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 10 d ) of the homologous expansion .
This observable , denoted here ET \equiv E _ { r } t 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 .