An application of symplectic integration for general relativistic planetary orbitography subject to non-gravitational forces. (English) Zbl 1482.70021
Summary: Spacecraft propagation tools describe the motion of near-Earth objects and interplanetary probes using Newton’s theory of gravity supplemented with the approximate general relativistic \(n\)-body Einstein-Infeld-Hoffmann equations of motion. With respect to the general theory of relativity and the long-standing recommendations of the International Astronomical Union for astrometry, celestial mechanics and metrology, we believe modern orbitography software is now reaching its limits in terms of complexity. In this paper, we present the first results of a prototype software titled General Relativistic Accelerometer-based Propagation Environment (GRAPE). We describe the motion of interplanetary probes and spacecraft using extended general relativistic equations of motion which account for non-gravitational forces using end-user supplied accelerometer data or approximate dynamical models. We exploit the unique general relativistic quadratic invariant associated with the orthogonality between four-velocity and acceleration and simulate the perturbed orbits for Molniya, Parker Solar Probe and Mercury Planetary Orbiter-like test particles subject to a radiation-like four-force. The accuracy of the numerical procedure is maintained using a 5-stage, \(10^{\mathrm{th}}\)-order structure-preserving Gauss collocation symplectic integration scheme. GRAPE preserves the norm of the tangent vector to the test particle worldline at the order of \(10^{-32}\).
MSC:
| 70H40 | Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics |
References:
| [1] | Acton, C. H.: Nasa’s spice system models the solar system. In I. M. Wytrzyszczak, J. H. Lieske, and R. A. Feldman, editors, Dynamics and Astrometry of Natural and Artificial Celestial Bodies, pages 257-262, Dordrecht, (1997). Springer Netherlands. ISBN 978-94-011-5534-2 |
| [2] | Balmino, G., Barriot, J.P.: Numerical integration techniques revisited. Manuscripta geodaetica. 15, 1-10 (1990) |
| [3] | Bertotti, B.; Iess, L.; Tortora, P., A test of general relativity using radio links with the Cassini spacecraft, Nature, 425, 6956, 374-376 (2003) · doi:10.1038/nature01997 |
| [4] | Biscani, F.; Izzo, D., Revisiting high-order taylor methods for astrodynamics and celestial mechanics, Mon. Not. Roy. Astronom. Soc., 504, 2, 2614-2628 (2021) · doi:10.1093/mnras/stab1032 |
| [5] | Brumberg, V.: Essential relativistic celestial mechanics. Routledge, (2017). ISBN 1351449699 · Zbl 0743.70001 |
| [6] | Brumberg, V., On relativistic equations of motion of an earth satellite, Celest. Mech. Dyn. Astron., 88, 2, 209-225 (2004) · Zbl 1131.70311 · doi:10.1023/B:CELE.0000016821.33627.77 |
| [7] | Brumberg, V., On derivation of EIH (Einstein-Infeld-Hoffman) equations of motion from the linearized metric of general relativity theory, Celest. Mech. Dyn. Astron., 99, 3, 245-252 (2007) · Zbl 1192.83016 · doi:10.1007/s10569-007-9094-5 |
| [8] | Brumberg, VA; Bretagnon, P.; Capitaine, N.; Damour, T.; Eubanks, TM; Fukushima, T.; Guinot, B.; Klioner, SA; Kopeikin, SM; Krivov, AV; Seidelmann, PK; Soffel, MH, General relativity and the IAU resolutions report of the IAU WGAS Sub-Working Group on Relativity in Celestial Mechanics and Astrometry (RCMA SWG), Highlights Astron., 11, 1, 194-199 (1998) · doi:10.1017/S1539299600020414 |
| [9] | Burcev, P., Non-gravitational force effect in general theory of relativity, Cechoslovackij fiziceskij zurnal B, 12, 10, 727-733 (1962) · Zbl 0108.40904 |
| [10] | Butcher, J. C., Goodwin, N.: Numerical methods for ordinary differential equations. Wiley Online Library, (2008) · Zbl 1167.65041 |
| [11] | Butcher, J. C.: Numerical methods for ordinary differential equations. Wiley, (2016). ISBN 1119121507 · Zbl 1354.65004 |
| [12] | Butcher, JC, Implicit Runge-Kutta processes, Math. Comput., 18, 85, 50-64 (1964) · Zbl 0123.11701 · doi:10.1090/S0025-5718-1964-0159424-9 |
| [13] | P. Cappuccio, I. di Stefano, G. Cascioli, and L. Iess. Comparison of light-time formulations in the post-newtonian framework for the bepicolombo more experiment. Class. Quant. Grav, (2021) |
| [14] | Charon, J. E.: Quinze leçons sur la relativité générale avec une introduction au calcul tensoriel. Kister, (1963) |
| [15] | Combrinck, L.: General Relativity and Space Geodesy. In G. Xu, editor, Sciences of Geodesy-II: Innovations and Future Developments, volume 2, chapter 2. Springer, (2012) |
| [16] | Cooper, G., Stability of Runge-Kutta methods for trajectory problems, IMA J. Num. Anal., 7, 1, 1-13 (1987) · Zbl 0624.65057 · doi:10.1093/imanum/7.1.1 |
| [17] | Damour, T.; Soffel, M.; Xu, C., General-relativistic celestial mechanics II. translational equations of motion, Phys. Rev. D, 45, 4, 1017 (1992) · doi:10.1103/PhysRevD.45.1017 |
| [18] | Damour, T.; Soffel, M.; Xu, C., General-relativistic celestial mechanics. III. rotational equations of motion, Phys. Rev. D, 47, 8, 3124 (1993) · doi:10.1103/PhysRevD.47.3124 |
| [19] | Damour, T.; Soffel, M.; Xu, C., General-relativistic celestial mechanics. IV. theory of satellite motion, Phys. Rev. D, 49, 2, 618 (1994) · doi:10.1103/PhysRevD.49.618 |
| [20] | Damour, T.; Soffel, M.; Xu, C., General-relativistic celestial mechanics. I. method and definition of reference systems, Phys. Rev. D, 43, 10, 3273 (1991) · doi:10.1103/PhysRevD.43.3273 |
| [21] | Damour, T., Soffel, M.: Chongming, and Xu. The general relativistic N-body problem. In: Hawking, S., Israel, W. (eds.) Relativistic Gravity Research With Emphasis on Experiments and Observations. Lecture Notes in Physics, pp. 46-69. Springer, Berlin, Heidelberg (1992) |
| [22] | Damour, T.: The problem of motion in Newtonian and Einsteinian gravity. In S. Hawking and W. Israel, editors, Three hundred years of gravitation, chapter 6, pages 128-198. Cambridge University Press, (1987) · Zbl 0966.83509 |
| [23] | Daquin, J.; Alessi, EM; O’Leary, J.; Lemaitre, A.; Buzzoni, A., Dynamical properties of the Molniya satellite constellation: long-term evolution of the semi-major axis, Nonlin. Dyn., 105, 3, 2081-2103 (2021) · doi:10.1007/s11071-021-06708-5 |
| [24] | d’Inverno, R.: Introducing Einstein’s relativity. Oxford University Press, USA (1992) · Zbl 0776.53046 |
| [25] | Donnelly, D.; Rogers, E., Symplectic integrators: an introduction, Am. J. Phys., 73, 10, 938-945 (2005) · Zbl 1130.37413 · doi:10.1119/1.2034523 |
| [26] | A. Einstein, L. Infeld, and B. Hoffmann. The gravitational equations and the problem of motion. Ann. Math., pp. 65-100, (1938). ISSN 0003-486X · Zbl 0018.28103 |
| [27] | Evans, S.; Taber, W.; Drain, T.; Smith, J.; Wu, H-C; Guevara, M.; Sunseri, R.; Evans, J., MONTE: the next generation of mission design and navigation software, CEAS Space J., 10, 1, 79-86 (2018) · doi:10.1007/s12567-017-0171-7 |
| [28] | Everitt, CF; DeBra, D.; Parkinson, B.; Turneaure, J.; Conklin, J.; Heifetz, M.; Keiser, G.; Silbergleit, A.; Holmes, T.; Kolodziejczak, J., Gravity probe b: final results of a space experiment to test general relativity, Phys. Rev. Lett., 106, 22, 221101 (2011) · doi:10.1103/PhysRevLett.106.221101 |
| [29] | Exertier, P.; Belli, A.; Samain, E.; Meng, W.; Zhang, H.; Tang, K.; Schlicht, A.; Schreiber, U.; Hugentobler, U.; Prochàzka, I.; Sun, X.; McGarry, JF; Mao, D.; Neumann, A., Time and laser ranging: a window of opportunity for geodesy, navigation, and metrology, J. Geod., 93, 11, 2389-2404 (2019) · doi:10.1007/s00190-018-1173-8 |
| [30] | Fox, NJ; Velli, MC; Bale, SD; Decker, R.; Driesman, A.; Howard, RA; Kasper, JC; Kinnison, J.; Kusterer, M.; Lario, D.; Lockwood, MK; McComas, DJ; Raouafi, NE; Szabo, A., The solar probe plus mission: humanity’s first visit to our star, Space Sci. Rev., 204, 1, 7-48 (2016) · doi:10.1007/s11214-015-0211-6 |
| [31] | Hairer, E., Lubich, C.,Wanner, G.: Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. Springer, (2006). ISBN 3540306668 · Zbl 1094.65125 |
| [32] | Huang, C., Ries, J.C., Tapley, B.D., Watkins, M.: Relativistic effects for near-earth satellite orbit determination. Celest. Mech. Dyn. Astronom. 48(2), 167-185 (1990). (ISSN 0923-2958) · Zbl 0704.70022 |
| [33] | Hugentobler, U.: Supporting material: Orbit perturbations due to relativistic corrections. In G. Petit and B. Luzum, editors, IERS conventions (2010) |
| [34] | Hughes, S. P.,Qureshi, R. H. , Cooley, S. D., Parker, J. J.: Verification and validation of the general mission analysis tool (gmat). In AIAA/AAS astrodynamics specialist conference, (2014) |
| [35] | Iess, L.; Asmar, S.; Cappuccio, P.; Cascioli, G.; De Marchi, F.; di Stefano, I.; Genova, A.; Ashby, N.; Barriot, J.; Bender, P., Gravity, geodesy and fundamental physics with bepicolombo’s more investigation, Space Sci. Rev., 217, 1, 1-39 (2021) · doi:10.1007/s11214-021-00800-3 |
| [36] | Kang, Z.; Tapley, B.; Bettadpur, S.; Ries, J.; Nagel, P., Precise orbit determination for grace using accelerometer data, Adv. Space Res., 38, 9, 2131-2136 (2006) · doi:10.1016/j.asr.2006.02.021 |
| [37] | Kinoshita, H.; Yoshida, H.; Nakai, H., Symplectic integrators and their application to dynamical astronomy, Celest. Mech. Dyn. Astron., 50, 59-71 (1991) · Zbl 0724.70019 · doi:10.1007/BF00048986 |
| [38] | Kopeikin, S. M.: Relativistic reference frames for astrometry and navigation in the solar system. In AIP Conference Proceedings, 886, 268-283. AIP, (2007). ISBN 0735403899 · Zbl 1183.85007 |
| [39] | Lange, B., The drag-free satellite, AIAA J., 2, 9, 1590-1606 (1964) · Zbl 0125.12503 · doi:10.2514/3.55086 |
| [40] | Lenoir, B.; Lévy, A.; Foulon, B.; Lamine, B.; Christophe, B.; Reynaud, S., Electrostatic accelerometer with bias rejection for gravitation and solar system physics, Adv. Space Res., 48, 7, 1248-1257 (2011) · doi:10.1016/j.asr.2011.06.005 |
| [41] | Lichnerowicz, A.; Teichmann, T., Théories relativistes de la gravitation et de l’électromagnétisme, Phys. Today, 8, 24 (1955) · Zbl 0065.20704 · doi:10.1063/1.3061795 |
| [42] | Lucchesi, DM; Iafolla, V., The non-gravitational perturbations impact on the bepicolombo radio science experiment and the key role of the isa accelerometer: direct solar radiation and albedo effects, Celest. Mech. Dyn. Astron., 96, 2, 99-127 (2006) · Zbl 1116.85300 · doi:10.1007/s10569-006-9034-9 |
| [43] | Marty, J.: Algorithmic documentation of the GINS software. GINS Algorithm Overview, (2013) https://www5.obs-mip.fr/wp-content-omp/uploads/sites/28/2017/11/GINS_Algo_2013.pdf |
| [44] | Misner, C. W., Thorne, K. S., Wheeler, J. A.: Gravitation. Macmillan, (1973) |
| [45] | Müller, T., Grave, F.: Catalogue of spacetimes. arXiv 0904, 4184 (2009) |
| [46] | Müller, J.; Soffel, M.; Klioner, SA, Geodesy and relativity, J. Geod., 82, 3, 133-145 (2008) · Zbl 1176.83028 · doi:10.1007/s00190-007-0168-7 |
| [47] | Novara, M., The BepiColombo ESA cornerstone mission to mercury, Acta Astronautica, 51, 1-9, 387-395 (2002) · doi:10.1016/S0094-5765(02)00065-6 |
| [48] | O’Leary, J.; Hill, JM; Bennett, JC, On the energy integral for first post-newtonian approximation, Celest. Mech. Dyn. Astron., 130, 7, 44 (2018) · Zbl 1396.83008 · doi:10.1007/s10569-018-9839-3 |
| [49] | Petit, G., Luzum, B.: IERS conventions (2010). Report, DTIC Document (2010) |
| [50] | Pireaux, S.; Barriot, J-P; Rosenblatt, P., SCRMI: A (S)emi-(C)lassical (R)elativistic (M)otion (I)ntegrator to model the orbits of space probes around the earth and other planets, Acta Astronautica, 59, 7, 517-523 (2006) · doi:10.1016/j.actaastro.2006.04.006 |
| [51] | Poisson, E., Will, C.M.: Gravity: Newtonian, Post-Newtonian. Cambridge University Press, Relativistic (2014) 1139952390 · Zbl 1334.83001 |
| [52] | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes: The Art of Scientific Computing, Vol. 3. Cambridge University Press, (2007) · Zbl 1132.65001 |
| [53] | Rodrigues, M.; Touboul, P., The lisa accelerometer, Adv. Space Res., 32, 7, 1251-1254 (2003) · doi:10.1016/S0273-1177(03)90326-7 |
| [54] | B. Schutz, B. Tapley, and G. H. Born. Statistical orbit determination. Elsevier, (2004). ISBN 0080541739 |
| [55] | Serra, D.; Di Pierri, V.; Schettino, G.; Tommei, G., Test of general relativity during the BepiColombo interplanetary cruise to Mercury, Phys. Rev. D, 98, 6, 064059 (2018) · doi:10.1103/PhysRevD.98.064059 |
| [56] | M. H. Soffel and W.-B. Han. Applied general relativity: theory and applications in astronomy, celestial mechanics and metrology. Springer, Cham, (2019). doi:10.1007/978-3-030-19673-8 |
| [57] | Soffel, M.; Langhans, R., Space-time reference systems (2012), New york: Springer, New york · Zbl 1255.85003 |
| [58] | M. Soffel, S. A. Klioner, G. Petit, P. Wolf, S. Kopeikin, P. Bretagnon, V. Brumberg, N. Capitaine, T. Damour, and T. Fukushima. The IAU 2000 resolutions for astrometry, celestial mechanics, and metrology in the relativistic framework: explanatory supplement. The Astronom. J., 126(6):2687, (2003). ISSN 1538-3881 |
| [59] | Soffel, M., Relativity in astrometry, celestial mechanics and geodesy (1989), Newyork: Springer, Newyork · doi:10.1007/978-3-642-73406-9 |
| [60] | Soffel, M., Report of the working group relativity for celestial mechanics and astrometry, Proc. IAU Colloquium, 180, 283-292 (2000) |
| [61] | E. M. Standish, J. G. Williams, et al. Orbital ephemerides of the sun, moon, and planets. Explanatory supplement to the astronomical almanac, pp. 279-323, (1992) |
| [62] | Stoer, J.; Bulirsch, R., Introduction to numerical analysis (2013), Newyork: Springer, Newyork · Zbl 0423.65002 |
| [63] | Touboul, P.; Willemenot, E.; Foulon, B.; Josselin, V., Accelerometers for champ, grace and goce space missions: synergy and evolution, Boll. Geof. Teor. Appl, 40, 3-4, 321-327 (1999) |
| [64] | D. Vallado, P. Crawford, R. Hujsak, and T. Kelso. Revisiting spacetrack report # 3. In AIAA/AAS Astrodynamics Specialist Conference and Exhibit, (2006) |
| [65] | Vallado, DA, Fundamentals of astrodynamics and applications (2001), Newyork: Springer, Newyork · Zbl 1191.70002 |
| [66] | T. Van Helleputte and P. Visser. Gps based orbit determination using accelerometer data. Aerospace Sci. Technol., 12(6):478-484, (2008). ISSN 1270-9638 |
| [67] | Weinberg, S., Gravitation and cosmology: principles and applications of the general theory of relativity (1972), New York: Wiley, New York |
| [68] | J. Yepez. Einstein’s vierbein field theory of curved space. arXiv preprint arXiv:1106.2037, (2011) |
| [69] | Yoshida, H., Construction of higher order symplectic integrators, Phys. Lett. A, 150, 5-7, 262-268 (1990) · doi:10.1016/0375-9601(90)90092-3 |
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