Chapter 78, Dark Matter Video Solutions, Pathways to Astronomy

Problem 1

Suppose we measure two galaxies to the farthest extent that any matter can be detected. One has a rotation velocity of $300 \mathrm{km} / \mathrm{sec}$ at $4 \mathrm{kpc},$ and the other has a rotation velocity of $150 \mathrm{km} / \mathrm{sec}$ at $20 \mathrm{kpc} .$ Which has the larger measured mass? What is the ratio of those masses?

Matthew Miranda

Numerade Educator

01:56

Problem 2

What is the mass-to-light ratio of a galaxy with $10^{11} M_{\odot}$ in stars and gas and $5 \times 10^{11} M_{\odot}$ of dark matter?

Sunny Guha

Numerade Educator

05:55

Problem 3

A spiral galaxy is observed in which the rotation speed remains $250 \mathrm{km} / \mathrm{sec}$ from $1 \mathrm{kpc}$ out to the largest distances at which we can detect anything. How much mass exists out to $10 \mathrm{kpc}$ ? 11 kpc? 12 kpc? Can you generalize your results as to how much additional matter there is for each kiloparsec farther out you look?

Alex Moore

Numerade Educator

01:56

Problem 4

In the Virgo Cluster there are galaxies measured to be traveling at about $1000 \mathrm{km} / \mathrm{sec}$ at a distance of $500 \mathrm{kpc}$ from the center of the cluster. If you apply the mass formula to this velocity and radius, what interior mass do you find?

Zachary Warner

Numerade Educator

01:55

Problem 5

Suppose the Milky Way contains 10 billion $M_{\odot}$ of dark matter at radii between 8 and 9 kpc from its center.
a. What is the density of matter in the "shell" between the radii of 8 and $9 \mathrm{kpc} ?$ (Hint: Subtract the volume of a sphere of radius $8 \mathrm{kpc}$ from the volume of a sphere of radius $9 \mathrm{kpc} .$
b. How much mass of dark matter would there be in a spherical volume with a radius equal to the Earth's radius? Compare your result to the mass of the Sun.

Sunny Guha

Numerade Educator

05:55

Problem 6

Most spiral galaxies show a decline in the amount of light they produce that drops by a factor of 2 every 2 kpc farther out. For example, $3 \times 10^{10} L_{\odot}$ are produced within $2 \mathrm{kpc}$ of the center; $1.5 \times 10^{10} L_{\odot}$ are produced between 2 and $4 \mathrm{kpc}$ of the center; half as much again between 4 and $6 \mathrm{kpc}$; and so on. Compare the amount of luminous matter found this way to the total amount of matter contained within these same regions if the galaxy has a flat rotation curve with a rotation speed of $250 \mathrm{km} / \mathrm{sec} .$ Find the ratio of dark matter to luminous matter within 2 kpc of the center; between 2 and 4 kpc from the center; between 10 and 12 kpc from the center.

Alex Moore

Numerade Educator