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 $E_{infty} approx M_g v_0^2 (1.5eta^{2/3} + beta f(rho_0))$, 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.
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