ASTR630 Planetary Science, Spring 2023
Homework Assignment #2


Read Chapters 2.2 - 2.5

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.


Return to ASTR630 Home Page