Quick help with equation P = a * T^4

[leonmate]

Hi all,

I have an exam tomorrow on the evolution of stars. Revision is going OK, but I've come across this equation. Some quick help would be very appreciated!

P = αT⁴

I'll give you some context:

I'm solving a question that states a star in Virial equilibrium is contracting quasi-statically (slowly). What effect would this have on the temperature of a very large star. (can assume gas pressure is a negligible fraction of the total pressure and temperature is uniform).
I've started the question with:

Ω = -3 ∫ P/ρ dm

Cancelled density ρ, by subbing dm = 4πr²ρ

Then the solution I have shows a substitution for pressure P as above:

P = αT⁴

Using this we can solve the problem by showing temperature T ∝ 1/R

I'm sure the equation is a correct one. I was just wondering if anyone knew a little more on it? When can/can't I use this substitution.

Thanks

 

[mfb]

This just looks like the Stefan–Boltzmann law, as radiated power and pressure are proportional to each other for radiation. If radiation pressure dominates, this is an approximation for the total pressure.