Question 40.4: A Star’s Radius Have you ever wondered how astronomers measu......
A Star’s Radius
Have you ever wondered how astronomers measure the size of distant stars? They are too far away to resolve into a disk, so their radius is inferred from their luminosity and temperature. The star’s temperature is easy to estimate from its spectrum, which looks like a black-body spectrum, but with some particular dark lines due to photon absorption in the star’s atmosphere. These lines are used to estimate the temperature. The star’s luminosity is another name for its total emitted power. Luminosity is harder to estimate than temperature because it comes from measuring the star’s brightness and estimating its distance. Consider a star that has the same temperature as the Sun but is 1.0 \times 10^4 times brighter than the Sun. Model the star and the Sun as black bodies, and find the star’s radius in terms of the Sun’s radius.
INTERPRET and ANTICIPATE
The luminosity or power emitted by a star is easily found from P=\sigma \varepsilon A T^4 \text { (Eq. 21.34). } By comparing the luminosity of the star to that of the Sun, we can find the ratio of their radii.
SOLVE
We are modeling the star and the Sun as black bodies, so we set the emissivity ε to 1.
Both the star and the Sun are spheres. Substitute the surface area of a sphere into the luminosity expression.
P=4 \pi R^2 \sigma T^4Write an expression for the ratio of the star’s power to that of the Sun. We have used the usual symbol \odot to stand for the Sun, and the symbol * to stand for the star.
\begin{aligned}& \frac{P_*}{P_{\odot}}=\frac{4 \pi R_*^2 \sigma T_*^4}{4 \pi R_{\odot}^2 \sigma T_{\odot}^4} \\& \frac{P_*}{P_{\odot}}=\left(\frac{R_*}{R_{\odot}}\right)^2\left(\frac{T_*}{T_{\odot}}\right)^4\end{aligned}Use the fact that the star has the same temperature as the Sun to write an expression for the ratio of their radii.
\begin{aligned}& \frac{P_*}{P_{\odot}}=\left(\frac{R_*}{R_{\odot}}\right)^2 \\& \frac{R_*}{R_{\odot}}=\sqrt{\frac{P_*}{P_{\odot}}} \end{aligned}Substitute values.
\begin{aligned}& \frac{R_*}{R_{\odot}}=\sqrt{1.0 \times 10^4}=1.0 \times 10^2 \\& R_*=1.0 \times 10^2 R_{\odot}\end{aligned}CHECK AND THINK
This very bright star is 100 times larger than the Sun. Such a large star is called a supergiant. Supergiants are “dying stars,” which means they have run out of the fuel in their core and are quickly fusing other material before their final demise