Temperature of black bodies

thephysicsman

1. The problem statement, all variables and given/known data

Two black bodies, a sphere and a disc, are situated at the same distance a from the sun. Calculate the ratio of their temperatures.

2. Relevant equations

Stefan-Boltzmann law:

[tex]U = \sigma T^4[/tex]

3. The attempt at a solution

At a distance a, the irradiance from the sun is

[tex]U = \frac{r_{sun}^2}{a^2}\cdot \sigma T_{sun}^4 [/tex]

The sphere absorbs

[tex]\pi r^2 \frac{r_{sun}^2}{a^2}\sigma T_{sun}^4[/tex]

and emits

[tex]4\pi r^2 \sigma T_{sphere}^4[/tex]

Equating the absorption and the emission, I get

[tex]T_{sphere}=T_{sun}\sqrt{\frac{r_{sun}}{2a}}[/tex]

The disc absorbs the same as the disc, and emits

[tex]2\pi r^2 \sigma T_{sphere}^4[/tex]

Equating the absorption and emission:

[tex]T_{disc}=T_{sun}\sqrt{\frac{r_{sun}}{a}}\cdot \frac{1}{2^{1/4}}[/tex]

[tex]\frac{T_{sphere}}{T_{disc}}=0.84[/tex]

Correct?