Collision probabilities at geosynchronous altitudes

Adv. 5pa~~~. Vol.5, No.2, pp.4757, 1983 02731177/83 $0.00 + .50Printed in Great Britain. All rights reserved. Copyright i~COSPAR COLLISION PROBABILITIES AT GEOSYNCHRONOUS ALTITUDES M. Hechier European Space Operations Centre. Robert- Bosch -Sir. 5, 6100 Darinsiadi. F. R. G. ABSTRACT A considerable collisional hazard to operational geostationary satellites is induced by a continuously increasing population of abandoned objects and related debris. During the past 5 years first measures have been taken to remove geostationary spacecraft from the geostatio- nary altitude at the end of their operational life. Another concern is the crowding of active satellites at some preferred longitude positions. This paper analyses the hazard due to abandoned objects and the probability of a collision between satellites maintained with- in the same longitudinal slot. The operational satellites are represented by their spatial probability density in the geostationary ring. A sample of orbit propagations based on a first-order perturbation theory represents the uncontrolled objects passing through the geo- stationary region of space. A great number of small debris particles turns out to be parti- cularly dangerous to large operational satellites. INTRODUCTION The geostationary orbit has become a resource of increasing importance fcc communication, broadcasting, and earth observation purposes. In the first two decades cf its utilization since SYNCQM2 and 3 in 1963 and 1964, about 170 satellites have been placed at geostationary positions. In April 1982 over 200 other unidentified objects, upper stages, etc., have been tracked by NORAD, not including some ‘classified’ satellites /1/, /2/. - In addition, about 1000-2000 currently untrackable objects with surface areas between 1cm’ and 1m2 are estimated to cross the geostationary region in various long-live orbits /1/. Presently, the geostatio- nary population increases at a rate of about 20 satellites per year. Up to now it has act been common practice to remove satellites at the end of their useful lifetimes from the geo- synchronous altitude. Also measures have not commonly been taken aimed a: limiting the increase of the number of other objects moving through geostationary altitudes, like apogee boost stages, lease caps and instrument covers, as well as objects in transfer orbits. In the late 70’s when the solar power station was under discussion, the :cllisional hazard due to debris became Sr. issue. Two measures for reducing the hazard have been proposed 1. reduce the debris population or at least do not increase it; 2. protect spacecraft against collisions. In 1977, the USSR were the first to remove one of their spacecraft from the geostationary orbit, followed by the US in 1981 /1/. Since then, a considerable number cf satellites has been brought to higher and lower orbits. Intelsat 3F—2, F-3, F—4, F-6 ha-.’e been inanoeuvred into orbits 400-4000 km above the geosynchronous orbit, ATS-6 is 600 km below it; NOAA has transferred its deactivated satellites SMS-1 (1981) and SMS-2 into higher orbits; the French- German Symphony has been placed 80 km above geosynchronous altitude /3/. Also NATO—i, Anik—Al, A2 have been moved to higher orbits and on 25 January 1984 ESOC has de—orbited ESA’s GEOS—2 satellite into an orbit 260 km above geostationary altitude /4/. Some of the USSR satellites (Raduga, Ekran, Horizon) seem to have been removed. Ekran-2 is moving in an elliptic orbit above geostationary height which is still crossing the gecstationary region /3/. It is fortunate that in many cases propellant reserves for station keeping tend to expire later than other resources on the spacecraft or on the ground (e.g. the financing of ground operations). This paves the way for the uneconomic, generous decision to terminate the active life of a satellite by boosting it into a higher orbit. Until no~:, however, it has not been general design practice to increase the propellant budget to al.cw for a final de- 47 48 i~i. Hechier orbiting of satellites. Within ESA this is currently being discUssed /3/. Active debris collection at the geostationary altitude will be a very costly measure, though some day it may become necessary because alternatives such as an active protection against collision, e.g. by bumpers, might be even more costly and evasion manoeuvres may be diffi- cult in a rather dense environment of small objects which are hardly trackable. When con- sidering direct satellite protection, it must be visualized that a piece of junk at typically 70 inclination will cross the geostationary orbit with a north—south velocity of about 370 m/s, which is over 1300 km/h. Thus, the only immediately feasible measure is to avoid a further growth of debris population in order to reduce the collision hazard to the minimum. Estimates of collision probabilities at the geostationary altitude have been published in /5/, /6/, and /7/. In the present paper the estimates of /5/ are reconsidered, taking into account in particular a larger amount of small debris which would reflect the actual situa- tion after the first collision has occured. Also, the collision risk between active satel- lites and the longitudinal distribution of dead objects is investigated in more detail on the basis of /8/, /9/. COLLISIONS BETWEEN ACTIVE SATELLITES Collision Risk We define the ‘geostationary ring’ as a region in space around the geostationary orbit, bound- ed by the envelope of station keeping volumes of geostationary satellites. Uniform station keeping intervals of ±0.10 in north-south and east-west directions would lead to a geosta- tionary ring extending about ±75 km north—south and about ±20 km in radial distance. Most operational satellites are maintained in this rather narrow ring which is part of the geo- synchronous altitude region populated by a few other active satellites on 24 hour orbits and by a large amount of abandoned objects. Three types of collisions may occur at the geosynchronous altitude 1. Collisions between two active satellites. 2. Collisions of an active satellite with an uncontrolled object crossing the station keeping area. 3. Collisions between abandoned objects. The latter type of collisions appears to be of no direct interest, but it may increase the number of small debris crossing the operational orbits and thus indirectly increase the hazard to active satellites. Active and deactivated satellites involved in the first two types of collisions naturally are represented by different mathematical models. Orbits of active satellites are difficult to predict over a longer time interval as they are ~ to frequent station keeping manoeuv— res, but their position is constrained to a tight volume in space around their ideal position on the geostationary orbit at a prescribed longitude. Station keeping strategies follow typical optimality features which allow to represent active satellites as a ‘cloud’ of dis- tributed objects. To each position within the geostationary ring a probability density can be attached describing the probability to encounter a satellite within an infinitesimal volume element at any fixed instant. Because of the repetitiveness of the daily motion the time can be eliminated from the considerations /5/, /8/. On the other hand, abandoned ob- jects move freely and in a well predictable manner following the laws of gravitation and, less predictably, under radiation pressure effects. They will undergo large oscillations in latitude, longitude and radius, but will permanently pass through the geosynchronous region crossing the geostationary ring up to twice per day. As they move through a much larger region of space than the satellites kept on station a representation by probability densi- ties appears to be less appropriate than a representation by a sample of propagated orbits of satellites with different area to mass ratio abandoned at different times and longitude posi- tions. The representative sample resembles all distributional properties induced by the long term orbit evolution of uncontrolled objects. Collision probabilities with active satellites can then be assessed by ‘counting’ the crossings with the geostationary ring of the above sample over a long time interval. Two active satellites may collide if their station keeping volumes intersect. As a distri- bution of a grid of prescribed ‘central’ longitudes of the station keeping intervals can be assumed, only the case of two satellites stationed on exactly the same reference longitude is further investigated. In this case both satellites must be represented by probability densities, so that the collision probability can be obtained by means of a rather complicated Geosynchronous Collision 49 integral over the station keeping volume. Once collision probabilities for pairs of satellites of unit cross Section have been estab- lished the collision hazard for a variety of populations with varying sizes and the evolu- tion of the hazard with time can be concluded. Collisions Between Satellites on the Same Longitude Some longitudes which are of ~articular interest for communications and broadcasting become increasingly crowded (e.g. 10 _3Q0 west, 700_1200 west, 750_gOD east, 1500_1700 east, /9/). Presently, longitudinal spacing of geostationary satellites is primarily regulated by the International Frequency Registration Board (IFRB) following guidelines of the International Telecommunications Union (ITU) on minimum separation of satellites operating on similar fre- quencies. Sophisticated telecommunications techniques and improved operations capabilities will allow closer spacing in narrower slots. Typically, station keeping manoeuvres every 2—7 days allow to maintain a satellite within ±O.O5~ from its reference longitude /10/. With the presently common technology the precision of station keeping is limited by opera- tional considerations, like the free drift time required to reach the necessary orbit deter- mination accuracy. The ccllision probability of several satellites maintained in the same longitudinal slot by different operators without coordination becomes an important issue with a growing number of satellites in a limited number of attractive slots. Combining satellite functions on large platforms or coordinating the motion of several satellites in the same slot may not be sufficient to accommodate all requests on usage for some longitudes. The fact that all operators manoeuvre their satellites according to similar optimality cri- teria, following the same physical laws leads to a key feature, a kind of synchronization of the motion of satellites in the same slot relative to each other. The motion of two satellites in the same longitudinal interval will be described by sets of Keplerian elements a =(C.,e,i,~), k=1,2 (1) —k K k K k These elements will slowly vary under the perturbing forces acting on both satellites in a similar fashion. Station keeping will repeatedly bring back the values of relative semi major axes 0k = ~ak/as (a~ = 42165 km) , the eccentricities ek and the inclinations ik such that the longitudinal drift and the north-south oscillation remains within the prescribed intervals. Typical bounds for these three elements which keep the motion in a ±0.050 slot are (satellite index k omitted from no~ on) 0.0302 e - m < 3 km/42165 km = 0 (2) - m i < 0.03’ = i - m The arguments of perigee ~. of both satelites will be close to each other under the assuxrpt- ion that both operators ~:ill point the eccentricity vector (for small area to mass ratio) e = (e.coaT~ + ~), e.sin(~3 + w)) (3) towards the sun to minimize the ma000uvoring effort required to Counteract the solar radia- tion pressure perturbation. Ar even stronger constraint on the selection of the right ascen- sion of the ascending node results frun the fact that close to the nominal value of the inclination i = 0 the earth oblateness and luni-solar perturbations will ‘drive’ the polar vector j (j.~g , i-sini3) (4) essentially in the vernal equinox direction (y-axis in Figure 1). Thus, station keeping naturally will reset the node ~ to a value close to 270°whenever the inclination leaves the station keeping interval with 12 close to 9QD• The long-term motion of the polar vector can be seen in Figure 1. A very comprehensive treatment of these station keeping considerations for geostationary satellites is given in /10/. As the node is ‘prescribed’ it is not expli- citly contained in the set of ‘probabilistic’ variables (1). The daily oscillation of the satellite notion may be described by O = —eCosT + 0(T) (5) A = ‘d 1’ + 2esinT (6) = isir(u +T) (7) 50 N. Hechler where 0 is the relative radius na 5, A is the geographical longitude deviation from the nomi- nal longitude A5, q, is the latitude and T is the fast variable corresponding to the local Hour Angle (i.e. one revolution per sidereal day) Under commonly adopted station keeping assumptions (namely setting the semi major axis back to a value such that the longitudinal drift remains within the Station keeping interval for a maximum time), the drift portion Xd(T) of the longitudinal motion and the relative semi major axis 0(1) between manoeUVreS roughly follow /8/, ~ (8) with ________ ° = 0 + G’.T, a = 2~/~l0’lA (9) m m %J3 m o’ stands for do/dT and is determined by the reference longitude A according to 0’ = 3.O5~lO 6.sin2 (A — 75°) (10)5 From this knowledge of the short term motion of the satellites and from the probability densities of the elements = ~ ~ ~ 1m (11) g (e) = , 0 < e~ < e (12) e e 2 — - m m g (0) = —i-- , —o ~ 0 ~ c (13)o 20 m m as concluded from their previously described typical evolution between station keeping manoeuvres /8/, spatial densities can be derived. To obtain the collision rate, the product fi(xIa;).f2(xlai) of these spatial densities of the two satellites conditioned upon their sets of elements a1,a2 has to be multiplied by the collisional cross section Ac and the relative velocity ~ (x,a1) - ~ (x,~) and integrated over the station keeping volume. Averaging over the element a~ and a2 by means of (1i)-(l3) finally leads to the mean collision rate ~ C C dI~ = J g(~.i)J~1(~.l~.1)-~-~(x)d°xd~a1 (14) V V(a) —i ~.—1 with dl I’ = A f (x~a) lv (x,a ) - v (x,aj g~a.)d 3a (15) dt c 2 — —2 —1 —1 —2 — —~ —~V (x) V(al) is the volume of positions x reached by satellite 1 for fixed elements a~, and !a is the volume of elements ai. g(a 1) and g(~) are the joint densities of the elements = (i1, e1, o~) and = (i2, e2, 02), respectively. An averaging over a uniformly distri- buted argument of perigee w may be included. The above 9-dimensional integral has been evaluated by means of a Monte-Carlo method. First, a random sample is generated with its distribution defined by (11)-(13), then uniformly distributed sample time points define a random sample x 3 by means of (E)-(7). Two of the remaining integrals, those in i 2 and e2, can be evaluated explicitly, the integral in 02 can be evaluated by a Monte-Carlo method again /8/. The final result is a surprisingly small probability of Py = 9.1o 7/year (16) for a pair of relatively large satellites with 400 m2 mutual collisional cross section placed in the same 0.1°slot to collide within 1 year. The mean relative velocity at collision is about 0.5 rn/s. This number takes account of the previously described synchronizing effect of standard station keeping practices. This may well be the reason why it does not compare well Geosynchronous Collision 51 with the corresponding number of /7/ which would be 2.7xlO~/year, and with simple ‘particle in box’ calculations. As final plausibility consideration one should imagine that ±0.050 station keeping as in (2) leads to a volume of 20 km in radius (r.em = 8.5 km. as.om = ~ kin) and 72 km in longi- tude and latitude which is of the order of i05 km3. Thus, in principle, 1012 satellite cubes of 100 m3 each could be contained at rest in this volume. In spite of the small collision probability found above, satellite owners are not likely to adopt a policy to maintain several satellites in the same slot without coordination. An ex- change of orbit prediction data would at least allow collision avoidance manoeuvres whenever deemed necessary. COLLISION RISK DUE TO ABANDONED OBJECTS The Motion of Abandoned Satellites It is well known that an uncontrolled object at geostationary altitude is subject to a typi- cal orbit evolution under the influence of the lunisolar, the earth non—sphericity and the sun—radiation pressure perturbations. A consistent first order perturbation theory of these effects which predicts the evolution of the Keplerian elements to sufficient precision over an interval of a few decades has been developed in /11/. It uses a set of non—singular equi- noctial elements and takes into account the first two parallactic terms of the moon, up to fourth degree zonal and tesseral harmonics of the earth’s gravitational field and constant reflectivity coefficients in the radiation pressure. Averaging over the fast variable has been executed by means of automated Poisson series manipulation on the computer. The result- ing averaged differential equations in the slow variables have been numerically integrated, and the results have been validated by a comparison with those from a numerical integration of the full equations. They display the well known three distinct essentially uncoupled long term evolutions of the polar vector, the eccentricity vector, and the seol-major axis versus longitude phase plane motion. Figures 1 to 3 give the result over a time interval of 53 years for a satellite with 0.04 m2/kg area to mass ratio abandoned on 29 June 1984 at 00 longitude under geostationary conditions. I,., — e —‘ 7 / -7 .--~--- --; rn, ryr rn yr -n try -rn -n-. ,,lr, r ri ,rl TY ny trY. .WJ .1. 0. I. 3. V. S. ~ L I. IV. IS. ~?. IV. II. IS. I”ifl5~ lD(f,I Fig. 1 53 years evolution of polar vector of an abandoned object 52 ~l,Hechier As displayed in Figure 1, the polar vector (4) precesses with a period of about 53 years around a point about 7~30 from the equatorial pole in the direction of the pole of the eclip- tic caused by the joint effects of earth oblateness and sun and moon attraction. The 6- months ripples are caused by the Sun’s attraction. ass .... - - - •020 - - z — 0_IS - --- - -_— — .. - -. - . -- Gb - 0.n, . -. - - 0.00 . . - -0 05 J - _- — -. - - - .0_I, - - - -0.15 -— ..... . . - - -o 20 - - - - - - — - - I nn rrrrnn, rrnyrrn ,nr,nn rrnrnn rrnrrn, ,,rnrrrr -Cm -020 -0.15 .n 0 -0.05 0.0* 0 03 0.10 0.15 0 20 0.25 .10 2 ErCOS Fig. 2 53 years evolution of eccentricity vector of an abandoned object The eccentricity vector (3) motion (Figure 2) is primarily caused by the solar radoation pressure which tends to rotate the vector away from the instantaneous earth-sun lone. Over a period of one year the eccentricity vector moves on an almost closed loop with curls introduced by the moon’s parallactic term. The main reason for the longitude drift in Figure 3 coupled with the semi-major axis varia- tion is the J 22 harmonic - the ellipticity of the earth’s equator. Under this effect the satellite oscillates with periods longer than 820 days around the nearest stable point, which lies above the equator’s minor axis at 750 east and 1050 west, respectively. Due to higher tesseral harmonics the minimum period is reduced to about 750 days. It should be noted that the satellite permanently passes through the geostationary region. Figure 4 gives the radii versus longitudes at which the representative abandoned satellite crosses the equatorial plane. As the active satellites all move in a narrow equatorial ‘lane’ around the geostationary or- bit of about 20 km width for a ±0.050 geostationary ring, Figure 4 gives an imrression on the frequency of crossings of this lane. Note that the example satellite has a rather large area to mass ratio. Figure 4 also shows that a satellite abandoned at Q will he a hazard to all satellites stationed between 00 and 150~ (symmetric around the stable poInt at 750); most of the time it will be close to the extrema of its oscillation so that it will especial- ly be a hazard to satellites stationed in that region. Geosynchronous Collision 53 0.10 -_._._———---- -— -- -—- (r,0IIV .. -. 0( It 0.00 -- - - - — — - - - -. - 00.01 -.- - - -V It -- - -.- - I,J0.OS - -.- - - - - - us 20.0I .. -- -- - - - - - Z 11.01 - — - — -~ —— — — — — - --0 I— 0.03 -_._ —— —- - — --- - - -- - . . - a >0.01 ~ —. — C, -0 00 - —— --- - -aol — - - -0.00 . - .. - -0.01 —- - . - — - -0_el -- -0.05 - —- -. -0.00 — —. - - -0.0? -— . — -0.01 — .—-- - -0.0I -- — — —. .- - - . - - -0.10 n -n-n- to t~ O~y,~iyy~ Tn -rrr, I -ID. 0. 10. 20. 30. 00. 50. 00. 10. 0*. 90. 00. 10. 20. ISO. ItO. ISO. lflNITIUD[ DRIFT 10(61 Fig. 3 53 years evolution of longitude versus seci—major axis of an abandoned object ~‘- 0 LONGITUOE 10(61 Fig. 4 53 years evolution of points at which abandoned objects cross the equatorial plane Collision Probability of Active Satellites with Abandoned Objects Since 20 years satellites and other objects such as burned—cut stages and satellite pieces have been left uncontrolled on geosynchronous or nearly synchronous orbits at different longitudes and different times. We represent the operatior.al satellites by their probability density in the geostationary ring and assume for the time being uniform distributimn in longitude. Then for each aban- doned object the orbit prcpagation is started with specified data, and a collision probabi- lity with the whole population of active satellites is obtained over the years. By repeat- ing this intersection process for a sample of satellites abandoned at different longitudes 54 M. Heobler and different times with different area to mass ratios a sample of collision probability values is obtained. These values turn out to be not heavily influenced by the choice of the initial parameters: the variation of resulting collision probabilities is not more than a factor 2. Also assumptions on the width of the geostationary ring (0.5°, 0.10) appear to be of minor importance for the resulting probability numbers. This means that a mean value of all the probability numbers for a set of sample motions of abandoned Objects will be a reasonable estimate of the collision hazard due to abandoned objects. The longitudinal modulation of this number will be investigated in more detail later on. For any of these abandoned objects moving through the ‘cloud’ of active satellites the coll- ision probability is proportional to its mean effective collisional cross section Ac with the distributed objects, the mean relative velocity “rel and the mean spatial density f(x) of the objects along the path x(t). = ~ (~~)~ (17) The probability density f(x) will be positive in the geostationary ring, it can be derived from the explicit representation of x = (p, A, 0) as a function of time as in (5)—(7) , start- ing from the time as uniformly distributed random variable and applying the usual probability calculus for functions of a random variable. The dependance on the slowly varying parameters (e, i, 0) can be removed by averaging as in (14) to obtain a mean density. The averaging over the collisional cross secticn Ac can be refined by subdividing both popu- lations of active and abandoned satellites in sub—classes of different areas and finally adding up the probabilities /5/, /8/. The probability for one sample motion becomes a path integral which splits into a sequence of paths through the geostationary ring where f(x’ is positive. Part of the evaluation of the integral is the calcuation of the entry and exit points of the geostationary ring. The mean relative velocity can be obtained by an expans:cn, since usually the velocity of active satellites in the fran.e rotating with the earth is small compared to that of ar abandoned object. The number which can be taken as a basis for a.l refined calculations concerning longitudes and cross sections is the probability that one active satellite and one abandoned object of a unit collisional cross section of 1 km 2 collide at least once in one year, ignoring any longitudinal distribution effects. For all variations of assumptions on initial data and size of the geostationary ring as described above this number lies within the interval 1.7x1O5 ~ Py ~ 3.1x105 18) This corresponds to an expected time of the first collision for a typical sample ci 100 act- ive and lOG abandoned spherical satellites with 50 in diameter (—8000 m’ collisicnal cross section) of 400 to 700 years /5/. From the basic ‘unit’ number Py defined above, collision probabilities for various extrapo- lations and assumptions on growth of populations and satellite sizes can be derived /5/. The effect of a large number of small debris which has not been studied in /5/ will be given in the last section of this paper. First, the derendance of the collisional hazard or, the longitude will be discussed in detail. The longitudinal distribution of the crossings of the geostationary rin~of the abandoned object’s population will provide the desired modula- tion of the number p as a function of longitude. Longitudinal Distribution of Deactivated Satellites As mentioned above, some longitudes tend to be far more populated by geostationary satellites than others. The dotted line in Figure 5 gives the relative density of active satellites as observed from September 1981 to January 1982 by NORAD /6/. On the basis of the assumption that this longitudinal preference will prevail in the future, a longitudinal distribution of abandoned satellites can be derived. The longitudinal motion of a satellite after deactivation is approximately described (com- pare Figure 3) as a function of time by the non—linear oscillation x(t) = —3n 0(t) tad/day, ~ c•sin (2xt) ) 1/day (19) Geosynchronous Collision 55 with the longi~~~~x = A - measured from the stable point nearest to the initial longi- tude Xl (X0 — A~). The rate of change of the relative semi-major axis C = ~a/a5 is a function of X~the constant c is about 3.O5x1cy 6 red/day. Elimination of the time t (measured in days) leads to o(X, X 0) = +\/I~’.VIi2 — sin 2x (20) and the period of oscillation for system (19) becomes the elliptical integral T(X ) = ~ f dx (21) 0 — sin2X 0sin 2x 0 If we apply the usual density transformation for functions of a random variable x(t) = ~(~x) I~I(X) (22) to the density function of the uniformly distributed time t ft(t) = , � < ‘E (23) we obtain the conditional density f (XIy-). A typical initial relative density g~ (~) is taken from /9/. It is empirically ~efir.ed by the number of satellites which have ~een ob- served within ±20 between Sept. 81 and January 82 (dotted line in Figure 5) . From this the relative density of the longitude at which a typical population of abandoned objects will be situated at any time instant is obtained by averaging over g (x 0)dX0 f (x) = Xo (24)X T(0).I/~i~x0— sin 2y lxkx 0 kr/2 with T(’:0) from (21). Above steps are typical for all probabilistic derivations of /8/. I-” cl_IS cI~ I I I11~A I ~- i~r~ ‘: C!I9C.11SPIII(AL 1.001,1*1101 10161110S 111311 Fig. 5 Relative longitudinal density of active and abandoned satellites The numerical evaluation of (24) is given by the solid line of Figure 5. 56 N. Hechier A comparison of this predicted relative density (solid line) with the observed longitudinal distribution of uncontrolled objects drifting at geostatjonary altitude as reported in /9/ (dotted line) is given in Figure 6. °“ ~ ~“?“~1~’”F” tIOr,00PIbl(51 tObIrIIUOI 101601(5 11151? Fig. 6 Relative longitudinal density of abandoned satellites The NORAD snapshot data of 20. September 1980 given in /9/ (dotted line in Figure 6) contains quite a few objects which are drifting at high rates (up to 5°/day). A drift rate of S~per day would imply according to (19) a semi—major axis deviation of about 400 kin, which could be the case for objects in near—synchronous orbits, e.g. AMF adaptors. In particular the peak in Figure 6 at one of the unstable points contains only objects at high drift rates, which can not be explained if they originated from initially geostationary orbits. The frequency of crossings of the geostationary ring of this kind of objects, many of them pro- bably in eccentric orbits, is difficult to assess, but it can be expected to be lower than that of objects originating from geosynchronous orbits. The conclusion from the longitudinal distribution of abandoned geostationary satellites is that at the stable points the collision hazard is about twice as high as on the average. Some reduction of this factor should be taken into account for a narrow geostationary ring because the satellites at large amplitude oscillations will pass the stable point above and below the geostationary altitude, e.g. from Figure 3 a satellite abandoned at o0 will pass the stable 750 point 34 km above and below the geostationary altitude provided it has a small area to mass ratio such that the eccentricity does not contribute significantly to the radial motion) Collision Hazard due to Large Debris Population Clearly, a large population of small debris produces a considerable hazard for large struc- tures at the geostationary altitude due to the feature that the collisional cross section with each of these debris particles is the full cross section of the operational satellite exposed to the debris flux. As a continuation of the analysis in /5/ one effect of a large debris cloud, e.g. after an explosion or a collision, in combination with a typcial popula- tion of active and abandoned satellites has been assessed. 40, 80, 60, 20 active satellites with cross sections of 5, 20, 100, 1000 m 2 respectively are assumed to be exposed to 140, 40, 20 dead objects with 5, 20, 100 m2 cross section and an additional debris cloud of 1000 or 10000 pieces of 1 cm2 cross section. The probabilities of at least one collision in 20 years are then number of debris collision probability 0 0.0051 1000 0.021 10000 0.16 TABLE I Collision probability for large number of debris Geosynchronous Collision 57 This indicates that a situation whiob is likely to develop after the first collisions, namely the presence of a few large active satellites and a large population of small junk crossing the geostationary ring, proves to be quite disastrous. CONCLUSIONS The dominating collisional hazard at geostationary altitudes is due to the population of freely moving objects originating from geostationary orbits. These objects frequently and permanently cross the ring occupied by active satellites at varying longitudes at a rather high relative speed, so that they expose the operational satellites to a permanent bombard- ment. With present populations the hazard still appears to be tolerable, but a continuous increase of dead objects at the geostationry altitude will further contaminate this valuable resource. This will become particularly critical when secondary ‘avalanche’ effects are generated by first collisions. As an immediate consequence satellites should not only be moved to a few hundred km higher circular orbits at the end of their useful life, but also the design of transfer and apogee stages, lense covers, and other removable satellite parts should be reconsidered. Acknowledgement: Dr. J. van der Ha has contributed substantially to this study by construct- ing an analytical model for the longterm satellite moticn. REFERENCES 1. N.L. Johnson, The Crowded Sky, Spaceflight, Vol. 24, December 1982. 2. L. Sehnal, Geostationary Orbit, United Nations Interregional Seminar on Space Applicat- ions, Addis Abeba, June 1982. 3. E.A. Roth, Geostationary Ring - keep it clean, SF14-B, SFCG Fourth Annual Meeting, London 1983. 4• p. Beech, M. Soop, J.C. van der Ha, The De—orbiting of GEOS2, ESA Bulletin No. 38, May 1984. 5. M. Hechler, J.C. van der Ha, Probability of Collisicros in the Geostationary Ring, Journal of Spacecraft and Rockets, Vol. 18, No. 4, July 1981. 6. V.A. Chobotov, the Collision Hazard in Space, The Journal of the Astronautical Sciences, Vol. 30, No. 3, July 1982. 7. K. Takahashi, Collision between Satellites in Stationary Orbits, lEE Trans. on Aero- space and Electronic Systems, Vol. 17, July 1981. 8. M. Hechler, The Probability of Collisions on the Geostationary Ring, MAO Working Paper No. 121, ESOC, Darmatadt, FRG, March 1980. 9. M.G. Wolfe, V.A. Chobotov, F.E. Bond, Man-made Space Debris - Implications for the Future, Earth-Oriented Applications of Space Technology, Vol. 3, No. 1, 1983. 10. E.M. Soop, Introduction of Geostationary Orbits, ESA SP—1053, ESA Scientific and Techn:- cal Publications Branch, ESTEC, Noordwijk, The Netherlands, November 1983. 11. J.C. van der Ha, Very Long Term Orbit Evolution of a Geostationary Satellite, MAO Wor~:ing Paper No. 122, ESOC, Darnistadt, FRG, March 1980.