Astrophysics: pressure in a supernova as in explodes

[beee]

Homework Statement

In its early phase, the supernova is an explosion within the star’s own envelope; about half the energy is thermal and half is kinetic. Estimate the ratio of radiation pressure to gas pressure in the star at the moment the shock reaches the surface. (Not in the pre-explosion star!)
Known values: radius = 50R⊙, M = 16M⊙, energy of the explosion = 10^{51} ergs, about M = 14M⊙ are ejected; the rest collapse.

Homework Equations

The equations for pressures that I know are:

P_{rad} = a/3 * T^{4}
P_{gas} = ρ/μ * kT

Where μ is mean molecular mass.

I also have E_{thermal} = 3/2 M/μ kT (where M/μ is to estimate the number of particles).

The Attempt at a Solution

I have substituted mass/volume for ρ and divided the two equations by each other getting:
P_{rad}/P_{gas} = 4πaμT^{3}R^{3} / (9Mk)

Evaluating it all I get the ratio to be 3.23 * 10^{-11} μT^{3} (with SI units used throughout, not cgs).

If I use E = 10^{51} ergs = 10^{44} J given in the setup of the question and equate half of that to E_{thermal} to try to get the value of T, I come up with some terribly small estimates, like 10^{-11}K. This is obviously wrong, but I don't quite get what should I change to get the correct result.

Any pointers appreciated.

 

[clamtrox]

If I use E = 10^{51} ergs = 10^{44} J given in the setup of the question and equate half of that to E_{thermal} to try to get the value of T, I come up with some terribly small estimates, like 10^{-11}K. This is obviously wrong, but I don't quite get what should I change to get the correct result.

Any pointers appreciated.

I think you are just calculating it wrong. Let's do a ballpark approximation.

The number of particles N ~ 10^31*10^27 = 10^58
Thermal energy E ~ 2*10^43 J ~ 10^66 K
-> Temperature ~ 10^66K/10^58 =10^8 K.

 

[beee]

I think you are just calculating it wrong. Let's do a ballpark approximation.

The number of particles N ~ 10^31*10^27 = 10^58
Thermal energy E ~ 2*10^43 J ~ 10^66 K
-> Temperature ~ 10^66K/10^58 =10^8 K.

This is called sleep is good for you! The next morning I redid my calculation and got something in that range - turns out I was accidentally multiplying by Boltzmann constant instead of dividing. :)