0.164 R∗ (R�) 1.084 0.127 Age (Gyrs) 7.64 2.08 RG IHZ (AU) ∼ 1.02 - MaG OHZ (AU) ∼ 1.77 - The values in Table 7.87 for mass and radius are quite similar to solar mass and radius. GSIC 35 is 3 billion years older and a little bit cooler than the Sun. Its hydrogen core mass fraction suggests that it is about to leave the main sequence, though the reason- ably large maximum frequency for oscillations would disagree with this. One planet was detected orbiting GSIC 35. GSIC 35b Table 7.88 – Orbital Parameters for GSIC 35b Parameter Value Error P (days) 45.2942007 2.29E – 05 a (AU) 0.2412 0.0145 τ (days) 0.3436 0.04 s (AU) 0.2116 0.0349 e > 0.123 0.154 i (◦) 89.712 0.108 b 0.2111 0.0897 72 Table 7.89 – Planetary Characteristics for GSIC 35b Parameter Value R (R⊕) 2.311± 0.270 Composition Gas dwarf A 0.3-0.5 Teq (K) 450-550 M (M⊕) 2.5-6 This is a gas dwarf orbiting at a period value which is relatively high for our study. Given the high period, the low minimum eccentricity is slightly surprising, as tidal forces will not be large at this distance, although it is of course possible that eccentricity is larger. The relatively low temperature constrains the mass value slightly, as the atmosphere will not be significantly in- flated. GSIC 36 The power spectrum from GSIC 36 had a high sig- nal to noise ratio, and as a result, both l = 0 and l = 2 peaks were clearly visible when the right area of the spectrum was enlarged. This made finding the large and small frequency spacing very simple. Table 7.90 – Stellar Parameters for GSIC 36 Parameter Value Error Δν0 (μHz) 103.260 0.071 νmax (mHz) 2.09566 0.00125 δν0,2 (μHz) 5.35 0.35 Teff (K) 5825 50 [Fe/H] 0.096 0.026 Xc - - M∗ (M�) 1.017 0.176 R∗ (R�) 1.225 0.140 Age (Gyrs) - - RG IHZ (AU) ∼ 1.22 - MaG OHZ (AU) ∼ 2.09 - The values in Table 7.90 for mass, radius and ef- fective temperature are similar to solar values. Unfor- tunately, due to a communication error at the end of the project, the metallicity for this star was not passed on to the computational analysis team and due to time constraints at the end of the project, this could not be rectified. Therefore, no age or hydrogen fraction could be obtained. No planets were detected orbiting GSIC 36. GSIC 38 The power spectrum of GSIC 38 required no fil- tering to find the l = 0 peaks, and therefore the large frequency spacing. However, the l = 2 peaks were too unclear after filtering to calculate the small frequency spacing, therefore there isn’t an age value for GSIC 38. Discussions led to the idea that GSIC 38’s power spectrum may have been hard to read due to G modes and P modes mixing, reducing the clarity of the l = 2 peaks. Table 7.91 – Stellar Parameters for GSIC 38 Parameter Value Error Δν0 (μHz) 23.525 1.706 νmax (mHz) 0.39608 0.00032 δν0,2 (μHz) 0 0 Teff (K) 5470 70 [Fe/H] –0.79 0.1 Xc - - M∗ (M�) 2.323 1.047 R∗ (R�) 4.325 1.104 Age (Gyrs) - - RG IHZ (AU) ∼ 3.87 - MaG OHZ (AU) ∼ 6.72 - The mass and radius values in Table 7.91 are sig- nificantly larger than solar values, therefore GSIC 38 is likely to have left the MS and may be a red giant. The l = 2 peaks seemed to be in the power spectra, but were located seemingly randomly, leading to the belief that they were actually mixed-mode oscillations, which are common in red giants. The lower-than-average value of the maximum power frequency backs up the claim that GSIC 38 is a strong candidate for a red gi- ant. No planets were detected orbiting the star. GSIC 39 GSIC 39’s power spectrum also needed no filtering to find the l = 0 peaks. Once again, the l = 2 peaks weren’t where they were expected to be, even after filtering. As with GSIC 38, it was thought that mixed- mode oscillations may be the cause of the lack of clear l = 2 peaks. Table 7.92 – Stellar Parameters for GSIC 39 Parameter Value Error Δν0 (μHz) 17.457 0.060 νmax (mHz) 0.24289 0.00035 δν0,2 (μHz) 0 0 Teff (K) 4840 97 [Fe/H] 0.2 0.16 Xc - - M∗ (M�) 1.471 0.284 R∗ (R�) 4.530 0.554 Age (Gyrs) - - RG IHZ (AU) ∼ 3.28 - MaG OHZ (AU) ∼ 5.82 - The radius value in Table 7.92 is approximately four and a half times larger than the radius of the Sun, leading to the belief that GSIC 39 is a red giant. The lower effective temperature backs up this theory. The maximum power frequency is, again, very low. This provides further evidence that the star may be a red giant. GSIC 39 also had a similar power spectrum to 73 GSIC 38, where the l = 2 peaks have been shifted due to suspected mixed-mode oscillations. One planet was detected orbiting the star. GSIC 39b Table 7.93 – Orbital Parameters for GSIC 39b Parameter Value Error P (days) 21.405289 1.33E – 04 a (AU) 0.1716 0.0111 τ (days) 0.4623 0.04 s (AU) 0.3106 0.0465 e > 0.811 0.295 i (◦) 86.890 0.0577 b 0.800 0.155 Table 7.94 – Planetary Characteristics for GSIC 39b Parameter Value R (R⊕) 8.352± 1.022 Composition Gas giant A 0.5 Teq (K) 950-1150 M (M⊕) 40-120 This is a close-orbiting gas giant with a very large eccentricity - in fact, it was calculated that the planet may well pass within 0.5R∗ of its host. The tidal forces on the planet will be very large, and it is likely that the orbit will eventually circularise. The temperature and radius of the planet are likely to fluctuate significantly throughout the orbit as the planet approaches the star. In comparison to the properties of Jupiter, this planet has a radius of 0.7409±0.0907RJ and a mass of 0.126-0.377MJ. GSIC 40 Much like GSIC 38 and GSIC 39, GSIC 40’s power spectrum displayed clear l = 0 peaks. However, the l = 2 peaks were actually found with ease, considering mixed-mode oscillations slightly obscured the power spectrum. Therefore, the mass and radius of GSIC 40 could be deduced. Due to the star no longer being on the MS, the asteroseismic diagrams could not be used to find its age. Table 7.95 – Stellar Parameters for GSIC 40 Parameter Value Error Δν0 (μHz) 18.567 0.0063 νmax (mHz) 0.25875 0.00348 δν0,2 (μHz) 2.25 0.09 Teff (K) 4995 78 [Fe/H] –0.07 0.1 Xc - - M∗ (M�) 1.446 0.302 R∗ (R�) 4.323 0.549 Age (Gyrs) - - RG IHZ (AU) ∼ 0.33 - MaG OHZ (AU) ∼ 0.59 - The radius value in Table 7.95 indicates that GSIC 40 may be a red giant. Its temperature is in the typ- ical range for red giants, and the lower-than-average maximum power frequency also points to the conclu- sion that GSIC 40 has left the MS. One planet was detected orbiting GSIC 40. GSIC 40b Table 7.96 – Orbital Parameters for GSIC 40b Parameter Value Error P (days) 52.5008334 5.14E – 05 a (AU) 0.3102 0.0216 τ (days) 0.6171 0.01 s (AU) 0.5446 0.0697 e > 0.756 0.256 i (◦) 89.819 0.141 b 0.0854 0.0684 Table 7.97 – Planetary Characteristics for GSIC 40b Parameter Value R (R⊕) 12.589± 1.599 Composition Gas Giant A 0.5 Teq (K) 700-800 M (M⊕) 350-1000 This gas giant follows a very eccentric orbit, espe- cially considering that it resides quite close to its host star. It is possible that the tidal forces acting on it are not sufficient to circularise this orbit. Due to the tail-off in the mass-radius relations caused by electron degeneracy pressure, the mass range for this planet is particularly generous, as planets above a certain ra- dius could theoretically be composed of the same mass of material. In comparison to the properties of Jupiter, this planet has a radius of 1.117 ± 0.142RJ and a mass of 1.101-3.145MJ. 74 7.2.2 Omissions (JG) GSIC 2 There is very little to say about this star due to the extremely low signal to noise ratio prohibiting the detection of the l = 0 peaks, so not even the large fre- quency spacing could be calculated. The only values known are the temperature of 6343 ± 85K and metal- licity of –0.04±0.1 which were given at the start of the project. As the temperature is higher than that of the Sun, we can expect this star to have a slightly greater mass and radius. GSIC 11 The power spectrum from GSIC 11 also had a very low signal to noise ratio, and despite using a lot of filtering, no values could be obtained. Therefore, the mass, radius and age of the star could not be calcu- lated. The only known values for this star are a tem- perature of 5046±74K and a metallicity of –0.55±0.07. The temperature here is significantly lower than that of the Sun, suggesting a lower mass and radius. GSIC 28 The power spectrum and light curve for this star were found to be identical to that of GSIC 1, indicating that these systems are in fact the same. As GSIC 1 has already been discussed above, there is no need for further analysis of GSIC 28. GSIC 37 The power spectrum and light curve for this star were found to be identical to that of GSIC 11, indicat- ing another duplicate system. As GSIC 11 has already been discussed above, there is no need for further anal- ysis of GSIC 37. 7.2.3 Results from Asteroseismic Diagrams (GM) The stars with computational values for both Δν and δν were plotted on asteroseismic diagrams, the re- sults of which are shown on the following pages along with Table 7.98, which shows the complete set of data extracted from all of the diagrams. Interestingly, GSIC 22 in Figure 7.2.4 seems to be a pre-MS star as it is above the ZAMS hydrogen mass fraction. We know that all of the stars investigated in this project are MS stars, which suggests that the computational value for either Δν or δν is incorrect for this star. Indeed, by using the value of δν calculated manually for this star it should in fact have a mass of around 1.4 M� and a hydrogen mass fraction of around 0.6, which is much more agreeable with expectations. The placement of the star with computational values means that no stel- lar properties have been extracted for this star and as such are unavailable in Table 7.98. Figure 7.2.8 – A more detailed asteroseismic diagram for GSIC 3. The red marker is the star position with accom- panying grey error bars. In Figure 7.2.6 GSIC 20 and GSIC 24 are the only two stars with masses outside of the scope of the dia- grams, however they are close enough that their masses can be estimated at 1.53± 0.10M� and 1.70± 0.20M� respectively. As determining the age and other stellar properties depended on matching the star to a model, and with no model available for these masses, most of the stellar properties are also unavailable in Table 7.98 for these two stars. Also, while there is available data to plot GSIC 13 and GSIC 36, due to a communication problem there was no available metallicity data for these stars when the asteroseismic diagrams were being constructed and hence the two stars do not appear on them. Unfortu- nately, due to the time constraints of the project, these have not been able to have been added to the existing diagrams and their stellar properties extracted. This could be done as a future investigation. The asteroseismic diagrams themselves are gener- ally the expected shape - matching those produced by previous studies. As with Figure 3.2.6, the less massive stars have a much broader range for δν which gradu- ally shrinks up to a mass value of around 1M�, from which point the range stays roughly the same across the hydrogen mass fraction range. More massive stars tend to have lower values for Δν and δν, which again agrees with expectations. The hydrogen mass fraction isopleths tend to follow the expected trend, with a few unexpected deviations such as the ‘kink’ in the ZAMS isopleth seen around 1.3M� in Figure 7.2.4. Apart from these minor deviations the asteroseismic diagrams should provide a relatively accurate tool to calculate stellar properties, presuming Δν and δν are relatively accurate. Figure 7.2.8 shows a more detailed asteroseismic diagram that was completed for GSIC 3. This allowed for a more precise value for the mass and hydrogen mass fraction to be extracted for each star, but due to time constraints was only completed for this star. 75 Figure 7.2.1 – Asteroseismic diagram of Metallicity Z=0.009 showing GSIC 18 Figure 7.2.2 – Asteroseismic diagram of Metallicity Z=0.011 showing GSIC 29 and 32 76 Figure 7.2.3 – Asteroseismic diagram of Metallicity Z=0.014 showing GSIC 3, 15, 16, 23, 26 and 35 Figure 7.2.4 – Asteroseismic diagram of Metallicity Z=0.017 showing GSIC 4, 6, 12, 22 and 33 77 Figure 7.2.5 – Asteroseismic diagram of Metallicity Z=0.023 showing GSIC 0, 9, 14, 25 and 31 Figure 7.2.6 – Asteroseismic diagram of Metallicity Z=0.029 showing GSIC 20 and 24 78 Figure 7.2.7 – Asteroseismic diagram of Metallicity Z=0.035 showing GSIC 28 Table 7.98 – Data extracted for available stars from the asteroseismic diagrams. Stellar hydrogen mass fraction and stellar age are included in the tables in Section 7.2.1 for these stars. GSIC Z JCD M (M�) σ(M) R (R�) σ(R) L (L�) σ(L) Teff σ(Teff) 0 0.021 0.023 1.06 0.05 1.14 0.11 1.48 0.37 5982 117 3 0.013 0.014 0.92 0.05 0.93 0.06 0.88 0.23 5803 181 4 0.018 0.017 1.50 0.05 1.97 0.11 6.85 1.03 6688 81 6 0.017 0.017 1.29 0.05 1.37 0.08 3.25 0.60 6667 116 9 0.023 0.023 1.38 0.05 1.46 0.07 3.83 0.63 6724 120 12 0.020 0.017 1.35 0.05 1.62 0.08 4.40 0.70 6604 77 14 0.023 0.023 1.43 0.05 1.48 0.06 4.33 0.70 6873 216 15 0.015 0.014 1.50 0.05 1.69 0.06 6.95 1.11 7248 173 16 0.014 0.014 1.28 0.05 1.39 0.08 3.50 0.65 6721 115 18 0.009 0.009 0.80 0.05 0.77 0.10 0.48 0.34 5496 410 20 0.028 0.029 1.53 0.10 N/A N/A N/A N/A N/A N/A 22 0.019 0.017 N/A N/A N/A N/A N/A N/A N/A N/A 23 0.014 0.014 1.50 0.05 1.83 0.09 7.22 1.14 7036 127 24 0.031 0.029 1.70 0.20 N/A N/A N/A N/A N/A N/A 25 0.023 0.023 0.88 0.05 0.87 0.06 0.59 0.17 5431 206 26 0.015 0.014 1.50 0.05 1.67 0.06 6.92 1.10 7269 177 28 0.034 0.035 1.50 0.05 1.99 0.17 5.41 1.31 6276 145 29 0.012 0.011 1.11 0.05 1.28 0.11 2.42 0.52 6405 90 31 0.025 0.023 1.13 0.05 1.30 0.13 2.01 0.41 6047 33 32 0.012 0.011 1.50 0.05 1.73 0.05 7.75 1.16 7366 191 33 0.019 0.017 1.50 0.05 1.61 0.06 6.23 0.98 7214 181 35 0.013 0.014 0.96 0.05 1.09 0.06 1.34 0.25 5963 124 79 Figure 7.2.9 – The diagram shows the distribution of lu- minosities of the stars, as extracted from the asteroseismic diagram data. Stellar Luminosity A stellar property that could not be found with the other methods was the luminosity of each of the stars. As this was given in the parameters for each model, by matching each star with its closest model this could be calculated from the asteroseismic diagrams. Figure 7.2.9 shows the distribution of luminosities for available stars. The diagram shows that the major- ity of the stars are much more luminous than the Sun, with only GSIC 3, 18 and 25 having a slightly lower luminosity. Most of the stars are between three and eight times as the sun, which would correspond with them being relatively solar-like, MS stars as expected. With a non-normalised light curve for each star, or from data about the apparent luminosity of each, it would perhaps be possible to calculate the distance each is from our solar system using a 1/r2 relationship. Without this, however, the luminosity data serves only as a comparison for luminosity of the stars against the Sun. 7.2.4 Comparison Between Scaling Relation and Asteroseismic Diagram Values (GM) Stellar Mass Figure 7.2.10 – Comparison of stellar mass values computed from scaling relations and extracted from asteroseismic di- agrams As can be seen in Figure 7.2.10, the masses ex- tracted from the asteroseismic diagrams agree with the vast majority of the mass values calculated from scal- ing relations, with the exception of GSIC 26 and GSIC 32. The values extracted from the diagrams also have a much smaller constraint to them than the scaling re- lation values. This is due to a relatively large error on Δν or δν not usually causing a significant change in mass and the relatively small errors on both calculated values. The only stars where the masses could not be constrained as accurately were GSIC 20 and GSIC 24, as both of these stars were slightly beyond the scale of the graph with masses greater than 1.5M�. However, as they were still relatively close to the available mass lines, an estimate of their masses could be made by extrapolating the data. There are a few reasons why the two methods could have disagreed on GSIC 26 and GSIC 32. One reason could be that the values derived from the scaling re- lations are incorrect, however this would suggest that either νmax or Δν for these values have been measured incorrectly, that these two stars were different types of stars than which the scaling relations work for or that the values given to us for the effective tempera- ture were incorrect for these two stars. The frequency- power spectra for these two stars had a fairly low SNR and were of a high enough quality for δν to be measured for them, which suggests that the measured value for Δν would have been to a higher accuracy than for other stars and that νmax would be fairly reliable also. It is also unlikely that these two stars alone were different 80 enough for the scaling relations to be inaccurate and the values for the effective temperature were calculated independent of this project so could be presumed to be correct. This, along with the large discrepancies be- tween the model effective temperatures and given val- ues for these stars, discussed in section 7.2.4, suggests that either there are inaccuracies in the model used to compute the frequency data for these stars or that they do not fit the model that they have been matched to. Needless to say, due to the majority of the mass data extracted from the asteroseismic diagrams agreeing to that from the scaling relation, asteroseismic diagrams can be seen as a fairly reliable source for this stellar property. It would have been interesting to investigate how the planetary properties calculated differed when using the asteroseismic diagram results, however due to the time it took to calculate Δν, δν and νmax; plot the diagrams; plot the stars and extract the data, it was not possible to do this within the time constraint of the project. Stellar Radius Figure 7.2.11 – Comparison of stellar radii values computed from scaling relations and extracted from asteroseismic di- agrams Figure 7.2.11 suggests that there is a very high de- gree of agreement between the stellar radii calculated with the two methods. Indeed, all but one of the stars in the two sets of data lie within each others constraints and the one that does not, GSIC 23, only marginally disagrees. The reasons for such high agreement are twofold. Firstly, the scaling relation for radius has a much lower power-dependency for νmax, Δν and the effective temperature. νmax/νmax,� has a linear depen- dency as opposed to a cubic dependency, Δν is only to the power of –2 as opposed to –4 and Teff is to the power of 1/2 compared to 3/2. This makes it much less sensitive to any errors in these values. Secondly, the radii values from the stellar models are also much less dependent on less understood stellar physics, namely surface effects, convection mixing and other free pa- rameters where published literature disagrees on the exact values, which often affects other intrinsic proper- ties of the star. This makes the radius calculation from nearly all stellar models very reliable and they tend to agree between different stellar evolution codes, mak- ing the asteroseismic diagrams a valid alternative to compute this property to the scaling relations. As the values were so similar, this probably would not have made any significant differences to the planetary prop- erties discussed earlier in this chapter. Effective Temperature Figure 7.2.12 – Comparison of effective stellar temperature values from given values and extracted from asteroseismic diagrams Unfortunately, as Figure 7.2.12 shows, there does not seem to be any degree of agreement between the effective temperature extracted from data from the as- teroseismic diagrams and the values given to use to use in the project. The main reason for this is probably the inaccuracy of stellar evolution models in modelling stellar surface physics. Due to the temperature gra- dient present in a star’s core, if the modelled surface temperature is wrong then this will affect the overall ef- fective temperature. From Figure 7.2.12, it seems that the majority of the stars have a much higher modelled effective temperature than the known data, with only GSIC 3, 18, 25 and 28 agreeing to any degree with the given values and the other temperatures being sig- nificantly higher than the actual values. Discrepancy with the given values also suggests that the luminosity data extracted from the asteroseismic diagrams may also disagree with values calculated using other meth- ods, but with no other data to compare the luminosity values to it is only possible to speculate about this based upon their relationship to one another. 81 It is obvious that other methods are significant bet- ter and more accurate at calculating effective temper- ature of stars and until the surface physics of stars is better understood and modelled they are not a feasible way of determining this property. 7.3 General Trends To comply with the overall question asked in the title of the project, a group of charts have been com- piled to display the relationship between the charac- teristics of an extrasolar system and the properties of its host star. This will include: the relationship be- tween the mass of the star and the age of the system, the relationship between the mass of the star and its luminosity, the relationship between the mass of a star and its radius, the relationship between the metallicity of a star and the composition of its orbiting planets, the relationship between the radius of a star and the radius of its orbiting planets, a simple histogram of the number of planets in our data set orbiting with a par- ticular period, and finally the relationship between the mass and radius of the discovered exoplanets. 7.3.1 Relationship Between Stellar Mass and System Age (GM) Figure 7.3.1 – This graph shows the masses and ages of the stars, as extracted from the asteroseismic diagrams. The blue points are the extracted values, with red error bars accompanying them. The black line is a second-order poly- nomial that has been fitted to the points. The age of the star has a very large dependence on its mass and Figure 7.3.1 shows this relationship for the stars plotted on the asteroseismic diagrams. The second-order polynomial that has been fitted to these stars suggests that as the stars increase in mass, they tend to decrease in age. This is in agreement with what would be expected, especially for MS stars: smaller stars burn through their hydrogen content at a significantly slower rate than more massive stars so a less massive star has a much greater probability of being older than a more massive one. More massive stars also leave the MS at a much younger age, so find- ing a more massive star older than around two billion years old that was not post-MS would be very unlikely. Our data correlates with this, with stars greater than 1.2M� having an age between a few hundred thousand and two billion years. For a greater range in stellar mass, you would ex- pect the relationship between the two to move away from a second-order polynomial, with the age of less massive stars curving to reach a cut-off limit (the age of the universe at an absolute maximum) and the age of more massive stars curving to either become more lin- ear or at least asymptotic to zero: there is a maximum rate at which a star could burn through its hydrogen fuel and also the consideration of a stellar mass limit to impose. However, for the small range of stars with this data the trend line fitted to the data is acceptable. 7.3.2 Relationship Between Stellar Mass and Luminosity (GM) For stars with a mass between 0.43M� and 2.0M� we would expect the luminosity to vary such that [142] L L� = ( M M� )4 (7.1) Therefore, if you plotted the mass against the lu- minosity of our stars you would expect a line of best fit to follow a fourth power-law. Figure 7.3.2 – The diagram shows the relationship between the mass and luminosity, as extracted from the asteroseis- mic diagram data. The blue circles are the positions of the stars while the red line is a line of best fit set to determine and follow a power-law relationship for the data. Figure 7.3.2 shows such a diagram with the data plotted and a power-law line of best fit added. The equation to the top left of the figure shows that the 82 line of best fit has the equation y = 1.235x4.099, sug- gesting that our data agrees with a fourth power-law to a high degree. This suggests that the luminosity data extracted from the asteroseismic diagram data should be reliable. However, without another method of cal- culating these values it is impossible to test the validity of this. 7.3.3 Relationship Between Stellar Mass and Radius (MH) Main-sequence stars are observed to adhere to a mass-radius power-law relation [143]: R ∝ Mα (7.2) where α = 0.8 for stars lower than solar mass and α = 0.6 for stars of mass greater than the Sun. This division is approximately the point below which a star can have a convective envelope, the presence of which allows energy to more easily escape the star, which reduces radiation pressure and causes a slightly lower radius than for a star without a convective zone, thus the steeper relationship. Unfortunately, as can be seen in Figure 7.3.3, our stars do not exhibit this relationship, possibly due to the small sample size and limited range of the vari- ables. The small number of stars all with mass just less than 1M� did not allow us to generate a fit for that region, and a fit on the region above 1M� gave a power of 2, which is not at all close to the expected value of 0.6. Despite this, it is clear that greater mass stars have a larger radius, which agrees, globally if not quantitatively, with Equation (7.2). Figure 7.3.3 – A plot of the radius against the mass of all the stars for which we have those values. 7.3.4 Relationship Between Stellar Metallicity and Planet Composition (MH) Figure 7.3.4 – This diagram shows the stellar metallicities associated with each type of planet composition of the plan- ets detected from our data. The red circles are rocky plan- ets, the black circles are gaseous planets, and the purple circles are those which could be either a large rocky planet or a small gas planet. Figure 7.3.4 allows us to visualise what type of planet would be expected from a star with a partic- ular metallicity. Clearly, there is not a strong relation; this is perhaps because the categories of composition are qualitative rather than quantitative. A useful mea- sure of composition might have been the mean density of the planet, where solid rocky planets would have a higher density than diffuse gas giants. Unfortunately, the large mass ranges for the planets, caused by not having a more direct method of measuring the mass such as the Doppler wobble method, meant that get- ting a reasonable value for the mean density was im- possible. It can, however, be noted that the average metal- licity of the stars with gas giants is higher than the average metallicity of the stars with rocky planets. This agrees with the findings of Fischer and Valenti (2005), who identified a subset of 850 Doppler method- detected planets. In this subset they found that the formation probability of gas giant planets rose with the square of the number of metal atoms [144]. They also determined a positive correlation between metal- licity and both the number of planets in a system and the total mass of the planets. The three of our sys- tems that have multiple planets (GSIC 6, 7, and 8) have metallicity ranging from Z = 0.015 to 0.022, in the middle of the range. With such a spread it is pos- sible that our sample size simply is not big enough to observe the same trend. 7.3.5 Relationship Between Stellar Radius and Planetary Radius (JG) Due to the information given about planet forma- tion in Section 4.1, the team decided to look for a trend between the radius of a star and the radius of the plan- ets in its system. Section 4.1 states that stars and plan- ets each form from the same Giant Molecular Clouds 83 (GMCs). A fair assumption to make is that a GMC with the ability to form a larger protostar will also leave a greater amount of matter in the protoplanetary disk, which in turn allows larger planets to form. Figure 7.3.5 – Scatter with straight line displaying the trend between the radius of a star and the radius of its orbiting planets. the uncertainty on the radius of the star has been omitted, as the error on planetary radius is dependent on this value and this accounts for both In Figure 7.3.5, all planets with a known radius have been used for the plot, giving a sample size of 28 planets. A straight line was fitted to the sample and this shows a clear positive trend. If the relationship were to be quantified, the gradient value of 0.1875 ± 0.0585 would be used, although there is no basis for the accuracy of this number due to the small sample size used to calculate it. This number is also taking into account the clear outlier in the sample to the far right of Figure 7.3.5, the gas giant in orbit around GSIC 39 which is one of the proposed red giants. As most stars on the MS will become larger as they evolve off the MS, the results of red giants do not relate to a trend. 7.3.6 Frequency of Orbital Period for Discovered Exoplanets (PS) It was decided to record the frequency of planets detected orbiting at various period ranges, to attempt to detect any particularly common values. For this, periods were divided into bins of width 2 days, and the number of planets with a period in each range was counted. The result is shown in Figure 7.3.6, and a clear peak can be seen around the 5 day mark. This peak can be explained by reference to the three-day pileup (4.6.5), and it is to be expected, given a higher sample size, that the centroid of the peak can be seen at a slightly lower point. Figure 7.3.6 – A histogram showing the periods of the de- tected planets, in bin sizes of 2 days. A peak around the 3-6 day mark is clearly visible. It is important to consider here that there is a heavy selection bias towards low period planets in transit de- tections. As the probability of observing a transit is in- versely proportional to the semi-major axis (see Equa- tion (4.34)), it can be can seen that by applying Ke- pler’s third law the probability of observing a transit will scale as P–2/3 , and thus many higher period plan- ets will not be observed. On the other hand, as planets in a system tend to orbit along a similar ‘ecliptic’ plane, if one planet in a system is observed to transit, there is a higher probability that other planets in the same system will also transit the star and be detected. This model assumes that the planets in our sample are a representative sample of extrasolar planets. This may well not be the case, as our study did not detect any planets smaller than about 1.5R⊕, and so if there are characteristic periods for small planets they will not have been observed here. The peak at around 5 days is significant, and clearly demonstrates a detection of the three-day pileup in our results. The second peak at around 42 days may have occurred by chance, but it is possible that there is some reason for an increased number of detections at this dis- tance. It may be, for example, that this is the distance at which tidal forces become too weak to significantly affect the orbit of a planet. A larger sample size would clarify whether this is the case. 7.3.7 The Mass-Radius Relation for Discovered Exoplanets (PS) It was decided to plot the masses and radii of the various planets detected by the study, in order to at- tempt to produce a means of deducing the composi- tion, mass or radius of a given planet when one or two of these values were known. The resultant graph is shown in Figure 7.3.7, and quite clearly shows group- ing of planets of similar compositions. It must, however, be noted that as the mass and composition of these planets were initially suggested by considering composition and radius, the groupings 84 shown in the graph are only useful for characteris- ing other planets provided our initial method is valid. However, the close grouping of planets described as similar supports our choice of this method, and gives weight to the distinctions that have been used in this study between planets of different composition. 7.3.8 Evidence for Tidal Circularisation (PS) It was expected that close orbiting planets would follow orbits of very low eccentricity, due to the large magnitude of the tidal forces acting upon them at this distance from their host stars. This may also result in tidal locking. Although we were only able to obtain a lower bound for eccentricity due to the nature of the transit method, it is still possible to observe some influence on the ec- centricity of a planet’s orbit caused by its period. Figure 7.3.8 – A graph displaying the minimum eccentricity and period (logarithmic scale) of the discovered planets. Note the lack of high eccentricity, low period data points. The uncertainty on period measurements is negligible, and has thus been omitted. This result is severely limited due to the high uncer- tainty on the calculation of minimum eccentricity, and it must be remembered that it is theoretically possible for all of these planets to orbit with a greater eccen- tricity than we have calculated. Nonetheless, a notice- able lack of close orbiting planets in our study have eccentricities which are constrained above around 0.2, whereas the minimum eccentricity seems to be more evenly distributed at higher orbital periods. From this we can infer that tidal effects are indeed being observed in our results. 7.4 Comparison of Stellar Limb- Darkening Coefficients (CL & OH) As was described in Section 6.2, stellar limb darken- ing was included in the computational obtention of the results. For the transit fitting code, discussed in Sec- tion 6.2.4, a quadratic limb-darkening transit model by Mandel & Agol (2002) [134] was fit to the data by an emcee program. As such, best fit values for the quadratic model LDCs were output by the program, namely LD1 and LD2, as can be seen listed in Table 7.99. As is described in Section 6.2.5, an independent numerical model was made on Python to compare to the analytical model, using Equation (6.1) to find the quadratic model LDCs as a function of effective tem- perature, namely a and b. As LD1 and LD2 represent the coefficients a and b respectively according to the second order limb-darkening model set forth in Equa- tion (4.59) one would expect, in an ideal case, that these values are equal for both models. It is easy to see from Table 7.99, that this is not the case, and that the values of LD1 & a and LD2 & b differ quite significantly for a large number of planets. Where the value for δ(a) or δ(b) is seen to be positive (shown in red), it is indicative of the difference between the two corresponding values being within the error on the fit value LD1 or LD2, which indicates a (relatively) accurate fitted value. However, for many planets this is not the case. Despite the transit fitting code providing very accurate results for the other output parameters, it is reasonable to assume that the best fits for the LDC values are (in some cases severely) inaccurate, if it is assumed that the LDCs a and b resulting from Equation (6.1) are the most accurate values available. This would indeed be the most reasonable assumption to make, as the values of a and b (and u for the linear limb-darkening model) were taken from literature and fit very closely to an equation (see Section 6.2.5) [138]. Although LDCs are a stellar property, it is seen from Table 7.99 that there are different values of LD1 and LD2 for GSIC 6b & c, 7b & c and 8b & c, as the transits to which the model was fit differed for each planet. This gives further confirmation that the LDCs a and b are the more accurate values. 85 Figure 7.3.7 – A log-log plot of the planetary mass and radius of the discovered planets, which are categorised by composition. Clear groups of similar planets can be seen. 86 Table 7.99 – LDCs LD1 and LD2 as found by the transit fitting code, and LDCs a and b as found by Equation (6.1) for the quadratic limb-darkening model. In red are marked those values for which a or b fall within the error on LD1 and LD2 respectively. δ is calculated by taking the difference between the corresponding LDCs and subtracting it from the error on the analytical LDC. GSIC LD1 σ(LD1) LD2 σ(LD2) a b δ(a) δ(b) 1b 0.281946 0.057581 0.953926 0.029235 0.307117 0.310235 0.032411 -0.614456 3b 0.747188 0.039034 0.429309 0.041779 0.412344 0.249885 -0.295811 -0.137646 5b 0.652608 0.131022 0.758749 0.163690 0.335458 0.295356 -0.186129 -0.299703 6b 0.388625 0.036454 0.833611 0.049687 0.317087 0.305136 -0.035083 -0.478787 6c 0.988516 0.064789 0.796393 0.038067 0.317087 0.305136 -0.606639 -0.453190 7b 0.882470 0.087867 0.104369 0.126693 0.380669 0.269413 -0.413934 -0.038351 7c 0.261469 0.074369 0.909432 0.088988 0.380669 0.269413 -0.044831 -0.551031 8b 0.451898 0.126590 0.849932 0.141064 0.362937 0.279881 0.037629 -0.428988 8c 0.618646 0.029096 0.967145 0.039145 0.362937 0.279881 -0.226613 -0.648119 9b 0.412785 0.021592 0.978656 0.028410 0.330556 0.298013 -0.060637 -0.652233 10b 0.329766 0.063529 0.881916 0.147350 0.376253 0.272053 0.017041 -0.462513 12b 0.301472 0.062617 0.867344 0.116023 0.384113 0.267339 -0.020024 -0.483982 13b 0.998064 0.006828 0.947266 0.007225 0.372072 0.274532 -0.619164 -0.665508 14b 0.379048 0.136541 0.463178 0.249297 0.294045 0.316677 0.051538 0.102796 16b 0.194638 0.064816 0.721760 0.112818 0.321116 0.303034 -0.061661 -0.305908 17b 0.096899 0.096755 0.715532 0.133208 0.406729 0.253418 -0.213075 -0.328906 18b 0.604716 0.006313 0.998855 0.001675 0.462686 0.216986 -0.135717 -0.780194 19b 0.866771 0.349176 0.858462 0.281290 0.334045 0.296125 -0.183551 -0.281047 20b 0.012750 0.012153 0.035237 0.046465 0.364062 0.279228 -0.339159 -0.197525 21b 0.994807 0.012118 0.990898 0.019791 0.374406 0.273151 -0.608283 -0.697955 22b 0.962142 0.022306 0.570369 0.039196 0.310167 0.308691 -0.629669 -0.222482 23b 0.709106 0.019197 0.994362 0.010838 0.319285 0.303992 -0.370624 -0.679531 24b 0.433532 0.149873 0.114893 0.097136 0.329866 0.298384 0.046207 -0.086355 28b 0.995511 0.058440 0.083856 0.105461 0.307117 0.310235 -0.629955 -0.120918 30b 0.493884 0.069008 0.948552 0.065305 0.358176 0.282630 -0.066700 -0.600617 31b 0.526037 0.022445 0.673590 0.030554 0.389707 0.263945 -0.113886 -0.379091 32b 0.992323 0.022641 0.974063 0.051706 0.369594 0.275993 -0.600088 -0.646365 34b 0.404859 0.037072 0.808262 0.049669 0.391831 0.262647 0.024045 -0.495946 35b 0.455911 0.059191 0.119908 0.086966 0.416515 0.247240 0.019795 -0.040367 39b 0.930814 0.063317 0.658300 0.077747 0.587960 0.128572 -0.279537 -0.451981 40b 0.653943 0.019171 0.019686 0.022956 0.554940 0.152412 -0.079832 -0.109770 Figure 7.4.1 – Comparison between Mandel & Agol’s quadratic limb-darkening model [134] and the numeri- cal Python model using both linear and quadratic limb- darkening models, for GSIC 22b. Following this, a comparison can be made between the analytical limb-darkened model using LD1 & LD2 and the numerical models using u, a and b, for cases where the LDCs are both similar and different. A com- parison of a case where the LDCs are significantly dif- ferent can be seen in Figure 7.4.1 for GSIC 22b. It is very clear that the three models do not resemble one another very closely. Most notable are the ‘spikes’ at the start and end of the transit for the analytical model. These are most likely the result of the quadratic LDC dominating the equation in the calculation of the transit, due to the values being inaccurate. This backs the assumption that the LDCs received from the transit fitting code are indeed less accurate than those found through Equation (6.1). 87 Figure 7.4.2 – Comparison between Mandel & Agol’s quadratic limb-darkening model [134] and the numeri- cal Python model using both linear and quadratic limb- darkening models, for GSIC 14b. A comparison of the different models for GSIC 14b can be seen in Figure 7.4.2, the only fit for which the values of a and b look to both be within the errors on the fitted values. As can be seen, the ‘spike’ features are no longer present. Looking at Figures 7.4.1 and 7.4.2 it is surprising how the individual relations be- tween the three separate models differ, especially con- sidering that all three use the same input parameters for values such as the period and the semi-major axis of the orbit (namely, the final results output by the limb darkened transit fitting code). Most striking is the significant difference in depth between the analyti- cal model and the numerical models. As the analytical model has been designed (and observed) to fit the tran- sit especially, one would assume that the depth would be especially well constrained. On the other hand, as it works with an array of pixels, the numerical model takes its value of depth directly from a calculation of the area of the planetary and stellar disks. This dis- crepancy could be due to the numerical model being a rough approximation, as the stellar array of pixels is constrained to finite dimensions, and does not have a continuous range of intensities but rather a discrete distribution. However, this is unlikely to cause a differ- ence that significant, especially for Figure 7.4.2, which used an array of more than 1000 by 1000 boxes to model the star. It is most likely therefore that this difference in depth between models is attributable to the inaccu- rate LDCs. Note how the difference in depth in Figure 7.4.1, for which the LDC values are expected to be very inaccurate, is extremely large. Compare this to Figure 7.4.2, where the LDC values are seen to be more accu- rate, and it can be seen that the difference between the analytical and numerical model depths decreases signif- icantly. In fact, the minimum intensity at the centre of the transit for the analytical model is now found between the first and second order models. While it could be said that the analytical model in Figure 7.4.2 appears to match the first order equation, it is worth noting that the shape of the curve matches the shape of the second order model, for the majority of the transit. Figure 7.4.3 – Comparison between Mandel & Agol’s quadratic limb-darkening model [134] and the numeri- cal Python model using both linear and quadratic limb- darkening models, for GSIC 14b. Both quadratic models use the same LDCs a & b. To further confirm the suspicion that the difference in depth is due to inaccurate LDCs, Figure 7.4.2 was recalculated and re-plot using the LDCs a and b for both the analytical model and the quadratic limb dark- ened model, as can be seen in Figure 7.4.3. Here it can clearly be seen that the second order model has lined up with the analytical model, most notably throughout the start and end of the transit. However, there is a small discrepancy in depth between the two models at the centre of the transit, which appears to remain rel- atively consistent throughout both models, with both curves resembling each other closely. This, however, is to be expected to a degree, as both models attempt to recreate an ideal representation of reality, which is difficult to obtain, through very different means. A possible reason for this difference, as explained before, would be the finite nature of the array used to visualise the stellar intensity. Thus it can be seen that the values of LD1 and LD2 affect the shape of Mandel & Agol’s model very significantly, and that these values of LD1 and LD2 are more often than not given accurately by the transit fitting code, as is seen in Table 7.99. It can clearly be seen from this analysis that the LDCs for the quadratic limb-darkening equation can significantly vary the depth of Mandel & Agol’s model used in the transit fitting code. It was previously as- sumed that, to fit the data, the transit fitting code var- ied the planetary radius parameter to get an accurate depth for the fit. However, now knowing that the val- ues of LD1 and LD2 can affect the depth of the transit, it is highly possible that emcee varied these values to 88 find a fit for the depth, thus possibly underestimating the best fit planetary radius output by the code. This naturally raises concerns about the nature of the errors on the final values obtained for the planetary radii, and whether using LDCs obtained from Equation (6.1) as input parameters for emcee would yield more accu- rate results than allowing the LDCs to ‘roam free’ as has been the case. An in depth study by Espinoza & Jordan (2015) [145] further explores the manner in which allowing the LDCs to roam free in similar transit fitting meth- ods affects the planetary radius. They conclude that, when using the quadratic limb-darkening model to fit to a light curve, it is better to allow the LDCs to vary to find a fit, as they argue that our understanding of the limb-darkening process is not sufficient to use fixed coefficients without further increasing the bias on the data. This preference for fitted values of LD1 and LD2 is shared by Csizmadia et al. (2012) [146], who also conclude that the inconsistencies in our understanding of LDCs do not allow them to be fixed in transit fit- ting methods. Furthermore, it was found that doing so did not allow them to obtain an accuracy in the plan- etary radius of better than 1-10%. They also mention that varying the LDCs as has been done in the tran- sit fitting code should provide an accuracy of less than 1%, in an ideal case. However, the data used naturally has many imperfections, which most likely encourage emcee to vary the LDCs away from their best fit val- ues. It would be very possible that a model which accounted for these imperfections would allow a better fit and more accurate values for LDCs, but this would require a more in-depth understanding of the process of stellar limb darkening, and is outside the scope of this project. As an aside, it is interesting to note the differ- ences between the first and second order light curves produced by the numerical model. The differences in depth here most likely result from the manner in which the total intensity was calculated. In Figure 7.4.2 it can be seen that the first order model appears to be tend- ing more towards a closer fit to the analytical model, as was mentioned above. This is slightly unexpected, as the temperature of GSIC 14 is 6463K, significantly outside the temperature range of the Sun. However, using the same LDCs, as can be seen in Figure 7.4.3, the first order model is left behind whilst the analyti- cal model lines up with the second order curve. A true gauge of the accuracy of the coefficients for first order limb-darkening model set forth in this report would be to compare it to the Sun, which is also outside the scope of this project. 89 Chapter 8 Conclusions 8.1 General Conclusions (JG) In this report we have introduced the fields of as- teroseismology and exoplanet detection, summarising their inception, history and current state. We then presented an extensive explanation of the theoretical aspects of each individual field. This included the data that are required to first observe and then quantify the expected results: stellar oscillations described by frequency-power spectra in the case of asteroseismol- ogy, and planetary transits described by luminosity- time graphs in the case of exoplanet detection. These sections also introduced the methods that would be used to interpret and analyse the data provided. Next, we detailed those methods, specifying pro- grams and models that were used, and sometimes cre- ated, in this project. The uses of these programs have been explained, along with how manual methods of data analysis have been used in conjunction with the computational methods to gain correct results. We have also discussed the accuracy of the results obtained by these methods and identified where they have fallen short in obtaining useful information. Finally, we provided suitable presentation and dis- cussion of all results that we were able to obtain through the variety of methods used. These results included properties of the stars and detectable plan- ets, derived through a combination of luminosity-time data and necessary stellar properties. We extracted and explained trends in the overall results, comparing these where possible to what should be expected from current knowledge. 8.2 Stellar Properties (MG) Investigating the properties of the stars as well as the methods required has been an important endeavour in this study. A star encompasses nearly all the mat- ter in its system so it is clear that examining the stars should provide great insight into the characteristics of extrasolar systems which, as we have found, is certainly the case. Having the ability to constrain the estimates of stellar parameters such as mass, radius and age not only improves our knowledge of the stars themselves but is essential in order to identify any bodies found orbiting using current techniques, notably the transit method used in this project and discussed in Section 4.3.1. Therefore, obtaining accurate results from as- teroseismology was essential for the progression of the rest of the project. By extension, the features of the systems as a whole have been evaluated. We began our investigation with the frequency- power spectra (FPS) of 41 stars, however we soon con- cluded that 2 out of the 41 were duplicates – GSIC 28 of GSIC 1, and GSIC 37 of GSIC 11. We also had the values of effective temperature and metallicity for each of the stars, determined independently. From the FPS we worked to extract the key stellar oscillation values of Δν0, νmax and, where possible, δν0,2 which were then used to derive information on the stars’ mass and radius, first and foremost, and age if δν0,2 was known too. Both of these stages were tackled with different methods detailed in Chapter 5. Once the FPS had been filtered, two different proce- dures were used to extract the values discussed above. The first was to manually analyse the FPS. This was done to get values quickly, which was a reasonable ap- proach for the small sample size, both for comparison with the coded, computational FPS analysis – which took much longer to develop and refine – and to pro- vide early estimates of stellar properties for use by the Planet-Finding Sub-Group. It was difficult attempting to find νmax manually as a function needed to be fitted. Once the FPS could be analysed much more accurately (using manually-determined values as a guide), values for Δν0 and νmax were found for all stars except for GSIC 2 and GSIC 11 (and, by extension, GSIC 37). We managed to find values of δν0,2 for about half of all stars, for which we were then able to create aster- oseismic HR diagrams (AHRDs). While it had been helpful in this investigation to manually decipher the FPS in the early stages, it is worth noting that we used a small sample size with only 41 FPS. With the vast collection of data by satellites such as Kepler and CoRoT many similar studies to our own may be tasked with analysing the FPS of thousands of stars or more, so being able to develop and run efficient and accurate programs or strings of code is paramount to keep to realistic time frames. With the oscillation values obtained, we again used two different approaches with the aim of constraining the parameters of each star. These values were first used to calculate stellar radius and mass from the scal- ing relations in Equations 3.22 and 3.23, respectively. As discussed in Section 3.2.4, these equations are, for the most part, quick and easy approximations with rea- 90 sonable associated uncertainties. Using stars modelled by the MESA evolution code we were also able to find values for stellar parameters by building δν0,2 against Δν0 AHRDs of differing metallicity. As is apparent, these diagrams required the values of δν0,2 for each star to be plotted, and as this was possible for only half of the stars the improved stellar values could not be evaluated for all the stars, unlike the scaling relation results which were found for nearly every star. Comparisons of the mass and radius values ob- tained by both methods (seen in Figures 7.2.10 and 7.2.11) show that using AHRDs systematically im- proves the constraints on these parameters and should be used where possible. Using AHRDs also enabled us to estimate the core hydrogen fraction, Xc , and con- sequently the age of the stars – another criterion vital for investigating the characteristics of extrasolar sys- tems. The effective temperatures of the stars were also re-evaluated using values from the AHRDs (see Figure 7.2.12). On the whole, the temperature values given and those calculated did not match and, while this may indicate underlying uncertainties in the method, there are a considerable number of subtle dependencies from various different values which could cause this re- sult. Unfortunately, due to the time required to build the different AHRDs (in Figures 7.2.2 through 7.2.7) the masses and radii used by the Planet-Finding Sub- Group to calculate their results were those evaluated using the scaling relations. Ideally we would have used the better-constrained, more accurate values from the AHRDs but we were unable to achieve this within the set time constraints. Overall, we have analysed 39 different stars which are on the MS and are solar-like, with a few excep- tions. These exceptions are GSIC 38, 39 and 40 whose radii are considerably greater than their masses from which we infer that they have reached the end of their time on the MS and have become subgiants. These stars could not be plotted on the AHRDs because δν0,2 could not be determined due to mixed modes interfer- ing with the FPS. Ironically, the mixed modes them- selves, though preventing the stars from being aged with the AHRDs, gave us the evidence to classify the stars as aged because the mixed modes only appear when p- and g-modes overlap, which occurs when a solar-like star begins to move off the MS. With this in mind, their ages could be conjectured when taking into consideration how the lifetime of a star on the MS changes with mass. The stars for which δν0,2 could not be determined were inhibited primarily by an un- favourably low signal to noise ratio, making it difficult to find the l = 2 peaks in the FPS. Excluding GSIC 13, for which we believe the oscillation values have been determined incorrectly which we have not been able to rectify, the masses of the stars were found to range be- tween 0.91±0.17 M� and 2.32±1.05 M� with the radii ranging between 0.83±0.10 R� and 4.53±0.55 R�. For the stars for which the age was ascertainable, the values spanned from 0.52±0.13 Gyr to 8.06±2.03 Gyr. Also, when the ages were compared with the given values of metallicity there was no emergent correlation. This may be because the stars in this study are of the same stellar population or perhaps the average metallicity in different parts of our galaxy increases at different rates. The relationships between stellar mass and sys- tem age, as well as stellar mass and luminosity (Figures 7.3.1 and 7.3.2, respectively) using the results from our data fit well with the expectations of their respective theories. These confirmations provide supportive evi- dence of the fact that our results are, at least for the most part, of reasonable accuracy. Therefore, amongst the other positive results of this project, the investiga- tion into the host stars of extrasolar systems has been successful. 8.3 Exoplanet Detection Methods and Properties (LP, FC, OH & PS) Over the course of the project a series of Python computer codes were developed which collectively cal- culated the period of an orbiting planet from raw tran- sit light curves. Orbital period values were perhaps the most important quantity to extract to a high ac- curacy from the light curves, as many other planetary properties can be calculated using the period of the planet. For this reason, a significant fraction of the code built by the planet-finding coding team was fo- cused on calculating the period. Time is one of the quantities astronomers can measure most precisely, so it was crucial to find the period values to a precision as high as possible. The coding in this project used to determine planet properties culminated in the production of two an- alytical transit model light curves to fit to the raw data. The first model treated the star as a uniform source, whilst the second more complex (and accu- rate) model incorporated quadratic stellar limb dark- ening. The model used was set forth by Mandel & Agol (2002) [134]. The best fit of each model with the phase folded data was acquired using an emcee random walk method to vary the parameters on which the shape of the model light curves were dependent. This was run until the closest match between the data and the models was obtained, which subsequently out- put precise values for the model parameters. These parameters were the orbital period, the planet radius and semi-major axis (both in units of stellar radius), and exclusively for the limb darkened model: the in- clination and two limb-darkening coefficients. Initial estimates for these parameters, i.e. the starting values for the random walks, were mostly obtained by mea- suring transits in the data manually. However, several stars with particularly noisy light curves were more difficult to analyse, thus requiring a different approach due to the first estimates needing to be relatively close to the actual values in order for emcee to successfully converge. A transit detection code was used to deter- mine the period values for these stars, which acquired higher accuracies, enabling the transit model to fit very closely to the phase folded light curve data. 91 The accuracy of the first estimates for the periods, as well as the period values output by the code, were tested by using them in a separate phase folding code, and observing whether a clear phase fold was produced. This allowed users to determine whether the manu- ally calculated values were accurate enough to enter into the transit modelling code, or if they needed im- provement via the transit detection code. This process demonstrates how the different codes worked together enabling us to obtain the most accurate values possible from the raw light curves. To test the accuracy of the model used and any effects it may have on the final results, an indepen- dent numerical model was developed using a Python code. This reproduced a light curve using first and second order limb-darkening equations and was cus- tomisable to many properties of star-planet systems. A comparison of this model to the analytical model by Mandel & Agol revealed that the depth of the an- alytical model was largely influenced by the value of the two limb-darkening coefficients, possibly underes- timating the best fit planetary radius output by the transit fitting program. Various studies have observed the same problem when using similar methods, but concluded that allowing the coefficients to ‘roam free’ during the fitting process yields a more accurate result than if they were fixed throughout [145] [146]. It can be concluded that the analytical model used does not fully account for some perturbations of the Kepler data, with inaccurate values for quadratic limb-darkening coefficients being produced as a result. Limb-darkening models with a better understanding of the underlying process must thus be used to get ac- curate fits for planetary values. Given more time, it would have been useful to study the effects that this imperfection in the model would have on the accuracy of the planetary radius more closely, and the manner in which other parameters influence the magnitude of this uncertainty, such as the effective temperature [147]. From computational and manual analysis of Kepler transits, results were extracted regarding the proper- ties of thirty detected planets. A picture of the var- ious characteristics of the planets was built. Period values were found by analysing the 39 Kepler light curves computationally. The majority of period val- ues were found to be less than 20 days with only three planets having periods greater than 100 days. Of the low-period planets, six displayed period values lower than 10 days, and evidence could be seen indicating a three-day pileup. From the Kepler data, four planets with hot Jupiter-like properties were detected, GSIC 1b, GSIC 21b, GSIC 39b and GSIC 40b. The range of period values extended from 2.204735299±0.000000139 days to 179.4304524±0.00213173 days. The errors ob- tained have a very high precision due to the computa- tional methods used. Every planet detected in the study had a radius greater than that of Earth. This may be because methods used in the study were not sensitive enough to detect smaller transiting planets, due to the pres- ence of noise in the light curves. Planets with small radii are particularly difficult to detect using the tran- sit method, as often the change in flux due to a transit is of lower magnitude than the noise in the system. Of the detected planets, eighteen were found to have radii between 1 and 3 R⊕, with only four, the appar- ent hot Jupiters, having a radius greater than 5 R⊕. GSIC 1b was the largest detected planet, with a ra- dius of 1.364±0.162 RJ. The smallest detected planet, GSIC 7c, which slightly surprisingly also possessed the largest period, had a radius of 1.407±0.180 R⊕. The errors in these results are large compared to that of the period, due to the uncertainty in the asteroseismic methods used to find stellar radii. Kepler’s third law was used to find semi-major axis values. All orbits were found to lie in relative close proximity to their host star, all with semi-major axes less than 0.7 AU. This meant that our data was useful for determining the properties of planets close to their host stars, but unhelpful when consider- ing more distant planets. Values for semi-major axis were collected in astronomical units and ranged from 0.03749±0.00225 AU (GSIC 1b) to 0.6582±0.0433 AU (GSIC 7c). The close proximity of the planets resulted in high values for equilibrium temperature, with the coolest planet (GSIC 7c once again) having an esti- mated temperature of around 400K and the hottest planet (GSIC 1b) having an estimated temperature of 1900K. Of the planets detected, only two were found with the possibility of harbouring liquid water; GSIC 7c and GSIC 18b (400-500K). The latter of which is far less likely to be able to harbour life, however since the equilibrium temperature values were estimates, there is still a slim possibility that compositions of liquid wa- ter may be present. GSIC 7c has a lower temperature and so there is a higher chance of liquid water being present. All other planets are too hot to be habitable. Values for transit duration and star-planet separa- tion during transits were used to find the lower limits of the eccentricities of the orbits of each planet, albeit with a very large error margin. These values varied from 0.012 (GSIC 12b) to 0.824 (GSIC 5b). As only minimum eccentricities could be derived, values for ec- centricity closer to one were of more interest. The distribution of minimum eccentricities provides some evidence for tidal circularisation of low period orbits. Analytical models describing the composition of planets gave an insight into the relations between masses and radii of planets. Since values of radius had been found to a reasonable degree of accuracy, mass values could be estimated by considering this and other results. From these techniques, planetary characteris- tics could be deduced. The majority of planets exhibit gaseous features with fourteen planets likely to be gas dwarfs. A further four planets displayed features in- dicating a rocky composition with some others being of indeterminate composition, where the data collected has not been sufficient to confidently categorise them. The four largest planets detected are hot Jupiters, in low period orbits, which are likely to have formed fur- ther from their host stars and then moved inwards by migration. 92 8.4 Relationships Between Characteristics of the Extrasolar Systems and the Properties of their Host Star (MH) As a result of the quantity of accurate, and in some cases extremely precise, results that we obtained from the data we received, we have been able to examine re- lationships between many of the properties of both the stars and planets, as we hoped to be able to do from the beginning of the project. Starting with the mass-age relationship of the stars for which we were able to obtain an age, we found a strong negative correlation between the mass and age of these stars, due to the much shorter main-sequence lifetime of higher mass stars which burn through their hydrogen more quickly. We also observed a larger spread at the lower mass end which should be expected because stars that can be old can also be young. These results reinforce our confidence in the asteroseismic diagrams we used to find the ages. Next we exam- ined the relationship between stellar mass and lumi- nosity. Our results corresponded impressively well to the expected relationship; we obtained a power-law in which luminosity increases with mass to the power of 4.099, when we would expect a fourth-power depen- dence. This demonstrated the potential accuracy of the method used to obtain the luminosity from the aster- oseismic diagram data. Together, these relationships suggest that our asteroseismic diagram methods were extremely well carried out. The relationship between the stellar mass and ra- dius did not appear to agree well quantitatively with what could be expected. While the data showed a pos- itive correlation, which is exactly what would be ex- pected, our attempts to fit the data did not result in the power-law relationship we expected, especially as the diagram should have two regions either side of so- lar mass, for the lower of which we only had five stars. Similarly, the diagram plotting the stellar metallicity against planet composition did not reveal an obvious relationship. Larger samples of data could have im- proved both of these, while the latter could also have been improved by a more quantitative measure of the composition of the planets and also by being able to detect more low-mass planets, though this would likely require a different method of exoplanet-detection to the transit method. The stellar radius-planet radius and planet mass- planet radius diagrams show strong relationships. There is a clear trend of increasing planet radius with increasing stellar radius. We deduced that a bigger star would have formed from a larger cloud, which would have provided more material to form bigger planets. The latter diagram demonstrates the obvious expecta- tion that more massive planets should have a larger radius, and also indicates that we were very consistent in our deductions of the planets’ composition, with the caveat that the constraints on the mass were in refer- ence to quantities that included the other two on the diagram. This second relationship in particular gives some confidence to our method for determining planet radius and mass. Finally, we examined the distribution of the orbital periods of the planets and the relation between the eccentricity and period of an orbit. In the former, we found possible evidence of the three-day pileup of hot Jupiter planets, however it cannot be discounted that this was due to the selection bias of the tran- sit detection method for planets that orbit close to their star. Latterly, in the diagram of minimum eccen- tricity against period, we see that the shorter period stars, those that take fewer than ten days to orbit their star, all have very low minimum values for eccentric- ity. While a maximum eccentricity would clearly have been useful to show that short-period stars have low eccentricities, by comparing to the spread in minimum eccentricity of stars with longer periods, we can sug- gest - though not confirm - that planets orbiting close to their star have low eccentricity orbits. In summary, our analysis of the data and the re- sults we obtained have allowed us to identify several relationships, which for the most part agree very well with current theories, shed light on the types of sys- tems we have observed, leaving us confident that the methods we have used throughout the project are valid. 93 Thanks and Acknowledgements (JG) We would like to thank Dr Andrea Miglio, Dr Daniel Reese, and Dr Guy Davies for their guidance, super- vision, and assistance throughout the duration of this project. Dr Miglio provided the group with a thorough introduction to the project and gave us a good head start into understanding the aims. His feedback on the initial worksheet was useful and advice on any unsure areas of theory ensured the accuracy of all scientific documentation produced in this project. Dr Miglio’s constant availability meant the group could continue to move forward without delay on important issues. Dr Reese was always on hand to provide the Asteroseismology Sub-Group advice on the practicalities of using the frequency-power spectra and provided the group with the MESA stellar model data. We would like to thank him for his help in ensuring that the methods used to deal with all of the data were as efficient as possible and his input saved a substantial amount of project time from being wasted. Dr Davies’s advice and help was particularly useful when it came to the problems faced by the Planet-Finding Sub-Group with our various Python codes. His expertise helped us make up valuable time we had lost due to our relatively slow coding progress at the outset of the project, ensuring we could obtain the high accuracy and relevant results we set out to achieve. Furthermore, Dr Davies contributed directly to data gathering by allowing us to use his (more powerful) computer cluster to run the transit fitting code simultaneously for all stars. 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In: Monthly Notices Of The Royal Astronomical Society 418 (2011), pp. 1165–1175. doi: 10.1111/j.1365-2966.2011.19568.x. 100 http://dx.doi.org/http://onlinelibrary.wiley.com/doi/10.1002/2014JA020778/abstract http://terpconnect.umd.edu/~toh/spectrum/ipeak.m http://www.mathworks.com/matlabcentral/fileexchange/35122-gaussian-fit/content/gaussfit.m http://www.mathworks.com/matlabcentral/fileexchange/35122-gaussian-fit/content/gaussfit.m http://www.astro.virginia.edu/class/majewski/astr551/lectures/ABUNDANCES/abundances.html http://www.astro.virginia.edu/class/majewski/astr551/lectures/ABUNDANCES/abundances.html http://astro.wsu.edu/models/calc/XYZ.html http://www.lpl.arizona.edu/~ianc/python/transit.html#transit.t2z http://dan.iel.fm/emcee/current/ http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html http://dx.doi.org/10.1553/cial147s76 http://www2.astro.psu.edu/users/rbc/a534/lec18.pdf http://dx.doi.org/10.1051/0004-6361/201219888 http://dx.doi.org/10.1111/j.1365-2966.2011.19568.x Introduction Introduction to the Report (AW & MH) Aims of the Report (JG) Introduction to Asteroseismology (MH) Discovery and Development (MH) MOST, CoRoT, and Kepler (MH) The Current State and Future of Asteroseismology (MH) Introduction to Planet-Detection (LP & AW) Transit Method (LP) Astrometric Wobble and Doppler Wobble Method (LP) Transit Timing Variation (TTV) Method (AE) The Kepler and PLATO Missions The Kepler Mission Mission Design (PS) Spacecraft Design (PS) The PLATO Mission (EM) Asteroseismology Theory Asteroseismic Oscillations The Description of Oscillations (MH) The Origin of Oscillations (MH) Analysis of Asteroseismic Data Frequency-Power Spectra (MH) Data Visualisation (MH) Spectrum Noise (MG & LS) Scaling Relations (MH) Asteroseismic Diagrams (MH, EM & MG) Results from Asteroseismic Data Stellar Properties (MH) Stellar Evolutionary Theory (MH) Planet-Finding Theory Planet Formation (AE & EM) Orbital Mechanics Kepler's Laws (AE) Two- and Three-Body Problem (LP) Planet-Finding Methods Transit Method (LP) Transit Timing Variations (TTV) (AE) Other Planet Detection Methods (CL) Stellar Limb Darkening (CL) Mass Constraining Methods (PS) Auxiliary Theory Stellar Variability (ML) The Effect of Starspots on Transits (CL) Hill Spheres (EM) Roche Limit (EM) Three-Day Pileup (EM) Exomoon Detection (AE) Circumstellar Habitable Zone (EM & MG) Asteroseismology Method Initial Data Analysis Plotting Raw Data (JG) Filtering and Removing Noise (GM) Isolating Regions of Interest (GM) Manually Measuring Stellar Oscillation Properties Measuring Large and Small Frequency Spacing (JG & ML) Computationally Measuring Stellar Oscillation Properties Measuring Large and Small Frequency Spacing (LS) Measuring Frequency of Maximum Power (LS & GM) Obtaining Asteroseismic Results Using Scaling Relations (MH) Using Asteroseismic Diagrams (GM) Planet-Finding Method Manual Methods (LP, CL & AE) Computational Methods Introduction (FC) Transit Detection Code (FC & AW) Phase Folding Code (FC) Transit Fitting Code (OH & FC) Modelling for Stellar Limb Darkening (CL & OH) Processing of Results (PS) Results and Discussion Introduction Key to Results Tables (MH & PS) Nomenclature (PS) Full Results Stellar, Orbital, and Planetary Results (PS, AW, JG & ML) Omissions (JG) Results from Asteroseismic Diagrams (GM) Comparison Between Scaling Relation and Asteroseismic Diagram Values (GM) General Trends Relationship Between Stellar Mass and System Age (GM) Relationship Between Stellar Mass and Luminosity (GM) Relationship Between Stellar Mass and Radius (MH) Relationship Between Stellar Metallicity and Planet Composition (MH) Relationship Between Stellar Radius and Planet Radius (JG) Frequency of Orbital Period for Discovered Exoplanets (PS) The Mass-Radius Relation for Discovered Exoplanets (PS) Evidence for Tidal Circularisation (PS) Comparison of Stellar Limb-Darkening Coefficients (CL & OH) Conclusions General Conclusions (JG) Stellar Properties (MG) Exoplanet Detection Methods and Properties (LP, FC & PS) Relationships Between Characteristics of the Extrasolar Systems and the Properties of their Host Star (MH)