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Electron cooling and the connection between expansion and flux-density evolution in radio supernovae

Radio supernovae (RSNe) are weak and rare events. Their typical maximum radio luminosities are of the order of only $10^{27}$\,erg\,s$^{-1}$\,Hz$^{-1}$. There are, however, very few cases of relatively bright (and/or close) RSNe, from which the expansion of the shock and the radio light curves at several frequencies have been monitored covering several years. Applying the standard model of radio emission from supernovae, it is possible to relate the defining parameters of the modelled expansion curve to those of the modelled light curves in a simple algebraic way, by assuming an evolution law for the magnetic field and for the energy density of the population of synchrotron-emitting electrons. However, cooling mechanisms of the electrons may affect considerably this connection between light curves and expansion curve, and lead to wrong conclusions on the details of the electron acceleration and/or on the CSM radial density profile. In this paper, we study how electron cooling modifies the flux-density decay rate of RSNe for a set of plausible/realistic values of the magnetic field and for different expansion regimes. We use these results to estimate the magnetic fields of different RSNe observed to date and compare them to those obtained by assuming energy equipartition between particles and magnetic fields. For some of the best monitored RSNe, for which deceleration measurements, optically thin spectral index, and power-law time decay have been observed (SN\,1979C, SN\,1986J, SN\,1993J, and SN\,2008iz), we find self-consistent solutions for the index of the power-law circumstellar density profile ($s=2$ for all cases), the index of the power-law relativistic electron population (rather steep values, $ p = 2.3 - 3.0$) and the initial magnetic field (ranging from $\sim 20$ to $> 100$\,G).