Improvement of Orbits of Geostationary Satellites from Observations Over a Time Interval of Days

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1 Improvement of Orbits of Geostationary Satellites from Observations Over a Time Interval of 15-2 Days * Konstantin V.Grigoriev, ** Felix R. Hoots * Pulkovo Observatory, St-Petersburg, Russia ** GRC International, Inc. Abstract. Improved modeling of the orbits of non-controlled resonant geostationary satellites is obtained by representation of the longitude by a power series. The coefficients of the series are calculated according to analytical formulas. These coefficients are obtained by use of the resonant longitude equation. The accuracy of the longitude representation has been investigated over different time intervals with different orders of the series. Examples of calculation of orbits for several resonant satellites and comparison with NASA Two-Line Elements are given. I. Introduction A commonly used method for modeling of the orbital motion of non-controlled geostationary satellites is to approximate the satellite longitude by a time polynomial and to determine the polynomial coefficients using observations as well as orbital elements. For the approximation of the longitude of geostationary satellites, the polynomial order is selected based on the length of the desired data span. A similar technique is commonly used to model the effects of atmospheric drag on satellite longitude (Ref. 1,2). In both cases, as the order of the polynomial increases, the normality condition becomes worse. To avoid this problem, we recommend deriving analytical formulas for the coefficients of the polynomial expressions. For testing the precision of the improved motion theory of geostationary satellites (Ref. 3) two kinds of osculating orbits, improved by optical observations, are used: Orbits determined over long time intervals (OLI) Two-Line-Elements of US Space Command (TLE) The OLI have shown better agreement with the actual orbital evolution than the TLE. The difference between these two types of data depends not only on the size of the time interval used, but also on the number of solved-for parameters.

2 The orbits of geostationary satellites are very nearly circular. Therefore, their eccentricities e and arguments of perigee ω are less well determined for shorter time spans. However, even for spans equal to a month the necessity arises to represent the longitude of the satellite by a time polynomial. If the coefficients of this polynomial are determined from observations empirically, naturally, then the condition of the normal equation becomes worse. Presently there are a large number of different motion theories that can be used for deriving analytical expressions for the coefficients of the secular terms. In addition, the various geopotential constants are known to a quite satisfactory precision. Therefore, they can be treated as constants and be excluding from the empirical solution process. This allows development of a precise and compact algorithm. II. Approximation of geostationary satellite longitudes. Usually OLI are improved using a time polynomial of ninth order. This allows representation of longitudes over time spans of 75-1 days. Such a long data span produces good precision for the orbit even when there are only a few observations. The longitude is represented as Taylor's series in the following way: = + 1 τ τ ( n ) n τ τ 2! 3! ( n 1)! where τ = t t t = Epoch for calculation t = Epoch of improved orbit R n (1) is determined from observations and is adopted as a constant for t. The derivatives of higher order are expressed with the help of the differential equation of the longitude (Ref. 4,5): = A sin( m m + ( k m) ) + LSP lmkpq lmkpq Ω Λ (2)

3 where A = 3m( a / a) J Q, lmkpq Q = D ( i ) E ( Λ) X ( e), C = J cos m, S = J sin m, e l lm lmkpq l 1, l 2 p lmkpq lpq Λ lmk l 2 p+ q lm lm lm lm lm lm ( 1+ ξ) π / 2 ( l m) even mlmkpq = mlm + pωλ if, ξπ / 2 ( l m) odd E l m = + 1 ( + + 1)/ 2 < ξ if {( 1) Qlmkpq} 1 > C lm, S parameters of the geopotential lm D ( i ), E ( Λ) inclination functions X lkp Λ l 1, l 2 p l 2 p+ q lmk ( e) Hansen coefficients (Ref. 6) The derivatives of higher order have been obtained by means of a package of analytical transformations called REDUCE. The formulas for derivatives up to fifth order are: = ma cos( m m ) lmkpq lmkpq, IV 2 2 = m A sin( m m ) ma cos( m m ), lmkpq lmkpq lmkpq lmkpq V 3 3 = m A cos( m m ) m A sin( m m ) ma cos( m m ). lmkpq lmkpq lmkpq lmkpq lmkpq lmkpq In these formulas, the orbital elements are referred to the Laplace plane. If the orbit is referred to the equator, then the angle Λ in equation (2) should be set equal to zero. This theory has been implemented using several procedures. The main procedure is called PROLONG, which calculates the longitude and rate of drift for each time of observation taking into account all the resonant harmonics up to 6 th order of the geopotential. The orbital model also includes the gravitational effects of the Moon and Sun as well as solar radiation pressure. (Ref. 7)

4 III. Results of calculations. The procedure PROLONG was used to determine the orbital element set (OLI) which best fit the observational data over a time interval of 15-2 days, where the epoch is in the middle of the span. This procedure is repeated for the next 15-2 day span of data to produce another OLI element set. Continuing in this manner, the resulting series of OLI data have been used for the testing of the improved motion theory of geostationary satellites. The main advantage of OLI data is its rather precise rate of drift, which permits calculating the ephemerides for some years with a mean error equal to.1 deg. OLI data are always calculated with a polynomial of ninth order. Nevertheless, for a shorter span the order of polynomial could be less. The order of the polynomial required is also dependent on the location of the satellite relative to the center of libration (where the value of rate of drift is maximum) and the turn points (where the rate of drift is minimum). In Table 1, the values of the derivatives and their contributions to the longitude for two different time points are given for satellite 8416A.

5 Table 1: Values of terms of the series for satellite 8416A, Raduga 14, for longitudes near the turn point and the center of libration. τ=t-t 25 days 1 days τ τ 2 2! τ 3 3! IV IV τ 4 4! V VI V τ VI.796 τ 5 5! ! VII VII.323 τ 7 7! VIII IX VIII.12 τ IX. τ 8 8! ! τ τ 2 2! τ 3 3! IV IV τ 4 4! V VI V τ VI.988 τ 5 5! ! VII VII τ 7 7! VIII IX VIII -.4 τ IX.1 τ 8 8! !

6 Figure 1: Derivatives of longitude for satellite 8416A, Raduga 14.1 I.12 II III 8E-8 IV E-6 4E-8-4E-6-6E-6-4E-8-8E-6-8E E-1 V 8E-12 VI 4E-1 4E-12 2E-1-4E-12-8E On April 26, 1994 (49468MJD) the satellite 814A, Ekran 6, changed its velocity markedly. Its rate of drift before this event was determined by means of osculating orbits, distributed over 392 days, to be deg/day. After the 'collision', the rate of drift became deg/day. This value was obtained with 155 osculating orbits, distributed over 1536 days. In Figure 2, we plot the difference between the

7 observed and computed longitudes, (O-C), during these two spans and for the whole time interval MJD containing this event. In Figure 3a, there are two curves. The first of them plots the (O-C) residuals of the longitudes based on prediction of the first osculating orbit TLE over the whole time interval. The second curve (O-C) has been obtained from the evolution orbit, improved by use of PROLONG over a short time interval equal to 8 days. The error of its rate of drift is less than.1 deg/day and this curve very sharply changes its direction at the moment of 'collision'. In Figure 3b, the same curves are given over the short span before the 'collision'. 4 (O-C) 2 IMPR 2 IMPR 1-2 TLE -4 MJD Figure 2: Representation of the longitude of satellite 814A, Ekran 6, by the improved orbits (IMPR 1 - before and IMPR 2 - after the 'collision') and by the evolution of the first osculating orbit TLE.

8 4 (O-C) 1 (O-C) 2 TLE.8.6 TLE -2-4 OLI MJD OLI MJD Fig.3a,b. The (O-C) representation of the longitudes by the evolution of the first osculating orbit TLE and the evolution orbit, improved by use of PROLONG over only 8 days of data for satellite 814A, Ekran 6. In Table 2 the representation of observations by the improved orbit is given. From this improvement of the orbit using only 13 days of data, the rate of drift at the epoch of is determined to be deg/day and barely differs from the value deg/day obtained with the use of osculating orbits during the span MJD. Table 2. The representation of optical observations by improved orbits of satellite 814A MJD UTC α δ (O-C) α (O-C) δ h 12 m 19 s.52 5 h 58 m 52 s.23 2 o h 22 m 32 s.52 7 h 9 m 42 s.23 4 o h 21 m 58 s.52 8 h 1 m 16 s.23 6 o h 1 m 18 s.52 8 h 59 m 45 s.79 6 o Collision h 3 m 31 s h 35 m 9 s.66-5 o h 32 m 14 s h 36 m 5 s.19-5 o h 59 m 31 s h 2 m 28 s.27-1 o h 1 m 6 s.4 16 h 4 m 1 s.69-1 o h 3 m 1 s h 31 m 58 s o h 3 m 35 s h 32 m 32 s o h 3 m 3 s h 11 m 41 s.77-7 o h 5 m 5 s h 13 m 41 s.77-7 o h 26 m 1 s h 33 m 14 s.2-11 o h 29 m 27 s h 36 m 36 s o

9 15 2 h 34 m 29 s h 41 m 8 s o h 36 m 37 s h 43 m 16 s o h 37 m 44 s h 47 m 56 s.33-5 o h 38 m 52 s h 47 m 52 s.3-8 o h 45 m 52 s h 53 m 5 s o In Table 3 the values of longitudes, calculated from observations during one tracking period are given. The results using three methods for approximating the orbit (circular, elliptical and mean) are shown. Additionally, the longitudes for the same dates, calculated with PROLONG (prolong) are given. Table 3. The comparison of calculated longitudes. DATA Longitude, (deg) (O-C) (MJD) circular Elliptical mean prolong (deg) t-t (days) t = As soon as the improved rate of drift is determined after the 'collision', the calculated value of longitude with PROLONG on can be used along with the difference of times to determine the change of rate of drift. From the comparison of these longitudes with the ones from primary orbits, we have that the change of velocity is equal -.16 deg/day, e.g. nearly the same value obtained from the treatment of osculating orbits ((O-C) equal deg for 1.3 days). In conclusion we note that the procedure PROLONG allows increasing the precision of orbits of geostationary objects even in the cases when the observations are distributed over short spans. The same principle can be used in the approximation of other perturbations and for other kinds of satellites such as the effect of atmospheric drag on the mean anomaly. In particular, Hoots (Ref. 8) has used analytical formulas to compute the higher order derivatives of mean motion and eccentricity to develop improved along track drag modeling in the SGP8 model.

10 IV. Acknowledgments The authors are grateful to Dr. A.S.Sochilina (Pulkovo Observatory, St.-Petersburg, Russia) for participation in discussion of the results, and to A.V.Didenko (AFI, Alma- Ata, Kazakhstan) and V.G.Vigon (NPO Kosmoten, Moscow, Russia) for providing the results of the observations. References 1 Hoots, F.R., A History of Analytical Orbit Modeling in the United States Space Surveillance System, Third U.S.-Russian Space Surveillance Workshop 2 23 October 1998, US Naval Observatory, Washington, 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, 66 15, Kiladze, R.I., Sochilina, A.S., Grigoriev, K.V., Vershkov, A.N., On Ways of Modernization of Geostationary Ring Surveillance, Fourth U.S.-Russian Space Surveillance Workshop October 2, US Naval Observatory., Washington, 2. 4 Gedeon, G.S., Terrestrial Resonance Effects on Satellite Orbits, Celest. Mech., Vol. 1, 2, , 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, , Gaposchkin, E.M., Smithsonian Standard Earth (III), SAO Spec.Rep. 353, Sochilina, A.S., Kiladze, R.I., Grigoriev, K.V., Vershkov, A.N., "On Occasional Changes of Velocity of Geostationary Uncontrolled Objects", Third U.S.-Russian Space Surveillance Workshop 2 23 October 1998, US Naval Observatory, Washington, Hoots, F.R. and Roehrich, R.L., "Project SPACE TRACK Report #3, Aerospace Defense Command, December 198.