Astronomy 102, Fall 2004

Homework Assignment #6

This homework set is due at the beginning of class on Friday, October 29. It must be turned in by 9:10AM that day. Late homework will not be accepted. This is includes your being late to class!

Staple. If you have more than one page, staple them together; do not just fold the corner. If you have multiple pages and do not staple, I will deduct one point from your score..

The first three problems are required. The remaining problems are optional, and will not be graded; they are here as additional review problems for you. After the homework has been graded, solutions will be posted to all problems.

Please write out the problem statement at the top of your solution. (This is for two reasons; it is so I can know which problems you answered, and that you answered the right problem from the bit. It also will make your graded homework more useful as a study aid later.)

You may consult with other students (as well as with the TAs and professor) on this homework set. However, your final answer should be your own. Do not write down an answer you don't understand, and do not "dictate" an answer to somebody else.

You want to escape from the Solar System! Waaaah! You strap yourself into your rocket and blast off from the vicinity of the Earth. You've already managed to escape from the Earth (but are still travelling along in the direction of Earth's orbit), and now only need to break free of the gravitational grip of the Sun. You can fire a single, short burst from your rocket to greatly increase, decrease, or change the direction of your velocity.

In which direction do you want to point your rocket (blasting the fuel out of the back) so as to escape the Solar System while using the minimum possible amount of fuel? Do you want to point it towards the Sun, away from the Sun, in the same direction as Earth's motion about it's orbit, in the direction opposite Earth's motion, or in some direction between one of these four "cardinal" directions? Note that escaping from the Sun means effectively putting yourself into an orbit that will take you a very, very long way from the Sun. Think about what we've talked about in terms of orbits and the conflict of gravity and motion, and justify the direction in which you want to point your rocket.
Is it easier to find planets around stars whose orbits are "face-on" (i.e. you are looking "down" on the plane of the orbit) or "edge-on" (i.e. you are looking at "the side" of the plane of the orbit)? Why?
You discover a binary star system that has a high proper motion. One of the stars is much brighter than the other one— the difference is enough that in the telescope you're using, you can only see the brighter star. You're observing the orbits of the binary star system "face-on" (all of the motion of the orbit is in the plane of the sky). The period of the binary star orbit is about a year. The stars are separated by about 2", and the system has a proper motion of about 4"/year.

If you obseve the binary star system for several years, make a sketch of the path across the sky that you would observe the visible star to follow (using a solid line) and that you would observe the fainter star to follow (using a dashed line).

The problems below are optional; they need not be turned in, and they will not be graded.

In class, we discussed the rotation curve of the Milky Way (and other spiral galaxies) as evidence for dark matter. Similarly, many (though perhaps not all!) elliptical galaxies show evidence for dark matter, and we can find evidence for dark matter in clusters of galaxies. Since these systems consist of objects (stars or galaxies) randomly moving about in all directions, and don't have an ordered rotation, the same "rotation curve" arguments we used for our Galaxy won't apply. Instead, we talk about velocity dispersion: the range of velocities (measured from the Doppler shift) that galaxies in a cluster show. Consider two hypothetical clusters of galaxies, each 10kpc in diameter, and each with an identical number of galaxies of identical brightnesses. One of these hypothetical clusters (from our Universe) has dark matter, but the other has absolutely none. Which cluster will show a greater range of velocities? (Hint: remember that these clusters are held together by gravity, and that their size is dominated by how far galaxies can get before they are pulled back to the cluster. Also think about what would happen if you dropped a ball on the Earth compared to dropping one on the Moon (where gravity is lower), or what would be different if you wanted to throw a ball to a height of 10m on Earth as compared to throwing a ball to a height of 10m on the Moon.)
Chapter 16, Question 10 in Pasachoff & Filippenko.