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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 radiationenergy in the ejecta, $E_r$, and the time since the explosion, $t$, at earlyphases ($t\lesssim 10d$) of the homologous expansion. This observable, denotedhere $ET \equiv E_rt$ is constant during that time and depends only on theprogenitor structure and explosion energy. We use a 1D hydrodynamic code tofind $ET$ of simulated explosions of 145 red supergiant progenitors obtainedusing the stellar evolution code MESA, and relate this observable to theproperties of the progenitor and the explosion energy. We show that $ET$ probesonly 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 $ET \approx 0.15E_{exp}^{1/2} R_* M_{ej}^{1/2}$, where $R_*$ is the progenitor radius, to anaccuracy better than $30\%$. We also provide relations between $ET$ and theenvelope properties that are accurate (to within 20\%) for all the progenitorsin our sample, including those that lost most of their envelope. We show thatwhen the envelope velocity can be reasonably measured by line shifts inobserved spectra, the envelope is directly constrained from the bolometriclight curve (independent of $E_{exp}$). We use that to compare observations of11 SNe with measured $ET$ and envelope velocity to our sample of numericalprogenitors. This comparison suggests that many SNe progenitors have radii thatare $\lesssim 500~R_\odot$. In the framework of our simulations this indicates,most likely, a rather high value of the mixing length parameter.

Type II supernovae progenitor and ejecta properties from the total emitted light, ET