American Institute of Mathematical Sciences

July  2013, 18(5): 1323-1344. doi: 10.3934/dcdsb.2013.18.1323

The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities

 1 Dipartimento di Matematica, Universitá di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy 2 Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127, Pisa, Italy

Received  June 2012 Revised  January 2013 Published  March 2013

We study the long term evolution of the distance between two Keplerian confocal trajectories in the framework of the averaged restricted 3-body problem. The bodies may represent the Sun, a solar system planet and an asteroid. The secular evolution of the orbital elements of the asteroid is computed by averaging the equations of motion over the mean anomalies of the asteroid and the planet. When an orbit crossing with the planet occurs the averaged equations become singular. However, it is possible to define piecewise differentiable solutions by extending the averaged vector field beyond the singularity from both sides of the orbit crossing set [8],[5]. In this paper we improve the previous results, concerning in particular the singularity extraction technique, and show that the extended vector fields are Lipschitz-continuous. Moreover, we consider the distance between the Keplerian trajectories of the small body and of the planet. Apart from exceptional cases, we can select a sign for this distance so that it becomes an analytic map of the orbital elements near to crossing configurations [11]. We prove that the evolution of the `signed' distance along the averaged vector field is more regular than that of the elements in a neighborhood of crossing times. A comparison between averaged and non-averaged evolutions and an application of these results are shown using orbits of near-Earth asteroids.
Citation: Giovanni F. Gronchi, Chiara Tardioli. The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1323-1344. doi: 10.3934/dcdsb.2013.18.1323
References:
 [1] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics," Springer, 1997. [2] R. V. Baluyev and K. V. Kholshevnikov, Distance between Two Arbitrary Unperturbed Orbits, Cel. Mech. Dyn. Ast., 91 (2005), 287-300. doi: 10.1007/s10569-004-3207-1. [3] E. Bowell and K. Muinonen, Earth-crossing asteroids and comets: Groundbased search strategies, in "Hazards due to Comets & Asteroids" (Ed. T. Gehrels), Tucson: The University of Arizona Press, (1994), 149-197. [4] R. A. Broucke and P. J. Cefola, On the equinoctial orbit elements, Cel. Mech. Dyn. Ast., 5 (1972), 303-310. [5] G. F. Gronchi, Generalized averaging principle and the secular evolution of planet crossing orbits, Cel. Mech. Dyn. Ast., 83 (2002), 97-120. doi: 10.1023/A:1020178613365. [6] G. F. Gronchi, On the stationary points of the squared distance between two ellipses with a common focus, SIAM Journ. Sci. Comp., 24 (2002), 61-80. doi: 10.1137/S1064827500374170. [7] G. F. Gronchi, An algebraic method to compute the critical points of the distance function between two Keplerian orbits, Cel. Mech. Dyn. Ast., 93 (2005), 297-332. doi: 10.1007/s10569-005-1623-5. [8] G. F. Gronchi and A. Milani, Averaging on Earth-crossing orbits, Cel. Mech. Dyn. Ast., 71 (1998), 109-136. doi: 10.1023/A:1008315321603. [9] G. F. Gronchi and A. Milani, Proper elements for Earth crossing asteroids, Icarus, 152 (2001), 58-69. [10] G. F. Gronchi and P. Michel, Secular orbital evolution, proper elements and proper frequencies for near-earth asteroids: A comparison between semianalytic theory and numerical integrations, Icarus, 152 (2001), 48-57. [11] G. F. Gronchi and G. Tommei, On the uncertainty of the minimal distance between two confocal Keplerian orbits, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 755-778. doi: 10.3934/dcdsb.2007.7.755. [12] K. V. Kholshevnikov and N. Vassiliev, On the distance function between two keplerian elliptic orbits, Cel. Mech. Dyn. Ast., 75 (1999), 75-83. doi: 10.1023/A:1008312521428. [13] H. Kinoshita and H. Nakai, General solution of the Kozai mechanism, Cel. Mech. Dyn. Ast., 98 (2007), 67-74. doi: 10.1007/s10569-007-9069-6. [14] Y. Kozai, Secular perturbation of asteroids with high inclination and eccentricity, Astron. Journ., 67 (1962), 591-598. [15] M. L. Lidov, The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies, Plan. Spa. Sci., 9 (1962), 719-759. [16] A. Milani, S. R. Chesley, M. E. Sansaturio, G. Tommei and G. Valsecchi, Nonlinear impact monitoring: Line of variation searches for impactors, Icarus, 173 (2005), 362-384. [17] A. Milani and G. F. Gronchi, "Theory of Orbit Determination," Cambridge Univ. Press, 2010. [18] G. B. Valsecchi, A. Milani, G. F. Gronchi and S. R. Chesley, Resonant returns to close approaches: Analytical theory, Astron. Astrophys., 408 (2003), 1179-1196. [19] A. Whipple, Lyapunov times of the inner asteroids, Icarus, 115 (1995), 347-353.

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References:
 [1] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics," Springer, 1997. [2] R. V. Baluyev and K. V. Kholshevnikov, Distance between Two Arbitrary Unperturbed Orbits, Cel. Mech. Dyn. Ast., 91 (2005), 287-300. doi: 10.1007/s10569-004-3207-1. [3] E. Bowell and K. Muinonen, Earth-crossing asteroids and comets: Groundbased search strategies, in "Hazards due to Comets & Asteroids" (Ed. T. Gehrels), Tucson: The University of Arizona Press, (1994), 149-197. [4] R. A. Broucke and P. J. Cefola, On the equinoctial orbit elements, Cel. Mech. Dyn. Ast., 5 (1972), 303-310. [5] G. F. Gronchi, Generalized averaging principle and the secular evolution of planet crossing orbits, Cel. Mech. Dyn. Ast., 83 (2002), 97-120. doi: 10.1023/A:1020178613365. [6] G. F. Gronchi, On the stationary points of the squared distance between two ellipses with a common focus, SIAM Journ. Sci. Comp., 24 (2002), 61-80. doi: 10.1137/S1064827500374170. [7] G. F. Gronchi, An algebraic method to compute the critical points of the distance function between two Keplerian orbits, Cel. Mech. Dyn. Ast., 93 (2005), 297-332. doi: 10.1007/s10569-005-1623-5. [8] G. F. Gronchi and A. Milani, Averaging on Earth-crossing orbits, Cel. Mech. Dyn. Ast., 71 (1998), 109-136. doi: 10.1023/A:1008315321603. [9] G. F. Gronchi and A. Milani, Proper elements for Earth crossing asteroids, Icarus, 152 (2001), 58-69. [10] G. F. Gronchi and P. Michel, Secular orbital evolution, proper elements and proper frequencies for near-earth asteroids: A comparison between semianalytic theory and numerical integrations, Icarus, 152 (2001), 48-57. [11] G. F. Gronchi and G. Tommei, On the uncertainty of the minimal distance between two confocal Keplerian orbits, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 755-778. doi: 10.3934/dcdsb.2007.7.755. [12] K. V. Kholshevnikov and N. Vassiliev, On the distance function between two keplerian elliptic orbits, Cel. Mech. Dyn. Ast., 75 (1999), 75-83. doi: 10.1023/A:1008312521428. [13] H. Kinoshita and H. Nakai, General solution of the Kozai mechanism, Cel. Mech. Dyn. Ast., 98 (2007), 67-74. doi: 10.1007/s10569-007-9069-6. [14] Y. Kozai, Secular perturbation of asteroids with high inclination and eccentricity, Astron. Journ., 67 (1962), 591-598. [15] M. L. Lidov, The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies, Plan. Spa. Sci., 9 (1962), 719-759. [16] A. Milani, S. R. Chesley, M. E. Sansaturio, G. Tommei and G. Valsecchi, Nonlinear impact monitoring: Line of variation searches for impactors, Icarus, 173 (2005), 362-384. [17] A. Milani and G. F. Gronchi, "Theory of Orbit Determination," Cambridge Univ. Press, 2010. [18] G. B. Valsecchi, A. Milani, G. F. Gronchi and S. R. Chesley, Resonant returns to close approaches: Analytical theory, Astron. Astrophys., 408 (2003), 1179-1196. [19] A. Whipple, Lyapunov times of the inner asteroids, Icarus, 115 (1995), 347-353.
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