**ASTR430 HW#4 (due 11/9/05)**

- Holy Shrinking Jupiters, Batman!
- Jupiter has an estimated emissivity of 0.64. Compute the mean
power in Watts emitted by Jupiter assuming the giant planet is
only reemitting light absorbed from the Sun.
- Jupiter actually glows like a blackbody of effective
temperature 124 K. Compute its emitted power on that basis
(recall, for a blackbody, the emitted
*flux*is ). Notice that the emitted power is larger that what you found in part (a)--Jupiter radiates more energy than it receives from the Sun! - The difference is attributable to the fact that Jupiter is
still contracting and converting gravitational potential energy
into heat. If Jupiter were to convert all its gravitational
binding energy into heat, how long would it take for the giant
planet to contract to a point, assuming it maintained the constant
blackbody temperature of part (b)? (In reality, the contraction
would be slowed to almost zero long before that--why?)

- Jupiter has an estimated emissivity of 0.64. Compute the mean
power in Watts emitted by Jupiter assuming the giant planet is
only reemitting light absorbed from the Sun.
- Everything You Didn't Want to Know about Moments of Inertia...
The angular momentum of a rigid body is related to its spin vector by

*inertia tensor*. For continuous bodies, the elements of are given by*coefficients of inertia*while the off-diagonal elements () are the*products of inertia*.- Compute the inertia tensor for a solid homogeneous sphere of
radius and mass .
*Hint:*compute and first, notice the pattern, and argue that the entire matrix reduces to the identity matrix times a single scalar of form --find . The quantity is the*moment of inertia*of a solid homogeneous sphere. - Compute the inertia tensor for a solid homogeneous spherical
*shell*of inner radius , outer radius , and total mass .*Hint:*all that's really changed from part (a) is the lower limit of the integral--use this to your advantage to avoid a lot of work! Again, express your answer as , the moment of inertia of a solid homogeneous spherical shell, and find (which will be a function of and but will still be dimensionless). Verify the expected behaviour as . *Bonus:*compute the inertia tensor of a*thin*shell of radius and mass .*Hint:*use your result from part (b)--you don't need to integrate!- It can be shown that the free precession frequency of a
symmetrical rigid body rotating at frequency around its
symmetry axis is given by
^{1}For the Earth, (not surprisingly, this value is close to the Earth's*oblateness*). Find the free precession*period*and compare it to the precession period of the equinoxes. (It turns out the free precession period is quite close and probably related to the so-called*Chandler wobble*period of the Earth--look it up with`google`!)

- Compute the inertia tensor for a solid homogeneous sphere of
radius and mass .
- Term Project
- Provide the title and a 2-3-sentence description of your
chosen term project (recall the essay is due Nov. 23 and the oral
presentations will be Dec. 7 & 9).

- Provide the title and a 2-3-sentence description of your
chosen term project (recall the essay is due Nov. 23 and the oral
presentations will be Dec. 7 & 9).