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.