Supernova remnant in a stratified medium: Explicit, analytical approximations for adiabatic expansion and radiative cooling

We propose simple, explicit, analytical approximations for the kinematics o f an adiabatic blast wave propagating in an exponentially stratified ambien t medium and for the onset of radiative cooling, which ends the adiabatic e ra. Our method, based on the Kompaneets implicit solution and the Kahn appr oximation for the radiative cooling coefficient, gives straightforward esti mates for the size, expansion velocity, and progression of cooling times ov er the surface when applied to supernova remnants (SNRs). The remnant shape is remarkably close to spherical for moderate density gradients, but even a small gradient in ambient density causes the cooling time to vary substan tially over the remnant's surface, so that for a considerable period there will be a cold dense expanding shell covering only a part of the remnant. O ur approximation provides an effective tool for identifying the approximate parameters when planning two-dimensional numerical models of SNRs, the exa mple of W44 being given in a subsequent paper.