Download Ay123 Fall 2011 STELLAR STRUCTURE AND EVOLUTION Problem Set 4

Ay123
Fall 2011
STELLAR STRUCTURE AND EVOLUTION
Problem Set 4
Due thursday Nov 3, 2011
1. (20 pts) The Opacity of H−
In this problem you will consider the opacity due to the negative hydrogen ion H− in
stellar atmospheres. In cool stars, bound-free (photoionization) of this ion provides most
of the opacity at wavelengths above the Balmer limit (λ = 3647 Å, below which opacity is
due to photoionization of hydrogen atoms in the n = 2 level). The ionization energy of
H− is χ(H − )− = 0.754 eV, meaning that this ion can absorb photons with wavelengths
λ < 16, 444 Å. H− has only one bound state.
a. First, use the Saha equation to write an expression for n(H − )/n(H), the ratio of the
hydrogen-ion to neutral-hydrogen abundances, in terms of χ(H − ), the temperature T , and
the electron pressure, Pe = ne kT (where ne is the free-electron number density; these free
electrons are provided in cool stellar atmospheres presumably by metals, which have lower
ionization thresholds than hydrogen or helium).
b. Next, calculate the ratio n(H; n = 2)/n(H; n = 1) of the abundance of hydrogen
atoms in the first excited state (n = 2) to the abundance in the ground state (n = 1); then
do the same for n(H; n = 3)/n(H; n = 1).
c. Assuming an electron pressure, Pe = 101.5 dyne/cm2 , at what temperature is n(H; n =
3) ≃ n(H − )? Assuming the bound-free absorption coefficients for H − and the n = 3 level of
hydrogen are comparable, this gives the temperature below which H − absorption dominates
the opacity above the Balmer break.
d. Write an expression for κ(3647+ )/κ(3647− ), the ratio of the opacities just above and
below the Balmer break for (i) low-temperature atmospheres where the opacity above the
Balmer break is due to the H − ion, and (ii) high-temperature atmospheres where the opacity
above the Balmer break is due to the n = 3 level of neutral hydrogen. Your results should
show that at low temperatures, the Balmer discontinuity depends on both temperature and
electron pressure, while at high temperature, it depends only on temperature. (You do not
need to find numerical values for this ratio, just a formula.)
2. Convection in the Sun.
We are going to try to find the convective zones in the Sun.
Use the interior model given in Allen’s Astrophysical Quantities, 4th edition, table 14.2.
We adopt the mixing length approximation for convection. Assume that for any surface
convection, the mixing length l is the pressure scale height, while for any core convection,
the mixing length is R∗ /10.
a) (10 pts) Calculate ∇(rad), ∇(ad) and ∇(actual) for each depth in the model for the
Sun.
b) (5 pts) What criterion is used to determine if a a zone is convectively unstable ? What
radial zone(s) in the Sun are convectively unstable ?
c) (5 pts) Now we consider one particular zone, the unstable zone with the smallest r.
Calculate w, the mean speed of the convective elements, vs , the sound speed there, the Mach
number, and the pressure scale height.
d) (5 pts) Compute also the cooling time for a rising column, the eddy turnover time
(τ ≈ l/w), and the ratio of the flux carried by convection to the total flux in this zone.
3 (5 pts) Energetic neutrinos from the Sun are produced by the decay of 8 B. Show that
the rate of the reaction producing 8 B, 7 Be(p, γ)8 B is approximately proportional to T 14 ,
when the temperature T is near 1.5×107 K. If one attempted to explain the discrepancy between the observed and predicted neutrino flux in Ray Davis’ experiment by
postulating an error in the central temperature of the Sun, what change in the central
temperature would be required ?
4 (15 pts) Assume only the main p + p reactions occur in a star composed initially only of
hydrogen. How long does it take for the whole chain to reach equilibrium so none of the
intermediate reactions is slower than the first reaction for T and ρ that of the center of
the Sun ? Plot the abundances relative to hydrogen (1 H) of the isotopes involved as a
function of time during the time it takes to reach equilibrium under these assumptions.
Please take any specific nuclear reaction rates needed from the Caughlan-Fowler (CN1988)
nuclear reaction rates available at http://www.phy.ornl.gov/astrophysics/ data/data.html.
2/3
In the notation used there to give the reaction rates as analytic functions, T923 = T9 , T913
1/3
is T9 . When the energy generation rate for a process (p + p or CNO) is required, use the
expressions given in HKT.