1. 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 from 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. e) 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?
2. Problem 2.15.E
3. 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.
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 (If you are using the first edition, you'll need to get Eq. 2.44a from a student with the second edition). 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.
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