Download astro340.f07.hw5

ASTRONOMY 340
Fall 2007 – HW #5
Due Thursday, November 29
Problem 1. – Planetary Rings
a) Calculate the total orbital energy and angular momentum contained in a very thin (δr)
circular ring of mass mring at a distance, r, from the planet.
b) Calculate the total orbital energy and angular momentum contained in two very thin rings
each of mass = mring/2, and at distances r1 and r2 from the planet.
c) Consider two ring particles on slightly different (zero inclination) circular orbits. Each
particle is 10m in radius. Derive an expression for the maximum speed at which they can
collide. Evaluate you answer numerically for ring particles with r=2RSaturn. How does the
collision velocity compare with the escape velocity from a ring particle?
d) A long, nasty Taylor expansion shows that E1 (one ring) has more energy than E2 (two
rings). You don’t have to do the math! What is the implication of this conclusion about
the energies for the evolution of ring systems? What are the limitations of this simple
model?
Problem 2.
a) Assume Jupiter formed at 6.0 AU and was dragged inward to its current 5.2 AU as
Jupiter interacted with icy planetesimals and threw them out of the solar system. Imagine
that the planetesimals all were in eccentric orbits at Jupiter’s distance to begin with, and
that they were thrown out of the solar system on parabolic trajectories. Use a
conservation law to estimate how much mass must have been ejected from the solar
system to account for the changes in Jupiter’s orbit.
b) Imagine that Jupiter was formed at 6.0 AU and was slowly dragged inward to 0.05 AU
(the semi-major axis of some of the extrasolar planets). Make a plot of its temperature as
a function of distance from the Sun. Assume it does not have an internal energy source.
At approximately what solar distances would the various cloud decks on Jupiter
vaporize? Check you book for information on Jupiter’s cloud decks.
Problem 3. This is a preview of our discussion of solar system formation. Assume the Earth is
made up entirely of Mg, Si, Fe, and O, and assume the relative abundances of Si, Mg, and Fe are
identical to those in the Sun. Also assume that when the Earth formed only the rocky solids made
it into the Earth while H, He, and most of the O remained in the solar nebula as gas. Assume
every atom of Si, Mg, and Fe in the “feeding zone” made it into the Earth along with a little O,
while all of the other elements were excluded. Finally, assume the ratio of O:Si in the Earth is
3:1. Calculate the fraction of mass in the Earth’s feeding zone that ended up inside of the Earth.
Problem 4. If you were to fly through Saturn’s A ring (with the appropriate space suit, of course),
how many collisions would you suffer through from particles with sizes less than 1 m but larger
than 1 cm? Assume you’re taking the short route perpendicular to the ring. Use the tables in
your book to help.