1. Problem 2.9.I: a) Find the (x,y) coordinates
of all points from statements made in the book. For L3,
find its location to first order in the small quantity
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 L4. 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 m1/m2 = 25 or near m1/m2 = 27?
2. Problem 2.15.E
3. Problem 2.17.E Do this problem by writing a short computer code in any language. 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 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 kT1 = 0.14 and Q1 =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 vr=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 vr=0 and vθ=.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.