Physics 701: Classical Mechanics
Thornton Hall 425, San Francisco State University
Fall 2016, MWF 5:10PM
Homework 5
Problems 1-2 Due 5:10PM 10/13; Mathematica Project due 5:10PM 10/20
While I may have consulted with other students in the class regarding this homework, the solutions presented
here are my own work. I understand that to get full credit, I have to show all the steps necessary to arrive
at the answer, and unless it is obvious, explain my reasoning using diagrams and/or complete sentences.
Name
Signature:
1. (50 points) Consider the attractive two-body potential U (r) = −(k/2)/r2 , with k > 0.
(a) (10 points) In class, we derived a formula for determining whether the orbits in a power law
potential are closed. How does this formula apply here, and can you make sense of the answer
it gives?
(b) (15 points) Solve for the general r(φ). Identify which orbits escape to infinity and which don’t.
(c) (15 points)Sketch or plot a reasonably accurate drawing of any non-circular orbit r(φ) in the
x-y plane, i.e. do not plot r vs φ, but x vs. y.
(d) (10 points) Solve for the general r(t) and φ(t).
2. (50 points) For the restricted, circularized three-body problem, two masses m1 and m2 are in a
circular orbit with r = D. Assume that the motion of a third, much smaller mass m3 is confined to
the plane of the orbit of m1 and m2 . Consider the frame whose origin is the center of mass of m1
and m2 . If the frame corotates with m1 and m2 , show that the Lagrangian in the rotating frame is
L=
i
m3 h
Gm1 m3
Gm2 m3
(ẋ − ωy)2 + (ẏ + ωx)2 + p
+p
2
2
2
(x − x1 ) + y
(x − x2 )2 + y 2
where ω is the angular frequency of the circular orbit for m1 and m2 . Write the Euler-Lagrange
equations in the co-rotating Cartesian coordinates. Find the Hamiltonian. Be sure to calculate the
values x1 , x2 , and ω from m1 , m2 , and the star separation D.
3. Mathematica Project. This is tracked separately for grading purposes. Consider two nonrotating stars of unequal mass M1 and M2 and unequal radii R1 and R2 in circular orbit about each
other. The centers of the stars are a distance D from each other. Move into a corotating frame
where the origin is the center-of-mass of the two stars and where the x-axis is the line connecting
the two stars.
(a) Define the 2D effective potential Ueff (x, y) as the Hamiltonian in the limit that ẋ and ẏ go to
zero. Calculate and plot it using ContourPlot and Plot3D in Mathematica. Mark the points of
stable and unstable equilibrium. Use stars with a 2:1 mass ratio, and renormalize your distance
units so that the distance between the stars is 1. You should convince yourself that with this
convention, you don’t actually have to enter values for G or the specific masses of the stars to
find the extrema of the effective potential (which are called Lagrange points, by the way). You
can therefore find the locations of the extrema in units of the separation distance. Which are
stable, and which are unstable points?
1
(b) Find the conditions under which a hydrogen atom on the surface of star 1 will begin travelling
to star 2, thus forming an “accretion stream” or a steady stream of material travelling from star
1 to star 2. Assume D > R1 + R2 . The cross-over region from non-accretion to accretion is
called the “Roche Lobe,” after 19th century French astronomer Edouard Roche.
(c) As time passes, star 2 is growing, and star 1 is shrinking. Assume the exchange of material is
proceeding at a constant rate Ṁ , and that the stars maintain a uniform density throughout the
process. How long will the accretion continue? What is the final size of the stars? This problem
is easier to solve numerically (in Mathematica). Hint: everything looks easier when cast in terms
of f = R1 /D.
(d) Use Mathematica or other software to solve for the motion of the hydrogen atom in the corotating
frame. Plot the path taken by the particle in the corotating X-Y plane for a suitable choice of
values of D, R1 , and R2 . Assume that the stream has zero initial velocity, and that it begins at
the point on star 1 that is closest to star 2. Use M2 = 10M1 , R2 = 0.01R1 . Be sure to use a
value of D for which accretion acctually happens. This model represent accretion streams from
puffy, low mass stars to high-mass black holes or neutron stars.
2