# Comet in a parabolic trajectory

A comet in a parabolic orbit around the sun has a least distance of kR, k < 1. Show that the time during which the comets distance is less than R is: (1/3*pi)[2(1-k)]^1/2 * (1+2k) years

I have derived the following expression for t as a function of r:

t = the integral from r to ro of [2/m (E - u(r) - l^2/2mr^2)]^-1/2 dr

I have the equation of a parabola L/r = 1 + cos(theta)

L = l^2 / Gm^2M

u(r) = -GmM / r

I have let E = 0 and substitute into equation for time, then using the fact that

1 year = [2*pi*R^(3/2)] / (GM)^(1/2)