14.7: Problems

1. Imagine a line whose intensity profile is \[f_\nu=1+\delta\left(\nu-\nu_0\right)\nonumber\]

   Calculate the observed line profile for a radially expanding atmosphere which exhibits a velocity gradient \[\frac{d v}{d \tau_c}=\text { const }=c \frac{\Delta \lambda_0}{\lambda_0}\nonumber\] State any assumptions that you make in solving the problem.

2. Consider a line generated by atoms constrained to move perpendicular to a radius vector from the center of the star. Find an expression for the atomic absorption coefficient due to Doppler broadening alone.
3. Find the natural width for
   1. Hβ
   2. Mg II (λ4481)
   3. FeI (λ3720).
4. Estimate the transition times from the natural widths of the lines in Problem 3, and compare them with a crude estimate of the collision rates for atoms in these states. State clearly any assumptions you make. In what kind of star would you expect to find these spectral lines?
5. If both the atoms of a radiating gas and the particles perturbing them are in statistical equilibrium, show that the average relative velocity between them is given by \[\langle v\rangle=\left[\left(\frac{8 k T}{\pi \mu_h}\right)\left(\frac{1}{A_1}+\frac{1}{A_2}\right)\right]^{1 / 2}\nonumber\] where \(\mu_h\) = the mass of a unit atomic weight and \(A_1\) and \(A_2\) are the atomic weight of the atom and perturber respectively.
6. Find the far-wing dependence of the line absorption coefficient of an atom having nondegenerate energy levels which are broadened by perturbers having only octopole moments of their charge configurations.
7. Compute a line profile for Si II(λλ6347.10) for an A0V star. Use a model atmosphere code if possible.
8. Show that \(\mathrm{W}_\lambda / \lambda=\mathrm{W}_\nu / \nu\)
9. Use a model atmosphere code such as ATLAS to generate "curves of growth" for Fe I(λλ4476), Mg II(λλ4481), and Si II(λλ4130). Include a microturbulent velocity of 2 km/s. Consider the reference atmosphere to be one with \(T_{\mathrm{e}}=10^4 \mathrm{~K}\), \(\log g=4.0\), and solar abundance (except for Fe, Mg, and Si). Compare your results with the classical curve of growth for a Schuster-Schwarzschild model atmosphere and obtain values for \(\Delta\lambda_{\mathrm{d}}\), the kinetic temperature, microturbulent velocity, and \(\Gamma\) for each line. Compare your results with the values used to generate the line profiles and discuss any differences.
10. Consider the following situation: A 1-mm beam of neutral hydrogen gas with an internal kinetic temperature of 10⁴ K is accelerated to an energy of 10^-3eV per atom. The beam enters a 10-m vacuum chamber and is directed toward a 1-cm bar located in the center of the chamber and oriented at right angles to the beam. The bar has been charged to 10⁷v. The beam passes through a 1-mm hole in the bar and proceeds out the opposite side of the chamber. A spectrograph is placed so that it "looks" along the beam and sees the beam against a 2 ×10⁴ K continuum blackbody source located near where the beam enters the chamber. Assuming that the beam density is sufficiently low to ensure that it is optically thin, but high enough to establish LTE, find the line profile for Hβ. Further assume that the central depth of the line is 0.6. Find the equivalent width of Hβ and the density of hydrogen. On the basis of your results, discuss the validity of the assumptions you used.
11. Consider a Schuster-Schwarzschild model atmosphere populated with several types of atoms having different atomic absorption coefficients. Find the theoretical curves of growth for each of these atoms.

    Compare with the classical solutions for the curve of growth.

    1. \(S_1(\nu)=a+\frac{b}{|\Delta \nu|}\)
    2. \(S_2(\nu)=\delta(\Delta \nu)\)

    3. \(S_3(\nu)=\left(\begin{array}{ll}
       a & \text { for }|\Delta \nu| \leq \Delta \nu_0 \\
       0 & \text { for }|\Delta \nu|>\Delta \nu_0
       \end{array}\right)\)

    4. \(S_4(\nu)=\left(\begin{array}{lll}
       a+b(\Delta \nu)^2 & \text { for }|\Delta \nu| \leq\left(\frac{-a}{b}\right)^{1 / 2} & b<0 \\
       0 & \text { for }|\Delta \nu|>\left(\frac{-a}{b}\right)^{1 / 2} &
       \end{array}\right)\)

    5. \(S_5(\nu)=\left(\begin{array}{lll}
       a+b|\Delta \nu| & \text { for }|\Delta \nu| \leq \frac{-a}{b} & \\
       0 & \text { for }|\Delta \nu|>\frac{-a}{b} & b<0
       \end{array}\right)\)

12. Let the probability of finding a value of the turbulent velocity projected along the line of sight v be uniform in the range -v₀ # v # v₀. The probability of finding a value of v outside this range is zero. In addition to turbulence, there are thermal Doppler motions present which correspond to a temperature T. Assuming that \(f\) and \(\Gamma\) are known, derive an expression for the atomic line absorption coefficient. Leave your answer in the form of a definite integral containing an error function.
13. Consider a certain atom in the solar atmosphere at a point where the hydrogen abundance N_h = 10¹⁷cm^-3 and T = 5500 K. The atom has a strong resonance line at l=5000Å with an Einstein A coefficient of 9.7 × 10⁷s^-1. The atom has interacted with a neutral hydrogen atom so that a frequency shift of ∆ω = 2 × 10⁶/r² s^-1 of the line frequency has resulted. Here, r is in angstroms.
    1. Make a reasonable estimate of how long the collision lasts.
    2. Qualitatively justify the type of broadening theory you would use to describe the atomic absorption coefficient.
    3. What is the approximate cross-section for this event?
    4. What is the value of \(\Gamma\) you would obtain from the impact phase-shift theory of line broadening?
14. Suppose the data below are observed in a certain star. They all pertain to the lines of the neutral state of the same element which has a partition function of 2.0. The parameter ε_i refers to the lower level of the transition.

    \(j\)	\(\lambda, \AA\)	\(\epsilon_i,\mathrm{eV}\)	\(g_i\)	\(W_\lambda,\AA\)	\(f_{ij}\)
    1	4,930	2.70	3	0.330	0.50
    2	3,470	2.70	3	0.0155	0.050
    3	5,676	2.70	3	0.0562	0.0667
    4	3,360	1.80	6	0.0190	0.00316
    5	10,050	1.80	6	1.349	1.060
    6	3,862	1.80	6	0.0617	----

    Using the Schuster-Schwarzschild model atmosphere, find

    1. the number of atoms per square centimeter above the photosphere,
    2. the missing \(f\) value, and
    3. the value for the Doppler velocity v₀.