Astronomy and Astrophysics – Astronomy
Scientific paper
Apr 2011
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2011dda....42.0603c&link_type=abstract
American Astronomical Society, DDA meeting #42, #6.03; Bulletin of the American Astronomical Society, Vol. 43, 2011
Astronomy and Astrophysics
Astronomy
Scientific paper
Horseshoe orbits are one of the possible stable configurations for two bodies sharing the same orbit. It is generally accepted that all horseshoe orbits have finite lifetimes, proportional to the primary-to-secondary mass ratio to the 5/3 power, greatly favouring small objects as horseshoe secondaries (Dermott and Murray, 1981). This would put the lifetime of Earth and Venus horseshoe companions at about 0.5-1 Gyr. However, my direct numerical integrations found the horseshoe coorbitals of both planets to be essentially stable at even longer timescales (Cuk 2009, DPS abstract). Further numerical experiments found that the stability boundary appears to be at mass ratios of about 1200 (with no stable horseshoes for even smaller ratios), assuming a massless ternary. An almost identical result was obtained by Laughlin and Chambers (2002) for equal-mass planet pairs in a horseshoe configuration. This is consistent with the instability of Jupiter's horseshoe coorbitals (Stacey and Connors, 2008), as Sun/Jupiter mass ratio is 1047. This stability boundary appears because the horns of the horseshoe have to be within 23.5 degrees from the secondary, in order to stay outside the L_3-level zero-velocity curve (Murray and Dermott, 1999). Depending on the secondary mass, this distance (fixed in degrees) corresponds to different numbers of Hill radii. Once the particle approaches within about 5.5-6 Hill radii of the primary, perturbations start accumulating and the horseshoe becomes unstable, producing the observed stability limit. I conclude that there is every reason to believe that horseshoe orbits are intrinsically stable for all larger mass ratios (i.e. all smaller secondaries). The result of Dermott and Murray (1981) that large mass ratios offer relatively more phase space to horseshoes still holds, explaining why the first (and only) observed horseshoe pair happen to be tiny Janus and Epimetheus.
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