Orbital energy required to change a comets orbit

[Deadstar]

Homework Statement

Use the information in Section A.3 and the data in Table A.19 to calculate the heliocentric radial distance (in AU) of Comet Halley at 12h UT on July 16, 1994.

- Done, Section A.3 just talked about finding dates using Epochs and such.
Table 19 had comet details which will appear below

(b) Assuming that the heliocentric semi-major axis of Comet Shoemaker-Levy 9 was the same as that of Jupiter (a approximately 5.20 AU), and that the masses of both comets are negligible compared to the mass of the Sun, estimate how much orbital energy per unit mass (in Joules per kg) would be required to change the orbit of Comet Shoemaker-Levy 9 to that of Comet Halley.

Homework Equations

Halleys comet.
time of perihelion passage - 1986 Feb 9.5
Perihelion distance q (AU) - 0.5871
e - 0.9673
Inclination - 162.24 degrees
argument of perihelion - 111.87 degrees
longitude of ascending node - 58.86 degrees (we have never discussed this or the argument of perihelion before and i don't they are involved in the answer)
Epoch - 46480

The Attempt at a Solution

I reeeeally want to figure this out on my own (it's not an assignment, just a tutorial question by the way) but it's sapping so much of my time right now that if I don't sort it soon other courses will suffer.

So in that respect I would like the slightest of hints as to which direction to proceed in this.

This is part of an exercise sheet that was given out after lectures on the 3-body problem so it makes sense that will be involved.

My questions are...

Are the three bodies the two comets and the sun? Or is it Shoemaker-Levy 9, Jupiter and the sun?

Do I start by using Tisserands relation to find out the eccentricity and Inclination of Shoemaker-Levy 9?

$$\frac{1}{2a} + \sqrt{a(1-e^2)} \cos(I) = \frac{1}{2a'} + \sqrt{a'(1-{e'}^2)} \cos(I')$$

Where a,e and I are parameters of a comet which then become...

a', e' and I' after a close approach to a planet.

When I have done this should I be trying to find velocities of each comet then will I compare them using the vis viva integral which is...

$$\frac{1}{2} v^2 - \frac{\mu}{r} = C$$

where $$C = -\frac{\mu}{2a}$$This equation came about during the 2-body problem which is why I'm skeptical as to whether it can be used here. It also states in my book that this formula shows that the orbital energy per unit mass is conserved but in the restricted 3-body problem I believe that doesn't happen.

So, anybody care to shove me in the right direction? Is the Jacobi integral/constant involved in this as well?EDIT: Some more thoughts:

I was having a little trouble with working out exactly what this question is asking but I think I have a better idea now.

I think the idea is that I have to calculate the energy required to 'save' Shoemaker-Levy 9 since on that day it crashed into Jupiter.

Given that I know 'r' in the vis viva integral for Halley (from part a), I need to find velocities and use the vis viva integral to compare energies (how exactly, I'm not sure... Is C the energy?)EDIT 2:

Will the velocity of Shoemaker-Levy 9 just be the escape velocity of Jupiter?

 

[NateD]

If so, I can use that in the vis viva equation and then compare it with Halley's velocity and energy.EDIT 3:Ok, so I think I've worked out what I have to do.I need to find the velocity of Shoemaker-Levy 9 at the time of its 'crash' into Jupiter. I'm assuming this will be the escape velocity of Jupiter at that time.Then I need to calculate the corresponding velocity and orbital energy of Halley at the same time.Once I have both velocities and energies I can compare them using the vis viva equation and work out the energy required to save Shoemaker-Levy 9 from its doom. So the only question now is how do I find the escape velocity of Jupiter at that time?