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A significant fraction of supernovae show signatures of dense circumstellarmaterial (CSM). While multiple scenarios for creating a dense CSM exist, masseruption due to injection of energy at the base of the outer envelope is alikely possibility. We carry out radiation hydrodynamical simulations oferuptive mass loss from a typical red supergiant progenitor with initial massof $15\ M_\odot$, for the first time focusing on the timescale of the injectionas well as energy. We find that not only sufficient injection energy but alsosufficient rate of energy injection per unit time, $L_{\rm{min}} \sim 8\times10^{40}$ erg s$^{-1}$ in this particular model, is required for eruption ofunbound CSM. This result suggests that the energy injection rate needs to begreater than the binding energy of the envelope divided by the dynamicaltimescale for the eruption. The density profile of the resulting CSM, whoseshape was analytically and numerically predicted in the limit of instantaneousenergy injection, similarly holds for a finite injection timescale. We discussour findings in the framework of proposed mass outburst scenarios, specificallywave-driven outbursts and common envelope ejection.

Eruption of the Envelope of Massive Stars by Energy Injection with Finite Duration