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

`r=5.1*10^(10)m`

Step-by-step 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*10^(-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{(P)/(4\pi \sigma T^4)}=\sqrt{(2.7*10^(31)W)/(4\pi (5.67*^(-8)Wm^(-2)K^(-4))(11000K)^4)}=5.1*10^(10)m`