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Ast 4001, 2015 November 17
Homework set 8 -- Two problems, turn in your solutions on Tuesday November 24.
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1. Let's estimate whether the center of a red dwarf star is degenerate or not. For simplicity,
pretend that it's pure ionized hydrogen throughout this problem. Electron density is n e .
(a) First we need a criterion for degeneracy. Use two facts:
 If the gas is extremely non-degenerate, its pressure is P  2 n e k T, the ideal gas law.
The factor of 2 accounts for electrons plus protons.
 If it's moderately degenerate, then P  ( 2.34 × 10 erg cm 2 ) n e 5 / 3 + n e k T.
The second term represents protons, which are certainly not degenerate.
Now imagine gas with some particular temperature T . For low densities we simply
use the ideal gas law, and for high densities we use the degenerate formula quoted above.
(Make a sketch-plot of log P vs. log  .) The crossing point of these two formulae gives
a rough, rule-of-thumb criterion for the onset of degeneracy. We can express it as
f (  , T ) > C implies that the gas is degenerate,
where  is the mass density of the gas, protons plus electrons.
Specify the function f (  , T ) and the numerical value of C in c.g.s. units.
(This can be written in various equivalent ways; choose a very simple form.)
(b) Roughly speaking, low-mass stars have central temperatures T 0  10 7 K, somewhat
cooler than in the Sun. Evaluate the central density  0 that would make the center of
the low-mass star marginally degenerate according to the criterion in part (a) above.
(c) Very low-mass stars are almost entirely convective, so they can be modeled as
polytropes with index n = 1.5. An n = 1.5 polytrope has central parameters
P 0  0.77 G M 2 / R 4
and
 0  1.43 M / R 3 .
Deduce the mass and radius of a red dwarf star that has the central density that you
estimated in part (b) above. (Regarding pressure, see part (a)).
Express these answers relative to the Sun: M / M sun and R / R sun .
(BTW, real stars are semi-degenerate at somewhat higher masses than this, partly
because they contain some helium.)
(d) The estimates in parts (b) and (c) refer to the center of the star. Consider some other
location inside the star, say r = 0.5 R . Based on our criterion and the structure of
an n = 1.5 polytrope, is that location less degenerate or more degenerate than the center?
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----- problem 2 is on next page -----
4001hwk08 - p2
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2. This problem concerns energy release by various forms of nuclear burning in a massive
star. A star with mass M = 10 M sun begins with luminosity L  4000 L sun , brightens
unsteadily as it evolves, and eventually reaches L  25000 L sun just before it becomes
a supernova.
(Meanwhile, earlier types of burning continue to occur outside the central core.
For example, a carbon-burning core is surrounded by a helium-burning shell and
a hydrogen-burning shell.)
For the purposes of this problem, suppose that 1 M sun of material is burned in the core
during each stage. This is a good rough guess because the Schoenberg-Chandrasekhar limit
is slightly more than 1 M sun for a 10 M sun star.
(a) Find out the precise masses of five special atoms: 1 H , 4 He , 12 C , 28 Si , and 56 Fe .
Use these to estimate the amounts of nuclear energy released by converting 1 M sun
of material in each of four stages: H  He , He  C , C  Si, and Si  Fe.
Express your answers in both Joules and ergs.
(Caution: High precision is needed in order to get the correct mass differences.
Use the masses of these atoms, not their bare nuclei, and think about why the
electrons should be included.)
(b) If the luminosity produced in the central core averages 5000 L sun on the main sequence
and 15000 L sun during each later stage, estimate the duration (a.k.a. nuclear timescale)
for each stage of nuclear burning. Pretend that all the energy escapes from the star as
photons. (In reality, neutrinos carry away a small fraction.)
If we randomly choose 1000 stars with mass = 10 M sun , roughly how many would
you expect to be in each stage of burning?
(c) For comparison, estimate the thermal timescale (a.k.a. Kelvin-Helmholtz timescale)
for a 10 M sun main-sequence star. Assume that it's a polytrope with n = 3,
radius  5 R sun , and average luminosity  5000 L sun . Clearly explain your
reasoning.
Caveat: Since most of the assumptions are shaky, this exercise gives only crude
order-of-magnitude suspicions. For example, silicon burning happens much faster
than we'd guess from this approach, and much of its energy does not promptly escape
in the form of radiative luminosity. Strictly speaking, 56 Fe is not the right isotope
of iron in this connection; and a 10 M sun star can become an "electron-capture
supernova" before the core is iron. But the simple assessments in (a), (b), (c) give us
a quick, simplified preview of the evolution of stars with M  10 M sun . Accurate
values are 100x or 1000x harder to calculate.
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