Non-linear Dynamical Stability of Magnetic Polytropes

Abstract

This work analyzes the non-linear dynamical stability of ideal-gas polytropes under homologous flow. A non-constant density profile requires the inclusion of magnetic fields, which is done by introducing a mean-field model that treats the spherically-averaged radial Lorentz force self-consistently and has the following properties: 1) The only essential simplifications are the Cowling approximation and a dominant radial flow. 2) The average radial Lorentz force due to an isotropic field is $-\frac13 dP_B/dr$, not $-dP_B/dr$ as is typically assumed. 3) A central peak in the magnetic field requires isotropy there; all other configurations are zero at the origin due to magnetic tension. 4) Solutions with negligible surface fields require $\gtrsim1/2$ of the magnetic energy to be in the radial component. 5) Solutions that resemble Lane-Emden solutions are restricted to $\gamma = 4/3$, where $\gamma$ is the material adiabatic index, and exhibit either collapse or escape. 6) Solutions for general $\gamma$ have a harmonic enthalpy profile and allow for non-linear radial pulsations. 7) A harmonic-enthalpy homologous flow becomes unbound when an overpressure satisfies $\delta = \Delta P_0/P_{\rm eq} > \frac{3\gamma - 4}{1 + 3(\gamma-1)\alpha_0}$, where $P$ is the total pressure, $P_{\rm eq}$ is its equilibrium value, $\alpha$ is the ratio of radiation to material pressure, and a zero subscript denotes minimum volume. This indicates that radiation pressure can unbind a linearly-stable polytrope in the presence of small but finite radial perturbations. The condition to unbind a fully-ionized $n = 3$ polytrope with $2/3$ of its magnetic energy in the radial component is $\delta \gtrsim 0.15\mu^{-1}\left(300M_\odot/M\right)^{1/2}$, where $\mu$ is the mean molecular weight. This non-linear dynamical instability threshold may have some relevance for mass loss in and dispersal of evolved high-mass stars.