J/A+A/605/A23Restricted planar three-body problem (Pichierri+, 2017)

Extreme secular excitation of eccentricity inside mean motion resonance: Small bodies driven into star-grazing orbits by planetary perturbations. Pichierri G., Morbidelli A., Lai D. <Astron. Astrophys. 605, A23 (2017)> =2017A&A...605A..23P (SIMBAD/NED BibCode)ADC_Keywords: Models ; Minor planetsKeywords: celestial mechanics - planets and satellites: dynamical evolution and stability - minor planets, asteroids: general - white dwarfs - methods: analyticalAbstract: 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 (planetesimals) 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 (e'≲0.1) 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 ∼1AU 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. Planetesimals can easily fall onto the central star even in the presence of a single moderately eccentric planet, but only from the vicinity of the 4:1 mean motion resonance. For sufficiently high planetary masses the General Relativity effect does not prevent the achievement of star-grazing orbits.Description: In this paper, we will focus mainly on resonances of order higher than 1, because they are much more efficient in pushing the eccentricity e from ∼0 to ∼1. In this case, for e<e' our approach is valid down to very small values of the eccentricity. We make available a Mathematica notebook which implements the calculations outlined in the paper, to allow the interested reader to examine the effect of secular dynamics inside mean motion resonances for other applications.File Summary:

FileName Lrecl Records Explanations

ReadMe 80 . This file SecResInMMR.nb 82 42469 Mathematica notebook (vnd.wolfram.nb application)

Acknowledgements: Gabriele Pichierri, Gabriele.Pichierri(at)oca.eu(End)Patricia Vannier [CDS] 12-Jun-2017

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