medium. The value of $\tau$ used is that which leads to the same mass of hot gas as for the simulation. We see that, while for much of the outer parts of the remnant, the radially averaged profile is similar to the WL profile, in the inner region the density and temperature stay flat for the simulation while the WL profile has a steeply rising temperature and falling density similar to a Sedov-Taylor type profile. In addition, the density variations, as well as averaged values, are important for the X-ray emission, as discussed below, since emissivity goes as density squared.

![3D rendering of the shock and clouds for one of our 3D runs. The cloud and shock surfaces are shown as temperature contours. The age of the remnant is $1.5\times 10^4$ yr. The box is 30 pc on a side. Note the strong distortions of the shock front. \[fig:3Dshock\]](3D_image-4){width="49.00000%"}

Shock Evolution
---------------

The similarity solution of WL results in expansion of the shock front with the same power law in time as for the Sedov-Taylor solution, i.e. $R_s \propto
t^{2/5}$. For our calculations, evaluating the shock radius is not entirely straightforward since in regions where a cloud is being encountered 