On Ways of Modernization of Geostationary Ring Surveillance

Transcription

1 On Ways of Modernization of Geostationary Ring Surveillance * Rolan I.Kiladze, ** Alla S.Sochilina, ** Konstantin V.Grigoriev, ** Andrei N.Vershkov * Abastumani Astrophysical Observatory, Abastumani, Georgia ** Pulkovo Observatory, St-Petersburg, Russia Abstract. On base of the new version of the long-periodic theory of motion of satellites, the package of algorithms and software for the surveillance of geostationary region has been prepared. This theory is based on the intermediate orbit and includes the perturbations of high orders. It permits to unite the osculating orbits of satellites over long time intervals. The comparison of the accuracy of two kinds of osculating orbits: NASA Two-Line Elements and the orbits determined over the long time intervals have been fulfilled. The additional arguments in favor of the occasional changes of velocity phenomenon are given. The method of looking for such changes according to the available orbital data may be realized in the automatic regime, due to the compactness of the new version of the software. The conclusion about the necessity of treatment of all accumulated orbits for uncontrolled geostationary satellites, to have the statistics of the occasional changes in the rate of drift, have been made. I. Introduction The continual increase of a number of satellites, moving in the near Earth space, makes the problem of their surveillance to be a very difficult matter. In future, to avoid the critical situation due to enormous quantity of calculations it is necessary to introduce more economical methods of the observation treatment. Therefore we began to elaborate algorithms, based on the long-term motion theory and allowed to watch non-controlled objects for a long span with a help of once improved orbits. The experimental surveillance has been developed for satellites moving in the vicinity of Geostationary Ring. The motion of these objects is commensurable with the Earth's rotation, what provoke the significant resonant perturbations from the geopotential. Their distances from the Earth's center vary from km up to km and their sizes can reach 40 square m and more. Therefore, they move under the influence of luni-solar attraction and the solar radiation pressure, beyond the Earth's oblateness. At present, the theoretical base for the solution of all necessary tasks and their software have been practically completed. This base contains the solution of following problems: the motion theory of geostationary satellites (GS) during a long time interval, the identifications

2 of GS, the determination of their preliminary orbits, the improvement of orbits with a help of observations, the creation and maintaining of GS catalog, the use of accumulated osculating orbits for the improvement of initial orbits, applied to ephemeris providing and the calculation of orbital evolution, and at last the search of lost objects. This work was continuing during some years and the all stages of it have been reported on the different conferences and published in the several papers (Ref. 1 9). For testing of the motion theory the series of osculating orbits OLI (determined with the help of photographic and electro-optical observations over long time intervals) and TLE (NASA Two-Line-Elements of US Space Commands) have been used. The all available series of osculating orbits have been treated. This treatment allowed to discover the occasional variations in the rate of drift among several objects, which could be explained by their collision with space debris (Ref. 1). In future such phenomenon we shall call as collision. The analysis of the discrepancies between the orbital elements (experiment) and their evolutions (theory) stimulated the new improvement of the theory motion and its software (Ref. 2). The recalculations of old collisions with improved theory and accumulated orbital data have confirmed previous results and have shown the existence of repeated collisions for several GS. Thus the prolongation of investigations of accumulated orbital data and their treatment could give the interesting information about the population of Geostationary Ring. The discovered types of GS, which change their regime of motion under the luni-solar attraction, could be useful for the improvement of geopotential parameters. II. The peculiarities of the developed motion theory The Earth oblateness exerts the main influence on the motion of close satellites, therefore the selection of the plane of equator as the basic coordinate plane, is quite reasonable. But for objects, remote from the Earth center, the luni-solar perturbations cause the long-term oscillation of orbital plane with the significant amplitudes. For GS this amplitude can be equal 6 8 but relative to equator, the plane of GS varies from 0 up to 15 with period of 54 years. In this case, the analytical motion theory becomes complicate and it is often replaced by the numerical integration of the motion equations. The principal distinction of the developed theory is in introducing of the Laplace plane as basic coordinate plane (Ref. 3, 9). Relative to this plane the inclination (i) of GS orbit oscillates with the amplitude 0. 6 for 28 years, and the coefficients of secular changes in the

3 longitude of node (Ω) and the argument of perigee (ω) become almost constant. Due to that it may be possible to integrate the Lagrange equations for Ω, ω, i and e (eccentricity) with adopted perturbating function in the linear approach. The resonant equation of GS for the longitude λ (λ=m+ω +ω - S, where M mean anomaly, S Siderial Greenwich Time) is also simplified and its solution reduces to the calculations of the hyperelliptic integrals (Ref. 5). Thus for a span (from 0.5 up to 1 year) the simplified and precise motion theory for GS, with taking into account short-period perturbations in orbital elements has been developed. More over, the resonant equation is integrated with the constant coefficients. For the long-term intervals the equations for elements i, Ω, ω, e are integrated with use of the intermediate orbit (Ref. 2) and their perturbations are taken into account under the solution of the resonant longitude equation. Both these theories (annual and long-term) have their practical applications. The annual theory is used for the improvement of osculating orbits by GS observations, but for the calculation of evolutions and representations of osculating orbits during the long time interval the second theory is applied. Under the constructions of these theories, the following simplifications have been made. The development of geopotential has been limited by terms of the sixth order. The numerical values of parameters have been taken from the Joint Gravity Model-3. The simplified motion models of the Moon and Sun have been used (Ref. 10). The development of the perturbing function of the Sun has been limited by terms from the second Legendre polynomial and terms from the third Legendre polynomial in the case of the Moon. III. The osculating orbits of geostationary satellites The series of the orbital elements improved by observations are very useful for the test of the long-term motion theory of these GS. However the orbits of GS, improved over the short span, have a poor precision because of different causes: small eccentricity, the ill-conditioned normal equations, the degree of used time polynomial, which approximates the longitudes of GS (Ref. 11). The span over which the observations are treated can be increased if the properties of equation for the resonant longitude are used. In the paper (Ref. 12) is shown, that the longitude can be represented by time polynomial of ninth order over span equal to days with constant coefficients. Therefore, if the epoch of the improved orbit is selected in the middle of the time interval, than the observations, distributed over days, can be treated and the

4 osculating orbits can be calculated on the every moment of observations. Such orbits we call orbits of long intervals or OLI. The quality of OLI, calculated from observations of several librating GS, are demonstrated in Fig. 1 by following way. OLI are used as the initial data for the calculation of evolutions of these GS. For these GS there are the series of osculating orbits, distributed over the long time interval. The longitudes of osculating orbits are adopted for observations (O), but the longitudes from the evolutions play role of calculations (C). In Fig.1 (O C) λ are depicted for following GS: 84016a (the least square estimate σ = on span ΔT = 3403 days), 84041А (σ = 0. 12, ΔT = 3847 days) and 85102А (σ = 0. 16, ΔТ = 3844 days). These (O C) λ show the usual accumulation of errors because of the erroneous initial rates of drift. It is necessary to note, that in reality the last two objects change their rates of drift by occasional way in the limits /day. Nevertheless, these GS can be identified with the help of one OLI during 10 years. The evolutions of TLE for these GS on the same intervals cause a larger accumulation of errors in the longitudes, therefore in Fig.1 the behavior of their (O C) λ has not shown. The orbital residuals, obtained with the time polynomials of the second and ninth orders (our software) correspond to the empirical deviations between TLE and OLI. In the Table 1, there are the results of the observation treatment of GS 84016A. The observations have been carried out during 16 days and the date of the first observation is adopted for the epoch of orbital improvement. In Table 1 the available TLE for this span are added for comparison. Table 1 GS 84016A: Osculating equatorial elements obtained from 61 observations, distributed during 16 days, with the approximate polynomials of the ninth and second orders Degree of polynomial T () e ω e λ e dλ e /dt ( /day) 1(9) (9) (2) (2) TLE TLE

5 The inclinations and longitudes of nodes are in good agreement, therefore their values are omitted hear. The elements е, ω, λ and dλ/dt (the rate of drift) are turned to be especially sensitive to the polynomial degree. The longitudes of GS in these two variants coincide only at the ends of the span. Their behavior inside of the span is shown in Fig (O-C) λ 64.2 λ A 84041A 84016A Fig. 1 GS 84016A, 84041A, 85102A: (О- С) λ, obtained from the comparison of the OLI evolution with the available osculating orbits Fig. 2 GS 84016A: the osculating longitudes, calculated over span equal to 16 days for every observations with the polynomial relative to time of the second and ninth orders. Thus, comparing the data obtained on the base of the motion theory with TLE data it is possible to expect that the maximum discrepancies will have the elements e, ω, and dλ/dt. In the longitudes of GS the errors will arise because of the ill-conditioned normal equations, the wrong values of e and ω and the incompatibility of a degree of the time polynomial with the size of span. It can be noted, that all OLI of librating GS are improved with the time polynomials of ninth order. In Fig. 3, for the comparison, the series (O C) λ for OLI and TLE are represented for the same time interval. The empirical data, obtained from the treatment of the nearly all observed satellites, show that the osculating longitudes from TLE of the most part of GS contain systematical errors with the period of year and amplitude of For 20 investigated GS the maximum of deviations happens to be in October November, the origin of which is not quite clear.

6 IV. The long-term orbital evolution of the geostationary satellite The software for calculation of the evolution of orbital elements i, Ω, ω and e on base of the analytical theory of the intermediate orbit (Ref. 2), with taking into account the rest perturbations and reductions for the adopted non-inertial coordinate system (because of the precession of equinox) has been realized. This software allows to calculate these elements for the moments of available osculating orbits (TLE or OLI) during 10 years and to improve the initial data with their help on the moment of the first or last used orbits. If the precision of used orbits is satisfactory, then the forward and back evolutions of the mentioned elements coincide (O-C) λ (1) (2) e Fig. 3 GS 84016A: (О-С) λ, improvement over span equal to 540 days for the orbits OLI (1) and TLE (2), separately Fig. 4 GS 85102A: the comparison of eccentricities (e) of OLI+TLE with their evolution, calculated with improved orbits forward and back. In Fig. 4 5 for GS 85102A the comparison of orbital elements e and ω of OLI with evolutions of the first and last improved orbits on the time interval have been shown. In Fig. 6 7 for the same satellite have been made the analogical comparison of TLE with evolutions on the time interval

7 300 ω Fig. 5 GS 85102A: the comparison of arguments of perigee (ω) of OLI+TLE with their evolution, calculated with improved orbits forward and back e Fig. 6 GS 85102A: the comparison of eccentricities (e) of TLE with their evolution, calculated with improved orbits forward and back. In Fig. 8 and 9 for GS 72041A have been shown the comparison of TLE with evolution of e and ω over the time interval of 2700 days. 390 ω Fig. 7 GS 85102A: the comparison of arguments of perigee (ω) of TLE with their evolution, calculated with improved orbits forward and back e Fig. 8 GS 72041A: the comparison of eccentricities (e) of TLE with their evolution, calculated with improved orbits forward and back. The discrepancy between the evolutions forward and back arises in elements e and ω of orbits with very small eccentricities and unsatisfactory precision. As a rule, orbits with e > give the full coincidence of the forward and back evolutions.

8 550 ω Fig. 9 GS 72041A: the comparison of arguments of perigee (ω) of TLE with their evolution, calculated with improved orbits forward and back e Fig. 10 GS 87091D: the comparison of eccentricities (e) of TLE with their evolution, calculated with improved orbits forward and back. In Fig for GS 87091D the comparison of TLE with evolution of e and ω over the time interval of 2400 days have been shown. 310 ω Fig. 11 GS 87091D: the comparison of arguments of perigee (ω) of TLE with their evolution, calculated with improved orbits forward and back. 6.9 i Ω Fig. 12 GS 87091D: the comparison of elements i and Ω TLE with their evolution, calculated with improved orbits forward and back.

9 8.2 i Ω Fig. 13 GS 72041A: the comparison of elements i and Ω TLE with their evolution, calculated with improved orbits forward and back (O-C) λ Fig. 14 GS 84016A: (О-С) λ, the improvement over the span equal to 3555 days (evolution forward and back). The satisfactory coincidence of the evolution with orbital elements show i and Ω. In Fig. 12 and 13 the evolutions of GS 87091D and 72041A in the axes i, Ω have been represented. In Fig. 13 the discrepancies between TLE (i and Ω) and their evolutions have been shown. It seems that the old orbital data of 1992 ( МJD), which correspond to values Ω near 100, are less precise. V. The long-term evolution of the longitude The values of the initial longitude and the rate of drift can be also improved on the first or last date of used osculating orbits. Usually the longitude evolutions forward and back coincide excluding the cases with noticeable collisions or unsatisfactory orbital data. For GS 84016A in Fig. 14 (O C) λ, obtained after the improvement on the epoch of the first and last dates of used osculating orbits during 3555 days have been represented. In these cases the mean square residuals are equal to and , because of errors of the observation treatment or the observations themselves. In Fig. 15 (O C) λ forward and back of GS 72041A, moving in the circular regime, during 2731 days with the mean square error σ = are given. In the behavior of these errors, the correlation with errors of other elements (Fig. 8, 9, 13) is seen. Hence, it leads to the conclusion about the connection of these systematical errors with the ill-conditioned normal equations. In this case, only TLE data were used.

10 0.10 (O-C) λ Fig. 15 GS 72041A: (О-С) λ, the improvement over the span equal to 2731 days (evolution forward and back). In Fig. 16 an other example on the systematical deviations in the (O C) λ behavior of GS 87091D during 2500 days is given. This satellite moving around two points of libration (Fig. 17) shows the ordinary values of σ for the evolutions forward and back, equal to σ f = and σ b = respectively, but in Fig. 16 the systematical discrepancies are seen (O-C) λ 300 λ Fig. 16 GS 87091D: (О-С) λ, the improvement over the span equal to 2500 days (evolution forward and back) Fig. 17 GS 87091D: the change of the longitude with time Comparing Fig. 16 and 17 with the time one can see that the systematical errors correlate with the moments of passages of unstable points with longitudes 348 and 161, in the

11 neighborhood of which the satellite moves very slowly. Hence, one can assume that these systematic residuals are connected with errors in the geopotential parameter and with the illconditioned equations. At last, the most frequent cases of systematic residuals are connected with the sudden stepped changes in the velocities of satellites. These changes are explained by the collisions of the space debris with satellites or their micro-explosions (Ref. 1). In such cases the form of curves (O C) λ depends on the type of GS motion and even on their place in the longitude at the moment of a phenomenon. The simplest pictures of such phenomenon we have for the circulating GS with the large rate of drift. In this case, the curve (O C) λ breaks at the moment of the event. In Fig. 18, the ordinary example of (O C) λ behavior is shown. There are annual oscillations due to the systematic errors of TLE (O-C) λ (O-C) λ Fig. 18 GS 86082A: (О-С) λ, caused by the collision of satellite moving in circulating regime Fig. 19 GS 81061F: (О-С) λ, caused by the collision of satellite moving in circulating regime. Another example of the sudden change of the velocity of moving in circular regime GS 81061F is shown in Fig. 19, with (O C) λ for whole time interval and for the span before the moment of collision. The annual waves of (O C) λ, hardly distinguished on the all curve, became clearly seen after division of the time interval in two parts. In Fig. 20, the behavior of curve (O C) λ for librating GS 83089B is shown, when the collision happens in the neighborhood of the stable point. The curve (O C) λ (bold line) during takes an unusual position above the abscissa. However, if data, contained between the turn points (49525, ), are treated then in the behavior of their (O C) λ the break with a

12 pit at the moment appears. After dividing of whole interval in two parts relative to and the separate treatment of data, the precision of the calculated (O C) λ is found to be in the usual limits of precision (thin line). It means that this satellite has been collided only once during the whole time interval. 0.7 (O-C) λ Fig. 20 GS 83089B: (О-С) λ, caused by the collision of librating satellite in neighborhood of the libration point. 0.2 (O-C) λ Fig. 21 GS 86090D: (О-С) λ, caused by the multiple collision of librating satellite.

13 0.06 (O-C) λ Fig. 22 (О-С) λ for a fictions GS with the change of the rate of drift on /day at the point of turn In the Fig. 21, we see the similar situation for GS 86090D with several collisions only. In Fig. 22 an example of (O C) λ for fictive librating GS is given, when the change of rate of drift, equal to /day, takes place at the point of turn. We have the analogical situation with GS 85102A (the libration is in the limits ), which twice has been collided near the point of turn 94 during the tested time interval (the corresponding (O C) λ are represented in Fig. 23 and 24) (O-C) λ 0.10 (O-C) λ Fig. 23 GS 85102A: (О-С) λ of the librating satellite with one collision and its modeling Fig. 24 GS 85102A: (О-С) λ of the librating satellite with two collisions and its modeling.

14 In Table 2 the results of analysis of GS 85102A collisions according to data OLI, distributed along the time interval of 2116 days, are given. Because of the insufficient number of data, the OLI series is supplemented by several TLE. Table 2 GS 85102A: the analysis of collisions with use of osculating orbits (OLI + TLE) T 0 () Time Interval () λ dλ/dt ( /day) σ Δ(dλ/dt) ( /day) date of event The mean error of the (O C) λ for the whole time interval is a rather large and reaches (line 1). Therefore, the calculations have been fulfilled with shorter time intervals, namely, before and after (the moment of the motion reverse). The results of these calculations are given in lines 2 3. The discrepancy of the rates of drift at the moment (at the point with longitude 94. 1) is equal to /day. Taking into account this variation of the velocity, the model of orbital evolution is constructed. (O C) λ, received with this model, are shown in Fig. 23 (bold curve). These models (O C) λ coincide with the behavior of (O C) λ of Fig. 22 qualitative, but differs from the real (O C) λ for osculating orbits (thin curve). The detail analysis of (O C) λ behavior for span has shown the existence of once more collision on the moment Therefore the additional calculations have been fulfilled (lines 4 6), which for the date have permitted to define the change of the rate of drift Δ(dλ/dt) = /day and for Δ(dλ/dt) = /day. With these Δ(dλ/dt) a new model has been constructed, (O C) λ of which (bold line in Fig. 24) agrees with the real (O C) λ (thin line). This agreement proves that Δ(dλ/dt) at is real. This change has been confirmed by the observations also. In Table 3 (O C) for the right ascension (α) and declination (δ), obtained from the improvement of OLI with intervals and are given. It can be seen here, that observations for date are compatible only for the span and show noticeable residuals for the span

15 Table 3 GS 85102A: Representation of photographic observations from the improvement of osculating orbits (OLI) in the right ascension (О С) α and declination (О С) δ DATA (O C) α (O C) δ DATA (O C) α (O C) δ σ = σ = On the base of the demonstrated examples we have concluded that the sudden change in GS motion not less than /day may be found over the span days from the improvement of orbits with the precise observations. For searching of a real cause of this phenomenon, it is necessary to have the statistics, based on the treatment of the all-available orbital data. For the automatization of the search of such events with the help of available osculating orbits the following method is proposed. If the mean error in the longitude representation of librating GS exceeds 0.05 /day, we divide the whole time interval into half-periods of libration (between the minimum and maximum longitudes). The sense of such operation is that during the half-period the behavior of (O C) λ, in the case of collision, is like the behavior of circular GS on the whole time interval. If the circular GS is moving with the rate of drift less than 2.5 /day, it must be better if the intervals are changed between the moments of passing of unstable points (161 and 348 ). After this, for each moment of passing by GS the boundary longitude the orbits are improved by means of data from the previous and following periods. If the values of rate of drift are different by more than /day, such change may be considered as real one.

16 In Table 4 the results of using of such method to the librating GS 86090D, the (O C) λ for which are shown in Fig. 21, are written. In the columns of Table 4 there are: the epoch of improving of orbit (T 0 ), the end of time interval (T e ), mean square error of representation of used osculating orbits during time interval (σ), longitude of GS (λ), the rate of drift (dλ/dt) and the change of rate of drift (Δ(dλ/dt)) at the epoch of T 0. Table 4 The analysis of changes in rate of drift for GS 86090D during the halfperiods of libration T 0 () T e () σ λ dλ/dt ( /day) Δ(dλ/dt) ( /day) TURN * TURN * TURN TURN (5) * * (8) * TURN In the first two lines the results of improving over whole time interval on the first and last data are given. The large value of error σ = shows the existence of collision during the treated 2323-days interval. Then, the analysis is fulfilled for half-periods of libration, between its boundaries ( ). In lines 3 10 there are given the results of calculations upon these intervals, which show two values of rate of drift for each boundary. We adopt their differences for the changes. Nevertheless, the analysis of (O C) λ shows that in lines 6 and 7 σ is still large and the collision was possibly happened in time interval , near the data (see Fig. 21 with the residuals).

17 The secondary calculations of rate of drift with the new dividing of intervals are given in lines 12 15, which show that on the data it happened the change of the motion equal to /day (lines 13 14). Then, at the moment dλ/dt = /day, instead of /day and accordingly the change of motion at this moment became /day (line 12). At the moment the change of the motion became /day, instead of /day (lines 8, 15). The residuals (O C) λ over the new spans have been shown in Fig. 21. Thus, the recalculation of the rates of drift with new spans gives the redistribution of their values, but the fact of the velocity changes in points of turn is indubitable. It is obvious that the accumulation of the observational data has to help us to find the true cause of this phenomenon. The most complicate cases of collision take place with the satellites, which change their regime of motion or are close to these conditions. The satellite 79035E is typical representative of such kind, because it moves in the circular regime with a very slow rate of drift. The changes of velocity on the dates 48368, and are given in Table 5. In Fig. 25 (O C) λ of this satellite during the all time interval and spans between collisions are shown. Table 5 GS 79035E: analysis of collisions according to TLE T 0 Time interval λ dλ/dt ( /day) σ Δ(dλ/dt) ( /day) In Fig. 26, the model and real (O C) λ have been represented for two time intervals (bold line is used for both models). The first model (M1) was calculated with one collision at the moment Two collisions at the moments and were took into account in the second model (M2). The (O C) λ, calculated for corresponding time intervals with real orbits, are shown in Fig. 26 by thin lines.

18 In Fig. 26 are clearly seen oscillations in the satellite longitude with the period about 460 days, correlating with the half period due to resonant perturbations. During this time GS passes the distance between two stable points. 0.9 (O-C) λ Fig. 25 GS 79035E: (О-С) λ of the satellite with the slow motion in the circular regime with thrice-repeated collisions. 0.6 (O-C) λ Fig. 26 GS 79035E: the model of (О-С) λ over different spans, for satellite moving in the circular regime with one and two collisions. These oscillations in (O C) λ are similar to the behavior of a fictional librating object, showed in Fig. 22. Their amplitudes are larger than amplitudes of annual oscillations typical for TLE data. The annual oscillations of (O C) λ display only after excluding the influence of collisions. Studying the changes of motion of GS 79035E and 85102A, we can conclude that the variations of velocity about /day or even less can be found from orbital data of objects, moving in circular regime for the long time intervals. VI. Conclusions The proposed the motion theory of the geostationary satellites, based on the intermediate orbit, allows to represent the available osculating orbits of any geostationary object during the time intervals equal to 10 years and more. The mean square error of representation of orbits TLE does not exceed If mean error is more than then, as a rule, the sudden change of the satellite velocity takes place. From the treatment of the available osculating orbits the statistics of the rate of drift changes and their conditions can be obtained which will be useful for the study of their origin

19 and the explanation of this phenomenon. The treatment all these data is quite real with the available software and can be fulfilled during a short time. The precision of the Two-Line-Elements for GS can be improve if the span, over which observations used in treatment, will be increased. The accumulated osculating orbits of the resonant geostationary satellites can be used for the estimate of geopotential parameter precision. For that purpose the satellites changed their motion regimes under the influence of the luni-solar perturbation are especially useful. The list of such objects has been proposed in the program Resonance (Ref. 6). References 1 Kiladze R.I., Sochilina A.S., Grigoriev K.V., Vershkov A.N., On investigation of longterm orbital evolution of geostationary satellite, Proceedings of 12th Symposium on Space Flight Dynamics, ESOC, Darmstadt, Germany, 2 6 June pp Kiladze R.I., Sochilina A.S., Grigoriev K.V., Vershkov A.N., On new investigations of geostationary satellite motion, Rev. Brasil. de Ciencias Mecanicas, v.21, , Grigoriev K.V., Sochilina A.S., Vershkov A.N., On catalogue of geostationary satellites, Proc. of First European Conference on Space Debris, Darmstadt, Germany, 5 7 April 1993, , Sochilina A.S., Grigoriev K.V., Kiladze R.I., Vershkov A.N., Malyshev V., Оn broadening of GEO catalogue contents, Space Forum, Vol.1, No 1 4, 51 56, Kiladze R.I., Sochilina A.S., On orbital evolution of geostationary satellites, U.S.- Russian Second Space Surveillance Workshop 4 6 July 1996, Poznan, Poland, , Kiladze R.I., Sochilina A.S., On the program «RESONANCE», Space Forum, Vol.1, No 1 4, 15 22, Sochilina A.S., Kiladze R.I., Grigoriev K.V., Vershkov A.N., On an improved geostationary catalog, Advances in Space Research, Vol.19, No 2, , Kiladze R.I., Sochilina A.S., On evolution of geostationary satellite orbits, Advances in Space Research, Vol.19, No 11, , Sochilina A.S., Kiladze R.I., Grigoriev K.V., Vershkov A.N., On occasional changes of velocities of geostationary satellites, Nav. Obs., Washington, Oct., Simon J.L., Bretagnon P., Chapront J., Chapron-Touze M., Francou G. and Laskar J., Numerical Expression for Precession and Mean Elements for the Moon and the Planets, Preprint no 9302, Bureau des Longetudes, July 1993.

20 11 Schumacher P.W. and Glover R.A., Analytical orbit model for U.S. naval space surveillance: an overview, U.S.-Russian Second Space Surveillance Workshop 4 6 July 1996, Poznan, Poland, , Grigoriev K.V., Hoots F.R., On Improvement of Orbits of Geostationary Satellite from Observations over the interval of time days, (in press).

21 Biographies Rolan I. Kiladze, Professor of Tbilisi University, Chief of department of Abastumani Observatory, Member of Academy of Science, Georgia. He is the specialist in dynamics of the Solar system bodies and since 1993 he is working with the geostationary objects dynamics. Address: app.52, korp.17, quart.6, Vaja Pshavela Av., , Tbilisi GEORGIA tel. (99532) home, (99532) off., Alla S. Sochilina is the leader scientific worker of the Pulkovo Observatory. She graduated from the Leningrad State University and worked at the Institute of Theoretical Astronomy up to Since 1957 she was engaged with the artificial satellite dynamics, but since 1984 is working with geostationary satellites. Address: fax(812) , app.102 h.135 prospect Engels, Saint-Petersburg Konstantin V. Grigoriev is the senior scientific worker of the Pulkovo Observatory. He graduated from the Leningrad State University and since 1989 during 10 years was working at the Institute of Theoretical Astronomy. His specialties are the geostationary object dynamics and software providing. Address: app.101, 47/2 Novocherkasskiy prospect, Saint-Petersburg, RUSSIA tel. (812) home, off., fax (812) Andrei N. Vershkov is the senior scientific worker of the Pulkovo Observatory. He graduated from the Leningrad State University and since 1989 during 10 years was working at the Institute of Theoretical Astronomy. His researches are concerning with the development geopotential models, satellite dynamics and software providing. Address: app. 1, 1 Krasnodonskaya st., , Saint-Petersburg, RUSSIA tel. (812) home, off., fax (812) , avershkov@mail.ru