Astronomy 2, Spring 2007

Homework set 3, due May 1 .

1. Newton’s law of Gravity.

(a) How many times smaller is the force of gravity of the earth at the moon than it is at the surface of the earth? The earth-moon distance is about 60 earth radii.

(b) A space shuttle orbits the earth at about 250 km above the earth’s surface. At that height the force of the earth’s gravity is about the same as it is at the surface of the earth. Why? Explain why astronauts float around in the shuttle weightlessly, if they are still under the same gravitational force as they were on the ground.

(c) Assume for this problem that the shuttle is one earth radius from the center of the earth and goes around the earth in 90 minutes. Calculate the radius of the orbit (in earth radii) required for a communication satellite to have a period of 24 hours, one day, using Kepler’s Third Law. [ Hint: You can restate Kepler's Third Law as "the ratios of the periods squared equals the ratios of the semi-axes cubed," where the ratios refer to the relevant parameters (p and a) for two objects orbiting the same central object.]

2. Jupiter’s moon Io orbits Jupiter with a period of 1.77 days in an orbit whose semi-major axis is 4.2x10⁵ km. Is it possible to find Jupiter’s mass from these two numbers? If so, how would you do it, that is, what formulae would you use? What quantities are involved? Set up the problem, but don’t actually work it out.

3. Parallax.

(a) The first accurate parallax was measured for the star 61 Cygni. The value was 0.3 arcseconds. Express the distance of this star in both parsecs and light years.

(b) The parallax of Sirius is 0.4 arcseconds. This is the brightest star in the sky, and it is prominent now low in the east in the early evening. Is Sirius closer or farther than 61 Cygni? Explain, and work out the ratio of their distances. Since Sirius appears to be many times brighter than 61 Cygni, would you expect it also to be intrinsically brighter? Explain.

(c) The smallest parallax we can reliably measure from the earth’s surface is about 0.005 arcseconds. How far away would a star be with this parallax? Would there be an advantage to being located on Pluto, the outermost planet, for measuring parallaxes? If so, explain what it is.