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Home , Stellar structure

24 — Simple Radiative Models [Revision : 1.1]

• Recapitulation of stellar structure equations – dP GM = −ρg = −ρ r dr r2 – Mass-radius dM r = 4πr2ρ dr – Energy conservation dL r = 4πr2ρ dr – Radiative diffusion 1 dP 4acT 3 dT F = − rad = − rad κρ¯ dr 3¯κρ dr 4 where Prad = aT /3 is radiation • Scaling relations for radiative models – Assume all energy is transported by radiation – Look for simple scaling relations between mass, radius and – Drop all numerical constants, assumeκ ¯,  are uniform in – Mass-radius M ∼ R2ρ R so M ρ ∼ R3 is mass divided by characteristic volume – Hydrostatic equilibrium & mass-radius P M GM ∼ R R3 R2 so GM 2 1 P ∼ R2 R2 pressure is charateristic gravitational force divided by characteristic area – Energy conservation & mass-radius L ∼ R2ρ R so L ∼ R3ρ ∼ M luminosity is mass times energy generation rate per unit mass – Radiative diffusion & mass-radius L acT 3 T F ∼ ∼ rad R2 κρ¯ R so  Rc  L ∼ aT 4 R3 κM¯ luminosity is radiant energy per unit volume times characteristic volume divided by dif- fusion timescale tdiff 2 ∗ Diffusion timescale for any process is tdiff ∼ R /D, where R is characteristic length, and D is diffusion coefficient ∗ D ∼ vhsi, where v is velocity and hsi is ∗ For diffusing in star v ∼ c and hsi ∼ 1/κρ¯ ∼ R3/κM¯ so D ∼ cR3/κM¯

∗ Hence, tdiff ∼ κM/Rc¯ – Final relation required to close above equations is equation-of-state ∗ ρkT kMT P ∼ ∼ 3 µmH µmHR ∗ Radiation P ∼ aT 4 – Focus on ideal gas case ∗ Eliminate pressure using hydrostatic and EOS

kMT GM 2 3 ∼ 4 µmHR R so GMµm TR ∼ H k ∗ Eliminate (TR)4 from radiative diffusion

GMµm 4 c L ∼ a H k κM¯ so ac Gµm 4 L ∼ H M 3 κ¯ k ∗ Big result: standard mass-luminosity relation L ∼ M 3 ∗ Radius does not enter into this relation ∗ Note dependence on µ4 — with higher molecular weight are more luminous for the same mass • Eddington standard model – Slightly more accurate model for a radiative star – Assume that all mass, energy generation concentrated right at center of star; so, away from center Mr = M and Lr = L – Also assume that opacity is constant – Radiative diffusion equation can now be written as dP κL¯ rad = −ρ dr 4πcr2 – Compare against hydrostatic equilibrium dP GM = −ρ dr r2 – Take ratio dP 4πGMc = dPrad κL¯ where right-hand side is constant – In fact, this constant is related to Eddington parameter: κL¯ Γ = 4πGMc – So, dP 1 = , dPrad Γ and so P P = rad Γ (set constant of integration to zero). – Define ratio of gas pressure to total pressure:

Pg = βP

so Prad = (1 − β)P

– Because P ∝ Prad in Eddington standard model, see that β is constant throughout, and equal to 1 − Γ – Use EOS to find pressure-density relation:

1 aT 4 P = 1 − β 3 but P µm T = g H ρk so 1 a P µm 4 P = g H 1 − β 3 ρk Resulting scaling: P ∼ ρ4/3

– One-to-one relationship between total pressure means Eddington standard model is a γ = 4/3 (n = 3)