Authors: Michele Bartuccelli, Jonathan Deane, Guido Gentile

Title: Numerics for the spin-orbit equation of Makarov with constant eccentricity
 
Abstract: We present an algorithm for the rapid numerical integration of a time-periodic ODE with a small dissipation term that is C¹ in the velocity. Such an ODE arises as a model of spin-orbit coupling in a star/planet system, and the motivation for devising a fast algorithm for its solution comes from the desire to estimate probability of capture in various solutions, via Monte Carlo simulation: the integration times are very long, since we are interested in phenomena occurring on times similar to the formation time of the planets. The proposed algorithm is based on the High-order Euler Method (HEM) which was described in a previou paper of ours [Celestial Mech. Dynam. Astronom. 121 (2015), 233-260], and it requires computer algebra to set up the code for its implementation. The pay-off is an overall increase in speed by a factor of about 7.5 compared to standard numerical methods. Means for accelerating the purely numerical computation are also discussed.

Keywords: Spin-orbit model; Dissipative systems; Forced systems; High-order Euler method; Fast numerical integration; Attractors.

Michele V. Bartuccelli
Department of Mathematics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: gentile@mat.uniroma3.it Jonathan Deane
Department of Mathematics
University of Surrey
Guildford, GU2 7HX, UK
e-mail: j.deane@surrey.ac.uk

Guido Gentile
Dipartimento di Matematica e Fisica
Università Roma Tre
Largo San Leonardo Murialdo 1 - 00146 Roma - Italy
e-mail: gentile@mat.uniroma3.it