## Comet in Parabolic Orbit

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

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

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

3. 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.
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 Tags comet, gravity, orbit dynamics

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