## Stellar Modelling

Stellar Models require:

• The fundamental stellar structure equations (FSSE)
• The constitutive relations (CR): “equations of state” describing the physical properties of the matter of the star.

If the structure of the star is changing, the influence of gravitational energy must be included. This introduces a time dependence not present in the static (time-independent) case.

Also, if acceleration is non zero, the acceleration term must be added to the pressure differential (first equation in FSSE):

The CR represent P, κ, and ϵ as a function of ρ, T, and composition.

Boundary Conditions: specify physical constraints to our equations.

• Interior mass Mr and interior luminosity Lr vanish in the center of the star (r=0).
• Another set of boundary conditions are required at a r=R* somewhere at the surface of the star. The simplest conditions are assuming that the temperature T, the pressure P and the density ρ vanish at r=R*. Strictly speaking, these conditions are never obtained in reality, so more sophisticated conditions may be needed.

How is it all used? The volume of the star is imagined to be constructed of spherically symmetric shells of width Δr. These shells separate the volume of the star into discrete Δr increments (…, ri-2, ri-1, ri, ri+1, ri+2, …, thus i is a “label” for one given shell), and the different properties are calculated via numerical integration for each r, using the Stellar Structure Equations, and the Constitutive Relations.

This numerical integration can be done starting from one boundary (either r=0 or r=R*) or somewhere in the middle point of one star radius, and integrate in both directions (i.e. toward the center and toward the surface). By doing this, for each value of i we can find the respective Pi, Mi, Li, and Ti.

In the end, the values obtained should match with the defined boundary conditions in both ends with a desired accuracy. Usually, it requires several iterations, which means that if the properties at the boundaries do not match the desired values, we must change the initial conditions and repeat the numerical integration.

In sum:

1. Decide a starting point, and a set initial conditions (guess).
2. Perform numerical integration toward both boundaries (center and surface).
3. Check accuracy of solution. If the solution is not accurate enough, return to step 1. Otherwise, you’re done.

Vogt-Russell Theorem: take as a general rule, more than a rigorous law.

The mass and composition of a star uniquely determine its radius, luminosity, and internal structure, as well as its subsequent evolution.

Changes of properties in Main sequence stars…

• Larger M mean larger central P and T.
• In low-M stars, the pp-chain dominates.
• In high-M stars, the CNO cycle dominates.
• Star lifetimes decrease with decreasing L.
• Stars of the main sequence have masses which range between:
– M < 0.08 M (no nuclear reactions taking place).
– M > 90 M (energy pulsation mechanism and unstable stars).
• A M change of 3 orders of magnitude corresponds to:
– A L change of 9 orders of magnitude (i.e. a damn huge change)
– Only moderate T change (factor of 20, 2,700K — 53,000K).
• A lower T involves a higher opacity (κ), i.e. low T favors convection domination.

Source (modified)