Reading: Chapter 30 of Carroll & Ostlie
The rules: Collaborative work is encouraged. This homework can be done in consultation with your fellow classmates, the AI, or the professor. However, everyone must submit their own solutions to get credit, and all help should be acknowledged (a single sentence mentioning the others in your study group is sufficient). Show your work.
1. [6 pts] Derive the formula for the bolometric flux for a universe with R = Ro(t/to)2/3 and k = 0.
2. [6 pts] Consider an Einstein-de Sitter universe. Show that the angular diameter of an extended object is a minimum when the redshift is z = 1.25. Hint: You have already worked out the luminosity distance in Problem 1. Simply multiply by (1+z)-2 to obtain angular diameter distance and proceed from there.
3. [9 pts] The deuterium nucleus is not very tightly bound. (a) Calculate the binding energy of the deuterium nucleus, using mH = 1.007825 amu, mn = 1.008665 amu, and mD = 2.014102 amu. (b) What is the wavelength of a photon with this energy? (c) From Wein's law, at what temperature is this the characteristic energy of a blackbody photon?
4. [6 pts] The relative line strengths of the carbon absorption lines in Q1331+70 indicate that the temperature of the intervening cloud is 7.4 +/- 0.8 K, and the lines are z = 1.776. How does the temperature of the cloud compare with the temperature of the CBR at that redshift? [Note: If there are sources of heating for the cloud in addition to the CBR, its temperature must be considered an upper limit.]
5. [5 pts] In 1941, microwave observations detected absorption lines due to CN in molecular clouds. A CN molecule has 3 first excited rotational states, each of which is degenerate and has an energy that is 4.8 x 10-4 eV above the ground state. An analysis of the absorption lines shows that for every 100 molecules in the ground state, there are 27 others that are in one of the three first excited states. Assuming that the molecular clouds are in thermal equilibrium with the CBR, use the Boltzmann equation to estimate the temperature of the CBR.
6. [9 pts] (a) Using the relativistic doppler shift (eq. 4.32 in Carroll & Ostlie) and Wein's law, derive the temperature measured by an observer with peculiar velocity, v, relative to Hubble flow. Hint: Keep track of negative signs. This problem is for a moving observer, not a moving source! (b) Show that this reduces to Tmoving = Trest(1 +v/c cos(theta)) for v much less than c. (c) Calculate the magnitude of the variation in the temperature of the CBR due to the Sun's peculiar velocity, which is approximately 370 +/- 10 km/s (i.e., our motion with respect to the standard of rest).
7.
[9 pts]
(a) Estimate the mean free path of a photon just before recombination, at
a redshift of z ~ 1100
(recall that the electron cross section = 6.65 x 10-29 m2
and 1/l = n sigma).
For convenience, use a composition of pure
hydrogen (one electron per proton; current baryon density=
density of hydrogen) and assume omega_o = 1.
(b)
Find the ratio of
the age of the universe to the time it takes a photon to travel
this mean free path (at z = 1100).
(c) For what value of the redshift would the age of the universe
equal the time for a photon to travel a mean free path? Neglect
the recombination of electrons and nuclei at z ~ 1100. This is
an estimate of when the universe would have become transparent due
solely to its expansion.