Astrophysics (Index) About

### stellar structure

(stellar structure model, stellar model, structure)
(the presumed internal structure of stars)

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) with conduction (transfer of heat by collision of particles) 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 affect the structure significantly.

Large mass stars (during their main sequence) have CNO cycle fusion in the core, with a region surrounding it conveying energy via radiative transfer, the inner part of which also has some proton-proton chain fusion, which can be triggered by somewhat lower temperatures. Small mass stars such as red dwarves have only proton-proton chain fusion in the core, and transfer energy through convection. Between are stars like the Sun, which have an inner portion much like a large star, with a convection layer surrounding it.

The basic mathematical model 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 is assumed constant at distance r from the center)

```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; Other equations are 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 and it is key that simple-but-effective approximate models have been developed (equations of state). 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 using difference equations, approximating the differential equations by calculating differences over a small value. A star with these equations, 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 this 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).

(stars,models,radiative transfer)
Further reading:
https://en.wikipedia.org/wiki/Stellar_structure
http://personal.psu.edu/rbc3/A534/lec2.pdf
http://www.vikdhillon.staff.shef.ac.uk/teaching/phy213/phy213_solution.html
https://www.astro.princeton.edu/~gk/A403/stellar.pdf

Referenced by pages:
binary star
binding energy
Boltzmann equation
BPASS
Eddington approximation
emission coefficient (j)
FGK star
flux-weighted gravity-luminosity relationship (FGLR)
giant planet
gray atmosphere
Lane-Emden equation
line blanketing
luminosity (L)
mass shell
mixing length theory
nuclear energy generation rate (ε)
planet structure
presolar grain
quantum tunneling
equation of radiative transfer (RTE)
RT instability
shell
specific heat
spectral class
stellar astronomy
stellar evolution
stellar temperature determination
subgrid-scale physics
Sun
variable star
Vogt-Russell theorem (VR theorem)

Index