The Hill stability of binary asteroid and binary Kuiper Belt systems

Abstract

The dynamical stability of a bound triple system composed of a binary asteroid system or Kuiper Belt binary system moving on an orbit inclined to a central third body, the Sun, is discussed in terms of Hill stability for the full three‐body problem. The regions of Hill stability of these triple systems, where the binary mass is very small compared with that of the third body, can be determined against the possibility of disruption, component exchange and capture. The critical Hill stability curves for the binary mass range of these types of systems are determined for different secondary‐to‐primary mass ratios as a function of their orbital eccentricity. The regions of stability are found to increase with increasing binary mass. The regions, however, decrease in size substantially with increasing orbital eccentricity and also decrease slightly as the secondary/primary mass ratio of the binary is decreased. The currently observed binary and multiple asteroid systems are discussed generally. In the majority of systems, the primary component is very much larger than the secondary component, forming an asteroid–satellite system. It was found that those systems where the binary mass is well determined would lie in stable regions if they moved on circular orbits, but when their eccentricity is taken into account, it is less clear that the systems are stable. The same is likely to be true for the systems where the masses are not well established. Upper mass limits could be placed on these systems that would ensure they are Hill stable. The currently observed Kuiper Belt binaries were also discussed generally. The majority of these binary systems have secondary components which are often comparable to the diameter of the primary component forming a true binary system. Similar to the asteroid binaries, it was found that binary systems where the mass was well determined were stable if they moved on circular orbits relative to the Sun. When the eccentricity is taken into account, it is less clear that the systems are stable. The same conclusions are also likely to be true for the systems with unknown masses. Upper mass limits again can be placed on these systems that would ensure they are Hill stable.

Limits on the Orbits of Possible Eccentric and Inclined Moons of Extrasolar Planets Orbiting Single Stars

“…For a hierarchical triple system, such as we are considering, the distance separation ratio a 2 =a 1 is large and can be considered to vary as x À3 0 (Li et al 2010). We have therefore separated out one of the terms of order x 4:5 0 ; as this term which varies as a 2 =a 1 ð Þ 1 2 x 4:5 0 can be considered as of order x 3 0 (see Donnison 2011), similarly the x 6 0 term, omitted by Li et al (2010), which varies as a 2 =a 1 ð Þx 6 0 can be considered of order x 3 0 and should also be retained By contrast the term involving a 1 =a 2 ð Þx 3 0 would be of order x 6 0 and all the other x 4:5 0 and x 6 0 terms in the expansion can be safely neglected. Therefore retaining only terms of up to order x 3 0 ; the critical condition S ac À S cr ¼ 0 can now be written in the slightly amended form, with decreasing powers of…”

The Stability and Fates of Hierarchical Two-Planet Systems

“…The Hill stability condition has been extended by Veras & Armitage (2004) and Donnison (2006Donnison ( , 2011 to arbitrary mutual inclinations i m . Even though the Hill stability might not determine the long-term stability of a two-planet system (see §2.1), we will use it as a benchmark (Barnes & Greenberg 2006, 2007.…”

Detectable close-in planets around white dwarfs through late unpacking

“…If that distance instead exceeds the Lagrange stability boundary, then the planets will forever remain bounded and ordered. The Hill stability boundary can be expressed in Jacobi coordinates for arbitrary eccentricities and inclinations (Donnison 2011) and is entirely analytical, except for a usually-negligible truncation in the expression of the energy (Fig. 19 of 3 .…”