Download Ay123 Fall 2011 STELLAR STRUCTURE AND EVOLUTION Problem Set 2

Ay123
Fall 2011
STELLAR STRUCTURE AND EVOLUTION
Problem Set 2
Due Thursday, October 13, 2011
1. (10 pts) The internal structure of a low mass hydrogen-burning star can be approximated
by a polytrope of index 5/3.
a. Assume the matter in the stars is 75% H by mass and 25% He. Calculate the mean
mass µ per particle and the mean mass µe per electron (assuming the gas is fully
ionized).
b. If gas pressure dominates, show that the central temperature is given by an expression
of the form Tc ∝ µM/R. Calculate the constant of proportionality and thus establish
an absolute scale using material from the lectures.
c. For low-mass stars it is observed that stars of mass M have radii R = R⊙ (M/M⊙ )0.08 .
Use the polytropic relations to calculate the run of central pressure Pc amd central
temperature Rc with mass for such stars.
d. For stars of what mass will the central radiation pressure, estaimted from part (b)
of this question, become equal to Pc ?
2. (10 pts) Show that when a self-gravitating body of polytropic gas shrinks homologously,
its thermal energy scales with the stellar radius R as Eth ∝ R3(1−γ) , where γ is the ratio
of principal specific heats. Hence, show that a polytropic star is unstable to gravitational
collapse if γ < 4/3.
3. (10 pts) Consider a family of stars in which the opacity is dominated by Thomson scattering by electrons, and in which the nuclear energy is generated by the carbon-nitrogen
cycle. Use your lecture notes to determine the density and temperature dependencies of
opacity and energy generation in such a case. In analogy with the homology relations
we derived in class, for this family of stars find a relation between the radius and the
mass, and a relation between the luminosity and the mass. Locate this population on
the Hertzprung-Russell diagram.
4. (10 pts) Estimate the ratio of the maximum mass of a neutron star to that of a pure
He white dwarf. Do the same for the radius of the neutron star. Evaluate both assuming that
the Chandrasekhar limit for a pure He white dwarf is 1.46 M⊙ and its radius is 0.01 R⊙ .
What is the mean density of the neutron star ? What is the mean separation between
neutrons in this star and how does it compare to the typical size of an atom ?
Show that, to within an order of magnitude, seetting degenerate energy density equal
to electostatic energy density gives the same density as taking one proton per Bohr radius
cube. Both calculations should only contain physical constants (do not substitute numbers).
Look up the density of liquid hydrogen, liquid helium, water, and iron. How well to they
satisfy the “same volume per atom” statement ? What is the representative value of the
atomic volume and of the typical atomic separation (the cube root of the former) ?
5. (20 pts) In this problem you will numerically integrate the equation of hydrostatic equilibrium to obtain the mass-radius relation for white dwarfs. You will assume that the
pressure in the white dwarf is provided entirely by electron degeneracy pressure, Eq.
(3.53) in Hansen, Kawaler. and Trimble.
a. Show that
me c2
x2F
dP
=
.
dρ
3µe mH (1 + x2F )1/2
b. Now devise a numerical scheme to integrate the two coupled differential equations,
dρ dP
dρ
=
,
dr
dP dr
and
dm(r)
= 4πr 2 ρ(r),
dr
using the equation of hydrostatic equilibrium for dP/dr. You can use Mathematica,
write a C or Fortran program, or any other numerical procedure you find handy.
c. As you change the central density, you should get a one-parameter family of stars
with a sequence of masses M and radii R. You should be able to understand the
qualitative behavior of your results for R → 0 and R → ∞.
d. Plot this mass-radius relation for µe = 56/26 for an iron white dwarf and µe = 2 for a
carbon white dwarf. Plot the following three white dwarfs on your graph: (1) Sirius
B, M = 1.053 M⊙ , R = 0.0074 R⊙ ; (2) 40 Eri B, M = 0.48 M⊙ , R = 0.0124 R⊙ ;
(3) Stein 2051, M = 0.50 M⊙ or 0.72 M⊙ , R = 0.0115 R⊙ , and try to infer their
compositions.
e. The results you have derived above should show that as M → 0, R → ∞. Clearly,
at some point this result must break down (think about where Jupiter would fall
on this plot !). This is because when the density becomes sufficiently low, Coulomb
interactions between the electrons and ions (neglected above) become important in
determining the equation of state. Estimate the Coulomb energy per electron. Use
this result to estimate the density at which Coulomb effects become important. You
can then estimate (very roughly) the maximum radius for a white dwarf and the
mass at which it occurs.