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 , radius , density , luminosity , typical temperature , typical pressure and photon mean free path .

From the condition of hydrostatic equilibrium

The luminosity can be estimated in the following way: the energy density is assumed to be that of black body where is the radiation constant. The volume of the star is . The time it take a photon to cross is . Hence the luminosity can be estimated as

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

If the material is dominated by gas pressure, then

If, instead, the material is dominated by radiation pressure

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

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

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

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

## Gas dominated Pressure, Thomson opacity Edit

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

## Radiation dominated Pressure, Kramer's opacity Edit

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

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

## Radiation dominated Pressure, Thomson opacity Edit

Similarly,