Assignment 6

Due: Monday, 25 November 2002

27. A planet of mass M is in an orbit of eccentricity 1 - b, where b << 1. The planet is struck by a comet of mass m where m << M, when it is at its greatest distance from the Sun (aphelion). The incoming comet is traveling tangential to the orbit at the moment of impact. If the collision is inelastic, find the minimum energy the comet must have to change the planet's orbit to a parabola.

28. Find the cross-section for scattering in the repulsive force field

f = kr^-3.

29. The solution for motion in the force field

V(r) = -k_or^-1 + k₁r^-2

was discussed in class. Apply these results to the precession of the apsides of Mercury's orbit. The perihelion of Mercury is observed to precess (after corrections accounting for the perturbations by other planets) at the rate of ~40 arc seconds per century. Show that this small precession can be accounted for if the strength of the extra perturbing potential, as measured by

A = k₁/(k_oa),

is as small as 7x10^-8. The eccentricity of Mercury's orbit is 0.206 and its orbital period is 0.24 year.

30. The addition of the potential, V(r) = k₁r^-2, to the r^-1 potential in the problem 29 looks like it simply augments the centrifugal potential, l²/(2mr²). Why does the additional potential cause the orbit to precess while changes in the centrifugal barrier (through changes in the angular momentum l) do not?