### Astro 405/505: Introduction to Astrophysics, Fall 2002

Assignment #1 Due: Wednesday, September 4, 2002

1. 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 self-gravitation and the outward-directed 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 self-excited pulsations can occur in certain stars, and we saw three examples of this in class:

StarPulsation PeriodMassTemperature
PG 13363 minutes0.5 Mo30,000K
Mira (o Cetus) ~330 days 2 Mo3,000K
delta Cepheus5.3 days 4 Mo6,000K

When stars undergo simple radial pulsation, the pulsation period is approximately equal to the free-fall time.

1. L et's rederive the "free-fall" timescale that corresponds to a temporary imbalance between pressure and gravity in a self-gravitating star. Assuming gravity takes over and the pressure gradient becomes inconsequential, show that the free-fall 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 free-fall time of a star in terms of its radius and mass in solar units. (Note that the mass of the Sun is 2x1033grams, and the radius is 7x1010cm).

2. Using the pulsation periods above, and assuming they represent radial pulsations, compute the approximate radius for the three pulsating stars above.

3. 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.

4. Sketch an H-R diagram with the main sequence. Very approximately, put the above three stars on the H-R diagram using the effective temperatures from the table and the radii you compute in part b).

2. 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 1.5x1013cm. Therefore, since there are 206,265 arc seconds in one radian, one parsec is 3.09x1018cm, or 3.26 lightyears.]

1. 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.
2.

3. 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
4.

5. 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?

6. 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?

7. 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!