(a)  A planet is moving in an elliptical orbit in the central gravitational field of the sun.   If e is the eccentricity of the ellipse, show that the ratio of the maximum and minimum speeds of the planet is given by (1+e)/(1-e).

(b)  Now let the planet move in a circular orbit, and assume a comet to be moving about the sun in a parabolic orbit.  Show that if the comet crosses the orbit of the planet, the speed of the comet at the crossing point is times that which the planet would have at the crossing point.  (Assume no collision occurs between the comet and the planet.)

A body moves in an elliptical orbit of eccentricity e under the action of a central force directed towards one focus.  When the body is at the pericenter, the center of force is transferred to the other focus.  Show that the eccentricity of the new orbit is e(3+e)/(1-e).

,

where a is the semi-major axis and e is the eccentricity. Prove that an inverse square law force (-k/r²) leads to

,

where E is the total energy of the particle and l is its angular momentum.

A point like comet of mass m moves in the gravitational field of a sun with mass M and Radius R.  What is the total cross section for the comet to crash on the sun?

A star of mass M and radius R is moving with velocity v through a cloud of particles of density r.  If all the particles which collide with the star are trapped by it, show that the mass of the star will increase at a rate

.

Find the maximum time a comet (C) of mass m following a parabolic trajectory around the Sun (S) can spend within the orbit of the Earth (E). Assume that the Earth’s orbit is circular and in the same plane as that of the comet.