stellar structure

The overall model of stellar structure (stellar-structure model or stellar model) has a hot region in the center where fusion is releasing energy (the stellar core), a region near the apparent surface of the star that generates the light that escapes (the photosphere), and regions in between that transfer the energy from core to photosphere via electromagnetic radiation (i.e., radiative transfer) and/or convection (transfer of heat by movement of bulk amounts the material holding the heat). Heat conduction (transfer of heat by collision of particles) is generally only a minor factor. The structural details depend largely on the mass and age of the star, with the initial chemical composition (characterized by its metallicity), the rotation rate, and nearby companions also as factors. A very high rotation rate or very close companion can affect the structure significantly.

Large-mass stars (during their main sequence) have CNO cycle fusion in the core, which is convective, surrounded by some proton-proton chain (PP-chain) fusion, which can be triggered by somewhat lower temperatures. Radiative transfer conveys the energy from this inner portion to the surface. Small mass stars such as red dwarfs have only PP-chain fusion in the core, and transfer their energy through convection. Between are stars like the Sun, which also have only PP-chain fusion in a non-convective core, but it is surrounded by layer of radiative transfer (radiation zone), in turn, surrounded by a convective layer (convection zone).

The basic mathematical model describing stellar structure includes four differential equations (stellar-structure equations) relating changes in mass, temperature, luminosity, and pressure to the distance from the center of the star. They presume local thermodynamic equilibrium and hydrostatic equilibrium.

dm
—— = 4πr²ρ
dr

(The mass continuity equation aka mass conservation equation: density varies smoothly with distance r from the center, by an inverse square law.)

dP     Gmρ
—— = - ———
dr     r²

(Pressure counteracts gravity at distance r from the center.)

dL
—— = 4πr²ε
dr

(The luminosity equation: energy is conserved, any addition is from fusion at that level.)

dT     3κρL
—— = ————————
dr   64πr²σT³

(Opacity directly affects the rate at which temperature changes with radius, i.e., the temperature gradient. This is the equation for radiative transfer, i.e., energy transfer via EMR; some alternate equation is needed if heat conduction is significant or if there is convection, which can happen if the temperature gradient is sufficiently high.)

• r - distance from the center of the star, i.e., radius of a spherical portion of the star centered at the star's center.
• m - mass of the star within distance r from the center.
• ρ - density, a function of r, i.e., the same at all points equidistant from the center.
• L - luminosity, the rate at which energy is flowing from inside r to outside r.
• T - temperature at r, also modeled as being the same at all points equidistant from the center.
• P - pressure at r, also modeled as being the same at all points equidistant from the center.
• ε - nuclear energy generation rate, the amount of energy generated by fusion per unit of volume at r.
• κ - opacity at r.
• G - gravitational constant.
• σ - Stefan-Boltzmann constant.

Opacity, density, and energy generation are functions of temperature and pressure (equations of state), and it is key that simple-but-effective approximate models have been developed. Among approximations used to make modeling the behavior of a star's atmosphere (stellar atmosphere) tractable are the plane-parallel atmosphere approximation (ignoring the curvature of its layers) and the gray atmosphere approximation (ignoring the wavelength-dependence by using values averaged over wavelength). Also used is the Eddington approximation.

To model a star, these are generally "solved" through numerical methods, using difference equations, approximating the differential equations by calculating differences over a small value. To do this, a set of consistent boundary conditions needs to be determined/selected. Some are obvious: m and L must be zero at the center (where r = 0), while ρ, P, and T must be (essentially) zero at the surface (the maximum value of r). Since any numerical calculations must begin at a point with values for all the variables, guessing is required and multiple calculation attempts are likely needed to satisfy the above five constraints. Codes using the above-described approach are called Eulerian codes: an alternative is Lagrangian codes, that specify (changes in) values in relation to dm rather than dr, i.e., mass rather than radius.

A stellar model specific to the Sun is a solar model, the current working model referred to as the standard solar model (SSM).