- Email: [email protected]
GALILEO
Advances in Space Research 34 (2004) 1221–1226 www.elsevier.com/locate/asr
Long-term evolution of navigation satellite orbits: GPS/GLONASS/GALILEO C.C. Chao *, R.A. Gick The Aerospace Corporation, P.O. Box 92957, M4-948 Los Angeles, CA 90009-2957, USA Received 19 October 2002; received in revised form 20 January 2003; accepted 25 January 2003
Abstract A recent study was performed to examine whether long-term growth in the eccentricity evolution exists for the disposal orbits of navigation satellite systems such as GPS, GLONASS, and GALILEO. Previous studies examined the orbit stability for GPS Block II satellites. The orbits of the non-operational GPS Block I satellites are included in this study because they are at 63.4° inclination, which is different from GPS Block II. Similar to earlier studies, long-term perturbations and stability of these orbits are understood through analytical and numerical investigations. Initially near circular, these types of orbits may evolve into orbits with large eccentricity (as much as 0.7 over 150 years). Analytical approximations through doubly-averaged equations reveal that the cause is due to the resonance induced by Sun/Moon and J2 secular perturbations. A total of 105 non-operational GLONASS satellites and upper stages and 10 GPS/Block I satellites were propagated for 200 years. Results show that the GLONASS satellites will start to enter the operating GPS constellation after 40 years. The uncovered resonance effect is strongly dependent on orbit inclination and altitude. The effect becomes more pronounced for GALILEO orbits due to their higher altitude, 3000 km above GPS. Strategies to minimize the significant eccentricity growth are identified in this study. These results directly impact the safety of future navigation satellites in the altitude region from 19,000 to 24,000 km. The dependence on initial inclination may help the designers of GPS III and GALILEO systems select the proper inclination for minimizing the large eccentricity growth. The maximum eccentricity growth for GPS and GALILEO can be significantly reduced by selecting operational inclinations a few degrees from the current nominal values for both programs. Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Space debris; Navigation satellite systems; GPS/GLONASS/GALILEO; Orbit eccentricity
1. Introduction Previous studies at The Aerospace Corporation (Chao, 1998, 2000; Gick and Chao, 2001) have considered the stability of GEO and MEO orbits. Results of the MEO disposal orbit stability studies (Chao, 2000; Gick and Chao, 2001) revealed the interesting fact of large eccentricity growth of the GPS disposal orbits. Analytical approximations through doubly-averaged equations showed that the cause is due to the resonance induced by Sun/Moon and J2 secular perturbations. This discovery has drawn considerable attention in the * Corresponding author. Tel.: +1-310-336-4295; fax: +1-310-3362831. E-mail addresses: [email protected] (C.C. Chao), [email protected] (R.A. Gick).
space community and may alter the disposal strategies of future GPS satellites and upper stages. Of interest in this paper is whether the long-term eccentricity growth exists for the disposal orbits of other navigation satellite systems such as GLONASS and GALILEO. The 10 GPS Block I satellites, which are out of service, are also included in this study. The GPS Block I satellites are in orbits with critical inclination (63.4°), the same as that of the GLONASS satellites. There are about 105 inoperable GLONASS satellites and upper stages drifting in their original orbits. As a result of eccentricity growth, these orbits, including the future disposal orbits of GALILEO, may intersect the orbits of the operational navigation constellations (GPS, GLONASS, and GALILEO) increasing the potential for collision (Jenkin and Gick, 2001) and may come close to both LEO and GEO operational altitudes. In
0273-1177/$30 Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2003.01.021
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this paper, we focus on the analysis of the GPS Block I, GLONASS, and GALILEO orbit stability and its impacts on future disposal orbit policies. This analysis is expected to provide considerable insight into the longterm behavior of eccentricity growth as well as suggest strategies for disposal orbit insertion. The objectives of this study are: (1) to understand the long-term orbit perturbations and stability of those outof-service GPS Block I and GLONASS satellites and future disposed satellites of GALILEO, considering all major perturbing forces; (2) to examine the impacts on future navigation systems in that orbit altitude region; (3) to recommend strategies and requirements for endof-life satellite/upper stage disposal maneuvers. This work is sponsored by the GPS Program Office and is a continuation of The Aerospace Corporation GPS Block II studies initiated in response to end-of-life disposal guidelines published by NASA, 1995.
2. Analytical investigation The long-term and secular variations of the orbital parameters can usually be related to the perturbing forces through the averaged equations of variation. The averaged equations may be categorized as singly averaged or doubly averaged. The singly-averaged equations are derived by averaging the equations of variation over the orbit period thus eliminating all short or orbit period terms. The resulting variational equations are significantly shorter and the dominant long-term and secular variations due to a particular force may be identified. The singly-averaged equations are again averaged over the orbit period of the third body, further shortening the equations of variation. The resulting equations are called the doubly-averaged equations. The doubly-averaged equation in eccentricity is derived for the above disposal orbits by removing the 6-month and 14-day terms from the singly-averaged equations of the third body (Chao, 1979). The nominal operational GPS Block I orbit was circular with a 20181.6 km altitude and a 63.4° (critical) inclination. A GLONASS orbit of 1000 km below the nominal GPS orbit with the same inclination of 63.4° is assumed in this paper. The future GALILEO system will have a circular orbit 3000 km above the GPS altitude with 56° inclination. The disposal orbits of GALILEO are assumed at 500 km above the operational altitude. The eccentricity of the GPS Block I orbit with 12 h repeating ground tracks is not affected, in the very long-term sense, by the resonant tesseral harmonics due to the fact that there are no stationkeeping maneuvers. A closed form doubly-averaged equation in eccentricity due to third body perturbations is derived via computerized series expansion in Chao (1979) and is given as:
de=dt ¼ ð15=8Þecs½C1 sin 2ðx DXÞ þ C2 sinð2x DXÞ þ C3 sin 2x þ C4 sinð2x þ DXÞ þ C5 sin 2ðx þ DXÞ;
ð1Þ
where e is the eccentricity of the orbit, c ¼ n23 Rm =n; s ¼ ð1 e2 Þ1=2 , n is the mean motion of the GPS orbit, n3 is the mean motion of the third body, Rm is the mass ratio [ ¼ 1 for solar perturbation, ¼ 182.3 for lunar perturbation], x is the argument of perigee, DX ¼ X X3 , and X is the right ascension of the ascending node (RAAN) of the satellite, X3 is the right ascension of the ascending node of the Sun or Moon’s orbit with respect to the Earth equator. The five coefficients in Eq. (1) are functions of the inclination with the following form: C1 ¼ 12 sin2 i3 ðcos i þ 1=2 sin2 i 1Þ; C2 ¼ 12 sin i sin 2i3 ðcos i 1Þ; C3 ¼ sin2 ið3=2 sin2 i3 1Þ; C4 ¼ 12 sin i sin 2i3 ð1 þ cos iÞ;
ð2Þ
C5 ¼ 12 sin2 i3 ð1=2 sin2 i cos i 1Þ; where i is the inclination of the satellite orbit, and i3 is the inclination of the third body. After substituting the critical inclination and the inclination of the third body (23.45° for both the Sun and the Moon), the series expansion assumes the following simple form: de=dt ¼ ð15=8Þecs½0:01215 sin 2ðx XÞ 0:1808 sinð2x XÞ 0:6092 sin 2x þ 0:4733 sinð2x þ XÞ 0:0833 sin 2ðx þ XÞ: ð3Þ The above equation has slightly different coefficients inside the bracket than that of the similar equation for inclination at 55° as shown below (from Chao, 2000; Gick and Chao, 2001). de=dt ¼ ð15=8Þecs½0:0072 sin 2ðx XÞ 0:1277 sinð2x XÞ 0:5110 sin 2x þ 0:4714 sinð2x þ XÞ 0:0984 sin 2ðx þ XÞ: ð4Þ Note that the orbit of the third body is assumed to be circular and, hence, the eccentricity and argument of perigee of the Sun and Moon’s orbits do not appear in the equation. This approximation significantly shortens the series equations of variation. Previous numerical integration results (Chao, 1979, 1998, 2000; Gick and Chao, 2001) show that this circular orbit approximation does not cause any noticeable degradation in the longterm propagation. Eq. (4) also applies to GALILEO orbits. As indicated in previous studies (Chao, 2000; Gick and Chao, 2001), the cause of the large eccentricity growth is due to the resonance in the angle (2x þ X) in
C.C. Chao, R.A. Gick / Advances in Space Research 34 (2004) 1221–1226
the above equation. At GPS altitude and 55° inclination, the rate of 2x nearly cancels the rate of X. For GLONASS and GPS Block I orbits, the term with sin 2x induces resonance at or near critical inclination where dx=dt vanishes. The secular rates of x and X, due to J2 and Sun–Moon effects, may be computed from the averaged equations (Chao, 1979). As indicated in previous studies (Chao, 2000; Gick and Chao, 2001), a GPS disposal orbit at 500 km above the nominal GPS operational altitude has a nodal regression rate of about )0.038°/day and an argument of perigee rate of about 0.021°/day. Of the five sinusoidal terms on the righthand side of Eq. (4), the fourth term has the largest period due to the resonance effect from the rate of the combined angle (2x X) which is very small, being about 0.0046°/day. For the orbits of GLONASS and GPS Block I, the third term in Eq. (3) induces resonance due to the critical inclination effect. Table 1 summarizes the angular rates of the dominant term in the equation for eccentricity rate. The resonance terms with the very small angular rates are identified in emphasized print. The term sinð2xÞ is responsible for the long-term eccentricity growth in GLONASS and GPS Block I orbits, while the term sinð2x þ XÞ is the cause of the similar growth in GPS Block II and GALILEO disposal orbits. The above analytical investigation clearly shows that the significant growth in the eccentricity evolution caused by resonance
1223
perturbations due to J2 and Sun/Moon exists in other navigation satellites in the MEO altitude region from 19,000 to 24,000 km.
3. Long-term numerical propagation studies This section describes the numerical analysis of the long-term eccentricity growth. The results were obtained using MEANPROP [a long-term orbit prediction software tool of The Aerospace Corporation (Peterson and Hart, 1998)] which employs a high-precision semianalytic orbit propagator (SAOP developed by Draper Laboratory) and assumes the following perturbing forces: 8 8 WGS 84 Earth gravity model, Sun/Moon gravitational attractions, solar radiation pressure and atmospheric drag. Perturbations due to the other planets and Earth tides are small and are therefore neglected in this analysis. The numerical precision of MEANPROP has been favorably compared [in 100-year propagations (Chao, 1998, 2000)] with TRACE (the most accurate and well-maintained orbit propagation/determination tool at The Aerospace Corporation) indicating the adequacy of using MEANPROP for 200-year integration. For GPS applications, the estimated uncertainty in predicting the long-term eccentricity is small, on the order of a few percent, when the eccentricity grows beyond 0.02. This verification was determined by
Table 1 Summary of angular rates (deg/day) due to J2 and Sun–Moon perturbations Navigation systems
x
X
2nd term 2x X
3rd term 2x
4th term 2x þ X
GLONASS GPS Block I GPS Block II GALILEO
0.0010a 0.0010a 0.0212 0.0132
)0.0368 )0.0314 )0.0378 )0.0262
0.0388 0.0334 0.0802 0.0526
0.002 0.002 0.0414 0.0264
)0.0348 )0.0294 0.0046 0.0002
a
Most orbits are near critical inclination (62°–65°) and 0.0010 is a representative value for this table.
Fig. 1. GPS Block I (non-operational) apogee altitude history.
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Fig. 2. GPS Block I (non-operational) perigee altitude history.
Fig. 3. GLONASS (non-operational) apogee altitude history.
comparing the results with and without the effects of solar radiation pressure which is believed to be the critical parameter contributing to the uncertainty over the 200-year integration period. Figs. 1 and 2 show the apogee and perigee histories for the 10 non-operational GPS Block I satellites. The long-term eccentricity growth of these orbits will cause apogee and perigee to approach the GEO and LEO operational altitudes, respectively. Fig. 3 shows the apogee histories for the 105 non-operational GLONASS satellites. It is clear from Fig. 3 that the GLONASS satellites will begin penetrating the GPS operational altitude within the next 40 years and will continue to intrude upon this space for the foreseeable future.
4. Dependence on inclination In order to develop strategies to alleviate long-term eccentricity growth and thereby improve the disposal orbit stability, we investigate the sensitivity of eccentricity variation to various orbital parameters. The
maximum (worst-case) eccentricity achieved over a (sufficiently) long period of time is used as a measure of orbital stability. In other words, we examine how the maximum eccentricity achieved over time depends on the orbital parameters in an effort to derive strategies to keep the maximum eccentricity small. As noted in our earlier work, a disposal orbit design strategy involves targeting the initial argument of perigee. This strategy is to determine, through analytical (Eqs. (3) and (4)) and numerical methods, the optimum window in initial argument of perigee for each orbit plane. By properly targeting the initial eccentricity and argument of perigee, the eccentricity of the disposed orbits will remain very small for the next 200 years, according to earlier studies (Chao, 2000; Gick and Chao, 2001). This strategy requires additional propellant and orbit adjust maneuvers which may not be desirable. A second recommended strategy to avoid potentially large eccentricity growth is to move the inclination of the operational orbit by a few degrees from the current nominal value, closer to 52° as can be seen in Figs. 4 and 5. A large number of 200-year numerical integrations
C.C. Chao, R.A. Gick / Advances in Space Research 34 (2004) 1221–1226
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Fig. 4. Maximum 200-year eccentricity growth of GPS disposal orbits versus inclination.
Fig. 5. Maximum 200-year eccentricity growth of GALILEO disposal orbits versus inclination.
were performed for inclinations from 50° to 70° for both the GPS and GALILEO constellation orbits. The results, shown in Figs. 4 and 5, reveal the strong dependence of maximum (worst-case) eccentricity growth on orbit inclination. By selecting the operational inclination a few degrees from the current nominal value for both programs, the maximum eccentricity growth for GPS and GALILEO can be significantly reduced. However, this small modification may have an impact on the achievable constellation coverage and hence the user accuracy and availability, which should be taken into consideration.
5. Conclusions The long-term growth in eccentricity of GLONASS, GALILEO and GPS Block I satellites was
studied both analytically and numerically. It was found that these types of orbits evolve into orbits with large eccentricity, as much as 0.7 over 150 years. Analytical approximations reveal that the eccentricity growth is due to Sun/Moon and J2 secular perturbations. The 200-year numerical orbit propagation studies indicate that the GLONASS satellites will begin to enter the operational GPS constellation within the next 40 years. Numerical studies show the strong dependence of eccentricity growth on inclination and altitude. The growth effects are more pronounced for GALILEO orbits due to their higher altitude. Strategies for minimizing the eccentricity growth are identified. Specifically, the maximum eccentricity and the long-term growth in the eccentricity evolution can be made small by changing the operational orbit inclination by a few degrees from its current nominal value.
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Acknowledgements This work reflects research conducted under US Air Force and Missile Center Contract F04701-93-C-0094 for the GPS Program Office. The authors thank H.J. Schraibman, A.B. Jenkin, and J.E. Gidney of The Aerospace Corporation, Lt.Col. M.N. Moya USAF/ CZS and the GPS Program Office for their review and comments on this paper. References Chao, C.C. An analytical integration of the averaged equations of variation due to Sun–Moon perturbations and its application. Technical Report, SD-TR-80-12, The Aerospace Corporation, El Segundo, CA, October, 1979.
Chao, C.C. Geosynchronous disposal orbit stability. AIAA Paper No. 98-4186, AIAA/AAS Astrodynamics Specialist Conference, Boston, MA, August 10–12, 1998. Chao, C.C. MEO disposal orbit stability and direct reentry strategy. AAS Paper No. 00–152, AAS/AIAA Space Flight Mechanics Meeting, Clearwater, FL, January 23–26, 2000. Gick, R.A., Chao, C.C. GPS disposal orbit stability and sensitivity study. AAS/AIAA 01-244, AAS/AIAA Flight Mechanics Meeting, Santa Barbara, CA, 11–14 February, 2001. Jenkin, A.B., Gick, R.A. Analysis of the collision risk associated with GPS disposal orbit instability. AAS/AIAA 01-115, AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara, CA, 11–14 February, 2001. NASA, guidelines and assessment procedures for limiting orbital debris. NASA Safety Standard 1740.14. Office of Safety and Mission Assurance, Washington, DC, August, 1995. Peterson, G.E., Hart, M.J. MEANPROP 1.1 Users Guide, Aerospace Technical Memorandum, ATM-98(3587-31)-1, El Segundo, CA, June 2, 1998.
Long-term evolution of navigation satellite orbits: GPS/GLONASS/GALILEO C.C. Chao *, R.A. Gick The Aerospace Corporation, P.O. Box 92957, M4-948 Los Angeles, CA 90009-2957, USA Received 19 October 2002; received in revised form 20 January 2003; accepted 25 January 2003
Abstract A recent study was performed to examine whether long-term growth in the eccentricity evolution exists for the disposal orbits of navigation satellite systems such as GPS, GLONASS, and GALILEO. Previous studies examined the orbit stability for GPS Block II satellites. The orbits of the non-operational GPS Block I satellites are included in this study because they are at 63.4° inclination, which is different from GPS Block II. Similar to earlier studies, long-term perturbations and stability of these orbits are understood through analytical and numerical investigations. Initially near circular, these types of orbits may evolve into orbits with large eccentricity (as much as 0.7 over 150 years). Analytical approximations through doubly-averaged equations reveal that the cause is due to the resonance induced by Sun/Moon and J2 secular perturbations. A total of 105 non-operational GLONASS satellites and upper stages and 10 GPS/Block I satellites were propagated for 200 years. Results show that the GLONASS satellites will start to enter the operating GPS constellation after 40 years. The uncovered resonance effect is strongly dependent on orbit inclination and altitude. The effect becomes more pronounced for GALILEO orbits due to their higher altitude, 3000 km above GPS. Strategies to minimize the significant eccentricity growth are identified in this study. These results directly impact the safety of future navigation satellites in the altitude region from 19,000 to 24,000 km. The dependence on initial inclination may help the designers of GPS III and GALILEO systems select the proper inclination for minimizing the large eccentricity growth. The maximum eccentricity growth for GPS and GALILEO can be significantly reduced by selecting operational inclinations a few degrees from the current nominal values for both programs. Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Space debris; Navigation satellite systems; GPS/GLONASS/GALILEO; Orbit eccentricity
1. Introduction Previous studies at The Aerospace Corporation (Chao, 1998, 2000; Gick and Chao, 2001) have considered the stability of GEO and MEO orbits. Results of the MEO disposal orbit stability studies (Chao, 2000; Gick and Chao, 2001) revealed the interesting fact of large eccentricity growth of the GPS disposal orbits. Analytical approximations through doubly-averaged equations showed that the cause is due to the resonance induced by Sun/Moon and J2 secular perturbations. This discovery has drawn considerable attention in the * Corresponding author. Tel.: +1-310-336-4295; fax: +1-310-3362831. E-mail addresses: [email protected] (C.C. Chao), [email protected] (R.A. Gick).
space community and may alter the disposal strategies of future GPS satellites and upper stages. Of interest in this paper is whether the long-term eccentricity growth exists for the disposal orbits of other navigation satellite systems such as GLONASS and GALILEO. The 10 GPS Block I satellites, which are out of service, are also included in this study. The GPS Block I satellites are in orbits with critical inclination (63.4°), the same as that of the GLONASS satellites. There are about 105 inoperable GLONASS satellites and upper stages drifting in their original orbits. As a result of eccentricity growth, these orbits, including the future disposal orbits of GALILEO, may intersect the orbits of the operational navigation constellations (GPS, GLONASS, and GALILEO) increasing the potential for collision (Jenkin and Gick, 2001) and may come close to both LEO and GEO operational altitudes. In
0273-1177/$30 Ó 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2003.01.021
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this paper, we focus on the analysis of the GPS Block I, GLONASS, and GALILEO orbit stability and its impacts on future disposal orbit policies. This analysis is expected to provide considerable insight into the longterm behavior of eccentricity growth as well as suggest strategies for disposal orbit insertion. The objectives of this study are: (1) to understand the long-term orbit perturbations and stability of those outof-service GPS Block I and GLONASS satellites and future disposed satellites of GALILEO, considering all major perturbing forces; (2) to examine the impacts on future navigation systems in that orbit altitude region; (3) to recommend strategies and requirements for endof-life satellite/upper stage disposal maneuvers. This work is sponsored by the GPS Program Office and is a continuation of The Aerospace Corporation GPS Block II studies initiated in response to end-of-life disposal guidelines published by NASA, 1995.
2. Analytical investigation The long-term and secular variations of the orbital parameters can usually be related to the perturbing forces through the averaged equations of variation. The averaged equations may be categorized as singly averaged or doubly averaged. The singly-averaged equations are derived by averaging the equations of variation over the orbit period thus eliminating all short or orbit period terms. The resulting variational equations are significantly shorter and the dominant long-term and secular variations due to a particular force may be identified. The singly-averaged equations are again averaged over the orbit period of the third body, further shortening the equations of variation. The resulting equations are called the doubly-averaged equations. The doubly-averaged equation in eccentricity is derived for the above disposal orbits by removing the 6-month and 14-day terms from the singly-averaged equations of the third body (Chao, 1979). The nominal operational GPS Block I orbit was circular with a 20181.6 km altitude and a 63.4° (critical) inclination. A GLONASS orbit of 1000 km below the nominal GPS orbit with the same inclination of 63.4° is assumed in this paper. The future GALILEO system will have a circular orbit 3000 km above the GPS altitude with 56° inclination. The disposal orbits of GALILEO are assumed at 500 km above the operational altitude. The eccentricity of the GPS Block I orbit with 12 h repeating ground tracks is not affected, in the very long-term sense, by the resonant tesseral harmonics due to the fact that there are no stationkeeping maneuvers. A closed form doubly-averaged equation in eccentricity due to third body perturbations is derived via computerized series expansion in Chao (1979) and is given as:
de=dt ¼ ð15=8Þecs½C1 sin 2ðx DXÞ þ C2 sinð2x DXÞ þ C3 sin 2x þ C4 sinð2x þ DXÞ þ C5 sin 2ðx þ DXÞ;
ð1Þ
where e is the eccentricity of the orbit, c ¼ n23 Rm =n; s ¼ ð1 e2 Þ1=2 , n is the mean motion of the GPS orbit, n3 is the mean motion of the third body, Rm is the mass ratio [ ¼ 1 for solar perturbation, ¼ 182.3 for lunar perturbation], x is the argument of perigee, DX ¼ X X3 , and X is the right ascension of the ascending node (RAAN) of the satellite, X3 is the right ascension of the ascending node of the Sun or Moon’s orbit with respect to the Earth equator. The five coefficients in Eq. (1) are functions of the inclination with the following form: C1 ¼ 12 sin2 i3 ðcos i þ 1=2 sin2 i 1Þ; C2 ¼ 12 sin i sin 2i3 ðcos i 1Þ; C3 ¼ sin2 ið3=2 sin2 i3 1Þ; C4 ¼ 12 sin i sin 2i3 ð1 þ cos iÞ;
ð2Þ
C5 ¼ 12 sin2 i3 ð1=2 sin2 i cos i 1Þ; where i is the inclination of the satellite orbit, and i3 is the inclination of the third body. After substituting the critical inclination and the inclination of the third body (23.45° for both the Sun and the Moon), the series expansion assumes the following simple form: de=dt ¼ ð15=8Þecs½0:01215 sin 2ðx XÞ 0:1808 sinð2x XÞ 0:6092 sin 2x þ 0:4733 sinð2x þ XÞ 0:0833 sin 2ðx þ XÞ: ð3Þ The above equation has slightly different coefficients inside the bracket than that of the similar equation for inclination at 55° as shown below (from Chao, 2000; Gick and Chao, 2001). de=dt ¼ ð15=8Þecs½0:0072 sin 2ðx XÞ 0:1277 sinð2x XÞ 0:5110 sin 2x þ 0:4714 sinð2x þ XÞ 0:0984 sin 2ðx þ XÞ: ð4Þ Note that the orbit of the third body is assumed to be circular and, hence, the eccentricity and argument of perigee of the Sun and Moon’s orbits do not appear in the equation. This approximation significantly shortens the series equations of variation. Previous numerical integration results (Chao, 1979, 1998, 2000; Gick and Chao, 2001) show that this circular orbit approximation does not cause any noticeable degradation in the longterm propagation. Eq. (4) also applies to GALILEO orbits. As indicated in previous studies (Chao, 2000; Gick and Chao, 2001), the cause of the large eccentricity growth is due to the resonance in the angle (2x þ X) in
C.C. Chao, R.A. Gick / Advances in Space Research 34 (2004) 1221–1226
the above equation. At GPS altitude and 55° inclination, the rate of 2x nearly cancels the rate of X. For GLONASS and GPS Block I orbits, the term with sin 2x induces resonance at or near critical inclination where dx=dt vanishes. The secular rates of x and X, due to J2 and Sun–Moon effects, may be computed from the averaged equations (Chao, 1979). As indicated in previous studies (Chao, 2000; Gick and Chao, 2001), a GPS disposal orbit at 500 km above the nominal GPS operational altitude has a nodal regression rate of about )0.038°/day and an argument of perigee rate of about 0.021°/day. Of the five sinusoidal terms on the righthand side of Eq. (4), the fourth term has the largest period due to the resonance effect from the rate of the combined angle (2x X) which is very small, being about 0.0046°/day. For the orbits of GLONASS and GPS Block I, the third term in Eq. (3) induces resonance due to the critical inclination effect. Table 1 summarizes the angular rates of the dominant term in the equation for eccentricity rate. The resonance terms with the very small angular rates are identified in emphasized print. The term sinð2xÞ is responsible for the long-term eccentricity growth in GLONASS and GPS Block I orbits, while the term sinð2x þ XÞ is the cause of the similar growth in GPS Block II and GALILEO disposal orbits. The above analytical investigation clearly shows that the significant growth in the eccentricity evolution caused by resonance
1223
perturbations due to J2 and Sun/Moon exists in other navigation satellites in the MEO altitude region from 19,000 to 24,000 km.
3. Long-term numerical propagation studies This section describes the numerical analysis of the long-term eccentricity growth. The results were obtained using MEANPROP [a long-term orbit prediction software tool of The Aerospace Corporation (Peterson and Hart, 1998)] which employs a high-precision semianalytic orbit propagator (SAOP developed by Draper Laboratory) and assumes the following perturbing forces: 8 8 WGS 84 Earth gravity model, Sun/Moon gravitational attractions, solar radiation pressure and atmospheric drag. Perturbations due to the other planets and Earth tides are small and are therefore neglected in this analysis. The numerical precision of MEANPROP has been favorably compared [in 100-year propagations (Chao, 1998, 2000)] with TRACE (the most accurate and well-maintained orbit propagation/determination tool at The Aerospace Corporation) indicating the adequacy of using MEANPROP for 200-year integration. For GPS applications, the estimated uncertainty in predicting the long-term eccentricity is small, on the order of a few percent, when the eccentricity grows beyond 0.02. This verification was determined by
Table 1 Summary of angular rates (deg/day) due to J2 and Sun–Moon perturbations Navigation systems
x
X
2nd term 2x X
3rd term 2x
4th term 2x þ X
GLONASS GPS Block I GPS Block II GALILEO
0.0010a 0.0010a 0.0212 0.0132
)0.0368 )0.0314 )0.0378 )0.0262
0.0388 0.0334 0.0802 0.0526
0.002 0.002 0.0414 0.0264
)0.0348 )0.0294 0.0046 0.0002
a
Most orbits are near critical inclination (62°–65°) and 0.0010 is a representative value for this table.
Fig. 1. GPS Block I (non-operational) apogee altitude history.
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Fig. 2. GPS Block I (non-operational) perigee altitude history.
Fig. 3. GLONASS (non-operational) apogee altitude history.
comparing the results with and without the effects of solar radiation pressure which is believed to be the critical parameter contributing to the uncertainty over the 200-year integration period. Figs. 1 and 2 show the apogee and perigee histories for the 10 non-operational GPS Block I satellites. The long-term eccentricity growth of these orbits will cause apogee and perigee to approach the GEO and LEO operational altitudes, respectively. Fig. 3 shows the apogee histories for the 105 non-operational GLONASS satellites. It is clear from Fig. 3 that the GLONASS satellites will begin penetrating the GPS operational altitude within the next 40 years and will continue to intrude upon this space for the foreseeable future.
4. Dependence on inclination In order to develop strategies to alleviate long-term eccentricity growth and thereby improve the disposal orbit stability, we investigate the sensitivity of eccentricity variation to various orbital parameters. The
maximum (worst-case) eccentricity achieved over a (sufficiently) long period of time is used as a measure of orbital stability. In other words, we examine how the maximum eccentricity achieved over time depends on the orbital parameters in an effort to derive strategies to keep the maximum eccentricity small. As noted in our earlier work, a disposal orbit design strategy involves targeting the initial argument of perigee. This strategy is to determine, through analytical (Eqs. (3) and (4)) and numerical methods, the optimum window in initial argument of perigee for each orbit plane. By properly targeting the initial eccentricity and argument of perigee, the eccentricity of the disposed orbits will remain very small for the next 200 years, according to earlier studies (Chao, 2000; Gick and Chao, 2001). This strategy requires additional propellant and orbit adjust maneuvers which may not be desirable. A second recommended strategy to avoid potentially large eccentricity growth is to move the inclination of the operational orbit by a few degrees from the current nominal value, closer to 52° as can be seen in Figs. 4 and 5. A large number of 200-year numerical integrations
C.C. Chao, R.A. Gick / Advances in Space Research 34 (2004) 1221–1226
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Fig. 4. Maximum 200-year eccentricity growth of GPS disposal orbits versus inclination.
Fig. 5. Maximum 200-year eccentricity growth of GALILEO disposal orbits versus inclination.
were performed for inclinations from 50° to 70° for both the GPS and GALILEO constellation orbits. The results, shown in Figs. 4 and 5, reveal the strong dependence of maximum (worst-case) eccentricity growth on orbit inclination. By selecting the operational inclination a few degrees from the current nominal value for both programs, the maximum eccentricity growth for GPS and GALILEO can be significantly reduced. However, this small modification may have an impact on the achievable constellation coverage and hence the user accuracy and availability, which should be taken into consideration.
5. Conclusions The long-term growth in eccentricity of GLONASS, GALILEO and GPS Block I satellites was
studied both analytically and numerically. It was found that these types of orbits evolve into orbits with large eccentricity, as much as 0.7 over 150 years. Analytical approximations reveal that the eccentricity growth is due to Sun/Moon and J2 secular perturbations. The 200-year numerical orbit propagation studies indicate that the GLONASS satellites will begin to enter the operational GPS constellation within the next 40 years. Numerical studies show the strong dependence of eccentricity growth on inclination and altitude. The growth effects are more pronounced for GALILEO orbits due to their higher altitude. Strategies for minimizing the eccentricity growth are identified. Specifically, the maximum eccentricity and the long-term growth in the eccentricity evolution can be made small by changing the operational orbit inclination by a few degrees from its current nominal value.
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Acknowledgements This work reflects research conducted under US Air Force and Missile Center Contract F04701-93-C-0094 for the GPS Program Office. The authors thank H.J. Schraibman, A.B. Jenkin, and J.E. Gidney of The Aerospace Corporation, Lt.Col. M.N. Moya USAF/ CZS and the GPS Program Office for their review and comments on this paper. References Chao, C.C. An analytical integration of the averaged equations of variation due to Sun–Moon perturbations and its application. Technical Report, SD-TR-80-12, The Aerospace Corporation, El Segundo, CA, October, 1979.
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