Question 44.5: Distance to a star using the H-R diagram and color. Suppose ......
Distance to a star using the H-R diagram and color. Suppose that detailed study of a certain star suggests that it most likely fits on the main sequence of an \mathrm{H}-\mathrm{R} diagram. Its measured apparent brightness is b=1.0 \times 10^{-12} \mathrm{~W} / \mathrm{m}^{2}, and the peak wavelength of its spectrum is \lambda_{\mathrm{P}} \approx 600 \mathrm{~nm}. Estimate its distance from us.
APPROACH We find the temperature using Wien’s law, Eq. 37-1. The luminosity is estimated for a main sequence star on the H-R diagram of Fig. 44-6, and then the distance is found using the relation between brightness and luminosity, Eq. 44-1.
\lambda_{\mathrm{P}} T=2.90 \times 10^{-3} \mathrm{~m} \cdot \mathrm{K} (37-1)
b=\frac{L}{4 \pi d^{2}} (44-1)
The star’s temperature, from Wien’s law (Eq. 37-1), is
T \approx \frac{2.90 \times 10^{-3} \mathrm{~m} \cdot \mathrm{K}}{600 \times 10^{-9} \mathrm{~m}} \approx 4800 \mathrm{~K}
A star on the main sequence of an \mathrm{H}-\mathrm{R} diagram at this temperature has intrinsic luminosity of about L \approx 1 \times 10^{26} \mathrm{~W}, read off of Fig. 44-6. Then, from Eq. 44-1,
d=\sqrt{\frac{L}{4 \pi b}} \approx \sqrt{\frac{1 \times 10^{26} \mathrm{~W}}{4(3.14)\left(1.0 \times 10^{-12} \mathrm{~W} / \mathrm{m}^{2}\right)}} \approx 3 \times 10^{18} \mathrm{~m}
Its distance from us in light-years is
d=\frac{3 \times 10^{18} \mathrm{~m}}{10^{16}\ \mathrm{~m} / \mathrm{ly}} \approx 300 \mathrm{ly}