Stability of the Moons orbits in Solar syste especially of Earth s Moon in the restricted three-body proble BP celestial echanics Sergey V. Ershkov Institute for Tie Nature Explorations M.V. Loonosov's Moscow State University Leninskie gory - Moscow 999 ussia e-ail: sergej-ershkov@yandex.ru Abstract: We consider the euations of otion of three-body proble in a Lagrange for which eans a consideration of relative otions of -bodies in regard to each other. Analyzing such a syste of euations we consider in details the case of oon s otion of negligible ass ₃ around the -nd of two giant-bodies ₁ ₂ which are rotating around their coon centre of asses on Kepler s trajectories the ass of which is assued to be less than the ass of central body. Under assuptions of BP we obtain the euations of otion which describe the relative utual otion of the centre of ass of -nd giant-body ₂ Planet and the centre of ass of -rd body Moon with additional effective ass ₂ placed in that centre of ass ₂ + ₃ where is the diensionless dynaical paraeter. They should be rotating around their coon centre of asses on Kepler s elliptic orbits. For negligible effective ass ₂ + ₃ it gives the euations of otion which should describe a uasi-elliptic orbit of -rd body Moon around the -nd body ₂ Planet for ost of the oons of the Planets in Solar syste. But the orbit of Earth s Moon should be considered as non-constant elliptic otion for the effective ass 0.078₂ placed in the centre of ass for the -rd body Moon. The position of their coon centre of asses should obviously differ for real ass ₃ = 0.0₂ and for the effective ass 0.0055+0.0₂ placed in the centre of ass of the Moon. Key Words: restricted three-body proble orbit of the Moon relative otion
Introduction. The stability of the otion of the Moon is the ancient proble which leading scientists have been trying to solve during last 400 years. A new derivation to estiate such a proble fro a point of view of relative otions in restricted three-body proble BP is proposed here. Systeatic approach to the proble above was suggested earlier in KAM- Kologorov-Arnold-Moser-theory [] in which the central KAM-theore is known to be applied for researches of stability of Solar syste in ters of restricted threebody proble [-5] especially if we consider photogravitational restricted three-body proble [6-8] with additional influence of Yarkovsky effect of non-gravitational nature [9]. KAM is the theory of stability of dynaical systes [] which should solve a very specific uestion in regard to the stability of orbits of so-called sall bodies in Solar syste in ters of restricted three-body proble []: indeed dynaics of all the planets is assued to satisfy to restrictions of restricted three-body proble such as infinitesial asses negligible deviations of the ain orbital eleents etc.. Nevertheless KAM also is known to assue the appropriate Hailton foralis in proof of the central KAM-theore []: the dynaical syste is assued to be Hailton syste as well as all the atheatical operations over such a dynaical syste are assued to be associated with a proper Hailton syste. According to the Bruns theore [5] there is no other invariants except well-known 0 integrals for three-body proble including integral of energy oentu etc. this is a classical exaple of Hailton syste. But in case of restricted three-body proble there is no other invariants except only one Jacobian-type integral of otion []. Such a contradiction is the ain paradox of KAM-theory: it adopts all the restrictions of restricted three-body proble but nevertheless it proves to use the Hailton foralis which assues the conservation of all other invariants the integral of energy oentu etc.. To avoid abiguity let us consider a relative otion in three-body proble [].
. Euations of otion. Let us consider the syste of ODE for restricted three-body proble in barycentric Cartesian co-ordinate syste at given initial conditions [-]: - here ₁ ₂ ₃ - ean the radius-vectors of bodies ₁ ₂ ₃ accordingly; - is the gravitational constant. Syste above could be represented for relative otion of three-bodies as shown below by the proper linear transforations:.
Let us designate as below: * Using of * above let us transfor the previous syste to another for:.. Analysing the syste. we should note that if we su all the above euations one to each other it would lead us to the result below: 0. If we also su all the eualities * one to each other we should obtain 0 ** 4
Under assuption of restricted three-body proble we assue that the ass of sall -rd body ₃ ₁ ₂ accordingly; besides for the case of oving of sall -rd body ₃ as a oon around the -nd body ₂ let us additionaly assue ₂₃ ₁₂. So taking into consideration ** we obtain fro the syste. as below: 0. 0 - where the -st euation of. describes the relative otion of assive bodies which are rotating around their coon centre of asses on Kepler s trajectories; the -nd describes the orbit of sall -rd body ₃ Moon relative to the -nd body ₂ Planet for which we could obtain according to the trigonoetric Law of Cosines [0]: cos cos. - here is the angle between the radius-vectors ₂₃ and ₁₂. Euation. could be siplified under additional assuption above ₂₃ ₁₂ for restricted utual otions of bodies ₁ ₂ in BP [] as below: 5
0.4 Moreover if we present E..4 in a for below 0.5 - then E..5 describes the relative otion of the centre of ass of -nd giant-body ₂ Planet and the centre of ass of -rd body Moon with the effective ass ₂ + ₃ which are rotating around their coon centre of asses on the stable Kepler s elliptic trajectories. Besides if the diensionless paraeters 0 then euation.5 should describe a uasi-circle otion of -rd body Moon around the -nd body ₂ Planet.. The coparison of the oons in Solar syste. As we can see fro E..5 is the key paraeter which deterines the character of oving of the sall -rd body ₃ the Moon relative to the -nd body ₂ Planet. Let us copare such a paraeter for all considerable known cases of orbital oving of the oons in Solar syste [] Tab.: 6
Table. Coparison of the averaged paraeters of the oons in Solar syste. Masses of the Planets Solar syste kg atio ₁ Sun to ass ₂ Planet Distance ₁₂ between Sun-Planet AU Paraeter ratio ₃ Moon to ass ₂ Planet Distance ₂₃ between Moon-Planet in 0³ k Paraeter Mercury.0²³ 946 0.055 0.87 AU Venus 4.870²⁴ 946 0.85 0.7 AU Earth 5.970²⁴ Earth = 946 kg AU = 49500000 k 000ˉ⁶ 8.4 Moon 550ˉ⁶ Phobos Phobos Phobos Mars 6.40²³ 946 0.07.54 AU 0.00ˉ⁶ Deios 9.8 Deios 0.0ˉ⁶ Deios 0.000ˉ⁶.46.40ˉ⁶ Ganyede Ganyede Ganyede 790ˉ⁶ 070.70ˉ⁶ Callisto Callisto Callisto Jupiter.90²⁷ 946 7.8 5. AU 580ˉ⁶ Io 88 Io 4.890ˉ⁶ Io 470ˉ⁶ 4 0.70ˉ⁶ 4 Europa 50ˉ⁶ 4 Europa 67 4 Europa 0.670ˉ⁶ 7
Titan Titan 400ˉ⁶ Titan.0ˉ⁶ hea 4.0ˉ⁶ Iapetus hea 57 Iapetus hea 0.80ˉ⁶ Iapetus Saturn 946 95.6 9.54 AU.40ˉ⁶ 4 Dione 56 4 Dione 54.460ˉ⁶ 4 Dione 5.690²⁶.90ˉ⁶ 77 0.070ˉ⁶ 5 Tethys 5 Tethys 5 Tethys.090ˉ⁶ 94.6 0.00ˉ⁶ 6 Enceladus 0.90ˉ⁶ 7 Mias 6 Enceladus 8 7 Mias 85.4 6 Enceladus 0.060ˉ⁶ 7 Mias 0.070ˉ⁶ 0.0080ˉ⁶ Titania Titania Titania 400ˉ⁶ 46 0.080ˉ⁶ 946 4.7 Oberon 50ˉ⁶ Oberon 584 Oberon 0.0ˉ⁶ Uranus 8.690²⁵ 9.9 AU Ariel: 60ˉ⁶ Ariel: 9 Ariel: 0.00ˉ⁶ 4 Ubriel:.490ˉ⁶ 5 Miranda: 4 Ubriel: 66. 5 Miranda: 9.4 4 Ubriel: 0.090ˉ⁶ 5 Miranda: 0.750ˉ⁶ 0.000ˉ⁶ 8
Triton Triton Triton 00ˉ⁶ 55 0.00ˉ⁶ Neptune.00²⁶ 946 7.5 0.07 AU Proteus 0.480ˉ⁶ Proteus 8 Proteus 0.00040ˉ⁶ Nereid Nereid Nereid 0.90ˉ⁶ 55 5.80ˉ⁶ Pluto.0²² 946 0.00 9.48 AU Charon 4600ˉ⁶ Charon 0 Charon 0.0060ˉ⁶. Discussion. As we can see fro the Tab. above the diensionless key paraeter which deterines the character of oving of the sall -rd body ₃ Moon relative to the -nd body ₂ Planet is varying for all variety of the oons of the Planets in Solar syste fro the eaning 0.00040ˉ⁶ for Proteus of Neptune to the eaning 54.460ˉ⁶ for Iapetus of Saturn; but it still reains to be negligible enough for adopting the stable oving of the effective ass ₂ + ₃ on uasi-elliptic Kepler s orbit around their coon centre of asses with the -nd body ₂. If the total su of diensionless paraeters + 0 is negligible then euation.5 should describe a stable uasi-circle orbit of -rd body Moon around the -nd body ₂ Planet. Let us consider the proper exaples which deviate differ to soe extent fro the negligibility case + 0 above Tab. []: 9
. Charon-Pluto: + = 0.006+4600ˉ⁶ eccentricity 0.00. Nereid-Neptune: + = 5.8+090ˉ⁶ eccentricity 0.7507. Triton-Neptune: + = 0.0+00ˉ⁶ eccentricity 0.000 06 4. Iapetus-Saturn: + = 54.46+.40ˉ⁶ eccentricity 0.086 5. Titan-Saturn: + =.+400ˉ⁶ eccentricity 0.088 6. Io-Jupiter: + = 0.68 + 470ˉ⁶ eccentricity 0.004 7. Callisto-Jupiter: + = 4.89+580ˉ⁶ eccentricity 0.0074 8. Ganyede-Jupiter: + =.7+790ˉ⁶ eccentricity 0.00 9. Phobos-Mars: + = 07+000ˉ⁶ eccentricity 0.05 0. Moon-Earth: + = 55+000ˉ⁶ eccentricity 0.0549 The obvious extree exception is the Nereid oon of Neptune fro this schee: Nereid orbits Neptune in the prograde direction at an average distance of 55400 k but its high eccentricity of 0.7507 takes it as close as 7000 k and as far as 9655000 k []. The unusual orbit suggests that it ay be either a captured asteroid or Kuiper belt object or that it was an inner oon in the past and was perturbed during the capture of Neptune's largest oon Triton []. One could suppose that the orbit of Nereid should be derived preferably fro the assuptions of 4BP the case of estricted Four-Body Proble or ore coplicated cases. As we can see fro consideration above in case of the Earth s Moon such a diensionless key paraeters increase siulteneously to the crucial eanings = 0.0055 and = 0.0 respectively + = 0.078 other case is the pair Charon- Pluto but only paraeter 0.5 is increased for the. It eans that the orbit for relative otion of the Moon in regard to the Earth could not be considered as uasielliptic orbit and should be considered as non-constant elliptic orbit with the effective ass ₂ + ₃ placed in the centre of ass for the Moon. 0
As we know [-4] the eleents of that elliptic orbit depend on the position of the coon centre of asses for -rd sall body Moon and the planet Earth. But such a position of their coon centre of ass should obviously differ for the real ass ₃ and the effective ass ₂ + ₃ placed in the centre of ass of the -rd body Moon. So the elliptic orbit of otion of the Moon derived fro the assutions of BP should differ fro the elliptic orbit which could be obtained fro the assutions of BP the case of estricted Two-Body Proble: it eans utual oving of gravitating asses without the influence of other central forces. 4. earks about the eccentricities of the orbits. According to the definition [] the orbital eccentricity of an astronoical object is a paraeter that deterines the aount by which its orbit around another body deviates fro a perfect orbit: e h - where - is the specific orbital energy; h - is the specific angular oentu; - is the su of the standard gravitational paraeters of the bodies = ₂ + + see.5. The specific orbital energy euals to the constant su of kinetic and potential energy in a -body ballistic trajectory []: a const - here a is the sei-ajor axis. For an elliptic orbit the specific orbital energy is the negative of the additional energy reuired to accelerate a ass of one kilogra to escape velocity parabolic orbit.
Thus assuing = t we should obtain fro the euality above: t const a t a 0 t a t 4. - where t is the periodic function depending on tie-paraeter t which is slowly varying during all the tie-period fro the inial eaning in > 0 to the axial eaning ax preferably ax - in 0. Besides we should note that in an elliptical orbit the specific angular oentu h is twice the area per unit tie swept out by a chord of ellipse i.e. the area which is totally covered by a chord of ellipse during it s otion per unit tie ultiplied by fro the priary to the secondary body [] according to the Kepler s -nd law of planetary otion. Since the area of the entire orbital ellipse is totally swept out in one orbital period the specific angular oentun h is eual to twice the area of the ellipse divided by the orbital period as represented by the euation: h b a - where b is the sei-inor axis. So fro E. 4. we should obtain that for the constant specific angular oentu h the sei-inor axis b should be constant also. Thus we could express the coponents of elliptic orbit as below: x t a 0 t cos t y t bsin t - which could be scheatically iagined as it is shown at Fig.-.
Fig.-. Orbits of the oon scheatically iagined.
Conclusion. We have considered the euations of otion of three-body proble in a Lagrange for which eans a consideration of relative otions of -bodies in regard to each other. Analyzing such a syste of euations we consider the case of oon s otion of negligible ass ₃ around the -nd of two giant-bodies ₁ ₂ which are rotating around their coon centre of asses on Kepler s trajectories the ass of which is assued to be less than the ass of central body. Besides we consider only the natural satellites which are assive enough to have achieved hydrostatic euilibriu; there are known of such a id-sized natural satellites for planets of Solar syste including Earth s Moon and Charon of Pluto Tab.. By aking several approxiations in the euations of otion of the three-body proble we derive a key paraeter that deterines the character of the oving of the sall -rd body ₃ the Moon relative to the -nd body ₂ Planet. Under assuptions of BP we obtain that the euations of otion should describe the relative utual otion of the centre of ass of -nd giant-body ₂ Planet and the centre of ass of -rd body Moon with additional effective ass ₂ placed in that centre of ass ₂ + ₃ where is the diensionless dynaical paraeter non-constant but negligible. They should be rotating around their coon centre of asses on Kepler s elliptic orbits. So the case BP of '-body proble' for the oon's orbit was elegantly reduced to the case BP of '-body proble' the last one is known to be stable for the relative otion of planet-satellite pairs [-4]. For negligible effective ass ₂ + ₃ it gives euations of otion which should describe a uasi-elliptic orbit of -rd body Moon around the -nd body ₂ Planet for ost of the oons of the Planets in Solar syste. But the orbit of Earth s Moon should be considered as non-constant elliptic otion for the effective ass 0.078₂ placed in the centre of ass for the -rd body Moon. The position of their coon centre of asses should obviously differ for the real ass ₃ = 0.0₂ and for the effective ass 0.0055+0.0₂ placed in the centre of ass of the Moon. 4
Conflict of interest The author declares that there is no conflict of interests regarding the publication of this article. Acknowledgeents I a thankful to CNews ussia project Science & Technology Foru Prof. L.Vladiirov-Paraligon - for valuable discussions in preparing of this anuscript. eferences: [] Arnold V. 978. Matheatical Methods of Classical Mechanics. Springer New York. [] Lagrange J. 87. 'OEuvres' M.J.A. Serret Ed.. Vol. 6 published by Gautier-Villars Paris. [] Szebehely V. 967. Theory of Orbits. The estricted Proble of Three Bodies. Yale University New Haven Connecticut. Acadeic Press New-York and London. [4] Duboshin G.N. 968. Nebesnaja ehanika. Osnovnye zadachi i etody. Moscow: Nauka handbook for Celestial Mechanics in russian. [5] Bruns H. 887. Uеber die Integrale der Vielkoerper-Probles. Acta ath. Bd. p. 5-96. [6] Shankaran Shara J.P. and Ishwar B. 0. Euilibriu points in the generalized photogravitational non-planar restricted three body proble. International Journal of Engineering Science and Technology Vol. pp. 6-67. [7] Chernikov Y.A. 970. The Photogravitational estricted Three-Body Proble. 5
Soviet Astronoy Vol. 4 p.76. [8] Jagadish Singh Oni Leke 00. Stability of the photogravitational restricted threebody proble with variable asses. Astrophys Space Sci 00 6: 05 4. [9] Ershkov S.V. 0. The Yarkovsky effect in generalized photogravitational -body proble. Planetary and Space Science Vol. 7 pp. -. [0] E. Kake 97. Hand-book for ODE. Science Moscow. [] Ershkov S.V. 04. Quasi-periodic solutions of a spiral type for Photogravitational estricted Three-Body Proble. The Open Astronoy Journal 04 vol.7: 9-. [] Sheppard Scott S. The Giant Planet Satellite and Moon Page. Departaent of Terrestrial Magnetis at Carniege Institution for science. etrieved 05-0-. See also: http://en.wikipedia.org/wiki/solar_syste 6