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{\Large ASTR430 Homework \#4 \\
Due Wednesday, October 29, 2003}
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1. 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?

2. Mass of the Oort cloud. In our Solar System, Jupiter formed at
about 6.0AU and was dragged inward to its current 5.2AU. This happened
as Jupiter interacted with icy planetesimals and threw them out of the
Solar System. Imagine that the planetesimals all were in eccentric
orbits at Jupiter's distance to begin with, and that they were thrown
out of the Solar System on parabolic trajectories. Use a conservation
law to estimate how much mass must have been ejected from the Solar
System to account for the changes in Jupiter's orbit.

3. Tidal forces between the Earth and the Moon are pushing the Moon's
orbit away from the Earth and slowing down the Earth's spin. What
happens in the end? \\ a) Calculate the Earth's spin angular momentum,
the Moon's spin angular momentum, and the Moon's orbital angular
momentum. Make the approximations that the lunar orbit is circular and
that the angular momenta are aligned.\\ b) Assume that only tidal
forces between the Earth and the Moon are operating and argue that the
total angular momentum of the Earth-Moon system is conserved. Is the
orbital energy conserved? Explain your reasoning.\\ c) In the end
state of orbital evolution, the Moon will be further away from the
Earth, than it is now, and the Earth will always keep one face toward
the Moon. This is the synchronous state (i.e. the ``month'' will equal
the ``day''). Use conservation of angular momentum to determine how
far the Moon will be from the Earth then, and how long the ``day''
will be. You may neglect the spin angular momentum of the Moon - use
your answer from a) to justify this. The final equation cannot be
solved analytically; try to get a solution with a numerical or
graphical method.\\d) That would be the end of it, but we've neglected
the effects of the Sun. Tides raised on Earth by the Sun slowly slow
Earth's rotation. How will the Moon's orbit react? (assume that the
Moon has already reached the synchronous state that you found in part c) -
because of the Sun's influence, the synchronous state actually will
occur with the Moon somewhat closer to the Earth). Describe what
happens to the Moon next using Angular Momentum and Energy arguments.



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