Equations of Stellar Structure

Reading Assignment

Equilibria of Stars

Hydrostatic
Thermal
Nuclear

Hydrostatic Equilibrium

The outward force due to the pressure gradient within the star is exactly balanced by the inward force due to gravity.

{{\mbox{d}}P \over {\mbox{d}}r}=-{GM_{r}\rho  \over r^{2}}

,

where $m(r)$ is the cumulative mass inside the shell at $r$ and G is the gravitational constant. The cumulative mass increases with radius according to the mass continuity equation:

{{\mbox{d}}M_{r} \over {\mbox{d}}r}=4\pi r^{2}\rho .

Integrating the mass continuity equation from the star center ( $r=0$ ) to the radius of the star ( $r=R$ ) yields the total mass of the star.

Energy Generation

Considering the energy leaving the spherical shell yields the energy equation:

{{\mbox{d}}L_{r} \over {\mbox{d}}r}=4\pi r^{2}\rho (\epsilon -\epsilon _{\nu })

,

where $\epsilon _{\nu }$ is the luminosity produced in the form of neutrinos (which usually escape the star without interacting with ordinary matter) per unit mass. Outside the core of the star, where nuclear reactions occur, no energy is generated, so the luminosity is constant.

Energy Transport

The energy transport equation takes differing forms depending upon the mode of energy transport. For conductive luminosity transport (appropriate for a white dwarf), the energy equation is

{{\mbox{d}}T \over {\mbox{d}}r}=-{1 \over k}{l \over 4\pi r^{2}},

where k is the thermal conductivity.

In the case of radiative energy transport, appropriate for the inner portion of a solar mass main sequence star and the outer envelope of a massive main sequence star,

{{\mbox{d}}T \over {\mbox{d}}r}=-{3\kappa \rho L_{r} \over 64\pi r^{2}\sigma T^{3}},

where $\kappa$ is the opacity of the matter, $\sigma$ is the Stefan-Boltzmann constant, and the Boltzmann constant is set to one.