Union College
Winter 2015
Astronomy 220
Final Exam Questions
Here are the questions to which I want you to know the answers, or at least how to answer, by
the end of the term. You may use whatever means you wish to figure out the answers. Please
feel free to talk to others, or even me, about these questions. At the end of the term, though, you
must be able to answer or show how to answer these questions off the top of your head in a
private, one-on-one oral exam. In the oral, I will only ask ten of these questions.
1. What is Olber’s paradox? What are two possible resolutions to the paradox? Is this paradox
solved in modern cosmology?
2. List and explain five assumptions made by cosmologists today. Which of these are
assumptions are weakest?
3. Consider the plot of Vrec vs. distance for a number of galaxies shown below.
What is the Hubble constant according to these data?
4. (a) If H0 = 70 km/s per Mpc, what is the distance, in Mpc, of a galaxy with z = 0.002?
(b) What is the distance of this galaxy in cm?
(c) If the flux of this galaxy is 2.0x10-14 J/m2.s, what is the luminosity of this galaxy in J/s ?
(d) What is the luminosity in solar luminosities?
5. If a galaxy is measured to have an apparent magnitude = +16 and a redshift z = 0.05, what is
the approximate M of the galaxy? (Use approximation for small z.)
6. Calculate the redshift of the radiation that was emitted at the epoch when the gravitational
force experienced by a galaxy due to any other distant galaxy was twice what it is today?
7. Discuss the observational evidence for the existence of Dark Matter. What are the
observations and how do they suggest that much Dark Matter exists? What is meant by Dark
Matter? How much Dark Matter does there seem to be?
8. List the typical M/L values (in solar units) for:
a) A typical population of stars, such as that in the solar neighborhood.
b) Galaxies.
c) Rich clusters of galaxies.
9. removed
10. Assume that observations of a cluster of galaxies have revealed the following data:
Recessional velocities of galaxies (in km/s):
900, 880, 800, 900, 320, 720, 360, 1450, 1470, 1510, 1040, 920, 1030, 1000
1/2=Angular size of circle inside of which 1/2 of the galaxies are located.
= 10o
a) Use the Hubble law to compute the distance to the cluster.
b) Compute the total mass of the cluster. What is the average mass per galaxy?
11. Using first principles, derive the approximate ratio of the number of photons in the CBR to
the number of baryons. What numerical value results?
12. (Not for exam) Show that the dark matter in the halos of galaxies cannot be due to neutrinos
by doing the following:
a) Calculate the number density of neutrinos needed to make up the galaxy halo as a
function of neutrino mass, m (use MHalo~1012MSun, RHalo ~ 50kpc).
b) Use the Virial Theorem to estimate the average energy per neutrino. Using the relativistic
expression for energy (E2=p2c2+m2c4 ~ p2c2 for small masses which the neutrino surely has)
calculate the average momentum per neutrino.
c) Since neutrinos are fermions, they obey the Pauli Exclusion Principle. The minimum
average distance between neutrinos can be determined, then, using the Heisenberg
Uncertainty Principle, which states that px>h/4, where p is the “uncertainty in p” and
x the uncertainty in position. The space between neutrinos must be at least as large as the
uncertainty in their location, so this gives a minimum separation distance. Likewise, the
uncertainty in the momentum cannot be larger than the momentum itself. So, you can use the
inferred average momentum per neutrino as an upper limit to p. Calculate the minimum x
allowed between neutrinos.
d) From the minimum spacing between the neutrinos, infer the maximum number density
(max. n ~ 1/x3).
e) By comparing your equation in (d) with that in (a) infer for what values of m it is possible
for the halo to be composed of neutrinos? How does this mass compare with the upper limit
(known from experiments) of 40eV? What does this say about the likelihood that neutrinos
are the source of dark matter in galaxy halos?
13. Derive the Cosmology equation for total energy per unit mass of the Universe using
Newtonian physics and then manipulate the equation to derive an equation for the density
parameter, .
14. a) Derive an equation for the dependence of peculiar velocity of a particle on the scale
factor, assuming no acceleration of the particle. Hint: consider a particle moving through
the Universe with velocity vpec that travels a distance dr and calculate what the velocity of
this particle is relative to the observers in the new region (at distance dr from the previous
location) using the Hubble Law. Manipulate the equation to determine how vpec depends on
the scale factor R.
b) Derive an equation for the dependence of the temperature of baryonic matter, ignoring
any influence by radiation, on the scale factor. Hint: Use 3/2 kTmatter=1/2 mH<v2> where
<v2> is the mean square of the peculiar velocity of the H atoms.
15. A Cepheid variable is discovered by the Hubble Space Telescope in a galaxy at a redshift of
0.005. The Cepheid is measured to have a flux of 8.8 x 10-16 ergs s-1 cm-2 and a period of
variability of 30 days. Use the graph below to answer the following questions.
Cepheid Period-Luminosity Relation, taken from
http://www.astro.virginia.edu/class/whittle/astr553/Topic01/t1_leavitt.html
a) What is the distance of this galaxy based on the Cepheid’s observed flux?
b) What is the inferred value of the Hubble Constant?
16. removed
17. A supernova type Ia is determined to have a redshift of z = 0.022, and a peak apparent ‘V’
magnitude of mV = 15.7. For the following questions, consider that type Ia Supernovae are
known to have light curves which peak at an absolute V magnitude of Mv ~ -19.3
a) What is the distance of the host galaxy in Mpc?
b) What value of H do these data imply?
c) Do these data provide any evidence in support of or in conflict with the existence of dark
matter or dark energy? Explain
18. The search for extraterrestrial intelligence has yielded an amazing discovery: a radio signal
from a distant galaxy has been received that has been interpreted and says, "Hey Dudes!
This is Wayne's World and we live in the fast lane. We see a dipole anisotropy of the cosmic
background radiation of T/T~0.01. What do you see?
a) What is their velocity relative to the cosmological rest frame at their location in the
Universe? What is the ratio of their velocity to ours?
b) The following day, another message is detected that says, "Hey this is cool. We've
determined that the temperature of this background radiation is only 8.1K. Is that what you
get?" What is the redshift of the signal sent by Wayne?
19. Explain Einstein’s model of a static Universe. Explain why the  term was required.
Explain why the  term did not violate existing physics experiments, and show/derive the
exact expression for  that was required (the one used to find a value for )?
20. How do the following parameters vary with the scale factor, R?
a) mass density of non-relativistic matter,
b) energy density of radiation,
c) temperature of black body radiation,
d) ,
e) acceleration of expansion due to dark energy,
f) temperature of non-relativistic matter,
g) k.
h) z.
21. a) What are the allowed geometries of space-time in the standard cosmological models?
b) What principle or assumption restricts the geometry of space-time to just those listed in
the answer to (a)?
c) Explain how all of these geometries are incorporated into just one metric. What is the
name of this metric? What other cosmological effect does this metric include?
22. Discuss the cosmological parameters , k, and . What are they, how are they defined,
what are their names, and how are they related to each other?
23. What are the equations for R(t) for all the relevant (in 2015) cosmological models?
24. What is the age of the Universe if m,0=1.0 and ,0=0? What is the age of the Universe in
the Benchmark model?
25. Assume that all galaxies formed at z~7.
a) For m=1 and =0, calculate the age of the Universe when galaxies formed.
b) What is the age of galaxies today?
c) Calculate the age of galaxies today if = 0.7 and M = 0.3.
d) Which of your solutions (to a, b, and c) assumed a reasonable model for correctly
inferring the real answer in our universe?
26. For a galaxy with a redshift of z = 0.5, how long did the light that we detect today travel to
reach us?
a) In an Einstein-deSitter universe (i.e. k= 0, m = 1).
b) If = 0.7 and M = 0.3.
27. While telling a story, Professor Simbad Gumbo accidentally says the magic words (which
are in Persian, like most magic words) that give him cosmological power and all the matter in
the Universe is spontaneously converted into radiation (while conserving mass/energy, of
course).
a) If = 0 and k = 0, how does the expansion of the Universe as a function of time, R(t),
change? How are the parameters in the Friedmann energy equation affected?
b) What is the energy density of the radiation immediately after Simbad's words? Assuming
that the radiation created is blackbody, what is the temperature of the Planck spectrum?
c) If he said a different set of Persian magic words which instead converted all the matter
into , how then would the functional form of R(t) change?
28. Consider a typical galaxy with bolometric (i.e. total) luminosity of 2x1037 J/s (1011 LSolar) and
a linear diameter of 50 kpc at a redshift of 3.
a) If p==0, and m,0=1.0, what is the proper distance to this galaxy today?
b) What is its luminosity distance?
c) What is its angular diameter distance?
d) What is the bolometric flux received at the Earth?
e) What is the observed angular size?
f) What was the proper distance to this galaxy at the time the radiation was emitted?
29. (Not for exam) Consider a galaxy at a redshift of 1.4 and the benchmark model (m=0.3
and =0.7).
a) Use the Mathematica routine in NumericalIntegdpvz_Mathematica.htm to calculate the
proper distance to the galaxy.
b) What is the luminosity distance?
c) What is the angular diameter distance?
30. Discuss in detail the recent observational evidence suggesting a non-zero value for .
31. For reasonable values of M, and , and ignoring the cosmic neutrinos, determine the time
of radiation-matter equipartition and of lambda-matter equipartition.
32. (not for exam) a) Consider the first law of thermodynamics (dE=dQPdV) and applying it
to our universe -- what should dQ equal?
b) Discuss the “equation of state” and how it relates to modeling the universe. What
variable in the equation of state is used to describe the different constituents of the universe.
c) Use the equation of state and apply the first law of thermodynamics to derive the
“acceleration equation.”
d) Starting with the energy equation, derive the acceleration equation leaving the power
index of the R-dependence of  as a variable.
e) Considering your answers to (c) and (d), how does the variable in the equation of state
relate to the power index of the R-dependence of  for the dominant constituent of the
universe?
33. Imagine if the universe were flat and dominated by a new particle, let’s call it the marron.
The marron has no mass and an equation of state given by P = u. Derive the following
relations from first principles:
a) How does the energy density, u, of the marrons depend on the scale factor, R?
b) How does the energy of each marron particle change as the universe expands, i.e what is
the dependence of E on R?
c) What is the form of the scale factor’s dependence on time, i.e. R(t)?
d) If non-rel’c matter,0=0.001, at what scale factor will or did the non-relativistic matter
dominate over the marrons?
e) What is the current age of the universe?
f) What was the age of the universe when a particular light signal was emitted, as a function
of the redshift, z, of that signal?
g) What is the equation for proper distance as a function of z?
34. Give order of magnitude estimates for the values (with proper units) of:
a) Ho
b) crit
c) galaxies
d) M/L of rich clusters of galaxies
e) the current age of the Universe
f) the temperature of the cosmic background radiation
g) the ratio of the number of photons to baryons
h) 
i) tUniverse when radiation and matter had equal mass/energy densities
j) tUniverse when  and matter had equal mass/energy densities
k) the redshift, z, of the nearest cluster of galaxies
35. For the following, imagine m ~ 0.01, r,0 = 0, and  = 0, and calculate the value of .
a) at recombination (T~3500 K)?
b) at the epoch of nucleosynthesis (T ~ 109 K)?
c) at the end of equilibrium era (T ~ 1010 K)?
d) Comment on the flatness problem.
36. Derive the expression for the horizon size if:
a) radiation,0=total=1.
b) m,0=total=1.
c) ,0=total=1.
d) Explain in each case, why the horizon size is not simply equal to the age of the universe
times the speed of light.
37. Consider the early universe just 10 years after the Big Bang
a) What is the horizon size at this time?
b) How much time must pass for the mass density of the matter to decrease by a factor of 8?
38. What is the ratio of the temperature of the CBR to the believed temperature of the
background neutrinos? What event caused this discrepancy?
39. Calculate the approximate number of positrons within an electron's horizon just before the
electrons and positrons annihilate.
40. a) Derive an approximate equation for the temperature of the Universe when the neutrinos
decoupled.
b) Calculate (approximately) the age of the Universe when the neutrinos decoupled.
c) Derive and calculate the temperature and time when electrons and positrons annihilated.
41. Imagine a homogeneous universe where antimatter is not allowed by the laws of physics so
that matter is never annihilated and converted into radiation. All other laws of physics are
identical to those in our Universe.
a) Assuming that this universe has an approximately equal number of photons and baryons
which were in thermodynamic equilibrium, write an equation that relates baryons to Tradiation
in this universe.
b) At some time to, ucrit=5200 MeV/m3 (as in our Universe), what must Tradiation be at to if
baryons=1?
c) What is the ratio radiation/baryons at to in this universe?
42. Imagine that there is a fifth force through which a particle called the moo-on interacts with
baryons. Assuming that moo-ons are about as numerous as the baryons (i.e. about one mooon per 109 photons), what would the cross-section for interaction between baryons and mooons have to be for moo-ons to have decoupled before the epoch of nucleosynthesis? (in
which case, the effect of the fifth force and moo-ons would have minimal effect on most of
our cosmological modeling).
43. Imagine that the laws of particle physics were discovered to allow for the existence of a
charged fermion with 1/10 the mass of an electron and two spin state.
a) At what temperature do these particles annihilate with their antiparticles?
b) What would the ratio of the temperatures of the background radiation and the background
neutrinos be?
c) Why do we not need to worry about the annihilation of baryons in the determination of
Tphotons/Tneutrinos?
d) Derive an equation for the ratio of Tph/T as a function of nf, where nf is the number of
species of charged fermions with mass less than or equal to the mass of an electron that
existed in the early Universe. Count one for each species of particles and their
antiparticles. Include the electrons, so that nf=1 if only electrons and positrons existed.
e) Explain why non-charged particles are not included in this equation.
44. a) What determines the temperature for the “freeze-in” point of the Nn/Np ratio (where
Nn=number of neutrons and Np=number of protons)?
b) Why is the final Nn/Np ratio smaller than the “freeze-in” ratio? What is the value of the
final Nn/Np ratio?
c) Show how this determines the ratio of N(4He)/N(H).
45. Why can't Carbon be produced in the early Universe?
46. Discuss the issue of how the abundance of 2H today relates to B,0 in terms of the theory of
the early Universe. Does a larger abundance of 2H imply a larger or smaller value of B,0?
Explain.
47. a) From the observed cosmic abundance of 4He relative to H, Gamow, Alpher, and
Hermann predicted the existence of a background of radiation with T<10K. Comment on
this by showing what effect a CBR temperature >10K has on primordial nucleosynthesis.
Which step in the analysis is affected and how?
b) Using the same analysis, but this time knowing the value of T0, show that the universe
that we see requires that b,0 > stars,0.
48. Using a back-of-the-envelope, order-of-magnitude calculation show that the existence of
some residual 2H left over from the Big Bang nucleosynthesis requires that B,0 < 1.
49. If the Universe is dominated by WIMPs, when do the WIMPs decouple, relative to the
epoch of nucleosynthesis? What effect do they have on nucleosynthesis arguments?
50. What are the problems with the standard Big Bang model and how does the inflationary
universe model solve them?
51. What did Alan Guth mean by "the Universe may be the ultimate free lunch?"
52. What is the physical mechanism for producing an exponential expansion in the inflation
epoch? Do the math to show why an exponential expansion results. Also show how
inflation sets the energy density of the Universe to equal the critical energy density.
53. Ryden 10.1
54. Ryden 10.2
55. Ryden 10.4
56. a) Derive the relation between the ratio of the number of Helium nuclei to Hydrogen nuclei
and the ratio of the number neutrons to protons.
b) Derive the relations between the mass fractions of Helium and Hydrogen (the percent of
the total mass in baryons that is in Helium and in Hydrogen) and the ratio of the number of
neutrons to protons.
57. a) Roughly calculate the horizon distance at the epoch of recombination.
b) Calculate the apparent angular size, to observers today, of the recombination-epoch
horizon-distance. Assume an Einstein-deSitter Universe (just to make the calculation
reasonable).
58. Discuss the physical processes responsible for the different size fluctuations in the cosmic
microwave background:
a) What process causes the fluctuations in the CMB on a scale of 180o?
b) What causes the fluctuations on angular scale of ~ 1o?
59—moved to “Not included in 2015”
60. a) What is meant by the “surface of last scattering?”
b) What determines the temperature when recombination occurs? What is the temperature?
c) How does kT at recombination compare to the binding energy of the H atom?
d) Show how the redshift of the surface of last scattering is determined?
61. Ryden 11.2
Not included in 2015:
59. Imagine that all the baryonic matter in the Universe becomes ionized.
a) Calculate the optical depth of the Universe at the present epoch in terms of . (Use the
Thomson scattering cross-section.)
b) Allowing for the change in the number density of electrons with time (or redshift), and
assuming an Einstein-deSitter universe, estimate at what z the universe becomes opaque, i.e.
what z corresponds to =1.
c) Compare your answer in (b) to the believed z of the CBR and explain why your
calculation does not argue that we can't see to the epoch of recombination. Comment, also,
on the effect of an epoch of re-ionization on the z of "last-scattering" of the CBR.
62. On a spherical 2-dimensional surface with radius of curvature = 1000 m, what is the
circumference of a circle of radius = 400m?
63. An observation of a cluster of galaxies reveals a perfectly circular arc around the cluster.
The cluster is at a redshift of z = 0.3 and the circular arc has an angular radius of 1 arcminute
(= 60 arcseconds). A spectrum of the light from the arc is taken and found to be that of a
distant quasar at a redshift of z = 0.6. Assume an Einstein-deSitter Universe (for simplicity)
and calculate the mass of the cluster.