# 質光關係

${\displaystyle {\frac {L}{L_{\odot }}}=\left({\frac {M}{M_{\odot }}}\right)^{a}}$

${\displaystyle {\frac {L}{L_{\odot }}}\approx .23\left({\frac {M}{M_{\odot }}}\right)^{2.3}\qquad (M<.43M_{\odot })}$
${\displaystyle {\frac {L}{L_{\odot }}}=\left({\frac {M}{M_{\odot }}}\right)^{4}\qquad \qquad (.43M_{\odot }
${\displaystyle {\frac {L}{L_{\odot }}}\approx 1.5\left({\frac {M}{M_{\odot }}}\right)^{3.5}\qquad (2M_{\odot }
${\displaystyle {\frac {L}{L_{\odot }}}\approx 3200{\frac {M}{M_{\odot }}}\qquad \qquad (M>20M_{\odot })}$

## 導算

${\displaystyle L=4\pi R^{2}\sigma T_{E}^{4}}$

${\displaystyle {\frac {dP}{dr}}=-{\frac {Gm(r)\rho (r)}{r^{2}}}}$

${\displaystyle \langle P\rangle =-{\frac {1}{3}}{\frac {E_{GR}}{V}}}$

${\displaystyle \langle P\rangle \approx {\frac {GM^{2}}{4\pi R^{4}}}}$

${\displaystyle \langle P\rangle ={\frac {\langle \rho \rangle }{\bar {m}}}kT}$
${\displaystyle kT={\frac {GM{\bar {m}}}{3R}}}$.

${\displaystyle R={\Big (}{\frac {3}{4}}{\frac {1}{\rho \pi }}M{\Big )}^{\frac {1}{3}}}$ to arrive at
${\displaystyle L\varpropto M^{3.33}}$

${\displaystyle \mathbf {D=l_{1}+l_{2}+\cdots +l_{n}} }$

${\displaystyle D^{2}=l_{1}^{2}+l_{2}^{2}+\cdots +l_{n}^{2}+2(\mathbf {l_{1}\cdot l_{2}+l_{1}\cdot l_{3}+\cdots )} }$

${\displaystyle D^{2}=l_{1}^{2}+l_{2}^{2}+\cdots +l_{n}^{2}=Nl^{2}}$

${\displaystyle T_{E}\approx {\Big [}{\frac {l}{R}}{\Big ]}^{\frac {1}{4}}T_{I}}$.

${\displaystyle L\approx 4\pi R^{2}\sigma T_{I}^{4}{\frac {l}{R}}\approx {\frac {(4\pi )^{2}}{3^{5}}}{\frac {\sigma }{k^{4}}}G^{4}{\bar {m}}^{4}\langle \rho \rangle lM^{3}}$

${\displaystyle l\varpropto \langle \rho \rangle ^{-1}}$

${\displaystyle L\varpropto M^{3}}$

### 大質量恆星和低質量恆星的區別

${\displaystyle {\frac {dP_{rad}}{dr}}=-{\frac {\kappa \rho }{c}}{\frac {L}{4\pi r^{2}}}}$

${\displaystyle {T_{I}}^{3}{\frac {T_{I}}{dr}}=-{\frac {3\kappa \rho }{16\sigma }}{\frac {L}{4\pi r^{2}}}}$

${\displaystyle L\varpropto {T_{I}}^{4}{\frac {R}{\rho }}\varpropto {T_{I}}^{4}{\frac {R^{4}}{M}}}$

${\displaystyle T_{I}\varpropto {\frac {M}{R}}}$

${\displaystyle L\varpropto M^{3}}$

${\displaystyle {T_{I}}^{4}\varpropto {\frac {M^{2}}{R^{4}}}}$

${\displaystyle L\varpropto M}$

## 參考資料

1. Salaris, Maurizio; Santi Cassisi. Evolution of stars and stellar populations. John Wiley & Sons. 2005: 138–140. ISBN 0470092203.
2. ^ Mass-luminosity relationship. Hyperphysics. [2009-08-23].
3. ^ Duric, Nebojsa. Advanced astrophysics. Cambridge University Press. 2004: 19. ISBN 9780521525718.
4. Mullaney, James. Double and multiple stars and how to observe them. Springer. 2005. ISBN 1852337516.
5. Phillips, A.C. The Physics of Stars. John Wiley & Sons. 1999. ISBN 9780471987987.
6. ^ Lecchini, Stefano. How Dwarfs Became Giants. The Discovery of the Mass-Luminosity Relation. Bern Studies in the History and Philosophy of Science. ISBN 9783952288269.[失效連結]