**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 C^{1} 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