Dynamical stability of giant planets: the critical adiabatic index in the presence of a solid core

The dissociation and ionization of hydrogen, during the formation of giant planets via core accretion, reduces the effective adiabatic index $\gamma$ of the gas and could trigger dynamical instability. We generalize the analysis of Chandrasekhar, who determined that the threshold for instability of a self-gravitating hydrostatic body lies at $\gamma=4/3$, to account for the presence of a planetary core, which we model as an incompressible fluid. We show that the dominant effect of the core is to stabilize the envelope to radial perturbations, in some cases completely (i.e. for all $\gamma > 1$). When instability is possible, unstable planetary configurations occupy a strip of $\gamma$ values whose upper boundary falls below $\gamma=4/3$. Fiducial evolutionary tracks of giant planets forming through core accretion appear unlikely to cross the dynamical instability strip that we define.