Supernova remnants is a collective name for changes in the environment of the progenitor due to the explosion. It is usually divided into three phases. The first is the ejecta dominated stage. At this stage the ejecta are in ballistic motion (constant velocity). It ends when the swept up mass is the same order of magnitude as the ejecta mass. For a supernova energy $ E_e = 10^{51} erg $, ejecta mass of $ M_e = 1 M_{\odot} $ and an environment where the density is uniform and equal to $ n \approx 1 \frac{1}{cm^3} $ , the end of the ejecta dominated stage is at a radius of about 3.4 parsec. Since the velocity is uniform and can be approximated as $ v_e \approx \sqrt{\frac{E_e}{M_e} } $, this phase lasts for about 500 years.

Next comes the Sedov Taylor stage. At this stage the interior of the shock wave evolves adiabatically. This stage ends when the energy loss through radiation becomes important. The cooling function is approximately constant in a wide range of temperatures and we will assume that it is equal to $ \Lambda \approx 10^{-22} erg \cdot cm^3 / s $^{[1]}. The density of the supernova is assumed to remain the same (because the explosion neither generates nor destroys particles). Equating the cooling time to the age of the supernova yields

$ t \approx \frac{E_e}{\Lambda n^2 V} \approx \frac{E_e}{\Lambda n^2 R^3} $

The relation between the radius and the time is given by

$ R \approx \left( \frac{E_e}{\rho} \right)^{1/5} t^{2/5} $

Where $ mu $ is the atomic mass and $ \rho = \mu n $. Combining these two equations and solving for the time yields

$ t \approx \frac{E_e^{2/11} \mu^{3/11}}{n^{7/11} \Lambda^{5/11} } \approx 200\cdot 10^3 years $

This corresponds to a radius of about 40 parsec. A more detailed calculation^{[2]} suggests that the end of the Sedov Taylor phase occurs an order of magnitude earlier.

The next phase is pressure driven snowplough. In this phase the outer, dense parts regions of the shock wave cool, but not the tenuous interior. In this phase the extra entropy generated by the shock is quickly radiated, so the entropy is dominated by the what was generated up to that point, so entropy is conserved. The trajectory of the shock is therefore given by

$ P = \rho \frac{R^2}{t^2} = S V^{5/3} = S R^5 \Rightarrow R \propto t^{2/7} $

As time evolves, radiative cooling intensifies, so at some most of the initial energy is radiated away. Radiation eliminates all the thermal energy, but not the momentum. Conservation of momentum dictates the trajectory of the shock

$ \rho R^3 \frac{R}{t} = {\rm const} \Rightarrow R \propto t^{1/4} $

The velocity of the shock continuous to decrease, until it becomes comparable with the speed of sound in the ambient medium. At this points the supernova just merges with the ambient medium.

## References Edit

- ↑ Sutherland, Ralph S & Dopita, M. A., Cooling Functions for low - density astrophysical plasmas, ApjS 88, 253 (1993)
- ↑ J. M. Blonding et al, Transition to the radiative phase in supernova remnants, ApJ 500:342-354 (1998)