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{\Large ASTR430 Homework \# 2 -- Orbital Motion\\
Due Thursday, March 1}
\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. Sketch the following orbit at four times, $t=0,1/8,1/4,3/8$ years, and
then describe, without using the names of the orbital elements
$a,e,i,\Omega,\omega$, how the orbit is changing in time.\\[270pt]

2. Atmospheric Drag.\\ Imagine a satellite orbiting above the Earth's
equator with orbital elements $a=5.2 R_E, e=0.8$ and $i=0$ where $R_E$
is the Earth's radius and $a,e,i$ are the orbit's semimajor axis,
eccentricity, and inclination, respectively. \\ a) How close does the
satellite approach the Earth's surface at pericenter, and what is its
speed there? Is it moving faster or slower that the Earth's
rotation?\\ b) The Earth's atmosphere decays exponentially with height
with a scale height of approximately 10 km, so it is an excellent
approximation to treat atmospheric drag on the satellite as an impulse
at pericenter.  Which way does this impulse point relative to the
velocity vector at pericenter? Show that the orbital semimajor axis
decreases in time and find out what happens to the orbital
eccentricity and inclination.\\ c) Sketch the original orbit, and
three or four of its future states.  What happens in the end?

3. 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.

4. 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.



5. EXTRA CREDIT CHALLENGE PROBLEM (Experts Only): 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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