Stellar Structure - Equations of Stellar Structure

Equations of Stellar Structure

The simplest commonly used model of stellar structure is the spherically symmetric quasi-static model, which assumes that a star is in a steady state and that it is spherically symmetric. It contains four basic first-order differential equations: two represent how matter and pressure vary with radius; two represent how temperature and luminosity vary with radius.

In forming the stellar structure equations (exploiting the assumed spherical symmetry), one considers the matter density, temperature, total pressure (matter plus radiation), luminosity, and energy generation rate per unit mass in a spherical shell of a thickness at a distance from the center of the star. The star is assumed to be in local thermodynamic equilibrium (LTE) so the temperature is identical for matter and photons. Although LTE does not strictly hold because the temperature a given shell "sees" below itself is always hotter than the temperature above, this approximation is normally excellent because the photon mean free path, is much smaller than the length over which the temperature varies considerably, i. e. .

First is a statement of hydrostatic equilibrium: the outward force due to the pressure gradient within the star is exactly balanced by the inward force due to gravity.

,

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

Integrating the mass continuity equation from the star center to the radius of the star yields the total mass of the star.

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

,

where 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.

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

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,

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

The case of convective luminosity transport (appropriate for non-radiative portions of main sequence stars and all of giants and low mass stars) does not have a known rigorous mathematical formulation, and involves turbulence in the gas. Convective energy transport is usually modeled using mixing length theory. This treats the gas in the star as containing discrete elements which roughly retain the temperature, density, and pressure of their surroundings but move through the star as far as a characteristic length, called the mixing length. For a monatomic ideal gas, when the convection is adiabatic, meaning that the convective gas bubbles don't exchange heat with their surroundings, mixing length theory yields

where is the adiabatic index, the ratio of specific heats in the gas. (For a fully ionized ideal gas, .) When the convection is not adiabatic, the true temperature gradient is not given by this equation. For example, in the Sun the convection at the base of the convection zone, near the core, is adiabatic but that near the surface is not. The mixing length theory contains two free parameters which must be set to make the model fit observations, so it is a phenomelogical theory rather than a rigorous mathematical formulation.

Also required are the equations of state, relating the pressure, opacity and energy generation rate to other local variables appropriate for the material, such as temperature, density, chemical composition, etc. Relevant equations of state for pressure may have to include the perfect gas law, radiation pressure, pressure due to degenerate electrons, etc. Opacity cannot be expressed exactly by a single formula. It is calculated for various compositions at specific densities and temperatures and presented in tabular form. Stellar structure codes (meaning computer programs calculating the model's variables) either interpolate in a density-temperature grid to obtain the opacity needed, or use a fitting function based on the tabulated values. A similar situation occurs for accurate calculations of the pressure equation of state. Finally, the nuclear energy generation rate is computed from particle physics experiments, using reaction networks to compute reaction rates for each individual reaction step and equilibrium abundances for each isotope in the gas.

Combined with a set of boundary conditions, a solution of these equations completely describes the behavior of the star. Typical boundary conditions set the values of the observable parameters appropriately at the surface and center of the star:, meaning the pressure at the surface of the star is zero;, there is no mass inside the center of the star, as required if the mass density remains finite;, the total mass of the star is the star's mass; and, the temperature at the surface is the effective temperature of the star.

Although nowadays stellar evolution models describes the main features of color magnitude diagrams, important improvements have to be made in order to remove uncertainties which are linked to the limited knowledge of transport phenomena. The most difficult challenge remains the numerical treatment of turbulence. Some research teams are developing simplified modelling of turbulence in 3D calculations.