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\begin{center}
{\Large ASTR450 Homework \# 10 -- Perturbations\\
Due Tuesday, May 2}
\end{center}


1. {\bf Impulse Approximation to a Drag Force.} Consider a satellite
on an inclined elliptical orbit acted on by a drag force of the form
${\bf F} = -k{\bf v}_{rel}$, where ${\bf v}_{rel}$ is the velocity of
the satellite relative to the atmosphere, and $k$ is a positive
constant. Recall that Earth's atmosphere decays exponentially with
height (scale height $\approx 10$km).\\ a) Approximate the range of
eccentricities for which the drag force can be approximated by an
impulse at pericenter. \\ b) Consider a rotating Earth. Start by
making a qualitative estimate of the error made in neglecting
rotation. How does the Earth's rotation affect an equatorial orbit
($i=\Omega=0$)? Describe how the orbital elements $a, e, i, \Omega,
\comega$ vary in time.\\ c) Now imagine orbits with $i\ne 0, \Omega
\ne 0$. Using the perturbation equations and other physical arguments,
describe qualitatively how these orbits will evolve (i.e.  how $a, e,
i, \Omega, \comega$ vary in time).

2. {\bf Radial Perturbation Forces.} Consider a radial perturbation
force of the form ${\bf F} = R \hat r$, where $R$ is a function of the
distance $r$.\\ a) Apply the perturbation equations to this force and
obtain simplified expressions for $da/dt$, $de/dt$, $di/dt$,
$d\Omega/dt$, and $\comega/dt$.\\ b) A radial perturbation to gravity,
which is itself a radial force, is an example of a central force. So
angular momentum must be conserved. Show that your equations conserve
the angular momentum vector and describe the constraints that this
imposes on these orbits. \\ c) Now let $R=Ar^n$ where $A$ is a
constant. Take the time average of your expressions over a single
unperturbed Keplerian orbit (this step assumes that the perturbation
is small). Show that $<r^n\sin\nu> = 0$ and argue, on physical ground,
that $<\cos\nu>$ is negative (or zero) and that $<r^{-2}\cos\nu> = 0$.
It can be shown that $<r^n\cos\nu>$ is negative for $n>-2$ and
positive for $n<-2$.  Use this fact to determine how the sign of your
time-averaged $d\comega/dt$ depends on $A$ and $n$. Use the Central
Force Integrator to check your results numerically.\\ d) Finally,
consider the General Relativistic (GR) Perturbation $R = A r^{-4}$
where $A$ is a small negative constant.  The integral $<r^{-4}\cos\nu>
= a^{-4}e(1-e^2)^{-5/2}$.  Solve, analytically, for the value of $A$
that will give 30 degrees of precession per orbit for $e=0.5$ (The
true effects of GR on Mercury's orbit are almost exactly a million
times weaker). Convert your prediction into the proper initial
conditions for the Central Force Integrator, and test it!  Start your
orbit at pericenter and turn in a copy of your plot.

3. Using the paper by Burns et al. (1979), fill in the missing steps
in the derivations of Eqs 24 and 28 for $da/dt$ and $de/dt$. Show
enough work that I can see that you understand the derivations!



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