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I am trying to answer a simple question: If a rocket performs a gravitational slingshot around a body of mass m, altitude h, and angle $\theta $ and initial velocity ${ v }_{ sun }$, what will be the resulting ${ v }_{ sun,f }$ of the rocket? This simple question has turned out to be extraordinary difficult to answer.

CONCEPUAL: I began by considering it conceptually; How does a gravity slingshot even work? In the planet's FOR, the rocket's $\triangle v=0$. But in the sun's FOR, the rocket gains some velocity from the planet's movement around the sun; a case of the conservation of energy.

MATHEMATICALLY: I know that a gravitational slingshot involves a hyperbolic orbit. Given that one focus of our hyperbolic orbit should be the planet of interest, it's understandable we seek the polar equation for a hyperbola, $r=\frac { p }{ 1+e\cos { f } } $, where $p = a(1-{e}^{2})$. The five parameters that uniquely define a hyperbola in the above two formulae are $r$ = distance from planet to rocket, $p$ = parameter, $e$ = eccentricity, $f$ = the True anomaly, or the angle is respect to the periapse (point when the planet and rocket and closest), and $a$ = semi-major axis of hyperbola. Other parameters include ${ v }_{ sun,i }$, the velocity of the rocket IRT the sun initially and ${ v }_{ sun,f }$, the velocity of the rocket IRT the sun after the assist. Let ${ v }_{ planet }$ be the velocity of the planet.

I've made a graphic consisting of all the constants I know so far.

enter image description here

I would greatly appreciate it if anyone could assist me in answering my question.

Qmechanic
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DarkRunner
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  • The question about the distance of the rocket doesn’t make sense. Distance when? – G. Smith Feb 11 '20 at 06:30
  • Why doesn’t your list of parameters include the velocity of the planet relative to the Sun? – G. Smith Feb 11 '20 at 06:31
  • Related question: https://physics.stackexchange.com/q/154854/ – G. Smith Feb 11 '20 at 06:34
  • @G.Smith Thanks; I revised my question, and removed the distance part (I realize it doesn't make sense, because since the rocket has enough excess velocity to escape the planet's gravitational tug, it will keep flying through space anyway). Yes, I've seen that question; I've scoured much of the web, but still haven't found a good mathematical explanation at my level. – DarkRunner Feb 11 '20 at 06:34
  • You didn't remove the distance part, and your edit has confused the velocity of the rocket and the velocity of the planet. – G. Smith Feb 11 '20 at 06:38
  • By ${ v }_{ sun }$, I'm referring to the velocity of the rocket IRT the sun, no? I believe if you refresh, you'll find my edit. – DarkRunner Feb 11 '20 at 06:39
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    If you want. Why doesn’t it have a subscript $i$ similar to $f$? Where is the velocity of the planet? You don’t even show that in the diagram, but without the planet’s velocity changing there can be no gravity assist. – G. Smith Feb 11 '20 at 06:42
  • Thank you; I made the suggested edits to the diagram and the question. – DarkRunner Feb 11 '20 at 06:48
  • It is inconsistent for v_planet to be the velocity of the planet but v_sun to be the velocity of the rocket. – G. Smith Feb 11 '20 at 06:51
  • Can you explain your need for this calculation? – G. Smith Feb 11 '20 at 06:52
  • This paper looks like it might be useful. – G. Smith Feb 11 '20 at 07:03

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