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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?)
- Everything You Didn't Want to Know about Moments of Inertia...
The angular momentum
of a rigid body is related to its
is a matrix called the inertia tensor. For
continuous bodies, the elements of
are given by
is a point inside the volume,
is the mass density at that point, and
is the Kronecker delta function defined such that
The diagonal elements () of
are called the
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
where we have set
taken the axis to be the symmetry axis (so that ).1 For the Earth,
(not surprisingly, this value is close to the
). 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
- 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).
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Derek C. Richardson