Time that a comet spends inside Earth's orbit

AI Thread Summary
The discussion focuses on calculating the time a comet spends inside Earth's orbit using angular momentum and areal velocity. The user derives the angular momentum L and attempts to find the area A under the comet's parabolic trajectory to calculate time T. However, they encounter difficulties with integration and the resulting value of T, which they believe is incorrect. Another participant points out potential errors in the integration process and suggests that the value of T should be significantly larger than initially calculated. The conversation emphasizes the need for correct mathematical formulation and integration techniques to arrive at accurate results.
matteo446
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Homework Statement
A comet of mass m is orbiting around the Sun in a parabolic orbit. Assume that Earth's orbit is circular with radius rT and that it's coplanar with the orbit of the comet.

Determine the time T that the comet spends inside Earth's orbit if the periaster (nearest point to the Sun) of the comet is rP=rT/3.

Determine the maximum time that the comet can spend inside Earth's orbit tMax.
Relevant Equations
U(r) = L^2/(2mr^2) - GmM/r where L is angular momentum of the body from P, m is the mass of the body orbiting r(θ) = ed/(1+ecos(θ)) where e is the eccentricity and d is the distance from the directrix
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I tried in the first place to use the effective potential of a parabolic orbit which is 0 to get the angular momentum L.

Evaluating the function U(r) at r = rP i get U(rP) = L^2/(2m(rP)^2) - GmM/rP = 0.

Here I get L = m√(2GMrP).

Now the relationship between angular momentum L and areal velocity α is L/2m = √((1/6)GMrT) which is a constant of motion.

My idea is to find an area and use this value of α to obtain time T.

With respect to a polar frame of reference centered at S i used the general equation for a conic in polar coordinates r(θ) = ed/(1+ecos(θ)) with e=1 for a parabola so r(θ)=d/(1+cos(θ)).

I know r(0) = rP so replacing rP = d/2 and d = 2rP.

So the comet follows the orbit of equation r(θ) = 2rP/(1+cos(θ)).

Now I want to find the angle λ to replace in the equation to get rT as the point of intersection between the orbits satisfies also the equation for the orbit of the Earth r(θ) = rT.

I get λ = arcos(-1/3) ≈ 1.9 rad.

So (here i think I made mistakes as i don't know polar integration) integrating from 0 to 1.9 and multiplying by 2 because of symmetry the area A is 2*(1/2 ∫(2rP/1+cosθ)^2dθ) ≈ 1.9rP^2 = (1.9/9)*rT^2.

Now simply T = A/α = ((1.9/9)*rT^2)/√((1/6)GMrT) ≈ 9.5*10^-3s which is very wrong.

I don't know how to start for the second question.
 
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I think your work is essentially correct.

However, you wrote
matteo446 said:
A is 2*(1/2 ∫(2rP/1+cosθ)^2dθ) ≈ 1.9rP^2
You need another set of parentheses to enclose the ##1 + \cos \theta## in the denominator. I don't get the factor of 1.9. I get a number between 4 and 5. (I'm lazy, so I used software to do the integration.)

I don't know how you got ##T## to be of the order of ##10^{-2}## seconds. Your equation for ##T## should yield a large number in seconds if you plug in the correct numbers.

I don't know how to start for the second question.
You'll need to let ##r_p## be a variable. You could let ##x = r_p/r_T##. Find ##\cos \theta##, ##L##, ##A##, and ##T## in terms of ##x##.
 
Thanks for your help. :smile:
 
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