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Suppose a three body scenario with an artificial satellite, a planet (much massive than the satellite), and a sun (much massive than the planet). There is no other significant force on these three bodies. We know the positions and velocities of the planet and the satellite relative to sun at a given instant ($t_0$). We need the satellite to be moving in a given direction, relative to the sun, at a given future instant ($t_1$). Any approximations that are valid for a gravity assist situation can be made (please mention in the answer). The objective here is to find a $\Delta V$ vector that can be applied at $t_0$ to achieve the needed final direction of the satellite. To make it really a gravity assist, most of the force on the satellite from $t_0$ to $t_1$ has to be from the planet's gravity, not the sun or the satellite's thrusters.

Can the objective be achieved without numerically solving differential equations? I am looking for an explicit equation (or a set of equations) that can be solved to get the $\Delta V$.

Is there any other data that if available, allows the use of any such method?

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    $\begingroup$ Pretty much anything that involves position as functions of time with orbital mechanics on orbits that aren't perfect circles or perfect parabolas is going to require the use of numerical methods. $\endgroup$
    – notovny
    Dec 3, 2023 at 20:23
  • $\begingroup$ Thanks @notovny $\endgroup$
    – paki eng
    Dec 3, 2023 at 20:27
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    $\begingroup$ @notovny with the approximation patched conics, can't both transfers and assist be done with pencil and (a few sheets of) paper? OP mentions "Any approximations..." $\endgroup$
    – uhoh
    Dec 3, 2023 at 21:16
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    $\begingroup$ @Kevin for the sake of simplicity, it would be OK to assume the satellite is already headed on a general direction to intercept the planet, and the maneuver you need to make is to trim the exact approach. I would take this as a valid assumption to consider it a gravity assist design problem. $\endgroup$
    – paki eng
    Dec 3, 2023 at 22:57
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    $\begingroup$ @Kevin only a very small fraction of folks posting questions here in Space SE are currently putting "real rockets into space"! (cf. What if my question about orbital mechanics is not a problem that I'm facing, but I'm simply interested in finding out the answer?) And once you read through the question again you'll see the OP emphasizes approximations are ok in answers. There were dynamical approximations involved in getting the Apollo folks on the Moon for example that one would not use today. But if you want to post a "No" answer go for it! $\endgroup$
    – uhoh
    Dec 4, 2023 at 1:50

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