  Recent multi-dimensional simulations of core-collapse supernovae are
producing successful explosions and explosion-energy predictions. In general,
the explosion-energy evolution is monotonic and relatively smooth, suggesting a
possible analytic solution. We derive analytic solutions for the expansion of
the gain region under the following assumptions: spherical symmetry, one-zone
shell, and powered by neutrinos and $\alpha$ particle recombination. We
consider two hypotheses: I) explosion energy is powered by neutrinos and
$\alpha$ recombination, II) explosion energy is powered by neutrinos alone.
Under these assumptions, we derive the fundamental dimensionless parameters and
analytic scalings. For the neutrino-only hypothesis (II), the asymptotic
explosion energy scales as $E_{\infty} \approx 1.5 M_g v_0^2 \eta^{2/3}$, where
$M_g$ is the gain mass, $v_0$ is the free-fall velocity at the shock, and
$\eta$ is a ratio of the heating and dynamical time scales. Including both
neutrinos and recombination (hypothesis I), the asymptotic explosion energy is
$E_{\infty} \approx M_g v_0^2 (1.5\eta^{2/3} + \beta f(\rho_0))$, where $\beta$
is the dimensionless recombination parameter. We use Bayesian inference to fit
these analytic models to simulations. Both hypotheses fit the simulations of
the lowest progenitor masses that tend to explode spherically. The fits do not
prefer hypothesis I or II; however, prior investigations suggest that $\alpha$
recombination is important. As expected, neither hypothesis fits the
higher-mass simulations that exhibit aspherical explosions. In summary, this
explosion-energy theory is consistent with the spherical explosions of low
progenitor masses; the inconsistency with higher progenitor-mass simulations
suggests that a theory for them must include aspherical dynamics.
