Outline

1. Introduction: Numbers and Estimations

Solar composition by mass: X = 0.71, Y = 0.27, Z = 0.02, μ = 0.6 (mean molecular weight).

1.2. Molecular weights

N_Av = Avogadro's number.
Define μ through n ≡ ρN_Av /μ = ρN_Av (1/μ_I +1/μ_e), where n = n_I + n_e.
For a fully ionized gas, μ ≈ 1 / (2 X + 3 Y /4 + Z /2).

These individual weights are not related to individual masses. “Weight” here really means weight in the statistical sense (unfortunately the same word in English as in the mass sense). For any constituent x, n_x ≡ ρN_Av /μ_x where ρ is the total density.

1.3. Important Relationships

L = 4πR ²σ_RT ⁴_eff

Main Sequence: L/L_o = (M/M_o) ^3.5, R/R_o = (M/M_o) ^0.75

Half of the sun’s mass is inside r = 0.3R. Half of the sun’s luminosity is generated inside r = 0.13R.

Hydrostatic equilibrium allows estimates of the central conditions as follows.
Divide the interior into a core containing half the mass and an envelope containing the other half. Let this boundary be at fR. The density rises sharply to the interior, so assume the entire envelope mass is concentrated at the boundary. In the core, the temperature is relatively uniform, so calculating T from mean values of the density and pressure is not too far off. The mean density of the core is then ρ = 3M/8πf ³R ³. The pressure must equal the weight / surface area of the envelope: ρN_AvkT/μ = GM ²/16πf ⁴R ⁴. Solving for kT gives kT = 4πμGM/N_AvfR (setting one π/3 = 1). For MS stars, the solar value f = 0.3 is OK; for giant stars, f << 0.3, while for white dwarfs the temperature is independent from the pressure.

We cannot specify the luminosity of a star without further information about nuclear rates and/or opacities. Once these quantities are specified, the radius and luminosity are determined by the mass and &mu(r).

Main Sequence life time:
Binding energy per proton = 0.007mc ²
Mass available as fuel is about 0.1M
t = 0.1 × 0.007 Mc ²/L ≈ 10¹⁰(M/M_o)^−2.5 years.