ASTR 545 Fall 2018 Homework 53 (XXX points) ALWAYS SHOW YOUR WORK

Throughout this assignment, we adopt the notation nijk, njk, and nk etc., where i is the excitation level, j is the ionization stage, and k is the species index.

1. (XX pts) Abundances. Using the table from Aspulund, Grevess, & Savaal (2004, astro-ph/0410214) given in the class notes...

(a) [X pts] If a given star has [Fe/H] = −0.63, what is the value of nFe/nH in this star. Show your work. (b) [X pts] If this same star has [Mg/H] = +0.26, then what is [Mg/Fe] for this star? Show all steps of your work. 2. (XX) Physics. Consider a gas of T = 10, 000 K and pressure of the nuclear particles −2 PN = 100 dynes cm . The gas comprises two elements, H and He, with relative mass fractions X = 0.7, and Y = 0.3. The ionization fractions of H are f11 = 0.15 and f21 = 0.85. The ionization fractions of He are f12 = 0.01, f22 = 0.89, and f32 = 0.10.

(a) [X pts] Compute the nuclear particle number density, nN.

(b) [X pts] Compute the H and He abundance fractions, αk.

(c) [X pts] Compute the H and He number densities, n1 and n2.

(d) [X pts] Compute the number densities of all ionization stages for H and He, n11, n21, n12, n22, and n32.

(e) [X pts] Compute the number density, ne. (f) [X pts] Compute the total particle number density n.

(g) [X pts] Compute the electron pressure, Pe, and the gas pressure, Pg.

(h) [X pts] Assuming thermal equilibrium, compute the radiation pressure, Pr. Compute the total pressure, P , and the ratio β = Pg/P .

(j) [X pts] Compute the molecular weights, µN, µe, and µ.

(k) [X pts] Compute the nuclear particles mass density, ρN. Compute the electron mass density, ρe. Show the total mass density obeys ρ ≃ ρN to high accuracy. (i) [X pts] At this temperature, which ion contributes the greatest number density of free to the gas? What percentage of the total electron density does this ion contribute to the gas.

3. (XX) Excitation and “Balmer ”.

(a) [X pts] Starting with the Boltzmann excitation equation for n211/n11, and the Saha ionization equation for n21/n11, combine them to derive the expression for the quantity n211/n1, i.e., the number density of neutral atoms in the n = 2 excitation sttate relative to the number density of all hydrogens.

(b) [X pts] Using your expression, for the gas condition given for Problem 1, compute n211/n11 and n211. 4. (XX) Ionization Fractions of Calcium. We will assume that there are only three ionization stages for Ca, denoted n1,20 for neutral, n2,20 for singly ionized, and n3,20 for doubly ionized, where n20 = n1,20+n2,20+n3,20.

(a) [X pts] In terms of the number densities, write out the expressions for ionization fractions, f1,20, f2,20, and f3,20

1 (b) [X pts] Apply the ’recursive’ Saha Equation and compute the caclium ionization fractions in the atmo- 13 −3 sphere of a G2V star with T = 6000 K and ne = 4 × 10 cm . 5. (XX pts) Equilbrium Gas Physics. Consider a gas structure in thermal equilibrium with T = 6000 K and −2 Pg = 100 dynes cm comprising hydrogen, helium, and one metal, calcium, with mass fractions X = 0.71, Y = 0.27, and Z = 0.02. Assume all ionization states for H and He and that Ca can be ionized twice (Ca0, Ca+, Ca+2). Compute the equilibrium conditions of the gas using particle-charge conservation,

Jk ne = (n − ne) X αk X(j − 1)fjk(ne,T ) (1) k j=1

+ − −5 + − In solving Eq. 1, apply a tolerance of at least ne − ne = 10 , where ne and ne are the values that bracket the root, ne. I have provided the solutions to this problem in some tables. Please present your solutions in the identical format.

(a) [XX pts] Table 1. For the given T and Pg, report the quantities n and the abundance fractions, αk.

(b) [XX pts] Table 2. Report the equilibrium values nN, ne, µN, µe, µ, ρN, ρe, ρ, Pe, and Pe/Pg.

(c) [XX pts] Table 3. Report the equilibirium values nk, njk, fjk,Φjk(T ), Ujk(T ) and fraction of electrons donated for ion j,k,, i.e., f(ne,j,k) = (j−1)njk/ne. (d) [XX pts] From Table 3, identify and rank which ionization stages of which species are the top three donors of free electrons to the gas. What do these data tell you about the role of metals in gas where hydrogen and helium are not highly ionized?

6. (XX pts) Equilibrium Gas Physics as a Function of Temperature. Expand your code to compute the equilib- rium conditions of this gas structure over the range 1500 ≤ T ≤ 25, 000 K.

(a) [XX pts] Produce fully labeled plots of the following versus T . Plot 1: log ne, log nN , and log ntot; Plot 2: log ρe, log ρN , and log ρtot; Plot 3: Pe/Pgas. Use different colors or line types when necessary. For Plots 1 and 2, limit your “y” axis to five decades in the log.

(b) [XX pts] Plot 4: fjk versus T (linear-linear: the fjk range between zero and unity).

(c) [XX pts] Plot 5: log nk, and log njk versus T . Make a separate panel for each element and use different colors or line types for each ionization stage. Limit your “y” axis to five decades in the log.

(d) [XX pts] Plot 6: log f(ne) versus T , where f(ne) = (j−1)njk/ne, for all jk on a single panel. Make the “y” axis of your plot over the range −6 ≤ log f(ne) ≤ +0.5.

(e) [XX pts] [i] Examining Plot 1, explain why ntot and ρtot decrease with increasing T , and why they have the slopes they do. [ii] From Plot 6, report which ionization species is the major electron donor and over which temperature range. [iii] Going back to your Plot 3, note that Pe/Pgas rises to 0.5, levels off, and then rises slightly above 0.5. Incorporating the data in Plot 6, explain physically why this is happening (why ≃ 0.5?) and explain the physics of why these values and changes happen at the they do.

YOUR CODES staple you codes to the end of your homework assignment. Clearly comment your codes.

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