Main Sequence Stars

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

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

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

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

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