Our Solar System contains eight planets and their respective natural satellites excepting the inner two planets Mercury and Venus. A satellite hosted by a given Planet is well protected by the gravitational pertubation of much heavier planets such as Jupiter and Saturn if the natural satellite lies deep inside the respective host Planet Hill sphere. Each planet has a Hill radius aH and planet mean radius RP and the ratio R1=RP/aH. Under very low R1 (less than 0.006) the approximation of CRTBP (centrally restricted three-body problem) to two-body problem is valid and planet has spacious Hill lobe to capture a satellite and retain it. This ensures a high probability of capture of natural satellite by the given planet and Sun’s perturbation on Planet-Satellite binary can be neglected. This is the case with Earth, Mars, Jupiter, Saturn, Neptune and Uranus. But Mercury and Venus has R1=RP/aH =0.01？and 5.9862 × 10-3 respectively hence they have no satellites. There is a limit to the dimension of the captured body. It must be a much smaller body both dimensionally as well masswise. The qantitative limit is a subject of an independent study.
References
[1]
Cohen, I.B. and Whitman, A. (1687) Newton, Isaac—The Principia. Mathematical Principles of Natural Philosophy, Assisted by Julia Budenz, 5th July.
Hill, G.W. (1898) Review of Darwin’s Periodic Orbits. Astronomical Journal, 18, 120. https://doi.org/10.1086/102833
[11]
Plummer, H.C. (1903) On Oscillating Satellites-1. Monthly Notices of the Royal Astronomical Society, 63, 436-443. https://doi.org/10.1093/mnras/63.8.436
[12]
Moulton, F.R. (1920) Priodic Orbits. Carnegie Institute of Washington Publications, Washington.
[13]
Stromgren, E. (1935) Connaissance actuelle des orbites dans le probleme des trios corps. Copenhagen Observatory Publications, No. 100.
[14]
Broucke, R.A. (1968) Periodic Orbits in the Restricted Three-Body Problem with Earth-Moon Masses. Tech. Rep. 32-1168, Jet Propulsion Laboratory, California Institute of Technology, Pasadena.
[15]
Du Problenon, M.H. (1965) Exploration Numeme des Trois Corps, (I), Masses Egales, Orbites Periodiques. Annales d’Astrophysique, 28, 499-511.
[16]
Du Problenon, M.H. (1965) Exploration Numeme des Trois Corps, (II), Masses Egales, Orbites Periodiques. Annales d’Astrophysique, 28, 992-1007.
[17]
Du Problenon, M.H. (1966) Exploration Numeme des Trois Corps, (III), Masses Egales, Orbites Non Periodiques. Bulletin Astronomique, 1, 57-80.
[18]
Henon, M. (1966) Exploration Numerique du Probleme des Trois Corps, (IV), Masses Egales, Orbites Non Periodiques. Bulletin Astronomique, 1, 49-66.
[19]
Henon, M. (1969) Numerical Exploration of the Restricted Problem. V., Hill’s Case: Periodic Orbits and Their Stability. Astronomy & Astrophysics, 1, 223-238.
[20]
Arenstorf, R.F. (1963) Existence of Periodic Solutions Passing near both Masses of the Restricted Three-Body Problem. AIAA Journal, 1, 238. https://doi.org/10.2514/3.1516
[21]
Goudas, C.L. (1963) Three Dimensional Periodic Orbits and Their Stability. Icarus, 2, 1-18. https://doi.org/10.1016/0019-1035(63)90003-4
[22]
Bray, T.A. and Goudas, C.L. (1967) Doubly Symmetric Orbits about the Collinear Lagrange Points. The Astronomical Journal, 72, 202-213. https://doi.org/10.1086/110218
[23]
Bray, T.A. and Goudas, C.L. (1967) Three Dimensional Periodic Oscillations about L1, L2, L3. Advances in Astronomy and Astrophysics, 5, 71-130. https://doi.org/10.1016/B978-1-4831-9923-8.50007-3
[24]
Kolenkiewicz, R. and Carpenter, L. (1968) Stable Periodic Orbits about the Sun Perturbed Earth-Moon Triangular Points. AIAA Journal, 6, 1301-1304. https://doi.org/10.2514/3.4738
[25]
Farquhar, R.W. (1968) The Control and Use of Libration-Point Satellites. PhD Thesis, Department of Aeronautics and Astronautics, Stanford University, Stanford.
[26]
Farquhar, R.W. and Kamel, A.A. (1973) Quasi-Periodic Orbits about the Translunar Libration Point. Celestial Mechanics, 7, 458-473. https://doi.org/10.1007/BF01227511
[27]
Breakwell, J.V. and Brown, J.V. (1979) The Halo Family of 3-Dimensional Periodic Orbits in the Earth-Moon Restricted 3-Body Problem. Celestial Mechanics, 20, 389-404. https://doi.org/10.1007/BF01230405
Moore, C. (1993) Braids in Classical Dynamics. Physical Review Letters, 70, 3675-3679. https://doi.org/10.1103/PhysRevLett.70.3675
[30]
Broucke, R. and Boggs, D. (1975) Periodic Orbits in the Planar General Three Body Problem. Celestial Mechanics, 11, 13-38. https://doi.org/10.1007/BF01228732
[31]
Hadjidemetrion, J.D. and Christides, Th. (1975) Families of Periodic Orbits in Three Body Problem. Celestial Mechanics, 12, 175-187. https://doi.org/10.1007/BF01230210
[32]
Hadjidemetrion, J.D. (1975) The Continuation of Periodic Orbits from the Restricted to the General Three Body Problem. Celestial Mechanics, 12, 255-276. https://doi.org/10.1007/BF01228563
[33]
Henon, M. (1976) A Family of Periodic Solutions of Planar Three Body Problem and Their Stability. Celestial Mechanics, 13, 267-285. https://doi.org/10.1007/BF01228647
[34]
Henon, M. (1977) Stability of Interplay Motions. Celestial Mechanics, 15, 243-261. https://doi.org/10.1007/BF01228465
[35]
Chenciner, A. and Montgomery, R. (2000) A Remarkable Periodic Solution of the Three Body Problem in the Case of Equal Masses. Annals of Mathematics, 152, 881-901. https://doi.org/10.2307/2661357
[36]
Simo, C. (2002) Celestial Mechanics. Contemporary Mathematics, 292, 209.
[37]
Chenciner, A., Fejoz, J. and Montgomery, R. (2005) The Rotating Eights: 1. The Three Families. Nonlinearity, 18, 1407-1424. https://doi.org/10.1088/0951-7715/18/3/024
[38]
Broucke, R., Elipe, A. and Riagus, A. (2006) On the Figure-8 Periodic Solutions in the Three Body Problem. Celestial Chaos, Solitons & Fractals, 30, 513-520. https://doi.org/10.1016/j.chaos.2005.11.082
[39]
Nauenberg, M. (2007) Continuity and Stability of Families of Figure Eight Orbits with Finite Angular Momentum. Celestial Mechanics, 97, 1-15. https://doi.org/10.1007/s10569-006-9044-7
[40]
Šuvakov, M. and Dmitrašinović, V. (2013) Three Classes of Newtonian Three Body Periodic Orbits. Physical Review Letters, 110, Article ID: 114301. https://doi.org/10.1103/PhysRevLett.110.114301
[41]
Richardson, D.L. (1980) Analytical Construction of Periodic Orbits about the Collinear Points. Celestial Mechanics, 22, 241-253. https://doi.org/10.1007/BF01229511
[42]
Howell, K.C. and Pernicka, H.J. (1988) Numerical Determination of Lissajous Trajectories in the Restricted Three-Body Problem. Celestial Mechanics, 41, 107-124. https://doi.org/10.1007/BF01238756
[43]
Gomez, G., Masdemont, J. and Simo, C. (1998) Quasihalo Orbits Associated with Libration Points. The Journal of the Astronautical Sciences, 46, 135-176. https://doi.org/10.1007/BF03546241
[44]
Gomez, G., Jorba, A., Llibre, J., Martinez, R., Masdemont, J. and Simo, C. (2001) Dynamicsand Mission Design near Libration Points. Vol. I-IV. World Scientific Publishing Co., Singapore. https://doi.org/10.1142/9789812794635
[45]
Clarke, A.C. (1950) Interplanetary Flight. Temple Press Books Ltd., London.
[46]
Farquhar, R.W. (1966) Station-Keeping in the Vicinity of Collinear Libration Points with an Application to a Lunar Communications Problem. In: Space Flight Mechanics, Vol. 11 of Science and Technology Series, American Astronautical Society, New York, 519-535.
[47]
Farquhar, R.W. (1967) Lunar Communications with Libration-Point Satellites. Journal of Spacecraft and Rockets, 4, 1383-1384. https://doi.org/10.2514/3.29095
[48]
Hill, K., Born, G.H. and Lo, M.W. (2005) Linked, Autonomous, Interplanetary Satellite Orbit Navigation (LiAISON) in Lunar Halo Orbits. Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, South Lake Tahoe, 7-11 August 2005, Vol. 123, Paper AAS 05-400.
[49]
Bond, V.R., Sponaugle, S.J., Fraietta, M.F. and Everett, S.F. (1991) Cislunar Libration Point as a Transportation Node for Lunar Exploration. Proceedings of the 1st AAS/AIAA Spaceflight Mechanics Meeting, Houston, 11-13 February 1991, Vol. 75, Paper AAS 91-103.
[50]
NASA Review (2009) Seeking a Human Spaceflight Program Worthy of a Great Nation. Review of U.S. Human Spaceflight Plans Committee, National Aeronautics and Space Administration, September 2009.
[51]
Hill, K. (2007) Autonomous Navigation in Libration Point Orbits. Ph.D. Thesis, University of Colorado, Boulder.
[52]
Hill, K., Lo, M.W. and Born, G.H. (2006) Linked, Autonomous, Interplanetary Satellite Orbit Navigation (LiAISON). Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, South Lake Tahoe, 7-11 August 2005, Vol. 123, Paper AAS 05-399.
[53]
Parker, J.S., Anderson, R.L., Born, G.H. and Fujimoto, K. (2012) Linked Autonomous Interplanetary Satellite Orbit Navigation (LiAISON) between Geosynchronous and Lunar Halo Orbits. Tech. Rep. D-72688 (Internal Document), Jet Propulsion Laboratory, California Institute of Technology, Pasadena.
[54]
Villac, B., Chow, C., Lo, M., Hintz, G. and Nazari, Z. (2010) Dynamic Optimization of Multi-Spacecraft Relative Navigation Configurations in the Earth-Moon System. Proceedings of the AAS George H. Born Symposium, Boulder, 13-14 May 2010.
[55]
Hill, K., Parker, J.S., Born, G.H. and Demandante, N. (2006) A Lunar L2 Navigation, Communication, and Gravity Mission. Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, Keystone, August 2006, Paper AIAA 2006-6662. https://doi.org/10.2514/6.2006-6662
[56]
Hill, K. and Born, G.H. (2007) Autonomous Interplanetary Orbit Determination Using Satellite-to-Satellite Tracking. AIAA Journal of Guidance, Control, and Dynamics, 30, 679-686. https://doi.org/10.2514/1.24574
[57]
Lagrange, J.-L. (1867-92) Tome 6, Chapitre II: Essai sur le problème des trois corps. In: Œuvres de Lagrange, Gauthier-Villars, Paris, 229-334. (In French)
[58]
Koon, W.S., Lo, M.W., Marsden, J.E. and Ross, S.D. (2006) Dynamical Systems, the Three-Body Problem, and Space Mission Design. 9.
[59]
Liu, C. and Dong, L. (2019) Stabilization of Lagrange Points in Circular Restricted Three-Body Problem: A Port-Hamiltonian Approach. Physics Letters A, 383, 1907-1914. https://doi.org/10.1016/j.physleta.2019.03.033
Laufer, R., Tost, W., Zielie, O., et al. (2007) The Karodylewski Clouds—An Example for a Cruise Phase Observation during the Lunar Mission BW1. 11th ISU Annual International Symposium, Strasbourg, 21-23 February 2007, 1-5.
[62]
Connors, M., Weigert, P. and Veillet, C. (2011) Earth’s Trojan Asteroid. Nature, 475, 481-483. https://doi.org/10.1038/nature10233
[63]
Kokubo, E., Canup, R.M. and Ida, S. (2000) Lunar Accretion from an Impact Generated Disk. In: Canup, R. and Righter, K., Eds., Origin of Earth and Moon, University of Arizona Press, Tucson, 145-163. https://doi.org/10.2307/j.ctv1v7zdrp.14
