Stellar Structure Section 3: Energy Balance Lecture 4 – Energy transport processes Why does radiation dominate? Simple derivation of transport equation Energy conservation equation Full sets of structure equations … … including boundary conditions Simple prescription for time evolution

Energy transport – from central energy source to surface emission Radiation – energy carried by photons Conduction – energy carried by electrons (mainly) Convection – energy carried by large-scale gas motions Convection: discussed later (Section 4) Radiation, conduction similar, so can be treated similarly, using a conduction coefficient λ (see blackboard) Which carries more energy? Normally (see blackboard) gas particles possess more energy than photons – but photons travel much farther: ℓ photon ≈ 10 5 ℓ electron. Exception: high density stars (e.g. WDs) (Pauli exclusion principle)

Energy carried by radiation: approximate argument (see blackboard) ℓ =  r r r +  r T T +  T T4T4  (T+  T) 4 ℓ = photon mean free path  = cross-sectional area for absorption (per unit volume)  = opacity (defined below)

Energy carried by radiation Approximate argument gives: Precise argument (see Handout) gives: Hence (see blackboard): (3.23) Finally, energy conservation in a spherical shell (see blackboard) gives: (3.24)

The four differential equations of stellar structure Assuming (i) steady state (ii) all energy carried by radiation: 4 equations 7 variables

Three relations to close the system Assuming that conditions in stellar interiors are close to thermodynamic equilibrium: P = P( , T, composition)  =  ( , T, composition) ε = ε( , T, composition). Using previous explicit expression for pressure: we also need  =  ( , T, composition).

Differential equations in terms of mass as independent variable Surface best defined by M = M s, so use mass as variable: (dividing (2.1) by (2.2)) (3.29) (3.30) (3.31) (3.32) Then central boundary conditions are: r = L = 0 at M = 0. (3.33)

Surface boundary conditions Vogt-Russell “theorem” Surface boundary conditions? A useful approximation is:  = T = 0 at M = M s. (3.34) Actually, T s ≠ 0 – and is one of the unknowns. But can find effective temperature from L s  4π R s 2  T eff 4 since L s, R s emerge from solution, and assume T s ≈ T eff. Vogt-Russell “theorem”: 7 equations of stellar structure, plus 4 boundary conditions, completely determine the structure of a star of given mass and composition. Unique for main-sequence stars, but not for evolved stars – may also depend on history.

Approximate treatment of stellar evolution Simple treatment uses sequence of static models, differing only in composition: Build initial model, with composition known as function of mass. Assume all energy release from nuclear reactions. Calculate resultant changes in composition (schematic):. (3.35) Integrate (3.35) for timestep Δt, starting from composition of previous model. Construct sequence of static models of gradually changing composition and structure.