## Temperature of black bodies

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:

$$U = \sigma T^4$$

3. The attempt at a solution

At a distance a, the irradiance from the sun is

$$U = \frac{r_{sun}^2}{a^2}\cdot \sigma T_{sun}^4$$

The sphere absorbs

$$\pi r^2 \frac{r_{sun}^2}{a^2}\sigma T_{sun}^4$$

and emits

$$4\pi r^2 \sigma T_{sphere}^4$$

Equating the absorption and the emission, I get

$$T_{sphere}=T_{sun}\sqrt{\frac{r_{sun}}{2a}}$$

The disc absorbs the same as the disc, and emits

$$2\pi r^2 \sigma T_{sphere}^4$$

Equating the absorption and emission:

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

$$\frac{T_{sphere}}{T_{disc}}=0.84$$

Correct?
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