Hydrodynamical simulations and similarity relations for eruptive mass loss from massive stars

Motivated by the eruptive mass loss inferred from Luminous Blue Variable (LBV) stars, we present 1D hydrodynamical simulations of the response from sudden energy injection into the interior of a very massive ($100 \, M_\odot$) star. For a fiducial case with total energy addition set to a factor $f=0.5$ of the net stellar binding energy, and applied within the stellar envelope, we detail the dynamical response that leads to ejection of the outermost $7.2 \, M_\odot$. We find that the ejecta's variations in time $t$ and radius $r$ for the velocity $v$, density $\rho$, and temperature $T$ are quite well fit by similarity forms in the variable $r/t \approx v$. Specifically the scaled density follows a simple exponential decline $\rho t^{3} \sim \exp (-r/v_{\rm o} t)$. This `exponential similarity' leads to analytic scaling relations for total ejecta mass $\Delta M$ and kinetic energy $\Delta K$ that agree well with the hydrodynamical simulations, with the specific-energy-averaged speed related to the exponential scale speed $v_{\rm o}$ through ${\bar v} \equiv \sqrt{2 \Delta K/\Delta M} = \sqrt{12} \, v_{\rm o}$, and a value comparable to the star's surface escape speed, $v_{\rm esc}$. Models with energy added in the core develop a surface shock breakout that propels an initial, higher-speed ejecta ($>$5000km s$^{-1}$), but the bulk of the ejected material still follows the same exponential similarity scalings with ${\bar v} \approx v_{\rm esc}$. A broader parameter study examines how the ejected mass and energy depends on the energy-addition factor $f$, for three distinct model series that locate the added energy in either the core, envelope, or near-surface. We conclude by discussing the relevance of these results for understanding LBV outbursts and other eruptive phenomena, such as failed supernovae and pulsational pair instability events.