Answer to Question #179922 in Physics for James

Question

Our sun has a wavelength of maximum intensity of emission approximately 510 nm.

b) Using the Stefan–Boltzmann law, and given that the luminosity of the sun is 3.9 x 10^26 W, calculate the radius of the sun, giving your answer to two significant figures.

Answer 1

From the Wien's displacement law:

$$\lambda_{max} =\dfrac{b}{T}$$

where $\lambda_{max} = 510nm = 5.1\times 10^{-7}m$ is the  wavelength of maximum intensity, $T$ is the absolute temperature of the Sun, and $b = 2.9\times 10^{-3}m\cdot K$ is the Wien's displacement constant. Thus, the temperature of the Sun is:

$$T = \dfrac{b}{\lambda_{max}}$$

According to the Stefan–Boltzmann law:

$$\dfrac{L}{A} = \sigma T^4$$

where $L = 3.9 \times 10^{26} W$ is the luminosity of the Sun, $A$ is the area of Sun's surface, $\sigma = 5.67\times 10^{-8}W/(m^2\cdot K^4)$ is the Stefan–Boltzmann constant. Substituting the expression for $T$ and expressing $A$, obtain:

$$A = \dfrac{L}{\sigma T^4} = \dfrac{L\lambda_{max}^4}{\sigma b^4}$$

On the other hand, the area of a sphere is:

$$A = \pi R^2$$

where $R$ is the radius of the Sun. Expressing R, find:

$$R = \sqrt{\dfrac{A}{\pi}} = \sqrt{ \dfrac{L\lambda_{max}^4}{\pi\sigma b^4}}= \dfrac{\lambda_{max}^2}{b^2}\sqrt{\dfrac{L}{\pi\sigma}}\\ R = \dfrac{(5.1\times 10^{-7})^2}{(2.9\times 10^{-3})^2}\sqrt{\dfrac{3.9 \times 10^{26}}{\pi\cdot 5.67\times 10^{-8}}} \approx 1.45\times 10^{9}m$$

Answer. $1.45\times 10^{9}m$