The Hill stability of low mass binaries in hierarchical triple systems

Abstract

The Hill stability of the low mass binary system in the presence of a massive third body moving on a wider inclined orbit is investigated analytically. It is found that, in the case of the third body being on a nearly circular orbit, the region of Hill stability expands as the binary/third body mass ratio increases and the inclination (i) decreases. This i-dependence decreases very quickly with increasing eccentricity (e 2 ) of the third body relative to the binary barycentre. In fact, if e 2 is not extremely small, the Hill stable region can be approximately expressed in a closed form by setting i = 90 • , and it contracts with increasing e 2 as e 2 2 for sufficiently low mass binary. Our analytic results are then applied to the observed triple star systems and the Kuiper belt binaries.

The Hill stability of binary asteroid and binary Kuiper Belt systems

“…In what follows, we will draw on some aspects of the previous work of Donnison (2010) and Li, Fu & Sun (2010). In this case, where M 3 ≫ M 1 + M 2 , a closed solution is possible.…”

“…However, for hierarchical triple systems, a 2 / a 1 is large, so that for the leading term in a 2 / a 1 in to be of order unity, we require that should be at most of order unity, so that can at most be of order x −3 0 . The assumption that a 2 / a 1 ∼ x −3 0 made by Li et al (2010) is therefore a possibility. In this case, the second term in which varies as is really only of order x 3 0 and should be retained, while the last term, which is the lowest order term in a 2 / a 1 , varies as and is of order x 6 0 and can be neglected.…”

“…If only terms of order x 3 0 or below are retained, the remaining six terms in can then be written in the form of a standard cubic equation in ( a 2 / a 1 ) 1/2 , which can be solved using Cardano’s method. Li et al (2010) found that this cubic equation has a real critical solution which could be written in the closed form as It is this equation which allows us to determine the critical a 2 / a 1 values for Hill stability in terms of the masses and orbital parameters e 2 and e 1 , and i .…”

Analytical criteria of Hill stability in the elliptic restricted three body problem

“…In the planar three body system, the angular momentum and energy of the system are determined by the masses of the three bodies and the semi-major axes and eccentricities of the two subsystems. Many researchers discussed the analytical criteria for different systems based on different assumption on the systems Williams 1983, 1985;Donnison 1984Donnison , 1988Donnison , 2010Donnison , 2011Walker et al 1980Walker et al , 1981Walker 1983;Walker and Roy 1983;Valsecchi et al 1984;Georgakarakos 2008;Li et al 2010;Liu et al 2012). The dependence of the stability on the orbital inclination in the spatial three body system was also studied widely Khodykin et al 2004;Donnison 2006Donnison , 2009Donnison , 2014Georgakarakos 2013).…”

Sundman stability of natural planet satellites

“…In a large series of works by Szebehely & Zare (1977), Walker, Emslie & Roy (1980), Donnison & Williams (1983), Donnison (2009), Li, Fu & Sun (2010), Donnison (2010) and other authors, the determination of the regions of possible motion is carried out in the six‐dimensional space of the Keplerian elements a 1 , a 2 , e 1 , e 2 , i 1 and i 2 or in the four‐dimensional space a 1 , a 2 , e 1 and e 2 for the planar problem.…”