\documentstyle[11pt]{article}

\textheight 22cm
\textwidth  16cm
\hoffset= -0.6in 
\voffset= -0.5in   
\setlength{\parindent}{0cm}
\setlength{\parskip}{15pt plus 2pt minus 2pt}
\pagenumbering{roman} 
\setcounter{page}{-9}

\newcommand{\dg}{^\circ}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bd}{\begin{displaymath}}
\newcommand{\ed}{\end{displaymath}}
\newcommand{\comega}{\varpi}

\begin{document}


\begin{center}
{\Large ASTR450 Homework \# 11 -- Planetary Oblateness\\
Due Tuesday, May 9}
\end{center}

The first two problems make use of the Planetary Satellite Integrator
(PSI). Be sure to include several plots of PSI's output with each of
these problems, label them Fig. 1, Fig. 2, etc., and describe what
features they show in your writeup.

1. a) Choose a moderately eccentric orbit ($e\approx 0.2)$ at about 6
Earth Radii and use the Planetary Satellite Integrator to test the
orbit-averaged equations for planetary oblateness. Toggle Earth's
equatorial bulge on and solar gravity off. Follow a set of orbits with
a given $a$ and $e$, but different $i$'s, for 0.1 years saving values
every 0.1 days.  Plot the $\delta\Omega$ and $\delta\omega$
experienced by your orbit over 0.1 years vs.  inclination and compare
to the expressions that we derived in class (Use the Orbital Elements
vs. Time plot).  Are the equations accurate? What features of the
solution (if any) do the orbit averaged equations miss?  \\ b) What
happens to the orbits as $e\rightarrow 0$ or $i\rightarrow 0$?\\ c)
This satellite has $a=3.84 R_p, e=0.22$ and evolves under the
influence of planetary oblateness only. What is the inclination of the
satellite's orbit, and which planet does it circle?\\[3in]

2. a) Take a moderately eccentric orbit with $i=30\dg$ and investigate
orbits of different sizes from about 6 to about 30 Earth Radii.
Consider orbits where oblateness alone is important and produce a plot
which tests the $a$ dependence of the orbit averaged equations.\\ 
b) Now toggle the equatorial bulge off and solar gravity on and repeat
the experiments. At what distance are the two effects comparable in
magnitude? How does solar gravity appear to depend on $a$? What
features does solar gravity add to the orbital evolution?\\ c) Try to
predict what you will see when both solar gravity and planetary
oblateness are active and test your guess with a few simulations.

3. Pluto and Neptune are linked by a 3:2 eccentricity-type resonance.
The resonant term in the disturbing function felt by Pluto (due to
Neptune) is ${\cal R}_N = M_N\beta e_P\cos(3\lambda_P-2\lambda_N
-\comega_P)$ while the term felt by Neptune (due to Pluto) is ${\cal
  R}_P = M_P\beta e_P\cos(3\lambda_P-2\lambda_N -\comega_P)$. Here
``P'' subscripts refer to Pluto, ``N'' subscripts refer to Neptune,
$M, e, \lambda,$ and $\comega$ are mass, eccentricity, mean longitude,
and longitude of pericenter, and $\beta$ is a constant.\\ a) List all
other first-order 2:1 resonant terms for Neptune's perturbations on
Pluto. List all other second-order resonant terms. \\ b) Find the time
rates of change of the orbital elements ($a,e,i,\Omega,\comega$) of
Pluto and Neptune due to disturbing functions ${\cal R}_N$ and ${\cal
  R}_P$. Remember that $d{\cal R}/d\epsilon = d{\cal R}/ d\lambda$ and
$p=a(1-e^2)$. Give your answer to lowest order in eccentricity and
inclination, and keep only the most perturbed element in each of these
sets: $(de_N/dt, di_N/dt, de_P/dt, di_P/dt)$ and ($d\Omega_N/dt,
d\comega_N/dt, d\Omega_P/dt, d\comega_P/dt)$. You can set the other
members of each set equal to zero.\\ c) Show that the sum of the
orbital energy of Pluto plus that of Neptune is conserved to lowest
order in $e,i$.  \\ 



\enddocument{}