  The early part of a supernova (SN) light-curve is dominated by radiation
escaping from the expanding shock-heated progenitor envelope. For polytropic
Hydrogen envelopes, the properties of the emitted radiation are described by
simple analytic expressions and are nearly independent of the polytropic index,
$n$. This analytic description holds at early time, $t<$~few days, during which
radiation escapes from shells initially lying near the stellar surface. We use
numerical solutions to address two issues. First, we show that the analytic
description holds at early time also for non-polytropic density profiles.
Second, we extend the solutions to later times, when the emission emerges from
deep within the envelope and depends on the progenitor's density profile.
Examining the late time behavior of polytropic envelopes with a wide range of
core to envelope mass and radius ratios, $0.1\le M_{\rm c}/M_{\rm env}\le10$
and $10^{-3}\le R_{\rm c}/R\le10^{-1}$, we find that the effective temperature
is well described by the analytic solution also at late time, while the
luminosity $L$ is suppressed by a factor, which may be approximated to better
than $20[30]\%$ accuracy up to $t=t_{\rm tr}/a$ by $A\exp[-(at/t_{\rm
tr})^\alpha]$ with $t_{\rm tr} = 15 (M_{\rm env}/M_\odot)^{3/4}(E/10^{51}{\rm
erg})^{-1/4}~\rm d$, $A=0.9[0.8]$, $a=1.7[4.6]$ and $\alpha=0.8[0.7]$ for
$n=3/2[3]$. This description holds as long as the opacity is approximately that
of a fully ionized gas, i.e. for $T>0.7~\rm eV$, $t<14(R/10^{13.5}{\rm
cm})^{0.55}~\rm d$. The suppression of $L$ at $t_{\rm tr}/a$ obtained for
standard polytropic envelopes may account for the first optical peak of
double-peaked SN light curves, with first peak at a few days for $M_{\rm
env}<1M_\odot$.
