ON A NEW THEORY OF THE GEOSTATIONARY SATELLITE MOTION AND ITS APPLICATIONS. R. I. Kiladze Abastumani Astrophysical Observatory, Georgia (FSU)

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1 ON A NEW THEORY OF THE GEOSTATIONARY SATEITE MOTION AND ITS APPICATIONS R. I. Kiladze Abastumani Astrophysical Observatory, Georgia (FSU) 1. INTRODUCTION aunching artificial Earth satellites into the geostationary orbits (GO) began about 40 years ago. At the present time the geostationary satellites (GS) moving in a sharp resonance with the Earth rotation have the most wide applications in resolving many practical problems, mainly those related to the TVcommunications, intercontinental communication and military purposes. After expiration of its active period a GS becomes a passive object and is transferred onto a higher or lower orbit in the neighbourhood of the GO. According to the statistics by January 005 there were 46 GS working in the GO and their total number was 114. Besides of observable objects there are ten thousand small fragments, smaller than 1m in size, which are small space debris (SD) moving in the vicinity of the GO. They are accompanying every launch of the GS and are created as a result of the GS explosions. According to the recent investigations the number of such explosions is 1; of them were the Russian GS (Ekran and Ekran 4) and 10 were the U.S. Transtages. Maybe this number is underestimated; therefore, in addition to elaboration of reliable methods for the detection of explosions by direct observations, it is necessary to dispose of the program for the search of all lost objects, and especially, of potentially dangerous ones for working GS. These fragments are endangering the existence of the GS because of the possibility of collisions. The number of small fragments is growing with time (their existence timespan in the region of the GO is unlimited) and the probability of their collision with the SD grows. Recent investigations of the evolution of 456 uncontrolled GS observed more than 1000 sharp changes in their drift, which may be explained in terms of collisions with small SD [1]. The main problem of the GS motion control is still related to the cataloguing of all observed small objects moving in the GO region. For elaborating a model of choking up the GO region by the SD it is necessary to have a reliable theory of the GS motion by which the available observations and orbital elements might be represented on the long timespan. It is also necessary to compile the software for calculation of the orbital evolution for hundreds of fragmens in a short time.

2 . THE THEORY OF A GEOSTATIONARY OBJECT MOTION 1. Introduction Our task was to construct a simple method to calculate the longperiod evolution of the real GS orbits. The elaborated method, based on observations made on the timespan of 1000 through 000 days with the mean square error in the range of 0.05 through 0.045, allows to represent the orbital evolution on the timespan of 7000 through days with an error of The theory has been tested by means of representation of long series of Two ine Orbital Elements (TE) of the NASA Goddard Space Flight Center and of the osculating elements (OE) determined by us directly from astrometric observations []. The principal peculiarity of the motion of a GS is the exact 1:1 commensurability of its mean daily orbital motion n with the Earth s rotation velocity S 1. In the case of absence of perturbations due to the nonsphericity of the Earth s shape and to the lunisolar attraction, the GS once launched at some geographical longitude would always rest there. However, under the influence of ellipticity of the Earth s equator, it begins to move like a pendulum near two stable points of libration with longitudes 75ºЕ and 55ºЕ. Because of the asymmetry of Earth s equator the periods of the small oscillations around these points differ by 110 days, being equal to 840 and 950 days, respectively. To simpify the representation of such a motion Gedeon [] has introduced a special system of coordinates, called by him stroboscopic. In this system instead of the continuous time the discrete timetimemoments are used as the argument of the equations of motion separated by daily intervals, and the phase plane of the stroboscopic longitude λ and drift dλ/dt is introduced in terms of orbital elements as follows: λ = M + Ω + ω S, dλ/dt = n + Ω 1 + ω 1 S 1, where M is the mean anomaly, Ω, the longitude of the ascending node, ω, the argument of the perigee, S, Greenwich sidereal time, Ω 1, ω 1, secular changes of the node and perigee longitudes. In the stroboscopic system of coordinates, in the case of an exact commensurability of motions, the central body and a satellite seem to be motionless, and in the case of an inexact commensurability the satellite would seem to slowly move along a trajectory. Such a picture may be observed if the system, the Earth a satellite, is in the darkness, being periodically illuminated by short lightflashes. The stroboscopic system has properties of the inertial and noninertial (rotating) coordinate systems. Introduction of this system essentially simplifies the equations describing the longperiod motion of the GS, but results in a loss of the shortperiod oscillations.

3 To describe the GS motion A. Sochilina in 1985 has used the aplace plane [4], being first applied by H. Struve [5] to the satellite motion. It passes through the vernal equinox and is inclined by some angle (depending on the semimajor axis of an orbit) to the geoequator. For the GS the angle is equal to The physical meaning of the aplace plane is the following. Due to the equatorial bulge of the Eartha couple of forces are created which are trying to make the plane of the GS orbit to coincide with that of the geoequator; as a result the plane of the orbit (in the case of absence of other forces) precesses, being constantly inclined to the geoequator.on the other hand the common attractions of the Moon and the Sun try to make this plane to coincide with the ecliptic. The resultant of these two couples of forces compels the GS orbit plane to precess by the constant inclination to a plane lying between the geoequator and the ecliptic and called the aplace s plane. The choise of the aplace plane gives some advantages over other planes of reference: relative to it the inclination of orbit i changes (because of inclination of lunar orbit to ecliptic) only in limits of 0.5 and the longitude of the node and the perigee argument become practically linear functions of time. The resonance equation for the stroboscopic longitude λ also becomes simpler. As a result the problem may be resolved by use of the first order theory with small corrections for the second order only. Usually the precision of some osculating orbital elements is not satisfactory, especially for the calculation of satellite s drift d/dt but the derivatives of with respect to time may be improved by use of the data of preliminary orbits combined with the longperiod motion theory. In the process of work with the Catalog of improved orbital elements [6] in 1996 it was pointed out that on the 1000day span some GS show considerable systematic differences in the longitude (in some cases reaching several degrees) between observed O and calculated C values. After estimation of the theory accuracy R. Kiladze [7] assumed that these discrepancies could result from accidental changes in d/dt which in their turn are caused by the collisions with small particles.. Perturbation forces Perturbations of the GS orbital elements may be found from the agrange equations. The perturbing function due to the geopotential U, the lunisolar perturbations R, R S and the radiation pressure R may be expressed in terms of orbital elements referred to the aplace plane [8]: U GE G a m lm e Re ( 1) ( 1) lm Elmk ( ) a a l m0 k 6 p0 q l1, l p D ( i) X ( e) exp 1 ( l p) ( l p q) M k ms, lkp l pq l (1) (

4 where GE is the gravitational constant, a e, the mean equatorial radius of the Earth, Glm Clm 1 Slm, C lm, S lm, the geopotential parameters, E lmk ( ), D ( lkp i ), the l1, l p inclination functions and X ( e), the Hansen coefficients calculated in [9], M,, l pq, e, i, a, the GS orbital elements. The perturbing function due to the Moon is expressed as follows: R Gm a l k a * ( 1) Elmk ( ) a l m k p0 p' 0 q q' l, l p l1, l p' D ( i) D ( i ) X ( e) X ( e ) exp 1 [( l p) lpk lmp' l pq l p' q' ( l p q) M k ( l p') ( l p' q') M m ], where E ( ) E ( l 1), 4 l1 1, m 0, m 0 lmk lmk m m (), is the inclination of the ecliptic to the geoequator,,, e, i, a, the orbital elements of the Moon. One can obtain the perturbing function due to the Sun by changing the index in () to the index S. According to Kepler s third law G( E m G( E m ) n a S m ) n a, S S. The mean radiation pressure function is as follows: 1 1 1s R Qae( 1) E11 s ( ) D1 sp ( i) exp 1(( 1 p) s S ), s1 p0 where S is longitude of the Sun, Q () P A 0 ( P m dyne cm 5. and A m, the ratio of the surface area to the mass expressed in cm /gr). The equation for calculation of may be obtained from the condition that the sum of terms of pertubing functions U, R, R S, with argument, be equal to zero: tg sin cos C a e n e e a where n ( 1 e ) 1 / ( sin i ) ns S ( 1 es ) / = , n, n n, S, are the mean motions of a satellite, the Moon and the Sun, respectively, expressed in radians per day: m ms S E m, E ms m. At last, all resonance perturbations are accounted for by the differential equation for []: d n Almkpq sin( m mlmkpq ( k m) ) SP, d t (5) 1, (4) 4

5 where A lmkpq is the function of a, e, i and of the geopotential parameters, lmkpq, the phase angle depending on lm, and the geopotential parameters Clm Jlm cosm lm, Slm Jlm sinm lm, SP meaning the lunisolar perturbations. The solution of this equation is described in [10]. 5. Integration of the equations of motion For simplification sake let us write the equation (5) as follows: where Choosing the aplace plane as the basic one for the coordinate system guarantees that the change of coefficients А i, m i, i and s i in (6) with time is very slow (i.e. that they are almost constant). If these coefficients and the longitude of node are constants, in the case of absence of the lunisolar perturbations, the equation (6) allows for the first integral: where С 1 is the integration constant, being analogous to the Jacobi s constant in the restricted problem of three bodies and in such a case the problem can be solved in terms of the quadrature: The behaviour of the function is shown on Fig.1. It is like the asymmetric double sinusoid with maximums and minimums of different heights. Accordingly, depending on the value of parameter С 1, the GS can move in accordance with one of the three modes the simple libration around one from the two stable points 75E or 105W, the compound libration (around both points), or in the mode of the circulation (the drift) around the Earth. 5

6 6 Fig.1. The function. Because of the smallness of d/dt, the solution of equation (6) may be expressed in the form (8), if one considers С 1, as a slowly changing function of time. In such a case one can obtain the derivative of С 1 with respect to time: introducing a new variable of integration d instead of dt by means of (8) we get: The exact calculation of the righthand side of the equation (1) is not feasible since the dependence of variables C 1 and on is unknown; but, with an accuracy acceptable for practical applications, they may be considered on short time intervals as constants equal to the average values of the quantities C 1 and, respectively. The variability of C 1 and inaccuracy of calculations by means of (10), connected with the variability of with time, may be significantly diminished by introduction of the oscillating system of coordinates, attached to the function. On Fig. different types of motion in the phase plane are schematically shown. If the value of the drift is less then 0. /day, then the GS move in the mode of a libration around one of the stable points with the amplitude depending on the initial longitude. The GS having maximum amplitudes of libration may periodically change their motion modes because of the lunisolar perturbations. 6

7 ddt ( /сут) Fig. Different types of the GS motion in the phase plane longitude λ vs. velocity of drift dλ/dt. 4. Formation of the intermediate orbit, the agrange s equations The method for calculation of the longitude just described allows to calculate ephemerides for a majority of the GS for the time intervals of several years with quite satisfactory accuracy for observers. Further improvement of the ephemerides accuracy is possible by improving the GS orbital elements. For all this it is important to have the intermediate orbit which is maximally close to the real one. The solution of this problem is published in [11] and [1]. Keeping the main terms in the righthand sides, the agrange equations for the 4 orbital elements e, i, ω and Ω, in terms of the coordinates used, may be reduced to the form: where dz m cos i cos z z, dt sin i di m cos i sin z, dt dr R m(sin icos i cos i tg dt (1) (14) de AR sin R Dx sin x; dt z R x k cos i., S,, i ) AR cos R Dx cos x, e, i, and e denote here the orbital elements (the argument of the latitude, the inclination, the longitude of the ascending node, and the eccentricity),,, the argument of the latitude and the longitude of the node of the Moon on the ecliptic, (15) (16) (17) respectively, S, the longitude of the Sun. D stands here for the light pressure, k, A, m, x, and R are considered as constants. 7

8 5. Solution of the agrange s equation system The equation system (116) is actually subdivided into two systems, each containing two equations. Multiplying equations (1) and (14) by expressions m sin i cos i sin z, ( z sin i mcos i cos z), respectively, and summing them up, one can obtain the equation in complete derivatives with respect to time, allowing for the first integral: k cos i msin icos( ) cos i C. (18) Using the integral (18) the solution of system (1) and (14) may be received in the form of the elliptic integral: (19) q by: t (0) and also where k 4 m cos 4 i k sini di cos i kc 8 m cos i C cos i C In order to resolve the system (15)(16) let us introduce new coordinates p and p ecos R A, q esin R; d (R )dt, (1) cos i 1 m sin i cos z cos i. cos i 1 () Then the system (15)(16) becomes: dp q D x sin(r x), d dq p D x cos(r x) A. d The solution of the linear system () isn t difficult, because the free terms in the righthand sides of these equations are the variables found from the equation of system (18)(19). At last, by integrating the expression (1) the time can be calculated as the function of variable τ: d t. R (4) The computer software compiled on the basis of the described solution of the problem copes successfully with the task of the orbit construction and of following its evolution on the timespan of tens of years for every known GS.. () 8

9 6. A new motion integral the Kren integral It is necessary to mark the scientific meaning of the first integral (18) obtained by integration of the agrange equations. After the epoch of resolving the twobody problem by Newton, in the scientific armoury of celestial mechanics there appeared only a single first integral, i.e. that of Jacobi found in XIX cy. By means of the integral (18) the inclination of the GS orbit to the aplace plane is given as the function of its longitude of the node and of the orientation parameters of the lunar orbit. Because the oscillations of the inclination of the GS orbit are described by (18), we named this integral by the Russian word Kren (the tilt, the banking angle) and the constant C by the Kren constant, respectively. It is shown in [11] and [1] that in the general case the change of the value of 8 C is not greater than C, i.e. C 10 is practically constant. Consequently, (18) is always highly accurate. Today the constants C 1 and C are used for the identification of unknown GS discovered in the process of the sky surveillance with objects lost in the past. But with the science progress each of the known first integrals has found a new, sometimes unexpected application. We should expect that the Kren integral will not be an exception of this rule. 9. THE MODE OF CHOKING THE NEIGHBOURHOOD OF GEOSTATIONARY ORBIT BY FRAGMENTS OF EXPODED SATEITE 1. Explosions on geostationary orbits As it was mentioned above the cosmic space developing is accompanyied by the choking of it with different fragments following the satellite launches. This phenomenon has caused anxiety since a long time, not only because of damage to working satellites by collisions, but also due to the potential start of a cascade process of their destruction. An especially tense situation is created on the GO zone, where the quantity of objects inaccessible for observation because of their smallness is estimated as hundreds of thousands. One of the sources of the GO choking are explosions of geostationary objects. From time to time this information apears in the scientific literature [1,14]. The explosion of only one satellite may generate several thousands of small fragments thrown out from the satellite with great or small velocities and remaining in the vicinity of the GO for a practically unlimited timespan. The velocity of the GS changes during this timespan by several m/s. The drift of GS changes, therefore, more than by 0.05/day. 9

10 Because of the smallness of these fragments they can be observed but with great technical difficulties, and the ascertainment of the explosion of each single GS is, as a rule, possible only by indirect methods, a long time after the event [15,16]. The method of determination of the timetimemoments and of dynamic characteristics of the explosion of an GS by using its orbital element changes is elaborated by the authors [17,18]. Today there are revealed 1 exploded GS by means of this method. Their real quantity can be greater. For these objects their designation numbers, the timetimemoments of the explosions T o, the orbital elements e, i, Ω, ω, λ and the drifts dλ/dt before and after the events are given in Table 1. The sharp change of the drift Δ(dλ/dt) given in the last column of Table 1 is considered as the principal indication of an explosion: if its value exceeds 0.1/day, the event is considered to be an explosion. The data about first 10 exploded objects are given in [18]. 10 Table 1. The orbital elements of 1 exploded GS. NOS. T 0 (MJD) J 587 e i d/dt ( o /day) o o o 4. o o (d/dt ) ( o /day) 0. o G 0/10/ /0/ E B 1/0/ D 08/0/

11 11 06/04/ С /0/ F * 760 J 7709 A 09/10/ D A 1/06/ /10/ /06/ B B A 17/09/ /1/ **

12 (*) Orbital elements of GS 760F and of its fragment GS 760J. (**)In different sources this GS is designated in a various way. For GS 760F we hadn t the orbital elements before the explosion, therefore the explosion timetimemoment is determined by establishing the identity of the object itself and its fragment GS 760J. The check for correctness of the timetimemoment T o of an explosion consists in identifying the GS positions in the threedimensional space as calculated by two orbital element systems (before and after the explosion). The mean discrepancy depends on the accuracy of the orbital elements used; for the TE data it is about 1 km. To get a clear view of the field of the space debris spreading after an explosion it is useful to construct the manifold of orbits for the fragments in the framework of the twobody problem for the spherically symmetrical explosion. The study of the orbital evolution allows to determine the size and the shape of the region of motion of the fragments. For determination of the shape of the space occupied by the fragments it is important to know the ratio Earth s attraction/ the lunisolar attraction and the radiation pressure. The orbital evolution of the fragments with a crosssection up to 40 m /kg has been studied by means of the numerical integration on the timespan of 100 yrs [19].. Determination of orbits of fragments generated after an explosion We use the Cartesian system of coordinates with equator of date as a fundamental plane; Xaxis is directed to the point of equinox of date [0]; the position of the GS at the timetimemoment of an explosion t o is defined by its radiusvector r o and the vector of its velocity V 0. Suppose that at this timetimemoment all fragments are ejected with the velocity V. 1 Fig.. Determination of the intersection point of orbits On Fig. the arc represents the celestial equator. The arcs 0 S and S are projections of orbits of the GS and of its fragment on the celestial sphere; S is their intersection point. By 0, i 0, u 0,, i and u are denoted the longitudes of nodes, the inclinations of orbits and the arguments of latitude of GS and its fragment, respectively; and are the right ascention and declination of the point S. 1

13 1 The equations relating arcs of great circles (Fig. ) to the orbital elements of the GS and its fragment are given in [17]. For a symmetrical explosion the absolute value of V is in limits of 1 50 m/sec depending on the fragment masses. The angle l is measured from the direction of the transversal velocity of the GS from 0 to 60, the angle is measured from its orbital plane in limits of 90. Small values of eccentricities are natural for the GS, and the ratio V V is less than 1/1. Furthermore, one can use simple formulae [17] to estimate the change of the orbital parameter p and that of i and as well: V p p 0 cos cos l, V V i sin cos u0, V V sin sin u0 / sin i0. V V e0 sin v0 e sin v cos sin l V V e cosv e0 cosv0 cos cos l. V (6) where v 0 is the true anomaly, 0, the perigee argument, e 0, the eccentricity, a 0, the semimajor axis, p 0, the orbit parameter, 0, the longitude of the ascending node, i 0, the inclination of the GS orbit at the timetimemoment of an explosion. In the process of investigation of the orbital evolution of the GS fragments it is necessary for each of them to use its own aplace plane, turning the coordinate system around the Xaxis by the angle 0. The data for the explosions of 9 objects and for their 5 fragments are given in Table containing the value of V, the longitude l relative to the GS transversal velocity and the latitude with respect to the orbital plane of the motion in the orbital system of coordinates, the argument of latitude u 0, the geocentric distance r 0 of the GS and the discrepancy r in distances expressed in km. (5) ( Table. Changes in velocities and directions of motion for exploded objects and differences in distances r, calculated from data of Table 1 NOS. T (MJD) V, m/sec 6605J l u 0 r 0, (in the R E units ) r, km G E B D A

14 7811D A B Fragment s 68081G H * 760J H C The detailed investigation of the GS 68081E explosion [1] shows that for the fragments of 18 0 magnitude the average velocity change ΔV is 70km/sec. Furthermore, ΔV is about 50m/sec. for smaller particles which are invisible for optical instruments. The example of calculation of the initial orbital elements of the fragments of the GS 760F referred to their own aplace planes for the ejection velocities of 50 km/sec, is illustrated by the Table. The Table clearly shows that the inclination of the aplace plane to the equator Λ exceeds 1 for the fragments having the great negative drift. The total inclinations of the orbits of small fragments to the equator, calculated as a sum of the orbit inclinations to the aplace plane and Λ, may reach 7 in the evolution process. For the GS moving in the equatorial plane the relative velocities of such fragments may reach 1.5 km/sec. It should be mentioned that at the timetimemoment of an explosion the longitude of the ascending node of the GS 760F orbit was 5. In the case of the explosion at the longitude of 180 the inclinations of the fragments orbits to the equator would reach 40. Table. Initial orbital elements of fragments generated by the explosion of the GS 760F for V = 50 km/sec 760F Transtage Т (MJD)= N i e v d/dt /d r in R E

15 Model calculations of the orbital evolution of the fragments after an explosion The problem of the study of the fragment dynamics is interesting from the point of view of choosing the program of search for table (i. e. of 18 0 mag.) objects. For this purpose let us investigate the evolution of an explosion point position in time and space. Obviously, at the timetimemoment of the explosion the orbits of all fragments are intersecting in this point. For representation of the dynamics of fragments generated by the GS explosion by use of the method described in [17] there were investigated the behaviour of GS fragments ejected with the same velocities in different directions directed to the vertices and along the sides of a rectilinear icosahedron (i. e. the polyhedron with 0 sides). Moreover, also the model consisting of 00 fragments with the same ejection velocities was investigated. For the timetimemoments of each GS explosion the initial orbits of their fragments referred to the corresponding aplace planes were constructed. It is postulated that at this timetimemoment the modules of the vectors of fragment velocity changes are the same, not exceeding 50 km/sec. After this timetimemoment each fragment begins to move along its orbit intersecting the orbital planes of other bodies, including the orbital plane of the 15

16 paternal body because it may be considered as the greatest fragment, in the point of the explosion. The inclination of the fragment orbit to the initial one is 16 V p i, (7) V where Δi is expressed in radians, V p is the polar component of the total velocity of the fragment and V τ is the tangential velocity of the GS. Accordingly, the initial poles of all fragment orbits are displaced along the great circle of the celestial sphere 90 o from the point of explosion (Fig. 4). On the other hand, the semimajor axes of the fragment orbits are differing from the semimajor axis of the initial orbit by: a a V o V. (8) After an explosion the motion of each fragment is referred to its own aplace plane. Their poles are disposed along the great circle of the celestial sphere passing through thе celestial pole and initial pole of the relevant aplace plane (Fig.4). Orbital pole North pole Poles of corresponding Orbital poles Orbital plane Equatorial plane aplace planes main body s pole fragment s pole Explosion Fig. 4. The location of the poles of fragment orbits and of those of their aplace planes just after the explosion. After the explosion the orbits of all fragments begin to precess around the poles of their aplace planes with different speeds. As a result, after some timeinterval these poles will be located along the curve slightly differing from a great circle. Consequently, the corresponding orbital planes intersect each other in almost the same line. We have called this phenomenon the regularization of fragment orbits. The problem concerned with calculation of the regularization timetimemoment (the most favourable epoch for observations of the fragments) and its numerical description by computation of spherical coordinates A and D of the intersection point of trajectories of the fragments and their dispersion σ is studied in [18]. 16

17 The evolution of the ensemble of fragments ejected from the GS with velocity of 75 m/sec is studied on the basis the theory of motion [1 7] for all 1 events, described in Table 1, over the timespan of 0,000 days (about 80 yrs). The results of these calculations are given in Table 4 containing designations of the exploded GS, the data for the most compact location of the intersection points of orbits of the fragments, the geocentric coordinates A, D of these points and the values of the average deviations σ of the fragment trajectories from the intersection points. Table 4. The timetimemoments, equatorial coordinates of points of orbits compact intersection and dispersions for 1 fragments of the exploded GS NOS. MJD Year A D σ 6605J h.4 0. o G E B D C F A D A B C For an illustration, the behaviour of the GS 68081E fragments on the timespans of 000 days is shown in Fig

18 18 4 At the moment of explosion 000 days later 4000 days later 6000 days later Fig. 5. The evolution of the poles of the GS 68081E fragments on the intervals of 000 days It is seen from Fig.5 that the poles of the fragment orbits 000 days after the explosion occupy a significant area on the sky, but after 899 days they are aligned along the great circle. Subsequently they occupy again more and more area on the sky. A majority of the GS investigated by us show the similar process of the orbit regularization with other characteristic times. In several cases (GS 6605J, 7811D), however, the process of the orbit regularization is faintly presented. 4 At the moment of explosion 000 days later 6000 days later 9000 days later 1000 days later Fig. 6. The evolution of the poles of the GS 6605J fragments on the timeintervals of 000 days. 18

19 19 As an example the orbital evolution of the GS 6605J fragments on the timeintervals of 000 days is represented on Fig 6. The minimum of the dispersion is reached 960 days later after the explosion timetimemoment but it is so faintly presented that Fig.6 is hardly differing from the picture of the stochastic distribution of the points on the plane. The cases for other ejection velocities (50, 0 and 10 m/sec) of fragments are also investigated but they don t influence significantlly the behaviour of the GS fragment orbits: the configurations on Figs. 5 and 6 are invariant with respect to the change of the fragment ejection conditions, and only the scale of each Figure changes proportionally to V. 4. Evolution of fragments orbits after explosion It is clear from above that the regularization process of the GS fragment orbits is mainly depending on the angles φ 1 and φ represented on Fig. 4, i.e. the angles between the great circle passing through the poles of the initial orbit and the aplace plane and the great circles on which the poles of the orbits of fragments and their aplace planes are located. To clarify this dependence a numerical experiment has been made: it has been calculated for each exploded GS what the orbital evolution would be if the explosion occurs with some delay with respect to its real timetimemoment (i. e. if it would happen at another point of the GS orbit). Some results of this experiment are represented in Table 5. In the first column of Table 5 the differences of mean anomalies ΔM with the step of 0 o (which corresponds to the explosion delay of h ) are given, in other columns the timeintervals between the timetimemoments of explosions and those of regularizations and corresponding values of dispersions σ for the intersection points of trajectories are given. Table 5 shows that for each GS there are two values of ΔM (separated by approximately 180 o ); for them the values of σ reach minimum. These values are written in the bold type. If the timetimemoments of the real explosions of the GS and the values of ΔM are known, it is easy to find that the GS whose right ascensions are about 9 or 1 hours at the timetimemoment of the explosion are characterized by the minimal values of σ. Table 5. The regularization models of the orbital planes for 1 exploded GS. The configurations with minimal values of σ are written in the bold type. \ NOS. ΔM\ 6605J ΔT 0 o o 57 σ 67066G ΔT o 0 σ 68081E ΔT o 070 σ 7040B ΔT o 11 σ 7100D ΔT o 07 σ 75118C ΔT o 091 σ 19

20 ~ ~ ~ ~ \ NOS. ΔM\ 760F ΔT σ 7709A ΔT σ 7811D ΔT σ 79087A ΔT σ 8019B ΔT σ 8419C ΔT σ 0 o o o o o o o ~ ~ ~ ~ ~ ~

21 The dynamics of this phenomenon are as follows. At the timemoment of explosion of the GS all fragment orbits intersect each other in two opposite points of the celestial sphere. Because of perturbations their orbits change and, according to the initial conditions, the intersection points of the visual fragment trajectories create two streams the centers of which are lying apart by 180 in the longitude. After a sufficient time the fragments are intersecting with the basic orbit practically at every point. If the ejection velocity of the fragments is 75 m/sec, the inclinations of their orbits don t exceed 5. After several (about ten) years of the evolution the orbit planes intersect each other nearly in one line, i. e. they have the common node. Such ordered configuration exists during 4 years after which the orbital planes take different orientations anew. The values of σ as a function of time passed after the explosion and its delay for the GS 6605J and 7040B are shown on Figs. 7 and 8. 1 Fig.7. Dependence of σ on the time passed after the explosion and its delay for the GS 6605J. 1

22 Fig. 8. Dependence of σ on the time passed after the explosion and its delay for the GS 7040B. Fig.7 shows that σ may reach minimum not only once, but every successive minimum always is less deep than previous one. This happens because the line of the poles of the fragment orbits placed eventually becomes more and more different from the great circle (Fig 9). Fig 9. Displacement of the poles of the GS 68081E fragment orbits at the second minimum of σ, days after the explosion The regularization phenomenon of the GS fragment orbits creates the most favourable situation for observations of fragments near their common node which all of them pass once a day. From this affinity of the GS fragment orbits is founded the barrier method of search for the fragments: in the years favourable for their detection (when σ has a minimum, according to Table 4) the permanent observations in the vicinity of both

23 nodes should be made as long as possible. By monitoring the enviroments of these points (of the radius σ) during a day it is possible to observe all fragments generated by an explosion of the same GS. By the way, the small particles ejected by the explosion with great velocities spread themselves along the GS orbit faster than those having the smaller relative velocities, the evolution of which runs slower. In Table 4 the geocentric values of the data (in particular those of σ) are given. When observing from the Earth s surface the deviation of individual fragments from the common center may grow because of the differential parallax. This effect of the vertical stretching depends linearly on the ejection velocity by the explosion. For velocity of 75m/sec it is equal to 0.º8 sin z and 0.º89 sin z (where z is the zenith distance) for the upper and lower directions, respectively. At last, the studies of the evolution of fragment orbits show that in their intersection points the distances between the orbits periodically become zero, which makes the favourable conditions for collisions. As it is shown in [15,16] many uncontrolled GS had repeated collisions, the exploded objects being frequently met among them. 5. Conclusions For elaboration of efficient precaution measures for safety of the controlled satellites in the GO zone it is necessary to clear up primarily the real situation about the littering and to process all orbital data in order to know the real quantity of explosions. The new longperiod theory of the GS motion allows to analyze the observation data of the uncontrolled objects for long timespans and to discover the accidental changes of orbits depending on explosions or collisions with the space debris. The model of creation of fragments for 1 exploded GS is constructed and their orbits evolution on the timespan of 80 yrs is investigated, the timemoments and dynamic parameters of explosions being determined. The maximal change of the inclination of fragment orbits (when the ejection velocity is 75 m/sec) is about 5 depending on the argument of latitude and the location of the explosion point on the GS orbit. In the process of the evolution the inclination of small particle orbits to the geoequator may reach 40. Such fragments are the most dangerous for the Earth satellites, because they can provoke the cascade process of continuous fragmentation [8, 9]. The search for the fragments becomes easier at time periods when the planes of the fragment orbits intersect themselves in the same line. During this period all fragments once a day pass two isolated points (the antipodes) on the sky. Our results may be used for the organization of observations of the spreading area of the smallsize debris using the groundbased instruments and for identification of the litter sources as well. The process of fillingup the cosmic space with the explosiongenerated fragments depends on relative velocities V: the velocities V< 75 m/sec are typical

24 for greater fragments which intersect the initial orbit of the GS at two parts about 10 о long, lying apart by 180 о. When the distances between the orbits in the intersection points become zero, the favourable conditions for collisions occur. The smallsized debris with great velocities (about 50 m/sec) fill up the neighbourhood of the exploded GS and soon begin to intersect all orbits. Moreover, there are created objects having masses of about 10 g and relative speeds of 11.5 km/sec. They can not only collide with the GS but also can provoke the cascade process of fragmentation. From this point of view also the process of permanent burial of the GS near the GO (staying km afar) which ceased to work, because the exploded GS can create fragments with great orbital eccentricities and inclinations of order of 0. In order to have the permanent and reliable information about the choking of the GO area it is necessary to put into operation more powerful means of groundbased technics for observations, and also to launch a satellite, with a telescope aboard, in the environments of the GO with the means for the permanent communication with the Earth REFERENCES 1. Sochilina A.S., R.I. Kiladze, I.E. Molotov, A.N. Vershkov, E. Kornilov, On investigation of TE precision of geostationary objects, Proc. of Fifth US/Russian Workshop, Pulkovo, Sept. 7, 00, ISBN , pp , 00.. Sochilina A.S., K.V. Grigoriev, R.I. Kiladze, A.N. Vershkov, V. Malyshev, On Broadening of GO Catalogue Contents, 1st International Workshop on Space Debris, Space Forum, October 1995, Moscow, Russia, Vol. 1, pp. 15, Gedeon, G.S., Tesseral resonance effects on satellite orbits, Celestial Mechanics, Vol. 1, No., pp , Сочилина А.С., Лунносолнечные возмущения в движении высоких спутников. Бюлл. ИТА, 15, 7(170) с. 895, Struve H., Über die age der Marsachse und die Konstanten im Marssystem, Sitzungsberichte d. Akad. d. Wiss., Berlin, pp , Cочилина А.С., Григорьев К.В., Киладзе Р.И., Вершков А.Н., Каталог орбит геостационарных спутников, ИТА РАН, СПетербург, 1996, 10 с. 7. Kiladze R.I., A.S. Sochilina, K.V. Grigoriev, A.N. Vershkov, On Investigation of ongterm Orbital Evolution of Geostationary Satellites, Proc. of 1th Symposium on Space Flight Dynamics, ESOC, Darmstadt, Germany, 6 June 1997, pp. 5 57, Grigoriev K.V., A.S. Sochilina, A.N. Vershkov, On Catalogue of Geostationary Satellites, Proceedings of the first European Conference on Space Debris, Darmstadt, Germany, 5 7 April 199, pp , Gaposchkin, E.M., 197 Smithsonian Standard Earth (III), SAO Special Rep., No. 5, 197, 88 p. 4

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