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{\Large ASTR430 Homework \# 3 -- Obital Motion\\
Due Wednesday, October 15, 2003}
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1. Sketch the following orbit at four times, $t=0,1/8,1/4,3/8$ years, and
then describe, without using the names of the orbital elements
$a,e,i,\Omega,\omega$, how the orbit is changing in time.\\[240pt]

2. a) Estimate the minimum rotation period of a star in hours by
equating gravity with the centrifugal force at the star's equator. How
does your answer differ for a planet?\\ b) Calculate the spin angular
momentum of the Sun (look up the moment of inertia for a uniform
sphere in a physics book), the spin angular momentum of Jupiter, and
the orbital angular momentum of Jupiter.\\ c) If, somehow, Jupiter
were absorbed by the Sun (assume that the angular momentum vectors are
parallel, that the total angular momentum is conserved, and that the
radius of the Sun is not changed), how fast would the Sun spin?  What
would happen?\\ d) Now imagine a one solar mass spherical cloud of gas
with uniform density and radius 1 light year. How fast does it spin if
it has the same angular momentum as the Sun-Jupiter system?

3. In this problem, you'll use conservation of energy and angular
momentum to solve for the radial turning points for orbits in central
force motion (i.e. you'll derive the blue curves seen in the Central
Force Integrator).  \\ a) For a general force, {\bf F}(r), write down
the expressions for angular momentum per unit mass, $h$, and energy
per unit mass, $C$, in terms of the position, $r$, and the velocity,
$v$. \\ b) Simply your expression(s) for a central force with the form
{\bf F} = -$Ar^n{\hat r}$. \\ c) Use your two equations and the
condition of being at a radial turning point to obtain an equation for
$r$ in terms of constants.  \\ d) For gravity ($n= -2$), use the
specialized formulas for $C$ and $h$ to solve for the radial turning
points in terms of the semimajor axis, $a$, and eccentricity, $e$.  \\ 
e) For the general case in c), argue that there are at most 2 positive
real solutions for $r$ (the only ones that matter physically).
Interpret the special cases of 0,1, and 2 solutions physically.\\ f)
How many physical solutions are there for the case $n=-3$? What does
this tell you about orbits in a $1/r^3$ force field?  Test your ideas
by investigating circular and near circular orbits with the central
force integrator. Show me some plots and summarize what you find!


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