2002ApJ...573..829M

Based on formulations by Heggie and by Eggleton, we present an efficient method for calculating self-consistently the tidal, spin, and dynamical evolution of a many-body system, here with particular emphasis on planetary systems. The star and innermost planet (or in general the closest pair of bodies in the system) are endowed with structure while the other bodies are treated as point masses. The evolution of the spin rates and obliquities of the extended bodies are calculated (for arbitrary initial obliquities), as is the tidal evolution of the innermost orbit. In addition, the radius of the innermost planet is evolved according to its ability to efficiently dissipate tidal energy. Relativistic effects are included to post-Newtonian order. For resonant systems such as GJ 876, the evolution equations must be integrated directly to allow for variation of the semimajor axes (other than from tidal damping) and for the possibility of instability. For systems such as Upsilon Andromedae in which the period ratio of the two inner planets is small, the innermost orbit may be averaged producing (in this case) a 50-fold reduction in the calculation time. In order to illustrate the versatility of the formulation, we consider three hypothetical primitive Earth-Moon-Sun-Jupiter systems. The parameters and initial conditions are identical in the first two models except for the Love number of the Earth, which results in dramatically different evolutionary paths. The third system is one studied by Touma & Wisdom and serves as a test of the numerical formulations presented here by reproducing two secular mean motion resonances (the evection and eviction resonances). The methods may be used for any system of bodies.