Application minimal orbit intersection distance calculation to studying young asteroid pairs
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ORIGINAL ARTICLE
Application minimal orbit intersection distance calculation to studying young asteroid pairs A. Rosaev1
Received: 7 August 2019 / Accepted: 8 November 2019 © Springer Nature B.V. 2019
Abstract A method for calculating the minimum distance between two almost overlapping elliptical orbits is proposed. An expression is obtained for the longitude of the minimum distance point as a function of the orbital elements. Applications are considered of the method for determining the age of pairs of asteroids in close-by orbits. Keywords Minimal orbital intersection distance · Asteroid families · Orbital evolution
1 Introduction Finding a common origin for various celestial bodies remains one of the important problems in Solar system astronomy. A very detailed review of orbital similarity criteria is given by Rozek et al. (2011). Different natural metrics on the five dimensional space of Keplerian elliptic orbits are introduced by Kholshevnikov with colleagues (Kholshevnikov and Vassiliev 2004; Kholshevnikov et al. 2016). On the other hand, the Minimum Orbit Intersection Distance (MOID) between two confocal Keplerian orbits is a useful tool to see when two celestial bodies can collide or undergo a very close approach (Gronchi 2005). Moreover, MOID can be used in the problem of the search for the epoch of breakup. Gronchi (2002), Gronchi et al. (2006) suppose an algorithm based on the algebraic elimination theory: through the computation of the resultant of two bivariate polynomials. They find a 16th degree univariate polynomial whose real roots give us one component of the critical points. This very
B A. Rosaev 1
Research and Educational Center “Nonlinear Dynamics”, Yaroslavl State University, Yaroslavl, Russian Federation
important result has independently been confirmed by Kholshevnikov and Vassiliev (1999) and in early our publication (Rosaev 2001). On the other hand, at present asteroid pairs on close orbits are studied very intensively by Pravec and Vokrouhlicky (2009), Novakovic et al. (2012), and other authors (see Pravec et al. 2019 for the present day state of the problem). These pairs are characterized by a very small relative velocity (about few m/s) and the most probable mechanism of they origin is rotational fission of rapidly rotated elongated body (Pravec et al. 2010). For this reason, the special case when the orbits are almost overlapping is interesting. In this paper, we describe an efficient straightforward (practically applicable) algorithm to compute all the critical points of the distance function between two almost overlapping elliptic Keplerian orbits with a common focus. The results are applied to the problem of age estimation of some minor bodies of the Solar system with a common origin. We do not have the goal to give a universal method, substituting existing methods for determinations of the age of the asteroid pair. However, we believe that, in combination with existing tools, the method presented can help to significantly reduce the errors in the pair age estimations. Moreov
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