Solutions for Assignment #14 (Chapter 22)


1.) Astronomers have found that dark matter resides in
a.) Spiral galaxies
b.) Elliptical galaxies
c.) Clusters of galaxies
d.) All of the above
e.) None of the above
Answer: "d" of the above

2.) Suppose that you were to apply the equation M = r v2 / G to our galaxy in order to calculate "the mass", and that you used the orbital velocity (v) of a star that is located at a distance (r) from the Galactic center. From this formula, which mass would you be calculating?
a.) The mass of stars in the volume within the sphere that is centered on the Galactic center and that has a radius of r
b.) The mass of stars outside of the sphere that is centered on the Galactic center and that has a radius of r
c.) The mass of stars along the path of length = r, which begins at the Galactic center and ends at the location of the star
d.) None of the above
Answer: "d" of the above

3.) How do astronomers measure the mass of a galaxy cluster?
a.) Measure the temperature of the intracluster gas, then calculate the velocities of the gas atoms from it, then calculate the mass from the velocities
b.) Measure the velocities of individual galaxies in the cluster, then calculate the mass from the velocities
c.) Measure how effectively the cluster has bent the light coming from more distant galaxies
d.) All of the above
e.) None of the above
Answer: "d" of the above

4.) Although we do not know if dark matter is little particles or big ones, let us
suppose that dark matter is many, many subatomic particles spread evenly throughout
the space of the galaxy. Assume that the diameter of the galaxy is 100,000 lightyears
(so the radius is 50,000 ly), and assume that the dark matter has 10 times as much mass
as the regular matter in the Galaxy (let us take the mass of the regular matter in the
Galaxy to be 100 billion times the mass of the Sun and the Sun's mass is 2.0 x 1033 grams)
a.) What is the density of dark matter in units of grams per km3?
(For comparison, note that a paperclip "weighs" about a gram.)
Answer: MDM = 10 x 108 x 2.0 x 1033 grams = 2.0 x 1042 grams
Volume of Galaxy = Vgal = (4/3) x pi x (50,000 ly x 9.46 x 1012 km/ly)3
= 4.4 x 1053 km3
DensityDM = MDM/Vgal = 2.0 x 1042 grams / 4.4 x 1053 km3 = 4.5 x 10-12 grams/km3
b.) Given the above assumption, how many grams of dark matter reside within the
solar system? (For the radius of the solar system, use the semi-major axis of
Pluto's orbit: 5.9 x 109 km .)
Answer: MDMss = Volume of solar system x DensityDM
Volume of solar system = (4/3) x pi x (5.9 x 109 km)3 = 8.6 x 1029 km3
MDMss = 8.6 x 1029 km3 x 4.5 x 10-12 grams/km3
= 3.9 x 1018 grams = 3.9 x 1015 kilograms
c.) Consider the mass of the Sun, Jupiter, Earth, and Pluto.
(The Sun's mass is 2.0 x 1030 kg, Jupiter's mass is 1.9 x 1027 kg,
the Earth's mass is 6.0 x 1024 kg, and Pluto's mass is 1.3 x 1022 kg). Which of these is
closest to the mass of the dark matter in the solar system?
Answer: Pluto's comes closest to the mass of dark matter in the solar system. Pluto is just
an oversized rock and yet it has 3.3 million times as much mass as the solar system's dark matter.
(If you are interested in comparing with the Sun's mass: the solar system's dark matter has
2.0 x 10-15 as much mass as the Sun.)
d.) I have never seen any mention of dark matter in the solar system. Why is the
solar system's dark matter unimportant?
Answer: Because dark matter
makes such a small contribution to the mass of the solar system, its effect on the
motions of the planets (our main way of detecting mass) would be unobservable.

e.) How can dark matter contribute a little to the mass of the solar system, but a lot
to the mass of the galaxy?
Answer: There is a lot of space between solar systems. That space contains dark matter, too.

5.) Which of these is on the list of hypothetical dark matter candidates?
a.) A diffuse gas of electrons
b.) Regular stars that were thrown out of the Galactic disk
c.) Planets, brown dwarfs, black dwarfs
d.) All of the above
e.) None of the above
Answer: "c"

6.) How does/did dark matter affect our galaxy?
a.) It has no effects on our galaxy
b.) It attracted the material that would eventually form the observable part of our galaxy
c.) It helps push the material in our galaxy out to further distances from the Galactic center
d.) All of the above
e.) None of the above
Answer: "b"

7.) According to models of the universe's evolution, the extremely dense regions where superclusters are
a.) Are currently gravitationally attracting nearby material
b.) Have been been extremely dense since the beginning of time
c.) Are impossible to reproduce with computer simulations
d.) All of the above
e.) None of the above
Answer: "a"

8.) If you were to make a 3 dimensional map of the observable part of the universe:
a.) Your map would have deserts where there are very few galaxy clusters
b.) Your map would have regions where galaxy clusters lined up along a curvy line or warped-sheet shape
c.) Both of the above
d.) Neither of the above
Answer: "c"

9.)
a.) How does the amount of dark matter in the universe affect the universe's final fate?
Answer: Dark matter has mass and so gravitationally attracts material. It slows the expansion of the universe. However, there is not enough mass in dark matter (even when combined with the mass in luminous matter) to prevent the universe from continuing to expand.
b.) Given our current understanding of the universe, what is the universe's final fate?
Answer: Continued expansion, forever and ever.

10.)
One of the traditional models for the universe is the "recollapsing universe". How does the data on distant supernova explosions rule-out the "recollapsing universe" model?
Answer: The data on distant supernova explosions allows us to determine the distances to far-away galaxies. We are able to determine the recessional velocities of these galaxies from their redshifts. From these two sets of information, it has been found that the universe's expansion has been accelerating. That means that in the contest between expansion and gravity, expansion has already won. As the universe expands, the material becomes more diffuse, so the gravitational pull (that attempts to counter expansion) decreases. So, gravity will not be able to overcome expansion at some point in the future. Thus, the universe cannot recollapse.