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{\Large ASTR430 Homework \# 3 -- Orbital Motion\\
Due Tuesday, March 13}
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1. a) Estimate the maximum rotation period of a star in hours by
equating gravity with the centrifugal force at the star's equator. How
does you 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?

2. Dimensional Analysis. In class, we looked at ballistic trajectories
and found an expression for the range of a projectile launched with
velocity $v$ at an angle $\theta$ from the horizontal: $D=(v^2/g)
f(\theta)$, where $f(\theta)$ is an undetermined dimensionless
function and $g$ is assumed to be constant. \\ a) Use dimensional
analysis to obtain a similar expression for the maximum height $H$ of
the projectile assuming constant $g$.\\ b) Use conservation of Energy
to solve for the maximum height exactly and check your answer against
part a)-- what is $f(\theta)$?\\ c) Now use dimensional analysis to
get an expression for $H$ using the true $1/r^2$ gravity of the Earth.
Neglect the Earth's spin, treat the Earth as a perfect sphere, and
consider a projectile launched at an arbitrary angle. Start by
identifying four variables that the solution will depend upon.  See if
you can form a dimensionless constant from the four variables and
remember that the most general solution is the one that is multiplied
by the most general dimensionless function.\\ d) Do the related
problem \#107 from the Physical Intuition in Mechanics handout (Note
that here the projectile is fired vertically with $\theta =
90^\circ$). \\ e) Finally, solve for the maximum height of a
projectile exactly and check your answer against part c).  Start by
arguing that trajectories will be segments of conic sections.  What
conservation laws can you use? Your answer will be algebraically messy
- simplify it as much as you can. Does your answer make sense?
Consider all of the limiting cases that you can think of as tests --
including your answers from c) and d). Write these down and use them
to argue whether your full answer is reasonable.


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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