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Most stars occupy a very small volume in luminosity - mass space. In this section we will derive the analytic relations between those variables. We will assume that stars are characterised by a mass $ M $, radius $ R $, density $ \rho $, luminosity $ L $, typical temperature $ T $, typical pressure $ P $ and photon mean free path $ l $.

From the condition of hydrostatic equilibrium

$ P \propto \frac{G M^2}{R^4} $

The luminosity can be estimated in the following way: the energy density is assumed to be that of black body $ \frac{E}{V} = a T^4 $ where $ a $ is the radiation constant. The volume of the star is $ V = \frac{4 \pi}{3} R^3 $. The time it take a photon to cross is $ \tau \propto \frac{R^2}{l c} $. Hence the luminosity can be estimated as

$ L \propto \frac{E}{\tau} \propto l R T^4 $

The density can be expressed in terms of the mass and radius

$ \rho \propto \frac{M}{R^3} $

If the material is dominated by gas pressure, then

$ P \propto \rho T $

If, instead, the material is dominated by radiation pressure

$ P \propto T^4 $

If the gas is fully ionised, then Thomson opacity can be used

$ l \propto 1/\rho $

If not, then Kramer's opacity gives a better approximation

$ l \propto T^{7/2}/\rho^2 $

In the next sections we will discuss each combination.

Gas dominated Pressure, Kramer's Opacity Edit

By substituting the equation of state in the expression for hydrodynamic equilibrium we can isolate the temperature

$ T \propto \frac{M^2}{R^4} / \rho \propto \frac{M}{R} $

Substituting this and Kramer's opacity to the expression for the luminosity

$ L \propto l R T^4 \propto T^{15/2} \rho^{-2} R \propto \left( \frac{M}{R} \right)^{15/2} \left(\frac{M}{R^3} \right)^{-2} R \propto M^{11/2} R^{-1/2} $

Gas dominated Pressure, Thomson opacity Edit

We can take the temperature from the previous section, and Thomson opacity

$ L \propto l R T^4 \propto R T^4 /\rho \propto R \left( \frac{M}{R} \right)^4 / \frac{M}{R^3} \propto M^3 $

Radiation dominated Pressure, Kramer's opacity Edit

We can retrieve the temperature in the same way as in the previous section

$ T \propto \frac{M^{1/2}}{R} $

Substituting into the expression for the luminosity, with Kramer's opacity

$ L \propto l R T^4 \propto T^{15/2} R / \rho^2 \propto \left( \frac{M^{1/2}}{R} \right)^{15/2} R / \left( \frac{M}{R^3} \right)^2 \propto \frac{M^{7/4}}{R^{5/2}} $

Radiation dominated Pressure, Thomson opacity Edit

Similarly,

$ L \propto l R T^4 \propto R T^4 / \rho \propto R \left( \frac{M^{1/2}}{R} \right)^4 / \left( \frac{M}{R^3} \right) \propto M $

Eddington Limit Edit

In the gas dominated case with Thompson Opacity, we saw that the luminosity scales with mass as $ L \propto M^3 $. At high enough masses, this luminosity will exceed the Eddington luminosity, which only increases linearly with mass. This happens at a about 20 $ M_{\odot} $, and above this mass the luminosity will only scale linearly with the mass.