  A significant fraction of supernovae show signatures of dense circumstellar
material (CSM). While multiple scenarios for creating a dense CSM exist, mass
eruption due to injection of energy at the base of the outer envelope is a
likely possibility. We carry out radiation hydrodynamical simulations of
eruptive mass loss from a typical red supergiant progenitor with initial mass
of $15\ M_\odot$, for the first time focusing on the timescale of the injection
as well as energy. We find that not only sufficient injection energy but also
sufficient rate of energy injection per unit time, $L_{\rm{min}} \sim 8\times
10^{40}$ erg s$^{-1}$ in this particular model, is required for eruption of
unbound CSM. This result suggests that the energy injection rate needs to be
greater than the binding energy of the envelope divided by the dynamical
timescale for the eruption. The density profile of the resulting CSM, whose
shape was analytically and numerically predicted in the limit of instantaneous
energy injection, similarly holds for a finite injection timescale. We discuss
our findings in the framework of proposed mass outburst scenarios, specifically
wave-driven outbursts and common envelope ejection.
