Outline

1D, Time-Independent Equations of Structure for Spherical Stars

3.i. Introduction
Since we know (and the star knows) how much mass is present, use m, the mass inside a shell of radius r, as the independent radial variable.
Write four differential equations for the four dependent variables
r (m): radius containing mass m
ρ(m): mass density (or equivalently pressure)
T(m): temperature
l (m): luminosity passing through surface at m
The radius R = r (M) as well as the emergent luminosity L = l (M) will be model results, not inputs.
In all but the most extreme cases we may assume LTE.

For non-spherical but still hydrostatic configurations (e.g. rapid rotators or close binary stars) one may still use 1D equations if one chooses the co-rotating gravitational potential as the independent variable.

3.1. Continuity

4πr 2ρdr = dm
boundary condition: r = 0 at m = 0.

3.2. Hydrostatic Equilibrium

dP = −Gmρ dr /r 2 or
4πr 4dP = −Gm dm
boundary condition: P = 0 at m = M.
Equation of State: P = P (ρ, T, f ); f = vector of atomic composition.

3.3. Energy Generation

dl = ε dm
boundary condition: l = 0 at m = 0.
Gravitational and Nuclear energy generation: ε = ε (ρ, T, f ) ergs/gram.

3.4. Energy Transport

3.4.1. Radiative layers
64π 2r 4acT 3dT = − 3κl dm
boundary condition: T = 0 at m = M.
Mean opacity: κ = κ (ρ, T, f ) cm2/gram.

3.4.2. Convective layers
P dT = T(1−1/γ) dP
boundary condition: set by radiative boundaries.
Adiabatic constant: γ = γ (ρ, T, f )