Solving for θ in a Parabolic Orbit

[pelmel92]

Homework Statement

A comet of mass m moves in a parabolic orbit in the ecliptic plane (the plane of Earth’s
orbit), so its perihelion distance ρ (its closest distance to the Sun) is less than Ro (the orbital distance of the Earth around the Sun) and occurs when θ = 0 for the comet. (The comet will cross the orbit of the Earth twice- once moving inward and once moving outward.) In terms of p and Ro, find an expression for θ at the two times when the comet crosses the orbit of the Earth. Assume: m << Msun.

Homework Equations

E = .5m(dr/dt)^2 + .5l^2/(μr^2) - GmM/r

where l is the angular momentum and μ is the reduced mass.

The Attempt at a Solution

Alrighty, so far I know that m is nothing in comparison to Msun, so μ≈m. At perihelion ρ, θ is zero, and dr/dt=0, so the velocity there Vmax = ρ(dθ/dt), and l = mrvsin(π/2)= mρ^2(dθ/dt). Plus, this being a parabolic orbit, E = 0.

I've been trying to put that all together given that l is conserved and E remains zero at all distances:

E = 0 = .5m(dθ/dt)^2 - GmM/ρ = .5m(dr/dt)^2 +.5m[ρ(dθ/dt)/Ro]^2 - GmM/Ro

but I end up with a bunch of dr/dt and dθ/dt variables that I can't figure out how to eliminate... I just don't see how I can isolate and solve for θ at Ro. It seems like my whole approach is probably wrong. :/

Any help from you good folk would be MUCH appreciated.

EDIT: Whoops this was actually super simple. I'd just forgotten that the semi latus rectum = 2p= Ro(1+εcosθ), and that epsilon for the parabola ia 1.

 

[grateful4dmb]

Assuming that your edit is correct, what ends up being the solution? I am working on the same problem, and I am lost myself as well! Some guidance would be wonderful! Thanks.