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 _ { 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 _ { o } through { \bar { v } } \equiv \sqrt { 2 \Delta K / \Delta M } = \sqrt { 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 } } \approx 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 .