PHAS3135 The Physics of Stars

Transcription

1 PHAS3135 The Physics of Stars Exam 2013 (Zane/Howarth)

2 Answer ALL SIX questions from Section A, and ANY TWO questions from Section B The numbers in square brackets in the right-hand margin indicate the provisional allocation of maximum possible marks for different parts of each question. Section A (Answer ALL SIX questions from this section) 1. Derive the equation of hydrostatic equilibrium in plane-parallel symmetry, explain- [3] ing its meaning. Explain the meaning of the condition of radiative equilibrium, and give its mathe- [4] matical formulation. (In constructing your answer, you may assume spherical symmetry, and consider an atmospheric slab with thickness much smaller than the stellar radius. You may further assume that all energy is transported by radiation, so that the monochromatic flux is the radiative flux F ν (x), where dx = dr and r is the radial co-ordinate.) 2. Explain what is meant by Local Thermodynamic Equilibrium (LTE). (Your an- [5] swer should include statements of the forms of the particle velocity distribution, radiation field, level populations, etc., but formulae are not required.) Under what general circumstances is LTE a reasonable approximation in stellar [2] atmospheres? 3. Explain the meaning of the extinction term in the transfer equation, and give its [3] mathematical expression. Explain the meanings of the terms opacity and optical depth, and give the [4] mathematical expression that links these two quantities. 4. Explain what is meant by a polytrope, including a statement of the appropriate [3] equation of state (defining all quantities involved). Given that the radiation pressure inside a star is given by P rad = at 4 /3, derive the [4] poytropic equation of state for a gas in which β, the ratio of gas pressure to total pressure, is constant. PHAS3135/ CONTINUED

3 5. The Schwarzschild criterion for stability against convection can be written as dt dr < γ 1 T dp rad γ P dr. Without recourse to additional equations, describe the physical interpretation of [3] this criterion (e.g., by reference to the superadiabatic temperature gradient). Given that the equation of radiative energy transport can be written in the form dt dr = 3 16π κ R ρ(r) r 2 L(r) act, 3 list three potential causes of convective instability. (Your answer should include [4] brief explanations and/or examples, but should not make use of equations.) 6. By consideration of the Virial Theorem, show that the minimum mass required for [6] a gas cloud to collapse (the Jeans mass) is M J defining all terms. ( ) 3/2 kt (ρ) 1/2, Gµm(H) PHAS3135/ CONTINUED

4 Section B (Answer ANY TWO questions from this Section) 7. A) When constructing a model atmosphere, an iterative correction scheme is often used in order to reach a self-consistent solution of the equations of the hydrostatic structure and those of the radiation field. a) The Lambda iteration scheme assumes that radiative equilibrium is not satisfied because the temperature used to evaluate the local Planck function is incorrect. The equation of radiative equilibrium can be expressed in terms of Milne s equation as 0 χ ν J ν (τ 0 ) dν = 0 χ ν B ν [T 0 (τ 0 )] dν where T 0 is the local temperature, τ 0 is the optical depth at frequency ν 0, J ν is the mean intensity, B ν is the Planck function, χ ν = k ν ρ, k ν is the opacity and ρ the density. Derive an expression for the error on the temperature, δt (τ 0 ). [3] Briefly explain why, when using this expression for the error on the temperature, [4] convergence becomes extremely slow; and outline why another source of error has to be taken into account. b) Starting with the Eddington approximation, [5] dj ν dτ ν = 3H ν derive the first Unsöld-Lucy equation J = 0 3 κ F κ P H dτ P + 2H(0), where κ F and κ P are the flux mean opacity and the Planck mean opacity, respectively. Show all workings and define all terms. c) Using the expression for J given above and assuming J = T exp[4τ p ], κ F = AT [5] and κ p = C exp[τ p ], show that, if the temperature changes from T 1 to T 2 from one iteration to the next, then [ 1 H = 2H(0)C 3A T 1 1 T 2 ]. QUESTION CONTINUES ON NEXT PAGE PHAS3135/ CONTINUED

5 B) Under the assumption of a grey atmosphere, the transfer equation can be written as cos θ di dτ = I S, where dτ = kρdx; k is the opacity; I = I 0 ν dν; and S = S 0 ν dν are the frequency-integrated specific intensity and the frequency-integrated source term, respectively. The equivalent forms of the radiative-equilibrium equations are: F = F 0 J = S dk dτ = F 0 4π where J and K are the zeroth and second moment of the frequency integrated specific intensity, F is the total flux, and F 0 is a constant. a) Make use of the Eddington approximation, by assuming that the specific inten- [10] sity is characterized by only two angle-independent terms, one propagating inwards and one outwards. Demonstrate that, under this approximation, a solution for the mean intensity is given by J(τ) = 3 4π (τ )F 0 = S(τ) b) A good approximation for the T -profile of a grey atmosphere is given by: T (τ) = [ 3 4 (τ + 2 ] 1/4 3 ) T eff Give an example of a numerical calculation in which the grey-atmosphere tem- [3] perature profile is often used (at least as a starting point), and explain why the approximation is a valid one. PHAS3135/ CONTINUED

6 8. Consider the process of electron scattering in the Thomson limit. Assume that the charge involved in the scattering initially oscillates with non-relativistic velocity v c, and that the incident wave is linearly polarized. The incident flux is S = c 8π E2 0 and the time-average emitting power, after the scattering, is dp dω = e4 E 2 0 8πm 2 c 3 sin2 θ. where E 0 is the electric field of the incident wave and θ is the angle between E 0 and the emitted photon direction. a) Derive (i) the differential cross-section per scattering in the polarized case; [5] (ii) the total cross-section; and [3] (iii) the differential and total cross-sections for unpolarized radiation. [5] Describe four of the most important properties of these cross-sections. [5] b) Describe the main limitations of the Thomson approximation and discuss under [7] which conditions the more general Compton expression must be used. c) With the help of a diagram, draw the profile of the Klein Nishina Compton [5] differential cross-section as a function of the scattering angle, for different values of incident photon energy. Discuss the most likely scattering angle in the Klein Nishina limit, and, qualitatively, the two main differences between the total Klein Nishina cross-section and the Thomson cross-section. PHAS3135/ CONTINUED

7 9. Explain the concept of homology, and describe the circumstances under which it [4] may reasonably be applied to stars. The density and pressure for two homologous stars of mass M 1 and M 2 are ρ 2 ρ 1 = ( M2 M 1 ) ( ) 3 R1 and R 2 P 2 P 1 = ( M2 M 1 ) 2 ( ) 4 R1. R 2 By using these equations, and starting with the equation of state for an ideal gas, [6] show that T 1 T 2 = µ 1 µ 2 M 1 M 2 R 2 R 1. (where µ is the mean molecular weight). Supposing that the energy-generation rate per unit mass is given by ε(r) = ε 0 ρ(r)t n (r) (where ε 0 is a constant), use the equation of continuity of energy to show that [8] L 2 L 1 = ( ε0,2 ε 0,1 ) ( ) n ( ) (n+2) ( µ2 M2 R1 µ 1 M 1 R 2 The equation of radiative energy transport is dt dr = 3 16π κ R (r)ρ(r) r 2 L(r) act. 3 ) (n+3) where the opacity may be assumed to be given by κ R (r) = κ 0 ρ(r)t m (r). Show that [8] ( ε0,1 ) ( ) ( ) (n m 4) ( ) (n m) ( ) (n m+6) κ0,1 µ1 M1 R2 = 1. ε 0,2 κ 0,2 µ 2 M 2 R 1 Hence obtain mass radius and mass luminosity relationships for low-mass stars, [4] for which the power-law exponents may be taken to be m = 3.5, n = 5. PHAS3135/ CONTINUED

8 10. The figure below shows selected stages of the calculated evolution of a star of initial mass 4M. Give a detailed qualitative description of the main stages along this evolutionary [30] track. (You may refer to points A H in your answer, if convenient, but it isn t necessary to reproduce the figure as part of your answer.) END OF PAPER PHAS3135/2013 7