Answer to Question #90279 in Astronomy | Astrophysics for Shivam Nishad

Question

Assume that sun radiates like a black body of temperature T. Calculate T using Stefan-Boltzmann law. Take Q=5.67 x 10^-8 Wm^-2K^-4 and Lo=4 x 10^26 W.

Answer 1

Stefan-Boltzmann law states, that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature T:

$$\frac{P}{A}=\sigma {{T}^{4}}$$

where $P=4\cdot {{10}^{26}}W$  is the power of Sun, $A=4\pi R_{\odot }^{2}$ is the surface area of Sun, ${{R}_{\odot }}=7\cdot {{10}^{8}}m$ is the Solar radius. Then we find

$${{T}^{4}}=\frac{P}{4\pi R_{\odot }^{2}\cdot \sigma }$$

or

$$T=\sqrt[4]{\frac{P}{4\pi R_{\odot }^{2}\cdot \sigma }}$$

Substitute known values

$$T=\sqrt[4]{\frac{4\cdot {{10}^{26}}W}{4\pi {{\left( 7\cdot {{10}^{8}}m \right)}^{2}}\cdot 5.67\cdot {{10}^{-8}}W\cdot {{m}^{-2}}\cdot {{K}^{-4}}}}\approx 5800K$$

So the Sun has a surface temperature of 5800 K.