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Two new equations of motion for a supernova remnant (SNR) are derived in theframework of energy conservation for the thin-layer approximation. The firstone is based on an inverse square law for the surrounding density and thesecond one on a non-cubic dependence of the swept mass. Under the assumptionthat the observed radio-flux scales as the flux of kinetic energy, two scalinglaws are derived for the temporal evolution of the surface brightness of SNRs.The astrophysical applications cover two galactic samples of surface brightnessand an extragalactic one.

Energy Conservation in the thin layer approximation: V. The surface brightness in supernova remnants