  The climate of a planet can be strongly affected by its eccentricity due to
variations in the stellar flux. There are two limits for the dependence of the
inner habitable zone boundary (IHZ) on eccentricity: (1) the mean-stellar flux
approximation ($S_{\mbox{IHZ}} \propto \sqrt{1-e^2}$), in which the temperature
is approximately constant throughout the orbit, and (2) the maximum-stellar
flux approximation ($S_{\mbox{IHZ}} \propto (1-e)^2$), in which the temperature
adjusts instantaneously to the stellar flux. Which limit is appropriate is
determined by the dimensionless parameter $\Pi = \frac{C}{BP}$, where $C$ is
the heat capacity of the planet, $P$ is the orbital period, and
$B=\frac{\partial \Omega}{\partial T_s}$, where $\Omega$ is the outgoing
longwave radiation and $T_s$ is the surface temperature. We use the Buckingham
$\Pi$ theorem to derive an analytical function for the IHZ in terms of
eccentricity and $\Pi$. We then build a time-dependent energy balance model to
resolve the surface temperature evolution and constrain our analytical result.
We find that $\Pi$ must be greater than about $\sim 1$ for the mean-stellar
flux approximation to be nearly exact and less than about $\sim 0.01$ for the
maximum-stellar flux approximation to be nearly exact. In addition to assuming
a constant heat capacity, we also consider the effective heat capacity
including latent heat (evaporation and precipitation). We find that for planets
with an Earth-like ocean, the IHZ should follow the mean-stellar flux limit for
all eccentricities. This work will aid in the prioritization of potentially
habitable exoplanets with non-zero eccentricity for follow-up characterization.
