Riemann surface
matlab

Magnitudes: 15 points
The absolute bolometric magnitude, M, of the Sun is 4.755 .
(a) Show that that the absolute magnitude of a star with luminosity $\mathrm{L}$ is given by
$$M=4.755-2.5 \log \left(\frac{\mathrm{L}}{\mathrm{L}_{\odot}}\right) \text {. }$$

)

Solution:
The relation between magnitudes and flux is given by Hershel’s calibration of 5 magnitudes as the equivalent, on a log scale, of a factor of 100 in flux. Defining flux as $L /\left(4 \pi d^2\right.$ and evaluating for two stars, 1 and 2 , both at $10 \mathrm{pc}$ and both bolometrically corrected:
$$M_{b o l}(1)-M_{b a l}(2)=2.5 \log \left(\frac{L_1 / 4 \pi\left(10^2\right)}{L_2 / 4 \pi\left(10^2\right)}\right)$$
Taking star 1 to be some arbitrary star with absolute magnitude, M, and star 2 to be the sun:
\begin{aligned} M_{b a l}(\text { sun })-M_{b o l}(\text { star }) & =2.5 \log \left(\frac{L}{L_{\odot}}\right) \ 4.755-M & =2.5 \log \left(\frac{L}{L_{\odot}}\right) \ \log \frac{L}{L_{\odot}} & =\frac{1}{2.5}(4.755-M) \end{aligned}

(b) Now solve this equation for $\mathrm{L} / \mathrm{L}_{\odot}$ given $\mathrm{M}$.

Solution:
This is given simply by taking the antilog of both sides
$$L=10^{1.9-0.4 M} \mathrm{~L}{\odot}=79.8 \mathrm{~L}{\odot} 10^{-0.4 M}$$

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