**ASTR430 HW#3 (due 10/26/05)**

- Toomre's . Analogous to the collapse of a gas cloud, a thin
rotating disk of particles may clump up if the random motions of the
particles, plus the differential shear of the disk, is insufficient
to prevent gravitational collapse. One measure of this is Toomre's
parameter:
*epicyclic*(radial) frequency (which for Keplerian orbits is just the orbital frequency, ), is the velocity*dispersion*(a measure of the randomness of motion), and is the surface mass density of the disk (in kg/m). If , the disk is unstable.- Compute as a function of for a disk of identical
1 km radius planetesimals of mass density 2000 kg/m orbiting
the Sun at 1 AU, assuming the velocity dispersion is roughly equal
to the escape speed from one of the planetesimals.
- For what value of is ``critical'' (i.e., equal to
1)?
- Roughly speaking, the scale height of the disk is
, where is the vertical frequency. Taking
to be , estimate the
*volume*mass density of the critical disk, assuming it's uniformly distributed. Comment.

- Compute as a function of for a disk of identical
1 km radius planetesimals of mass density 2000 kg/m orbiting
the Sun at 1 AU, assuming the velocity dispersion is roughly equal
to the escape speed from one of the planetesimals.
- Textbook problems.
- Problem #15 from Ch. 6 (p. 153).
- Problem #14 from Ch. 7 (p. 189).

- Problem #15 from Ch. 6 (p. 153).
- Hazard mitigation. Suppose a 10 km radius comet of bulk density
0.5 g/cc traveling at 20 km/s relative to Earth while still far away
is on a direct (straight-line) collision course with our planet...
- Using the solution strategy of Homework 1, Problem 3, compute
the minimum required energy as a function of distance to deflect
the comet so that it just grazes the Earth.
- Using the result of Homework 2, Problem 2, compute the minimum
required energy to fully disrupt the comet, assuming an explosive
efficiency of only 0.1% (which is probably optimistic).
- At what critical distance is it more energy efficient to just
blow up the comet rather than deflect it? Express your answer in
Earth radii.

- Using the solution strategy of Homework 1, Problem 3, compute
the minimum required energy as a function of distance to deflect
the comet so that it just grazes the Earth.