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{\Large ASTR688S Homework \# 2 - due Thursday, March 14.}

1.  Apollo 13 is on a circular orbit 100km above the Earth's surface.
It fires its engines and produces a tangential velocity which will
take it to the Moon. In a two-body approximation (Earth, spacecraft)
how much extra velocity must be added to put the spacecraft onto an
orbit whose apocenter is at the Moon? What is the minimum extra
velocity that might enable the spacecraft to get to the Moon in the
three-body approximation (Earth, Moon, spacecraft)? Jacobi's integral
could be useful here ...

2. Hyperbolic Orbits. a) A particle is projected at great distance from a
star of mass M and radius $r_0$ with velocity V, such that, neglecting
the attraction of the star it would approach the star at a minimum
distance (impact parameter) $b$. Find $b_{crit}$ such that the particle
just strikes the Sun. Now let the star move through a stationary field of
particles with $n_0$ particles per unit volume. Derive Eddington's
formula for accretion:

\be
  A= \pi r_0^2 n_0 \biggr(V + {2 M G \over r_0 V}\biggr)
\ee

Which term dominates for the Sun (assume $V\approx 26$km/s)?

b) For a hyperbolic orbit, show that the incoming speed (far from the
Sun) equals the outgoing speed (far from the Sun). Find the deflection
angle (the angle between the incoming and outgoing velocity vectors)
as a function of the Sun's mass, the velocity $V$, and the impact
parameter $b$. This is the first step in calculating gravity assist
trajectories for spacecraft.

3. Parabolic Orbits. a) Tisserand's criteria is applicable to both
bound and unbound orbits.  Write down a simplified version of the
Tisserand Criteria for an object (a comet, say) on a nearly parabolic
orbit. Show that a parabolic prograde orbit (with $I < 90\deg$) cannot
be perturbed to a parabolic retrograde orbit ($I>90\deg$) by the action
of Jupiter alone. Can you explain physically why this is so? Can a
parabolic prograde comet be perturbed onto a retrograde ellipse? \\

b) Consider an $I=0$ parabolic comet with a pericenter approach
distance of 1AU. Make a plot showing the possible elliptical $I=0$ and
$I=180\deg$ orbits that can result (include the orientation of the new
orbit's pericenter if you can).

4. Learn about Xephem! Use it to answer the following
questions. During which years is Pluto is closer to the Sun than
Neptune? What is the angular distance between Mars and Saturn on
the sky on March 22, 1996 at 0:00 UT? When is the first time after
March 22 that the Apollo 11 landing site comes into sunlight? At what
times on March 22 are the various planets observable from Maryland? At
what time is Jupiter's red spot first visible from the Earth on March
22? Tell me what you like and dislike about Xephem!

5. - 8. Do problems 12.1, 12.2, 12.3, and 12.4 from the Planet
Formation handout. E=Easy, I=Intermediate, A=Advanced.

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