質光關係

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

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

導算

$L = 4\pi R^2\sigma T_E^4$

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

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

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

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

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

$\mathbf{D = l_1 + l_2 + \cdots + l_n}$

$D^2 = l_1^2 + l_2^2 + \cdots + l_n^2 + 2(\mathbf{l_1\cdot l_2 + l_1\cdot l_3 + \cdots)}$

$D^2 = l_1^2 + l_2^2 + \cdots + l_n^2 = Nl^2$

$T_E\approx \Big[ \frac{l}{R}\Big]^{\frac{1}{4}}T_I$.

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

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

$L \varpropto M^3$

大質量恆星和低質量恆星的區別

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

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

$L \varpropto {T_I}^4 \frac {R}{\rho} \varpropto {T_I}^4 \frac {R^4}{M}$

$T_I \varpropto \frac {M}{R}$

$L \varpropto M^3$

${T_I}^4\varpropto \frac{M^2}{R^4}$

$L \varpropto M$

參考資料

1. ^ 1.0 1.1 1.2 1.3 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. ^ 4.0 4.1 Mullaney, James. Double and multiple stars and how to observe them. Springer. 2005. ISBN 1852337516.
5. ^ 5.0 5.1 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.