1. Stellar Structure Section 4: Structure of Stars Lecture 9 - Improvement of surface boundary conditions (part 1) Definition of optical depth Simple form of improved surface conditions

2. Surface boundary conditions and optical depth (re-cap) One obvious better condition is T = T eff at the surface.(4.53) “the surface” (“photosphere”) is the “visible surface” = surface from which radiation just escapes: photon mean free path is infinite Mean free path (m.f.p.) ≡ ‘e-folding distance’ of the radiation Define monochromatic m.f.p. by Integrating over frequency, and taking the frequency-integrated mean free path to be infinite, we call this integral the optical depth,  : Then: photosphere, or visible surface, is equivalent to  = 1.

3. Radiative transfer and optical depth (see blackboard for detail) In terms of intensity of radiation, mean free path of photon corresponds to “e-folding distance” of the intensity. Writing  as the monochromatic absorption coefficient, we can write down a formal expression for the monochromatic intensity that shows why the e-folding distance is a useful concept, and define a monochromatic mean free path by an integral expression. Integrating over frequency, and taking the frequency-integrated mean free path to be infinite, we call this integral the optical depth, . Then: photosphere, or visible surface, is equivalent to  = 1.

4. Improved surface boundary conditions Need new conditions when zero conditions give a model in which the radiation escapes from a surface either much hotter or much cooler than the effective temperature: New conditions? See blackboard  = 1 T = T eff

5. Useful approximation to surface boundary conditions Boundary conditions involve an integral – awkward to use. Approximate integral by using fact that main contribution comes from region just above photosphere, where only density changes rapidly (Sun: e-folding distance ~300 km << R Sun ). Assume every other variable is constant, neglect radiation pressure, and evaluate the integral (see blackboard) for this “isothermal atmosphere”. Finally leads to simple boundary conditions: T = T eff and P  = g at M = M s (4.60) Vital to use these for cool giant stars – see Section 6.

6. Stellar Structure Section 5: The Physics of Stellar Interiors Lecture 9 – Limited thermodynamic equilibrium (part 2)Composition and molecular weight … … for fully ionised gas Deviations from simple pressure laws: … no changes needed for radiation pressure … need relativistic and quantum effects for gas pressure (next lecture)

7. Limited thermodynamic equilibrium Is it true that P,  and  are functions only of , T and composition? Strictly true only for complete thermodynamic equilibrium Good approximation provided interactions occur either very fast or very slowly compared to timescale of problem of interest Nuclear interactions – very slow – “never” reach equilibrium Atomic interactions – very fast – “instantaneously” reach equilibrium Limited thermodynamic equilibrium: t nuclear >> t problem >> t atomic => P, ,  = P, ,  ( ,T,composition) Valid for most (not all) problems of interest

8. Composition and molecular weight: approximation and definitions Pre-main-sequence stars fully convective (Section 6), so expect ‘zero-age’ main-sequence stars to have uniform composition. Molecular weight  depends on abundances and on whether gas is molecular, atomic or ionised (or some combination). Outer layers of stars – partially ionised:  very complicated. Through bulk of star – gas essentially fully ionised: can make useful approximation to find expression for . Define: X = fraction of material by mass in form of hydrogen (5.1) Y = fraction of material by mass in form of helium (5.2) Z = fraction of material by mass in form of other elements. (5.3) Note thatX + Y + Z = 1. (5.4)

9. Composition and molecular weight: formula for fully ionised gases Number of particles per hydrogen atom mass: (a)Hydrogen:2(2 per m H – 1 proton, 1 electron) (b)Helium:¾(3 per 4 m H – 1 He nucleus, 2 electrons) (c)“Metals”:~½(Z+1 per A m H if fully ionised) Then calculate: N = total number of particles per unit volume, starting by finding total number of H atom masses per unit volume =  /m H Do this by adding up the numbers for each of (a) to (c) and using definitions of X, Y and Z (see blackboard). Finally obtain: (5.6)

10. Value for molecular weight X, Y, Z found from observation of surface layers of stars Assume they’re the same in the interiors Similar for different stars – Handout 4 Decline towards large mass number consistent with all heavy elements being formed inside stars – so older stars have fewer heavy elements Taking solar abundances, find:  ≈ 0.62. Only really valid in deep interior, but that is most of mass of star:~90% of mass is within ~50% of radius Solar convection zone: ~30% of radius, only ~1% of mass

11. Pressure – do we need to modify our simple expressions?P rad (and P gas ) (a) Radiation pressure – simple expression follows if intensity of radiation nearly equals Planck function: Two potentially important deviations: (i)Anisotropic radiation field: needs tensor pressure, but only important near surface P rad tensor effects normally unimportant (ii)Plasma effects: EM waves cannot propagate if their frequency is less than the natural oscillation frequency of the plasma (see blackboard), but: P rad plasma effects normally unimportant (b) Gas pressure – do need to modify: see next lecture