  Intermediate mass stars and stellar remnants often host planets, and these
dynamical systems evolve because of mass loss and tides. This paper considers
the combined action of stellar mass loss and tidal dissipation on planetary
orbits in order to determine the conditions required for planetary survival.
Stellar mass loss is included using a so-called Jeans model, described by a
dimensionless mass loss rate \gamma and an index \beta. We use an analogous
prescription to model tidal effects, described here by a dimensionless
dissipation rate \Gamma and two indices (q,p). The initial conditions are
determined 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 the
critical dissipation rate \Gamma, which marks the boundary between orbits that
spiral outward due to stellar mass loss and those that spiral inward due to
tidal dissipation. This analytic result
\Gamma=\Gamma(\gamma,\beta,q,p,\eta,\theta) is essentially exact for initially
circular orbits and holds to within an accuracy of 50% over the entire
multi-dimensional parameter space, where the individual parameters vary by
several orders of magnitude. For stars that experience mass loss, the stellar
radius often displays quasi-periodic variations, which produce corresponding
variations in tidal forcing; we generalize the calculation to include such
pulsations using a semi-analytic treatment that holds to the same accuracy as
the non-pulsating case. These results can be used in many applications, e.g.,
to predict/constrain properties of planetary systems orbiting white dwarfs.
