Transcribed Text
Problem set
These problems are meant to reenforce the discussion in class. Some trigonometry/algebra is
required. If you have difficulty, consult classmates or me during office hours. It is helpful to work
through the mathematics to understand the concepts discussed in this class.
1) parallax and proper motion:
a) A star has a parallax angle of 0.3". How far away is this star in parsecs, kilometers, and light
years?
b) The same star has a proper motion of 8 km/s. Proper motion is the velocity in the direction
perpendicular to the direction viewed the apparent motion on the sky. By what angle does the
star move on the sky in 1000 yr?
Hint: For both parta use the small angle approcimation.
2) A crude estimate for the number of galaxies in the observable Universe:
The Universe is 14 billion years old. Therefore, the direction we can see in any direction the
radius of the observable Universe is approximately 14 billion light years. (The expansion of space
makes it so that the observable size is somewhat larger, but please ignore this effect here.)
The distance between the Milky Way and the nearest Milky Waysized galaxy, Andromeda, is 2.5
million light years. Use this distance as an estimate for the mean distance between galaxies.
From these numbers for the size of the Universe and the mean distance between galaxies, estimate
the number of galaxies in the entire observable Universe.
Hint: 1 jual want a Tough eatimate!
3) The age of the Universe:
You look out your window and observe that the velocity of galaxies obeys the following relation:
For every million parsecs away, they are moving another 70 km/s faster. (This value is similar to
what is found in our Universe.) This is Hubble's law, which can be expressed algebraically as
= Hd,
(1)
where v is the galaxies' velocities, d their distance, and H is 'Hubble's constant'. For this problem,
H = 70 km where Mpc stands for a million parsees. There are 3.1 x 1019 km in a Mpc.
Estimate the age of the Universe in billions of years. This same technique was first used by Edwin
Hubble to estimate the age of our Universe (although he used a value of H that was well off the
modern value).
Hint: Estimate how long it has been since all galaries were at the same point, i.e. lage = d/v = H
So. all you have to do for this problem is convert the unita of H1 into gears.

4) Dark matter:
We are orbiting around the center of our galaxy with a velocity of 220 km/s in a circle with radius
8 kiloparsecs. A kiloparsee is equal to a thousand parsees or 3.1 X 1016 km.
a) Estimate how long it takes to orbit the Galaxy in years.
Hint: Use that there are 3.15 X 107 acconds xit a year and remember T = 2mr/v. Be careful with
units!
b) If we measure R in astronomical units (A.U.), T in years and M in the Sun's mass [called 'a
solar mass'] then Newton's version of Kepler's third law can be written as:
R°
M.
T2
Newton's version of Kepler's third law still applies for our galaxy where R is our distance from the
center of the Galaxy, M is the mass between us and the center, and T is our period around the
galaxy. Using that 1A.U.= 5 X 109kiloparsec (or equivalently 1 kpe = 2 X 108 SA.U.), estimate how
much mass is between us and the center of the galaxy in units of solar masses. Compare this to
the estimated mass of our galaxy in stars of 5 X 1010 solar masses.
c) Stars at even a few times the distance from the center of the galaxy (at 30 kiloparsee or,
equivalently, 90 million light years) still orbit at 2 220 km/s (see figure). This is what astronomers
mean when they say that galaxy rotation curves are 'flat'. Argue that this flatness cannot be
explained by the gravitational pull of stars in our galaxy and the required gravity necessitates a
significant amount of some other source of mass. This fact (true in our galaxy as well as others) is
one of the first lines of evidence that there is dark matter in the Universe. In fact, we think there is
five times more mass in dark matter in the Universe compared to regular matter. The dark matter
is more diffusely distributed around our galaxy than the atomic matter we are familiar with. Our
best guess is that the dark matter is a stable subatomic particle that was produced in the hot Big
Bang.
300
Sun
200
100
10 20 30 40 50 60 70 80 90 100
distance from center of Milky Way
(thousands of lightyears)
Fig. 1 The measured rotation curve of the Milky Way.
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