Astronomy 142: Practice Final Exam
Spring 2013
This practice exam will do you very little good unless (1) you have mastered the
homework and workshop problems, and tried to understand everything discussed in
the online solutions for all these problems; (2) you have already composed your twopage Cheat Sheet and want to see if it really has everything you need on it, and nothing
you don’t. If you’ve done (1) and (2), feel free to proceed.
If this were a real exam, you would be advised here of the exam rules:
"You may consult only two pages of formulas and constants and a calculator while
taking this test. You may not consult any books, online or digitally-stored resources, nor
each other. All of your work must be written on the attached pages, using the reverse
sides if necessary. The final answers, and any formulas you use or derive, must be
indicated clearly. Exams are due 2.5 hours after we start, and will be available to be
reclaimed on Tuesday. Good luck.
"Note:
•
"Work first on the problems you find easiest, and come back to harder or less
familiar material later. Don’t get stuck.
•
"The amount of space left for each problem is not necessarily an indication of the
amount of writing it takes to solve it.
•
"If you need a physical or astronomical constant that is not on your cheat sheet,
don’t give up: estimate its value. If your estimate is reasonable you will lose few or
no points.
•
"On that same theme: remember that you can earn partial credit for being on the
right track. Be sure to show enough of your reasoning that we can figure out which
track you’re on."
Name: _______________________
Problem 1. (30 points)
Consider a cylindrical, rotating cluster of stars. It is gravitationally bound. Its radius R
and thickness H  R are much larger than the typical distance between stars, and the
mass density ρ of the cluster is uniform on scales much larger than the typical distance
between stars.
a.
What is the rotation speed V of stars in the cluster, as a function of cylindrical
radius r?
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Problem 1. (continued)
b.
What is the typical random stellar speed, v?
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Problem 2. (30 points)
At galactic longitude 20°, a series of H I 21 cm line spectra are obtained; a typical result
is shown in Figure 1.
Line intensity
Cloud 1
Cloud 2
100
200
LSR velocity (km/s)
Figure 1: H I line intensity as a function of radial velocity with respect to the local
standard of rest.
a.
The highest Doppler velocities correspond to a peak at 180 km s-1, and belong to
a diffuse cloud we will refer to as Cloud 1 (see Figure 1). Calculate the distance r
from Cloud 1 to the galactic center, and the distance d from the solar system to
Cloud 1. Express your answers in kiloparsecs.
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Problem 2. (continued)
b.
Calculate the galactic rotation velocity of Cloud 1, and the galactic mass enclosed
by its orbit.
c.
What is the galactocentric radius of Cloud 2?
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Problem 3. (30 points)
A spherical, molecular clump with mass 10M is observed to have temperature T = 10
K and to be r = 0.1 pc in radius.
a.
Show that the clump is hydrostatically unstable, and must collapse.
b.
How long, in seconds, will it take the core to collapse to stellar dimensions?
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Problem 3. (continued)
c.
Suppose that we have luckily found a clump with very small rotational motion,
and the material all collapses right into a spherical protostar which eventually
becomes a 10 M star. Estimate the radius of such a star, and from this estimate
and the previous results estimate the average luminosity of the object during the
collapse that forms the star.
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Problem 4. (30 points)
A star with mass M and outer radius R has density given as a function of radius r by

r2
ρ (r ) ρ (0)  1 − 2
=

R

a.



Find an expression for the central density ρ ( 0 ) in terms of M and R.
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Problem 4. (continued)
b.
Find an expression for the central pressure P ( 0 ) in terms of M and R.
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Problem 4. (continued)
c.
Find an expression for the temperature at the center of the star.
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Problem 5. (30 points)
a.
Suppose that the Universe were flat – as observed – but besides a very small
amount of normal matter it contains only dark energy. Starting from the
Friedmann equation, derive an expression for the relation between time t and the
normalized scale factor a.
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Problem 5. (continued)
b.
Repeat for a flat matter-dominated universe: one in which there’s negligible dark
energy but lots of matter. Which universe expands faster?
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Problem 6. (30 points)
Here is an H-R diagram for M 3; the apparent V magnitude is plotted against
B − V = mB − mV .
a.
RR Lyrae variables are omitted from this diagram, as is traditional. Indicate on
the diagram where they would lie if their maximum magnitudes were included,
indicate roughly the range they cover in the diagram as they pulsate, and
estimate the average maximum apparent V magnitude of the RR Lyrae stars in M
3.
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Problem 6. (continued)
b.
RR Lyrae itself – the star in Lyra, not the class of variable stars – has maximum V
c.
The Sun is estimated to have a total main-sequence lifetime of about 15 billion
years. Use this information, the information on main-sequence stars provided in
the list on the last page of the exam, and the HR diagram of M 3, to estimate the
age of M 3.
= 7.13 and a trigonometric parallax of =
p 4.4 × 10 −3 arcsec. Assume the variables
in M 3 are just like RR Lyrae; what are the distances to RR Lyrae and M 3?
(Neglect extinction, and express your answer in parsecs or kiloparsecs.)
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Problem 7. (30 points)
Magnitudes of stars in an open cluster are measured, and plotted along with the zeroage main sequenced derived from observations of nearby stars. Here they are:
-5
-1.50
-1.00
0
-0.50
0.50
V
U-B
0.00
5
1.00
10
1.50
2.00
15
2.50
-0.5
0.0
0.5
1.0
1.5
2.0
B-V
a.
-0.5
0.0
0.5
1.0
1.5
2.0
B-V
Estimate the selective extinction E ( B − V ) , and the total visual extinction AV ,
between us and this open cluster.
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Problem 7. (continued)
b.
Estimate the distance modulus µ, and the distance r to this cluster, in parsecs.
c.
A classical Cepheid variable in this cluster has V = 8.2 mag. What is its absolute
V magnitude?
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Problem 8. (30 points)
Short answers. Answer any six of these nine questions.
a.
Why is it that time told on a sundial differs in general by several minutes from
time told on a clock?
b.
How is it possible for knots within quasar jets to appear to move faster than
light?
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Problem 8. (continued)
c.
Why is the cosmic background radiation essentially isotropic?
d.
What is the average mass density of a thirty-day classical Cepheid?
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Problem 8. (continued)
e.
How would you tell an Sa galaxy from other types of spirals?
f.
How old is the Milky Way Galaxy, and how do we know?
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Problem 8. (continued)
g.
What characteristics are shared by all types of active galaxies?
h.
Give one difference between supernovae of type Ia and II.
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Problem 8. (continued)
i.
After the Big Bang, the normal matter in the Universe consisted of hydrogen and
helium atoms and hardly anything else. Out of every hundred atoms, 92 were H,
8 were He. What was the value (in proton masses) of the mean molecular mass,
µ, in the centers of the first stars?
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