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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 Surface boundary conditions and optical depth (re-cap) • One obvious better condition is T = Teff 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 r ds 1 r • Integrating over frequency, and taking the frequency-integrated mean free path to be infinite, we call this integral the optical depth, : ds • Then: r photosphere, or visible surface, is equivalent to = 1. 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: =1 =1 T = Teff • New conditions? See blackboard T = Teff 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 << RSun). • 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 = Teff and P = g at M = Ms • Vital to use these for cool giant stars – see Section 6. (4.60) 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) 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: tnuclear >> tproblem >> tatomic => P, , = P, , (,T,composition) • Valid for most (not all) problems of interest 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 that X + Y + Z = 1. (5.4) Composition and molecular weight: formula for fully ionised gases • Number of particles per hydrogen atom mass: (a) Hydrogen: 2 (2 per mH – 1 proton, 1 electron) (b) Helium: ¾ (3 per 4 mH – 1 He nucleus, 2 electrons) (c) “Metals”: ~½ (Z+1 per A mH 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 = /mH • 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: 1 N / mH 3 1 2 X Y Z . 4 2 (5.6) 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 Pressure – do we need to modify our simple expressions? Prad (and Pgas) • • (a) Radiation pressure – simple expression follows if intensity of radiation nearly equals Planck function: 1 4 I B Prad aT . 3 Two potentially important deviations: (i) Anisotropic radiation field: needs tensor pressure, but only important near surface Prad << Pgas near surface => 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: Prad << Pgas near surface => plasma effects normally unimportant • (b) Gas pressure – do need to modify: see next lecture