ASTR221 HW #3 - due Oct 23rd 2008

1.  The Roche Limit

In class we calculated the Roche limit, which showed how close a moon could get to its parent planet before it will be tidally disrupted.

Saturn's moons are mostly icy bodies, with densities of about 1.2 g/cm³. Calculate the Roche limit for these moons. Compare this to the size of Saturn's rings, most of which lie within 120,000 km from the planet's center. Noting that all of Saturn's moons lie outside of 130,000 km from the planet' center, what does this suggest as one possible formation mechanism for Saturn's rings?

Calculate the Roche limit for the Moon. Is it in any danger of being disrupted any time soon?

2. Jupiter's Heat

Jupiter is giving off heat through gravitational contraction. We're going to calculate how much, and how this has changed throughout the history of the solar system. When an gravitating object contracts, half of the lost gravitational energy is radiated away (the other half goes into heating up the object). For simplicity, assume that throughout the collapse, Jupiter can always be considered a sphere of uniform density, for which the gravitation energy can be written U=-(3/5)GM²/R.

• First, estimate the total amount of energy radiated by Jupiter by gravitational contraction over the last 4.5 billion years. Assume Jupiter started out as a ball of gas with its current mass, except much, much bigger in size.

• Now estimate the average rate of energy output from Jupiter over the last 4.5 billion years from gravitational contraction alone.

• Calculate the current rate Jupiter is giving off energy due to gravitational contraction. Remember that if Jupiter was at its equilibrium temperature, it would simply be radiating back the energy it got from the Sun. Since Jupiter is hotter than this, it must be emitting more energy than what it receives from the Sun. Calculate how much more.

• Compare the current rate (in part 3) with the average rate (in part 2). What does this tell you about how the energy output from Jupiter has changed over time? How might this affect the conditions in which Jupiter's moons formed?

3. Saturn's Rings, Again

We will make a rough estimate of the mass contained in Saturn's rings. Assume that the rings have a constant mass density and that the rings are 30 meters thick with an inner radius of 1.5 R_Saturn and an outer radius of 3 R_Saturn. Assume also that the ring particles are water-ice spheres of radius 1 cm and that the optical depth of the rings is unity (i.e., tau=1, so you can just barely see through them).

• What is the number density of particles in the rings (how many particles per cubic meter)? So how many particles total are there?
• If water-ice particles have a density of 1 g/cm³, what is the mass of each particle? So how much total mass is in the rings?

• If you were to take these ring particles and merge them together to make a spherical moon, how big would that moon be? How does this compare to the size of Mimas, one of Saturn's inner moons?