Chapter 11, Problem 54EAP

Re sure to show all calculations clearly and state your final answers in complete sentences.

1. Disappearing Moon. Io loses about a ton (1000 kilograms) of sulfur dioxide per second to Jupiter’s magnetosphere, a. At this rate, what fraction of its mass would lo lose in 4.5 billion years? b. Suppose sulfur dioxide currently makes
up 1% of lo's mass. When will lo run out of this gas at the current loss rate?

2. Ring Particle Collisions. Each ring particle in the densest part of Saturn's ringB collides with another about every 5 hours. If a ring particle survived for the age of the solar system, how many collisions would it undergo?
3. Prometheus and Pandora. These two moons orbit Saturn at 139,350 and 141,700 kilometers, respectively.
a. Using Newton's version of Kepler’s third law, find their two orbital periods. Find the percent difference in their distances from Saturn and in their orbital periods, b. Consider the two in a race around Saturn: In one Prometheus orbit, how far behind is Pandora (in units of time)? In how many Prometheus orbits will Pandora have fallen behind by one of its own orbital periods? Convert this number of periods back into units of time. This is how often the satellites pass by each other.

4. Orbital Resonances. Using the data in Appendix E, identify the orbital resonance relationship between Titan and Hyperion. (Hint: If the orbital period of one were 1.5 times the orbital period of the other, we would say that they were in a 3:2 resonance.) Which medium-size moon is in a 2:1 resonance with Enceladus?
5. Titanic Titan. What is the ratio of Titan’s mass to that of the other satellites of Saturn whose masses are listed in Appendix E? Calculate the strength of gravity on Titan compared to that on Mimas. Comment on how this affects the possibility of an atmosphere on each.
6. Titan's Evolving Atmosphere. Titan's exosphere lies nearly 1400 kilometers above its surface. What is the escape velocity from this altitude? What is the thermal speed of a hydrogen atom at the exospheric temperature of about 200 K? Use these answers (and the method of Mathematical Insight 10.2) to comment on whether thermal escape of hydrogen is likely to be important for Titan.
7. Saturn's Thin Rings. Saturn’s ring system is more than 270,000 kilometers wide and only a few tens of meters thick; let’s assume 50 meters thick for this problem. Assuming the rings could be shrunk down so that their diameter was the width of a dollar bill (6.6 cm), how thick would the rings be? Compare your answer to the actual thickness of a dollar bill (0.01 cm).