The post-Newtonian motion around an oblate spheroid: the mixed orbital effects due to the Newtonian oblateness and the post-Newtonian mass monopole accelerations

When a test particle moves about an oblate spheroid, it is acted upon, among other things, by two standard perturbing accelerations. One, of Newtonian origin, is due to the quadrupole mass moment J 2 of the orbited body. The other one, of order O 1 / c 2 , is caused by the static, post-Newtonian fie...

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Veröffentlicht in:	General relativity and gravitation Jg. 55; H. 12; S. 136
1. Verfasser:	Iorio, Lorenzo
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.12.2023 Springer Nature B.V
Schlagworte:	Astronomy Astrophysics and Cosmology Classical and Quantum Gravitation Differential Geometry Equations of motion Mathematical and Computational Physics Oblate spheroids Orbital elements Physics Physics and Astronomy Quadrupoles Quantum Physics Relativity Theory Theoretical Fundamental problems and general formalism Classical general relativity Experimental studies of gravity Satellite orbits Experimental tests of gravitational theories
ISSN:	0001-7701, 1572-9532
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Abstract	When a test particle moves about an oblate spheroid, it is acted upon, among other things, by two standard perturbing accelerations. One, of Newtonian origin, is due to the quadrupole mass moment J 2 of the orbited body. The other one, of order O 1 / c 2 , is caused by the static, post-Newtonian field arising solely from the mass of the central object. Both of them concur to induce indirect, mixed orbital effects of order O J 2 / c 2 . They are of the same order of magnitude of the direct ones induced by the post-Newtonian acceleration arising in presence of an oblate source, not treated here. We calculate these less known features of motion in their full generality in terms of the osculating Keplerian orbital elements. Subtleties pertaining the correct calculation of their mixed net precessions per orbit to the full order of O J 2 / c 2 are elucidated. The obtained results hold for arbitrary orbital geometries and for any orientation of the body’s spin axis k ^ in space. The method presented is completely general, and can be extended to any pair of post-Keplerian accelerations entering the equations of motion of the satellite, irrespectively of their physical nature.
AbstractList	When a test particle moves about an oblate spheroid, it is acted upon, among other things, by two standard perturbing accelerations. One, of Newtonian origin, is due to the quadrupole mass moment J 2 of the orbited body. The other one, of order O 1 / c 2 , is caused by the static, post-Newtonian field arising solely from the mass of the central object. Both of them concur to induce indirect, mixed orbital effects of order O J 2 / c 2 . They are of the same order of magnitude of the direct ones induced by the post-Newtonian acceleration arising in presence of an oblate source, not treated here. We calculate these less known features of motion in their full generality in terms of the osculating Keplerian orbital elements. Subtleties pertaining the correct calculation of their mixed net precessions per orbit to the full order of O J 2 / c 2 are elucidated. The obtained results hold for arbitrary orbital geometries and for any orientation of the body’s spin axis k ^ in space. The method presented is completely general, and can be extended to any pair of post-Keplerian accelerations entering the equations of motion of the satellite, irrespectively of their physical nature. When a test particle moves about an oblate spheroid, it is acted upon, among other things, by two standard perturbing accelerations. One, of Newtonian origin, is due to the quadrupole mass moment J2 of the orbited body. The other one, of order O1/c2, is caused by the static, post-Newtonian field arising solely from the mass of the central object. Both of them concur to induce indirect, mixed orbital effects of order OJ2/c2. They are of the same order of magnitude of the direct ones induced by the post-Newtonian acceleration arising in presence of an oblate source, not treated here. We calculate these less known features of motion in their full generality in terms of the osculating Keplerian orbital elements. Subtleties pertaining the correct calculation of their mixed net precessions per orbit to the full order of OJ2/c2 are elucidated. The obtained results hold for arbitrary orbital geometries and for any orientation of the body’s spin axis k^ in space. The method presented is completely general, and can be extended to any pair of post-Keplerian accelerations entering the equations of motion of the satellite, irrespectively of their physical nature.
ArticleNumber	136
Author	Iorio, Lorenzo
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References	RenzettiGHistory of the attempts to measure orbital frame-dragging with artificial satellitesCentr. Eur. J. Phys.20131155315442013CEJPh..11..531R10.2478/s11534-013-0189-1 Bertotti, B., Farinella, P., Vokrouhlický, D.: Physics of the solar system. Kluwer, Dordrecht, Aug 2003. https://doi.org/10.1007/978-94-010-0233-2 PitjevaEVPitjevNPMasses of the main asteroid belt and the kuiper belt from the motions of planets and spacecraftAstron. Lett.2018448–95545662018AstL...44..554P10.1134/S1063773718090050 Capderou, M.: Satellites. Orbits and Missions. Springer-Verlag France, Paris, (2005). https://doi.org/10.1007/b139118 KlionerSAKopeikinSMThe post-keplerian orbital representations of the relativistic two-body problemAstrophys. J.19944279511994ApJ...427..951K10.1086/174201 BrumbergVAEssential Relativistic Celestial Mechanics1991BristolAdam Hilger EinsteinAErklärung der Perihelbewegung des Merkur aus der allgemeinen RelativitätstheorieSitzber. Preuss. Akad.191547831839 HeimbergerJSoffelMRuderHRelativistic effects in the motion of artificial satellites—The oblateness of the central body IICelest. Mech. Dyn. Astr.19894722052171989CeMDA..47..205H105033910.1007/BF00051205 Iorio, L.: Post-Newtonian orbital effects induced by the mass quadrupole and spin octupole moments of an axisymmetric body. arXiv e-prints, art. arXiv:2310.02834, (2023). https://doi.org/10.48550/arXiv.2310.02834 IorioLPost-Keplerian corrections to the orbital periods of a two-body system and their measurabilityMon. Not. Roy. Astron. Soc.20164603244524522016MNRAS.460.2445I10.1093/mnras/stw1155 GurfilPEfroimskyMAnalysis of the PPN two-Body Problem using non-osculating orbital elementsAdv. Space Res.20226915385532022AdSpR..69..538G10.1016/j.asr.2021.09.009 Poisson, E., Will, C.M.: Gravity. Cambridge University Press, Cambridge, (2014). https://doi.org/10.1017/CBO9781139507486 WillCMIncorporating post-Newtonian effects in N-body dynamicsPhys. Rev. D20148940440432014PhRvD..89d4043W10.1103/PhysRevD.89.044043 PearlmanMArnoldDDavisMBarlierFBiancaleRVasilievVCiufoliniIPaolozziAPavlisECSośnicaKBloßfeldMLaser geodetic satellites: a high-accuracy scientific toolJ. Geod.20199311218121942019JGeod..93.2181P10.1007/s00190-019-01228-y Kopeikin, S.M., Efroimsky, M., Kaplan, G.: Relativistic celestial mechanics of the solar System. Wiley-VCH, Weinheim, (2011). https://doi.org/10.1002/9783527634569 Le Verrier, U.: Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète. Cr. Hebd. Acad. Sci., 49:379–383, (1859) HuangCLiuLAnalytical solutions to the four post-Newtonian effects in a near Earth satellite orbitCelest. Mech. Dyn. Astr.19925332933071992CeMDA..53..293H10.1007/BF00052615 King-HeleDGThe effect of the earth’s oblateness on the orbit of a near satelliteProc. R. Soc. A1958247124849721958RSPSA.247...49K9924210.1098/rspa.1958.0169 SoffelMHWirrerRSchastokJRuderHSchneiderMRelativistic effects in the motion of artificial satellites—The oblateness of the central body iCelest. Mech. Dyn. Astr.1987421–481891987CeMec..42...81S105033410.1007/BF01232949 DebonoISmootGFGeneral relativity and cosmology: unsolved questions and future directionsUniverse20162232016Univ....2...23D10.3390/universe2040023 BoltonSJLunineJStevensonDConnerneyJEPLevinSOwenTCBagenalFGautierDIngersollAPOrtonGSGuillotTHubbardWBloxhamJCoradiniAStephensSKMokashiPThorneRThorpeRThe Juno MissionSpace Sci. Rev.20172131–45372017SSRv..213....5B10.1007/s11214-017-0429-6 IorioLPost-Newtonian direct and mixed orbital effects due to the oblateness of the central bodyInt. J. Mod. Phys. D20152481550067592015IJMPD..2450067I336824910.1142/S0218271815500674 NobiliAMWillCMThe real value of Mercury’s perihelion advanceNature198632039411986Natur.320...39N10.1038/320039a0 IorioLOn the mean anomaly and the mean longitude in tests of post-Newtonian gravityEur. Phys. J. C201979108162019EPJC...79..816I10.1140/epjc/s10052-019-7337-8 IorioLCalculation of the Uncertainties in the Planetary Precessions with the Recent EPM2017 Ephemerides and their Use in Fundamental Physics and BeyondAstron J.201915762202019AJ....157..220I10.3847/1538-3881/ab19bf CiufoliniIPaolozziAKoenigRPavlisECRiesJMatznerRGurzadyanVPenroseRSindoniGParisCFundamental Physics and General Relativity with the LARES and LAGEOS satellitesNucl. Phys. B Proc. Suppl.20132431801932013NuPhS.243..180C10.1016/j.nuclphysbps.2013.09.005 Soffel, M.H.: Relativity in Astrometry, Celestial Mechanics and Geodesy. 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References_xml	– reference: EinsteinAErklärung der Perihelbewegung des Merkur aus der allgemeinen RelativitätstheorieSitzber. Preuss. Akad.191547831839 – reference: IorioLCalculation of the Uncertainties in the Planetary Precessions with the Recent EPM2017 Ephemerides and their Use in Fundamental Physics and BeyondAstron J.201915762202019AJ....157..220I10.3847/1538-3881/ab19bf – reference: IorioLOn the mean anomaly and the mean longitude in tests of post-Newtonian gravityEur. Phys. J. C201979108162019EPJC...79..816I10.1140/epjc/s10052-019-7337-8 – reference: IorioLPost-Keplerian corrections to the orbital periods of a two-body system and their measurabilityMon. Not. Roy. Astron. Soc.20164603244524522016MNRAS.460.2445I10.1093/mnras/stw1155 – reference: Bertotti, B., Farinella, P., Vokrouhlický, D.: Physics of the solar system. Kluwer, Dordrecht, Aug 2003. https://doi.org/10.1007/978-94-010-0233-2 – reference: HuangCLiuLAnalytical solutions to the four post-Newtonian effects in a near Earth satellite orbitCelest. Mech. Dyn. Astr.19925332933071992CeMDA..53..293H10.1007/BF00052615 – reference: Iorio, L.: Post-Newtonian orbital effects induced by the mass quadrupole and spin octupole moments of an axisymmetric body. arXiv e-prints, art. arXiv:2310.02834, (2023). https://doi.org/10.48550/arXiv.2310.02834 – reference: KlionerSAKopeikinSMThe post-keplerian orbital representations of the relativistic two-body problemAstrophys. J.19944279511994ApJ...427..951K10.1086/174201 – reference: Kopeikin, S.M., Efroimsky, M., Kaplan, G.: Relativistic celestial mechanics of the solar System. Wiley-VCH, Weinheim, (2011). https://doi.org/10.1002/9783527634569 – reference: CiufoliniIPaolozziAKoenigRPavlisECRiesJMatznerRGurzadyanVPenroseRSindoniGParisCFundamental Physics and General Relativity with the LARES and LAGEOS satellitesNucl. Phys. B Proc. Suppl.20132431801932013NuPhS.243..180C10.1016/j.nuclphysbps.2013.09.005 – reference: IorioLPost-Newtonian direct and mixed orbital effects due to the oblateness of the central bodyInt. J. Mod. Phys. D20152481550067592015IJMPD..2450067I336824910.1142/S0218271815500674 – reference: DebonoISmootGFGeneral relativity and cosmology: unsolved questions and future directionsUniverse20162232016Univ....2...23D10.3390/universe2040023 – reference: SoffelMHWirrerRSchastokJRuderHSchneiderMRelativistic effects in the motion of artificial satellites—The oblateness of the central body iCelest. Mech. Dyn. Astr.1987421–481891987CeMec..42...81S105033410.1007/BF01232949 – reference: King-HeleDGThe effect of the earth’s oblateness on the orbit of a near satelliteProc. R. Soc. A1958247124849721958RSPSA.247...49K9924210.1098/rspa.1958.0169 – reference: PitjevaEVPitjevNPMasses of the main asteroid belt and the kuiper belt from the motions of planets and spacecraftAstron. Lett.2018448–95545662018AstL...44..554P10.1134/S1063773718090050 – reference: Le Verrier, U.: Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète. Cr. Hebd. Acad. Sci., 49:379–383, (1859) – reference: BrumbergVAEssential Relativistic Celestial Mechanics1991BristolAdam Hilger – reference: GurfilPEfroimskyMAnalysis of the PPN two-Body Problem using non-osculating orbital elementsAdv. 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Snippet	When a test particle moves about an oblate spheroid, it is acted upon, among other things, by two standard perturbing accelerations. One, of Newtonian origin,...
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SubjectTerms	Astronomy Astrophysics and Cosmology Classical and Quantum Gravitation Differential Geometry Equations of motion Mathematical and Computational Physics Oblate spheroids Orbital elements Physics Physics and Astronomy Quadrupoles Quantum Physics Relativity Theory Theoretical
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Title	The post-Newtonian motion around an oblate spheroid: the mixed orbital effects due to the Newtonian oblateness and the post-Newtonian mass monopole accelerations
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