  It was recently shown that the bolometric light curves of type II supernovae
(SNe) allow an accurate and robust measurement of the product of the radiation
energy in the ejecta, $E_r$, and the time since the explosion, $t$, at early
phases ($t\lesssim 10d$) of the homologous expansion. This observable, denoted
here $ET \equiv E_rt$ is constant during that time and depends only on the
progenitor structure and explosion energy. We use a 1D hydrodynamic code to
find $ET$ of simulated explosions of 145 red supergiant progenitors obtained
using the stellar evolution code MESA, and relate this observable to the
properties of the progenitor and the explosion energy. We show that $ET$ probes
only the properties of the envelope (velocity, mass and initial structure),
similarly to other observables that rely on the photospheric phase emission.
Nevertheless, for explosions where the envelope dominates the ejected mass,
$M_{env}/M_{ej} \gtrsim 0.6$, $ET$ is directly related to the explosion energy
$E_{exp}$ and ejected mass $M_{ej}$ through the relation $ET \approx 0.15
E_{exp}^{1/2} R_* M_{ej}^{1/2}$, where $R_*$ is the progenitor radius, to an
accuracy better than $30\%$. We also provide relations between $ET$ and the
envelope properties that are accurate (to within 20\%) for all the progenitors
in our sample, including those that lost most of their envelope. We show that
when the envelope velocity can be reasonably measured by line shifts in
observed spectra, the envelope is directly constrained from the bolometric
light curve (independent of $E_{exp}$). We use that to compare observations of
11 SNe with measured $ET$ and envelope velocity to our sample of numerical
progenitors. This comparison suggests that many SNe progenitors have radii that
are $\lesssim 500~R_\odot$. In the framework of our simulations this indicates,
most likely, a rather high value of the mixing length parameter.

High Energy Astrophysical Phenomena