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Intermediate mass stars and stellar remnants often host planets, and thesedynamical systems evolve because of mass loss and tides. This paper considersthe combined action of stellar mass loss and tidal dissipation on planetaryorbits in order to determine the conditions required for planetary survival.Stellar mass loss is included using a so-called Jeans model, described by adimensionless mass loss rate \gamma and an index \beta. We use an analogousprescription to model tidal effects, described here by a dimensionlessdissipation rate \Gamma and two indices (q,p). The initial conditions aredetermined by the starting value of angular momentum parameter \eta(equivalently, the initial eccentricity) and the phase \theta of the orbit.Within the context of this model, we derive an analytic formula for thecritical dissipation rate \Gamma, which marks the boundary between orbits thatspiral outward due to stellar mass loss and those that spiral inward due totidal dissipation. This analytic result\Gamma=\Gamma(\gamma,\beta,q,p,\eta,\theta) is essentially exact for initiallycircular orbits and holds to within an accuracy of 50% over the entiremulti-dimensional parameter space, where the individual parameters vary byseveral orders of magnitude. For stars that experience mass loss, the stellarradius often displays quasi-periodic variations, which produce correspondingvariations in tidal forcing; we generalize the calculation to include suchpulsations using a semi-analytic treatment that holds to the same accuracy asthe non-pulsating case. These results can be used in many applications, e.g.,to predict/constrain properties of planetary systems orbiting white dwarfs.

Evolution of Planetary Orbits with Stellar Mass Loss and Tidal Dissipation