Synchrotron cooling
= Synchrotron cooling
{title2=$\dot E\propto-E^2$}
At fixed <magnetic field>, ultrarelativistic <electrons> lose <energy> through <synchrotron radiation> at a rate proportional to $E^2$. For steady injection $Q(E)\propto E^{-s}$, negligible escape and $s>1$, integrating the energy-space continuity equation gives $N(E)=|\dot E|^{-1}\int_E^\infty Q(E')\,dE'\propto E^{-(s+1)}$. Above the cooling break, the optically thin <radiation spectral index> therefore steepens from $(s-1)/2$ to $s/2$.