1. (a) The Saha equation may be written in the form N + n e N = C u+ u T 3/2 exp ( ) χ kt where C = 4.83 1 21 m 3. Discuss its importance in the study of stellar atmospheres. Carefully explain the meaning of each term and explain how u, u + can be obtained. [2.5] (b) A B-type star with a pure hydrogen photosphere has a surface temperature of 15,K and a gas pressure of 1 N m 2. i. What is the electron density and electron pressure? [2] ii. What is the ratio of ionized to neutral hydrogen? [1] (c) Sketch the ratio of electron to gas pressure for main sequence stars versus effective temperature. [1.5] (d) If the B-type star of surface temperature 15,K instead has a pure helium photosphere, and an electron pressure of 5 N m 2, what is the ratio of electron to gas pressure in this case? You will need to use the Saha equation to calculate N(He 2+ )/N(He + ) and N(He + )/N(He). [3] Note: The ionization energies of neutral hydrogen, neutral helium, ionized helium are 13.6eV, 24.6eV and 54.4eV, respectively. The partition functions for neutral, ionized and doubly ionized helium are 1, 2 and 1. 2 CONTINUED
2. (a) For an absorption line, define the line depth R λ and equivalent width W λ. Sketch the line profile for an optically thin and an optically thick line. What is the line depth in the core of the thick line? Does LTE hold for the line core or wings of optically thick lines? [2] (b) Briefly discuss the following broadening mechanisms for absorption lines: i. Doppler broadening; ii. Natural broadening; iii. Pressure broadening. Be sure to include a discussion of their associated line profiles. [2.5] (c) The Mg ii (atomic mass 24) line at 448.1 nm is observed in an late B star with an effetive temperature of T =12,5 K. i. The full width at half maximum (FWHM) for a thermally broadened spectral line, λ D 1/2 in nm, is given by λ D 1/2/λ = 7.16 1 7 (T/µ) where T is the temperature in K, λ is the line wavelength in nm, and µ is the atomic mass in atomic mass units (amu). Calculate the Doppler broadening FWHM of this line in velocity space (m s 1 ). [1] The observed FWHM greatly exceeds the predicted Doppler width. Which other broadening mechanism might be responsible for this observed FWHM? [.5] Would your answers to λ D 1/2 and the alternative broadening mechanism be the same, if instead we were considering the Hβ at 486.1nm in the B star? [1] (d) What is the curve of growth, and how can it be used to determine elemental abundances in stellar photospheres? Explain the abundance dependence of the three distinct parts of the curve of growth. Show graphically examples of line profiles in each of these domains. [3] 3 TURN OVER
3. (a) Describe the physical properties of Local Thermodynamic Equilibrium (LTE). Give two astrophysical examples where it is necessary to consider non-lte. [1.5] (b) Energy levels of hydrogen lie 13.6eV/n 2 below the ionization limit. Calculate the threshold energies and wavelengths of the Paschen and Balmer continua. [1.5] Compare the relative number of H atoms and H ions that contribute to the continuous opacity in the photosphere of a star with T =7K on both sides of the Balmer jump. Assume LTE, an electron pressure in the stellar photosphere of 5 N m 2, and identical bound-free cross-sections for atomic hydrogen and H for simplicity. Which absorption or scattering processes dominate at wavelengths shortward and longward of the Balmer jump? [3.5] Sketch the absorption coefficient versus wavelength in the vicinity of the Balmer jump, together with the emergent continuum flux. Is the strength of the Balmer jump sensitive to stellar temperatures and/or electron densities in cool stars? Is your answer the same for hot stars? [2] (c) What is the meaning of a grey atmosphere? Identify one form of continuous opacity in stellar photospheres which is grey. For which types of star is this the dominant opacity source? [1.5] Note: The ionization energy for the negative hydrogen ion is.75ev and neutral hydrogen is 13.6eV. Saha s equation is: log N + N = log u+ u + 5 54 log T 2 T χ log P e 1.176 4 CONTINUED
4. (a) Define the terms effective temperature and surface gravity. [1] (b) The transfer equation for a plane-parallel stellar atmosphere is cos θ di λ dτ λ = I λ S λ Define each term in this equation, and derive the equation for the surface intensity I(, θ) = If we adopt a linear source function, S λ e τ λ sec θ d(τ λ sec θ) [2.5] S λ (τ λ ) = a λ + b λ τ λ show that I(, θ) = S λ (τ λ = cos θ) You may use the standard integral [1.5] u n e u du = n! Using the above relationship, or otherwise, explain the concept of limb darkening in stellar atmospheres. [1.5] (c) Explain how limb darkening observations can be used to obtain the optical depth dependence of the temperature of the photosphere. How important is this information with regard to identifying the source of continuous opacity in stars? [2] Give two methods used to derive limb darkening information for stars other than the Sun? [1.5] 5 TURN OVER
5. (a) Write down the formal definitions of the mean intensity and flux. Why can we only measure flux rather than intensity for most stars? Does either obey the inverse square law? [1.5] If there is no azimuthal dependence on I λ, the surface flux F λ () is given by 1 F λ () = 2π I λ (, θ) cos θd(cos θ). Assuming a linear source function, S λ (τ λ ) = a λ + b λ τ λ, we obtain I λ (, θ) = a λ + b λ cos θ. Hence, derive the Eddington-Barbier relation, and explain its significance. [2] (b) For a grey atmosphere in LTE, using the Eddington approximation, the source function becomes S τ = 3 4π (τ + 3 )F () 4 Derive the surface temperature in terms of the effective temperature. [2.5] (c) The radiation pressure, P ν, at frequency ν can be expressed as P ν = 1 I ν cos 2 θdω c Use the Eddington approximation to show that for a grey atmosphere in LTE the total radiation pressure, P R, is equal to P R = 4σ 3c T 4 Briefly discuss how radiation pressure can drive winds in earlytype stars. [2.5] (d) The Eddington parameter, Γ e, can be written as Γ e = 1 4.5 q L/L M/M where q is the number of free electrons per atomic mass unit. Explain the meaning of Γ e and derive the Eddington luminosity for a completely ionized hydrogen atmosphere of mass 1M. [1.5] END OF QUESTION PAPER 6