Extreme Secular Excitation of Eccentricity Inside Mean Motion Resonance: Driving Small Bodies into Star-Grazing Orbits by Planetary Perturbations

Abstract

It is well known that asteroids and comets fall into the Sun. Metal pollution of white dwarfs and transient spectroscopic signatures of young stars like $\beta$-Pic provide growing evidence that extra solar planetesimals can attain extreme orbital eccentricities and fall onto their parent stars. We aim to develop a general, practically implementable, semi-analytical theory of secular eccentricity excitation of small bodies in mean motion resonances with an eccentric planet valid for arbitrary values of the eccentricities and including the short-range force due to General Relativity. Our semi-analytic model for the restricted planar three-body problem does not make use of any series expansion and therefore is valid for any values of eccentricities and semi-major axes ratios. The model is based on the application of the adiabatic principle, which is valid when the precession period of the longitude of pericenter of the planetesimal is much longer than the libration period in the mean motion resonance. This holds down to vanishingly small eccentricities in resonances of order larger than 1. We provide a Mathematica notebook with the implementation of the model allowing direct use to the interested reader. We confirm that the 4:1 mean motion resonance with a moderately eccentric planet is the most powerful one to lift the eccentricity of planetesimals from nearly circular orbits to star-grazing ones. However, if the planet is too eccentric, we find that this resonances becomes unable to pump the planetesimal's eccentricity to very high value. The inclusion of the General Relativity effect imposes a condition on the mass of the planet to drive the planetsimals into star-grazing orbits. For a planetesimal at $\sim$1 AU around a solar-mass star (or white dwarf), we find a threshold planetary mass of about 17 Earth masses. We finally derive an analytical formula for this critical mass.

arXiv commentIn press in Astronomy & Astrophysics

References in corpus (6)

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• The frequency of planetary debris around young white dwarfs
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• Fomalhaut b could be massive and sculpting the narrow, eccentric debris disc, if in mean-motion resonance with it
• K2-19b and c are in a 3:2 Commensurability but out of Resonance: A Challenge to Planet Assembly by Convergent Migration
• Exocomet Orbit Fitting: Accelerating Coma Absorption During Transits of $\beta$ Pictoris
• The critical binary star separation for a planetary system origin of white dwarf pollution