1. Problem 2.9.**I**: a) Find the (x,y) coordinates
of all points from statements made in the book. For L_{3},
find its location to first order in the small quantity
m_{2}/m_{1}.

b) State the stabilty of each point
from the discussion in the book.

After finishing the book problem
(don't forget part c!), go to the Lagrange Point
Explorer from the class webpage.

d) Check your answers to
part b) by starting an orbit with zero speed at each of the Lagrange
Points; describe the resulting five orbits in both the rotating and
inertial frames.

e) Consider particles starting from
L_{4}. Change the mass ratio to determine when this point is
stable and when it is unstable. Compare with the predictions of
section 2.2 and discuss the reasons for any differences. Does
instability set in near m_{1}/m_{2} = 25 or near
m_{1}/m_{2} = 27?

2. Problem 2.15.**E**

3. Problem 2.17.**E** Do this problem by writing a
short computer code in any language that gives its results in SI units.
Turn in a copy of your code and its output. Check your answers in as
many ways as you can think of and discuss your results.

4. Problem 2.23.**I** All satellites near planets
have nearly circular orbits, so you can assume e=0. Before starting
this problem, make a careful sketch of Eq. 2.44a vs. distance for orbits
within 10 radii of Mars' surface; indicate Phobos and Deimos on your
sketch. What are the main features of your sketch and do they make
sense physically? Note that a(0) is simply a(t=0). After
solving the problem, calculate the impact timescale for Mars' moon
Phobos. Assume k_{T1} = 0.14 and Q_{1} =86, and get
other values that you need from the Satellite Calculator at
http://janus.astro.umd.edu/astro/calculators/scalc.html. Comment
on your numerical result.

5. Problem 2.32.**I** Instead of parts a) and b),
derive the general result for launch from a distance r along an elliptical
orbit. Show that the special cases a) and b) follow from your more
general result.

6. Go to the Central Force
Integrator from the class webpage.

a) Change the defaults to
v_{r}=0.3, B=0.05, and n=-4. This adds the General
relativistic perturbation (greatly magnified) to Newtonian
gravity. Does the orbital pericenter precess (rotate in the same
direction as the orbital motion) or regress (go in the opposite
direction)?

b) Now change the power on the perturbation force.
Which forces lead to precession and which to regression? What happens
when you change the sign and/or the magnitude of the perturbation?

c) Repeat part b) for the harmonic oscillator (m=1), subject to
various perturbation forces, changing the other parameters as
needed.

d) Generalize your observations. What is the condition on
the perturbation force that leads to precession? Print out and turn in
a few plots to support your conclusions.

e) (Optional) Want to
explore more? See which force laws allow stable circular orbits (put
v_{r}=0 and v_{θ}=.99999999). See if you can
find other force laws that lead to closed orbits. See if you can
create an orbit that looks like a 5-pointed star.

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