
Time scales are the first things that you should think of when encountering an
astronomical phenomenon. The question "is that fast" or "is that slow" requires
understanding of the underlying physics of the problem.
The example we looked at in class concerned radial pulsation of stars. That
is, stars are on average stably balanced between the inward force of
selfgravitation and the outwarddirected force of gas (and radiation)
pressure. But if the radius of the star is perturbed, gravity (for an
outward push) or pressure (for an inward push) acts as a restoring force, and
in fact the result of the perturbation can be a sustained oscillation. Such
selfexcited pulsations can occur in certain stars, and we saw three examples
of this in class:
When stars undergo simple radial pulsation, the pulsation period is
approximately equal to the freefall time.
 L et's rederive the "freefall" timescale that corresponds to a temporary
imbalance between pressure and gravity in a selfgravitating star. Assuming
gravity takes over and the pressure gradient becomes inconsequential, show
that the freefall time scale (also called the dynamical time scale) scales
as the mean density to the 1/2 power. Plug in values for the Sun to derive
the freefall time of a star in terms of its radius and mass in solar units.
(Note that the mass of the Sun is 2x10^{33}grams, and the radius is
7x10^{10}cm).
 Using the pulsation periods above, and assuming they represent radial
pulsations, compute the approximate radius for the three pulsating stars
above.
 For PG1336, is this radius consistent with the radius you find from the
eclipse light curve shown in class? If
you don't recall the discussion, you can figure the radius given that the
pulsator is the bright star in the system, the mass in the table above, and
the orbital period. With those facts, you can use Kepler's third law to find
the orbital velocity. Then the duration of the eclipse tells you the stellar
radius.
 Sketch an HR diagram with the main sequence. Very approximately, put the
above three stars on the HR diagram using the effective temperatures from the
table and the radii you compute in part b).

As we saw in class, radial veolcity measurements show that objects in spiral
galaxies
revolve about the galactic center at roughly constant velocities beyond
about 10,000 parsecs from the galactic center. Within that distance, velocity
increases linearly with distance from the center. . However, the starlight from
these galaxies is concentrated in the inner 10,000 parsecs.
[Note that a parsec (pc) is a convenient astronomical distance unit  it is
defined as the distance at which 1 astronomical unit would subtend 1 second
of arc. The astronomical unit (au) is, itself, defined as the mean distance
between the Earth and the Sun. For these purposes, we can define the au as
1.5x10^{13}cm. Therefore, since there are 206,265 arc seconds in one
radian, one parsec is 3.09x10^{18}cm, or 3.26 lightyears.]

Our Milky Way is a pretty typical spiral galaxy. Assuming that the light
traces the mass of the Galaxy, use Kepler's third law (or some other simple
contortion of Newton's laws applied to circular motion)to compute the mass of
the Milky Way that lies within the position of the Sun. The Sun lies 9,000 pc
from the center of the Milky Way, and moves at 220 km/s in its orbit.

Rotation curves for galaxies can be measured using radio emission from
neutral hydrogen clouds and molecular clouds. With these objects, the
rotation curve measurements extend to 45,000 pc and more from the galactic
center. Material there has the same orbital velocity as material at 10,000
pc. Use a general form of Kepler's law as above, compute M, the mass of the
Galaxy that lies within the orbit material at 45,000 pc from the galactic
center if the material is also orbiting with a velocity of 220 km/s

Since the extra mass from part b) is invisible, assume that it is distributed
in a spherically symmetric halo centered at the center of the galaxy. To
maintain all objects orbiting the galaxy at the same velocity, how must the
density change with increasing distance to maintain a constant rotation
velocity with increasing distance... that is, what is the relationship
between halo density and radius (it will be a power law, but what power?).
The farthest objects that we have measured lie 10 times farther out than the
visible (stellar) disk. Material there also moves at the same velocity (about
220 km/s in a galaxy like the Milky Way). What fraction of the mass of a
galaxy is therefore dark matter?
The mean density of interstellar neutral hydrogen is about 10 atoms per cubic
centimeter; observations show that the neutral hydrogen clouds are confined
to the disk of the galaxy with a disk thickness of about 1000 pc. Can this
neutral hydrogen be the dark matter? Why or why not?

In the galaxy M87, astronomers have measured the rotation velocity near its
center with the Hubble Space Telescope. They find a rotation
velocity of 550 km/s (see the HST press release
on M87 for details) at a distance of 60 light years from the center.
Compute the mass of the galaxy that lies within 60 light years of the
center. The average distance between the Sun and the stars that are visible
to the eye is about 60 light years. Think about your answer to this question
in this context!