  We present a self-consistent model for the tidal evolution of circumbinary
planets. Based on the weak-friction model, we derive expressions of the
resulting forces and torques considering complete tidal interactions between
all the bodies of the system. Although the tidal deformation suffered by each
extended mass must take into account the combined gravitational effects of the
other two bodies, the only tidal forces that have a net effect on the dynamic
are those that are applied on the same body that exerts the deformation, as
long as no mean-motion resonance exists between the masses. We apply the model
to the Kepler-38 binary system. The evolution of the spin equations shows that
the planet reaches a stationary solution much faster than the stars, and the
equilibrium spin frequency is sub-synchronous. The binary components evolve on
a longer timescale, reaching a super-synchronous solution very close to that
derived for the 2-body problem. After reaching spin stationarity, the
eccentricity is damped in all bodies and for all the parameters analyzed here.
A similar effect is noted for the binary separation. The semimajor axis of the
planet, on the other hand, may migrate inwards or outwards, depending on the
masses and orbital parameters. In some cases the secular evolution of the
system may also exhibit an alignment of the pericenters, requiring to include
additional terms in the tidal model. Finally, we derived analytical expressions
for the variational equations of the orbital evolution and spin rates based on
low-order elliptical expansions in the semimajor axis ratio and the
eccentricities. These are found to reduce to the 2-body case when one of the
masses is taken equal to zero. This model allow us to find a close and simple
analytical expression for the stationary spin rates of all the bodies, as well
as predicting the direction and magnitude of the orbital migration.

Earth and Planetary Astrophysics