1. a) Problem 2.16.I. This is a classic problem,
the answer to which can be found in many places. Try to do it on your
own, but you can search for clues online if you get stuck.
b)
Now, work out the potential at any point interior to a spherical
shell.
c) (Optional) Can you work out the potential at any point interior to an ellipsoidal shell?
Very Challenging
2. In this problem, you will find the (x,y,z) coordinates of the
five Lagrange points for a system with a primary of mass m1
= M(1-μ) and a secondary of mass m2 = Mμ where M is
the total mass of the system. The primary is on the negative x-axis,
it is separated from the secondary by distance R, and the center of
mass is at (0,0,0).
a) Find the coordinates of the primary and the
secondary.
b) find the coordinates of L4 and L5 by using the fact
that these point form equilateral triangles with the larger bodies.
c)
Find r, the distance of L3 from the origin for μ << 1 by balancing
the appropriate forces.
d) Find rH, the distance of L1 from the
secondary valid to lowest order in μ for μ << 1 by balancing
forces. Compare your result with the radius of the Hill sphere given
in the book.
Next, go to the Lagrange Point
Explorer from the class webpage. Check the stability of the
Lagrange Points for μ=0.001 by starting an orbit with zero speed
near each of them describe the resulting five orbits in both the
rotating and inertial frames.
e) Finally, consider particles
starting from L4. Conduct a careful numerical experiment
by raising the mass ratio μ in several steps to determine when this
point is stable and when it is unstable. State whether your orbit is a
point, an extended tadpole or an escape orbit in the rotating frame.
Compare with the conflicting predictions of section 2.2.1 in the book
where m1/m2 = 25 or near
m1/m2 = 27 are both given for the stability
limit. What can you conclude about the critical mass ratio?
3. Problem 2.15.E You might try the Planetary Calculator (http://janus.astro.umd.edu/astro/calculators/pcalc.html) and Satellite Calculator (http://janus.astro.umd.edu/astro/calculators/scalc.html) to do the math for all planets and satellites simultaneously!
4. Problem 2.17.E Do this problem by writing a short computer code in any language (python or C recommended) that gives its results in SI units. Use the equatorial radius R=60330km here and 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.