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\begin{center}
{\Large ASTR430 Homework \# 2 -- Orbital Motion\\
Due Wednesday, October 8, 2003}
\end{center}

Read the rest of the ``Building Physical Intuition in Mechanics''
handout and try some examples from each section. In the following
problems - and in all future homeworks - make sure to check units,
limits, symmetry, and your common sense! You will catch many errors
this way. Show me that you have thought about your answers rather than
just deriving them.


1. Show that $C = -GM_\odot/2a$ in the energy equation $v^2/2 -
GM_\odot/r = C$ by using equations for the specific angular momentum
$h$. Hint: Since the left side of the energy equation is constant all
along the orbit, evaluate it at a convenient point. Use your result to
obtain an expression for the speed of a planet at an arbitrary point
along its orbit. What are the maximum and minimum speeds for each of
the planets? Use the Planetary Calculator to find out (follow
Astronomy Workshop/Solar System Calculators links from the class
webpage).

2. At apocenter, a 10-ton satellite on an $a=4 R_E$, $e=0.5, i=0$
orbit uses a large coiled spring to launch a 1-ton communication
satellite onto an $a=6 R_E$, $e=0.0, i=0$ orbit ($R_E$ is the Earth's
radius). Find the new orbit of the (now 9-ton) parent satellite. Are
satellite orbital energy and angular momentum lost, gained, or
conserved?  Explain your answer.

3. As the interstellar cloud collapsed to form our Solar System, the
gas was heated significantly. In this problem, you will estimate how
hot the disk was at 1AU assuming that no energy was lost via radiation
during collapse. In reality, lots of energy is radiated away so this
will be a serious overestimate! Imagine an $H_2$ molecule which is at
rest at infinity and falls inward toward the Sun.\\ a) Calculate the
circular velocity at 1AU around a 1 Solar Mass star, and determine the
total (kinetic + potential) energy of an $H_2$ molecule on a circular
orbit at 1AU.\\ b) Noting that the total energy of the molecule at
rest at infinity is zero, calculate the temperature increase of the
hydrogen gas assuming that it has not suffered radiative losses.



4. EXTRA CREDIT CHALLENGE PROBLEMS (Experts Only!): \\
Do both, one or none.


a) Show that the vector equation
\begin{displaymath}
{d^2{\bf r} \over dt^2} = F(r) \hat {\bf r}
\end{displaymath}

reduces to two scalar equations

\begin{displaymath}
{d^2 r \over dt^2} - r \biggr({d\theta \over dt}\biggr)^2 = F(r)
~~~~~~~~ {\rm and} ~~~~~~~
2 {dr \over dt} {d\theta \over dt} + r {d^2\theta \over dt^2}= 0
\end{displaymath}
and show that the second secular equation expresses the conservation of angular momentum. 

b) Convert the scalar equation

\begin{displaymath}
{d^2r \over dt^2} - {h^2 \over r^3} = - {GM_\odot \over r^2}
\end{displaymath}

that we derived in class into one for $d^2u/d\theta^2$ using the
transformation $u=1/r$ and the expression for angular momentum $h=r^2
d\theta/dt$. Solve for $u$ and show that the solution can be written
as

\begin{displaymath}
r = {h^2/GM_\odot \over 1+e\cos(\theta - \varpi)} 
\end{displaymath}

where $e$ and $\varpi$ are integration constants. These solutions are
ellipses with the Sun at one focus.


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