Radiation pressure in terms of luminosity?

Question

My class slides have given me the following equation for $P_{rad}$:

$P_{rad}=\frac{4\sigma T^4}{3c}$, where $\sigma$ is the Stefan-Boltzmann Constant.

I know that radiation pressure is pressure on a surface from emitted light, so I wanted to try rewriting that equation in terms of L, luminosity. I know that $L=4\pi R^2\sigma T^4$. So I thought, I'd try substituting that into the first equation:

$P_{rad} = \frac{L}{3\pi c R^2}$

It looked alright to me at first, but then when I checked with other lecture slides floating around on the internet, they all give me an equation with a 4 in the denominator rather than a 3. (This 4 that I'm missing also appears in the equation for Eddington's Luminosity so I know it should definitely still exist in my equation for radiation pressure.) I'm quite confused about where I'm going wrong. Did I use the wrong equation for luminosity?

Answer 1

The relationship between pressure and luminosity depends on the isotropy of the radiation field.

The expression $P = L/4\pi R^2c$ is approximately correct for an object a long way from the source of luminosity, where the radiation is essentially a parallel beam from one direction.

The expression $P = 4\sigma T^4/3c$ applies in the interior of the star when the radiation field is almost isotropic.