Physics 556 Stellar Astrophysics Prof. James Buckley Lecture 9 Energy Production and Scaling Laws
Equations of Stellar Structure Hydrostatic Equilibrium : dp Mass Continuity : dm(r) dr (r) dr =4πr 2 ρ(r) = ρ(r) GM(r) r 2 Eqn. of State : P gas = nkt(r) = ρ(r) µm u kt(r) P rad (r) = 1 at (r)4 3 Radiative transport equation : T 3 κρ L r = 16πac r 2 T 3 Energy dl(r) Production/Conservation= T γ 4πr 2 ρ(r)ɛ(r) dr Convective Energy Transport: P Energy Production: dl dm r =(E n E ν ) dt dp = γ 1
Energy Production Nuclear fusion essentially during core hydrogen burning is essentially: p + p + p + p γ + γ +... + ν + ν +... Let E n denote the amount of energy produced per unit mass, per unit time (note that this is a function of density and temperature, i.e., E n = E n (ρ,t) Let E ν denote the amount of energy lost to neutrinos per unit mass per unit time The nuclear interaction cross section is a function of velocity σ n = σ n (v) and the rate of nuclear interactions is R = 1 t σ n(vt)n = σ n vρ/µm u and the energy released per unit nuclear interaction sequence is E n =(4m p m He )c 2 E n = σ n v ρ/µm u (4m p m He)c 2
Energy Production If there is also radiative energy transport, then each spherical shell of volume will have a flux F (r)4πr 2 entering from the bottom and F (r + dr)4π(r + dr) 2 entering from the top. Since the thermal emission is isotropic, the flux is related to the luminosity as F (r) = L(r) 4πr 2 Rate of energy transfered to the shell per unit time = L(r) L(r + dr) Rate of energy transfer to shell per unit time per unit mass L(r) L(r + dr) ρ 4πr 2 dr = 1 4πr 2 ρ From the equation of continuity of mass dm r =4πr 2 dr ρ So the total amount of heat delivered per unit mass at radius r (less the energy loss through neutrinos) can be written δq dm dt =(E n E ν ) dl dm r dl dr
Energy Production From the previous result, taking care to distinguish between total differentials of state variables (e.g., du), the amount of mass in a thin shell dm and differentials corresponding to changes in non-state variables over time, e.g., the heat added to the system of constant mass in time δt is denoted δq/δt Say that we are considering the expansion of a shell at constant pressure, and that the mass of the shell dm does not change. If the density is ρ = dm/dv where dv =4πr 2 dr, then 1/ρ = dv/dm and the change in volume of a shell with fixed mass is δv = δdv dm δq dm δt =(E n E ν ) dl dm r From the first law of thermodynamics (energy conservation) δq dm δt = du dm δt + δw dm δt dm = δ dv dm dm = δ 1 dm ρ Substituting δw = P δv and for δq into the first law of thermodynamics du dm δt =(E n E ν ) dl + P δ 1 dm r δt ρ
Energy Production In equilibrium, no net heat is added per unit time, and the internal energy does not change with time 0 0 du dm δt =(E n E ν ) dl + P δ 1 dm r δt ρ dl dm r =(E n E ν )
Birthplaces of Stars
Kelvin-Helmholtz Contraction For a spherically symmetric cloud, Gauss law (for gravity) implies that the gravitational force and potential only depends on the enclosed mass M(r) = r 0 U G = GM(r)m r ρ 4πr dr v m M r K = 1 2 mv2 = 3 2 kt if K <U G particles can not escape, instability resulting in gravitational collapse Gas collapses, pressure causes temperature to rise When temperatures are high enough (millions K), the average kinetic energy of the ionized helium nuclei is enough to overcome the Coulomb barrier and nuclear fusion begins.
Main Sequence Thermostat T core const
Scaling: Hydrostatic Equilibrium P surface 0 Hydrostatic Equilibrium : dp dr = GM rρ r 2 A P core A F P,b dm F P,t dr P core P surface GM (M /R ) 3 R R 2 R P core, gravity M 2 R 4 P core, thermal ρkt core M R 3 In hydrostatic equilibrium, P gravitational = P thermal and we obtain: M 2 R 4 M R 3 giving the final result that the radius of a main sequence star is roughly proportional to its mass: R M P thermal M R 3
Scaling: Radiative Transport In radiation zone : l mfp 1 ρ R3 M l Time to diffuse out of radiation zone : t diffusion R2 l mfp Rate energy transported = amount of energy in rad zone time for energy to diffuse out R L rad at 4 radiation zone volume t diffusion T 4 R 3 R 2 /l mfp RT 4 l mfp L fusion T large power T core constant for different stars L rad RT 4 core R 3 M R4 M Substituting the relationship R M we obtain L M 3 which, despite gross oversimplification, is close to the correct empirical relationship: L M 3.5
Main Sequence Lifetime Nuclear fuel M H c 2 M L M 3.5 Main sequence lifetime t ms Total energy available for Hydrogen fusion Rate energy released t ms M M 3.5 t ms M 2.5 More massive stars burn much more brightly, but live for a much shorter period of time