Physics 103 Homework #13
Due Thursday, December 8, 2005
Submit by 5:00 p.m. outside the Physics Department Office, Park 348.
Look for a plastic bracket on the wall labeled with the course number.
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Reading for this week:
Greene, The Elegant Universe, §2, 3, and 14.
Reiss & Turner, “From Slowdown to Speedup,” Scientific American, 290 (2), 62 (Feb. 2004).
Perlmutter, “Supernovae, Dark Energy, and the Accelerating Universe,” Physics Today, 56
(4), 3 (Apr. 2003).
The articles by Reiss & Turner and by Perlmutter will be distributed in class.
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Problems to turn in. Unlike previous homework sets, several of these problems are
non-quantitative questions. In these cases, a short answer—a couple sentences—will
suffice.
1. In his seminal 1929 paper “A Relation between Distance and Radial Velocity of
Extra-Galactic Nebulae,” Edwin Hubble presented the plot of Galaxy distances and
velocities reproduced below. Distances, along the x−axis, are given in parsecs. (The
parsec is a unit of distance, abbreviated “pc.” 1 pc=3.09×1016 m. Use parsecs
and megaparsecs throughout this problem; there is no need to convert to meters.)
Velocities, along the y−axis, are given in km/sec. (The y−axis is mislabeled “km”).
Hubble’s paper doesn’t actually give a value of the Hubble constant, H0 , derived
from these data. What would that value be? Give your answer in (km/s)/pc, and
then convert to (km/s)/Mpc, where 1 Mpc = 106 pc. How does this value compare
to the best modern measurement, H0 = 71 (km/s)/Mpc? The problem with Hubble’s
original data is that he had incorrect distances to the Galaxies. Based on the value of
H0 you found from his data, must his Galaxy distance measurements have been too
large, or too small? Very briefly explain your reasoning.
Continued on other side....
2. Why do we think the universe is expanding?
accelerating?
Why do we think the universe is
3. How, specifically, do we know that “dark matter” is present in our Galaxy?
4. If Einstein’s cosmological constant provides an accurate model of the “dark energy”
in the universe, then the equation of state parameter is w = −1. What would be
different about the large-scale motions of Galaxies (and the future of the universe) if
w = +1 for dark energy instead?
5. How can we say that the universe is homogeneous when there are obvious
inhomogeneities from one place to the next? For example, the Sun is obviously very
different from the Earth, and the space in between the Sun and the Earth is very
different from either.
6. The density of mass or energy in the universe is surprisingly small compared to the
densities of things in our everyday experience.
Estimate the average densities of the following things, giving your answers in kg/m3 .
Although the term “energy density” is often used when talking about the content of
the universe, for this problem it will be easier for you to use “mass density”, ρ, which
is mass, m, per unit volume, V : ρ = m
V .
(Reminder: the volume of a sphere of radius r is V = 43 πr3 .)
(a) You. (Hint: you are basically water.)
(b) The solar system. Nearly all of the mass in the solar system is contained within
the Sun. So, to calculate the average density of the solar system, use the Sun’s
mass, m = 2 × 1030 kg, but assume that it is spread throughout the volume of
the solar system. To estimate the volume of the solar system, treat it as a sphere
with radius the size of Pluto’s orbit, 5 × 1012 m. (Yes, this is highly arbitrary...
but it will give you a sense for how little stuff there really is in the solar system.)
(c) The Milky Way Galaxy.
i. To get a (very approximate) value of the mass of the Galaxy, note that the
Sun follows an orbital path of radius r = 8 kiloparsecs = 2.5 × 1020 m and
velocity v = 200 km/s = 2 × 105 m/s. If all the mass in the Galaxy were
concentrated at a single point at the center of the Sun’s orbit, what value
of M would give these values of v and r? Of course, the mass isn’t really
concentrated at the center, but this still gives a good order-of-magnitude
estimate of the mass of the Galaxy within the radius of the Sun’s orbit.
ii. To get the average density of the Galaxy, assume that the mass you just
calculated is not concentrated at the center of the Galaxy after all, but
instead is spread uniformly throughout a sphere of radius r = 8 kiloparsecs =
2.5 × 1020 m. What is the average density?
(d) Average density of the universe. I’ll give you the answer to this one: it is
ρ = 1 × 10−27 kg/m3 .
7. How is it that motions of objects on small scales (Galaxies of size 0.1 Megaparsecs)
can be dominated by one thing, dark matter, while motions of objects on larger
scales (widely separated galaxies, at distances of 1000 Megaparsecs) are dominated by
a completely different thing, dark energy? Shouldn’t dark energy affect motions in
galaxies, and dark matter affect motions between widely separated galaxies?