Download PHYS-633: Problem set #2

PHYS-633: Problem set #2
Due classtime Tuesday Sept. 23
Please write neatly and show your work. Please put a box around all your final
answers. At the top of your front page, near your own name, please list the name of all
persons with whom you consulted in solving the problems.
1. Temperature to bring protons to nuclear scale
For protons colliding with thermal energy kT the associated electric potential for minimum approach distance b is given by kT = e2 /b. Compute the temperature T needed
for approach to the scale of an atomic nucleus, i.e. b = 1 f m = 10−15 . Why is
the temperature so much higher than the temperature for proton-proton fusion, i.e.
T ≈ 15 MK.
2. Effect of Radiation Pressure on Mass-Luminosity relation
a. For a star of luminosity L, mass M , and opacity κ, derive an expression for the ratio
Γ of the radiative acceleration to gravitational acceleration. How does this depend on
radial distance r?
b. First generalize the equation for hydrostatic equilibrium to include the total (gas
+ radiation) pressure P = Pgas + Prad ; then use the equation for radiative diffusion to
derive a hydrostatic equilibrium equation in terms of just the gas pressure by including
a correction to the gravitation term in terms of the ratio Γ defined in part a.
c. With this correction, now repeat the scaling analysis in sec. 16.1 of DocOnotesstars2 to derive a revised mass-luminosity relation. How does L scale with M for large
M?
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3. Hydrogen fusion burning
a. Using Einstein’s famous formula E = mc2 , compute the mass reduction rate dM/dt
associated with the sun’s radiative luminosity L . Give your answer in both kg/s and
M /yr.
b. Now using the energy release efficiency ≈ 0.007 for H burning to He, compute the
associated rates of reduction dMH /dt in the mass of Hydrogen in the sun. Again give
your answer in kg/s and M /yr.
c. Now compute the total decrease in the mass of solar H (in units of M ) over its
main sequence lifetime. What happened to this H?
d. The sun started out with a mass fraction X = 0.72 of H, with the rest mostly
Helium, with a mass fraction Y ≈ 0.26. Use the above to compute the average mass
fractions Xtams and Ytams at the end of the sun’s main sequence life (known as “TAMS”,
for “terminal age main sequence”).
e. After the main sequence, the sun will evolve into a Red Giant with L ≈ 5000L , by
burning H in the shell around the hot He core. How long (in Myr) can it last in this
stage before it doubles the amount given in part c for H consumed by core burning
during the full main sequence.
4. Planetary nebula emission and expansion
Suppose we observe the Hα line from a spherical planetary nebula, and find it has a
maximum wavelength of 656.32 nm and minimum wavelength of 656.24.
a. Estimate how fast the nebula is expanding? Give your answer in both km/s and
au/year.
b. Now suppose that over a time of 10 years the nebula’s angular diameter is observed
to expand from 10 arcsec to 10.1 arcsec. What is the distance to the nebula, in pc?
c. What is actual physical diameter of the nebula, in au?
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5. Suppose a planetary nebula emits radiation at a constant emissivity η (energy/time/
volume/solid angle) within a spherical shell whose thickness ∆R is 10% of its radius
R. (Assume there is no absorption opacity, and the the center of the shell is hollow
and so emits no radiation.)
a. Using the fact that the thickness ∆R R, derive an approximate expression for
the total luminosity L of the shell in terms of η and R.
b. Next derive an expression for the length of the longest line of sight through the
shell, and compare this with length of a radial line through the shell.
c. Now compute the ratio of the intensity (i.e. surface brightness) along this longest
limb line to that along the radial line. (Note that the radial line cuts though both the
front and back of the shell.)
d. How does this result help explain the “ring-like” appearance of planetary nebula?
6. White dwarf cooling time
a. How many Carbon nuclei are there in a pure Carbon white dwarf of mass 1 M ?
b. If the Carbon is full ionized, how many electrons are there?
c. Each particle in a gas (even a degenerate gas) of temperature T has a thermal
energy (3/2)kT . If this fully ionized Carbon white dwarf has a constant temperature
of 107 K through nearly all of its interior mass, what is its total thermal energy (in J)?
d. Assuming it has a radius of 0.01 R and radiates like a 30, 000 K blackbody, compute
its luminosity, Lwd (in L ).
e. Now use these results to estimate this white dwarf’s cooling lifetime, tcool (in yr).