Abstract
We study the post-encounter evolution of fictitious small bodies belonging to the so-called Line of Variations (LoV) in the framework of the analytic theory of close encounters. We show the consequences of the encounter on the local minimum of the distance between the orbit of the planet and that of the small body and get a global picture of the way in which the planetocentric velocity vector is affected by the encounter. The analytical results are compared with those of numerical integrations of the restricted three-body problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The components of \(\mathbf {U}'\) in the X–Y–Z reference frame are (\(U\sin \theta '\sin \phi ',~U\cos \theta ',~U\sin \theta '\cos \phi '\)).
References
Carusi, A., Valsecchi, G.B., Greenberg, R.: Planetary close encounters - geometry of approach and post-encounter orbital parameters. Celest. Mech. Dyn. Astron. 49, 111–131 (1990)
Everhart, E.: An efficient integrator that uses Gauss–Radau spacings. In: Carusi, A., Valsecchi, G.B. (eds.) Dynamics of Comets: Their Origin and Evolution, p. 185. Reidel, Dordrecht (1985)
Greenberg, R., Carusi, A., Valsecchi, G.B.: Outcomes of planetary close encounters: a systematic comparison of methodologies. Icarus 75, 1–29 (1988)
Kizner, W.: A method of describing miss distances for lunar and interplanetary trajectories. Planet. Space Sci. 7, 125–131 (1961)
Milani, A.: The asteroid identification problem. I. Recovery of lost asteroids. Icarus 137, 269–292 (1999)
Milani, A., Chesley, S.R., Sansaturio, M.E., Tommei, G., Valsecchi, G.B.: Nonlinear impact monitoring: line of variation searches for impactors. Icarus 173, 362–384 (2005)
Öpik, E.J.: Interplanetary Encounters: Close-Range Gravitational Interactions. Elsevier, Amsterdam (1976)
Valsecchi, G.B.: Geometric conditions for Quasi-collisions in Öpik’s Theory. In: Souchay, J. (ed.) Dynamics of Extended Celestial Bodies and Rings. Lecture Notes in Physics, vol. 682, p. 145. Springer, Berlin (2006)
Valsecchi, G.B., Milani, A., Gronchi, G.F., Chesley, S.R.: The distribution of energy perturbations at planetary close encounters. Celest. Mech. Dyn. Astron. 78, 83–91 (2000)
Valsecchi, G.B., Milani, A., Gronchi, G.F., Chesley, S.R.: Resonant returns to close approaches: analytical theory. Astron. Astrophys. 408, 1179–1196 (2003)
Valsecchi, G.B., Alessi, E.M., Rossi, A.: An analytical solution for the swing-by problem. Celest. Mech. Dyn. Astron. 123, 151–166 (2015a)
Valsecchi, G.B., Alessi, E.M., Rossi, A.: Erratum to: an analytical solution for the swing-by problem. Celest. Mech. Dyn. Astron. 123, 167–167 (2015b)
Valsecchi, G.B., Alessi, E.M., Rossi, A.: Cartography of the b-plane of a close encounter I: semimajor axes of post-encounter orbits. Celest. Mech. Dyn. Astron. 130, 8 (2018)
Acknowledgements
We are grateful to D. Farnocchia for his very useful comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Valsecchi, G.B., Del Vigna, A. & Ceccaroni, M. The evolution of the Line of Variations at close encounters: an analytic approach. Celest Mech Dyn Astr 131, 47 (2019). https://doi.org/10.1007/s10569-019-9924-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10569-019-9924-2