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

given that:

radiated power, \(\mathrm{P}=2.7 \times 10^{31} \mathrm{~W}\)

temperature, \(T=11000 \mathrm{~K}\)

Radius of the star Rigel which is given as ::

using an equation, \(\quad P=\sigma A T^{4}\)

where, \(\mathrm{A}=\) surface area of sphere \(=4 \pi \mathrm{R}^{2}\)

\(\mathrm{P}=\sigma\left(4 \pi \mathrm{R}^{2}\right) \mathrm{T}^{4}\)

or \(\mathrm{R}^{2}=\mathrm{P} / \sigma(4 \pi) \mathrm{T}^{4} \quad\{\) eq.2 \(\}\)

where, \(\sigma=\) Stefan-Boltzmann constant \(=5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^{4}\)

inserting the values in eq.2,

\(\mathrm{R}^{2}=\left(2.7 \times 10^{31} \mathrm{~W}\right) /\left(5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^{4}\right)(4 \times 3.14)(11000 \mathrm{~K})^{4}\)

\(\mathrm{R}^{2}=\left(2.7 \times 10^{31} \mathrm{~W}\right) /\left(71.2152 \times 10^{-8} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^{4}\right)\left(1.46 \times 10^{16} \mathrm{~K}^{4}\right)\)

\(\mathrm{R}^{2}=\left(2.7 \times 10^{31} \mathrm{~W}\right) /\left(103.97 \times 10^{8} \mathrm{~W} / \mathrm{m}^{2}\right)\)

\(\mathrm{R}=\sqrt{0.0259 \times 10^{23}}\)

\(\mathrm{R}=5.089 \times 10^{10} \mathrm{~m}\)

Answer in meters to two significant figures -

\(R=5.1 \times 10^{10} \mathrm{~m}\)