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_⊙) 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_⊙. We find that the ejecta's variations in time t and radius r for the velocity v, density ρ, and temperature T are quite well fit by similarity forms in the variable r/t ≈ v. Specifically the scaled density follows a simple exponential decline ρt³ ∼ exp (- r/v_ot). This `exponential similarity' leads to analytic scaling relations for total ejecta mass ΔM and kinetic energy ΔK that agree well with the hydrodynamical simulations, with the specific-energy-averaged speed related to the exponential scale speed v_o through \bar{v} ≡ √{2 Δ K/Δ M} = √{12} v_o, and a value comparable to the star's surface escape speed, v_esc. Models with energy added in the core develop a surface shock breakout that propels an initial, higher speed ejecta (>5000 km s^-1), but the bulk of the ejected material still follows the same exponential similarity scalings with {\bar{v}} ≈ v_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.