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{\it Astronomy 45 problem set 3}\hfill {\it Spring 1997}\newline
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{\bf Astronomy 45.  Introduction to Astrophysics}
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\noindent\underline{Problem set 3} \hfill Due Monday, March 17
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\noindent {\bf 1. Elliptical Orbits}  
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\noindent
Show that the total energy $E$ and angular momentum $L$
for a bound two-body system in an elliptical orbit with an eccentricity $e$ 
and semimajor axis $a$ are given by the following expressions:
$$
E = -\frac{G M \mu}{2 a}\ \ \ \ {\rm and}\ \ \ \ L= \mu \sqrt{G M a (1-e^2)}
= \mu \sqrt{GMr_0},
$$
where $M$ is the sum of the masses of the two bodies, and $\mu$ is the 
reduced mass.  Thus, $E$ depends only on the length $a$, and $L$ depends only 
on the length $r_0$, which has an equally simple geometrical interpretation
as the semi-latus rectum. Hint: use the fact that both of these quantities 
are conserved at any point in the orbit.

\vskip12pt\noindent{\bf 2. Properties of Orbits} 
\begin{itemize}
\item[a)] Consider the family of elliptical orbits with constant energy,
and show that the circular orbit has the most angular momentum.
\item[b)] Consider the family of elliptical orbits with constant 
angular momentum and show that the circular orbit has the most binding energy.
\item[c)] Spacecraft are typically able to make short concentrated momentum 
changes by turning on their thruster (rocket).  They are not usually able to 
thrust continuously at a low level.  Describe the sequence of events used by a 
spacecraft to change from one circular orbit to a higher circular orbit (say
at twice the semi-major axis).  Be sure to describe when a thrust occurs, and 
in what direction.
\item[d)] A spacecraft of mass $M$ is in a highly elliptical Earth orbit, and 
it has enough fuel for a single momentum change $\Delta p$.  Its goal is to 
break loose from Earth and coast off to infinity using the smallest possible
$\Delta p$.  Is it better for it to make its $\Delta p$ thrust when it is at
perigee (close to Earth) or apogee (far from Earth)?  Explain why? Hint:
think about {\em energy}.
\item[d)] Communications and weather satellites are often placed in {\em
geosynchronous} ``parking'' orbits above Earth.  These are orbits where 
satellites can remain fixed above a specific point on the surface of Earth.  
At what altitude must these satellites be located?  Is it possible for a 
satellite in a geosynchronous orbit to remain parked over {\em any} location
on the surface of Earth?  Why or why not?
\end{itemize}

\vskip12pt\noindent {\bf 3. Sun-Jupiter System} 
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\noindent
Jupiter is the largest planet in the solar system with mass $M_J = 0.95\times 
10^{-3} M_{\odot}$ and radius $R_J = 7.14\times 10^9\,{\rm cm}$.
The semimajor axis of Jupiter's orbit is $a=5.2\,{\rm AU}$, its orbital 
eccentricity is $e=0.048$, and its orbital period is $P=11.86\,{\rm yr}$.

\begin{itemize}
\item[a)] Calculate the total angular momentum of the Sun-Jupiter system,
using the result from Problem 1.
\item[b)] Estimate the contribution the Sun makes to the total orbital 
angular momentum of the Sun-Jupiter system. For simplicity assume that the
Sun's orbital eccentricity is $e=0$.  Hint: First find the distance of the 
center of the Sun from the center of mass.
\item[c)] Estimate the contribution that the orbit of Jupiter makes to the 
total angular momentum, also assuming that $e=0$.
\item[d)] Recall that the moment of inertia of the solid uniform 
sphere of mass $m$ and radius $r$ is given by $I=\frac{2}{5} m r^3$.  
Assuming (incorrectly) that both the Sun and Jupiter rotate as solid 
uniform spheres, calculate the approximate 
values for the rotational angular momentum of the Sun and Jupiter.  Take the
rotation periods of the Sun and Jupiter to be 26 days and 10 hours 
respectively.
\item[e)] What part of the Sun-Jupiter system makes the largest contribution 
to the {\em total} angular momentum (total orbital angular momentum + total 
rotational angular momentum)?
\end{itemize}

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