Question

Find the radius RRigel of the star Rigel, the bright blue star in the constellation Orion that radiates energy at a rate of 2.7×1031W and has a surface temperature of 11,000 K. Assume that the star is spherical. Use σ=5.67×10−8W/m2⋅K4 for the Stefan-Boltzmann constant and express your answer numerically in meters to two significant figures.

Answer 1

Answer:

$r=5.1\times10^{10}m$

Explanation:

The Stefan–Boltzmann law for a black body (as stars are treated) can be written as $j^*=\sigma T^4$, where $j^*$ is the total energy radiated per unit surface area across all wavelengths per unit time, $T$ the absolute temperature and $\sigma=5.67\times10^{-8} Wm^{-2}K^{-4}$ is the Stefan–Boltzmann constant.

If we multiply $j*$ by the surface area $A$ of the star we get the total energy radiated across all wavelengths per unit time, which is the total power radiated, so we can write:

$P=Aj^*=A\sigma T^4=4\pi r^2\sigma T^4$

where we have used the formula for the surface area of a sphere $A=4\pi r^2$

Solving for r we have:

$r=\sqrt{\frac{P}{4\pi \sigma T^4}}=\sqrt{\frac{2.7\times10^{31}W}{4\pi (5.67\times^{-8}Wm^{-2}K^{-4})(11000K)^4}}=5.1\times10^{10}m$