# What is the relationship between the periapsis altitude and the change in velocity in a gravity assist?

In a gravity assist maneuver if I were to decrease the distance from the rocket to the periapsis of the planet it is orbiting (I believe this is called the periapsis altitude but I may be wrong) it would cause a greater change in velocity (meaning a more effective gravity assist) assuming all other variables are constant.

Is this relationship between change in velocity and periapsis altitude linear? Because I originally assumed it would be linear but then I read something that made me think otherwise:

http://maths.dur.ac.uk/~dma0rcj/Psling/sling.pdf

(in the paragraph exactly after equation 10)

From what I understood it says that the change in velocity would be equal at a periapsis altitude of both 2 million km and 87500 km which would indicate that it would not be a linear relationship (maybe a parabola for example).

I am asking because for an essay I am writing I think it would be cool to find the optimal distance the rocket should fly from the planet (ignoring affects from the atmosphere), which I would be able to find if there is a clear maximum on the graph of change in velocity vs periapsis altitude.

• What do you mean with "the distance from the rocket to the periapsis of the planet it is orbiting"? Are you trying to figure out a difference if the planet is near its periapsis in its orbit around sun. Or do you mean the distance the probe passes the planet, where the swing by is made? – CallMeTom Jun 18 at 7:54
• @CallMeTom So by periapsis altitude I mean the distance from the rocket at periapsis to the planet (the closest distance the rocket is to the planet). And so I would make this distance smaller and see the effect it has on the acceleration of the rocket. In this image the distance is r and thats what I would decrease: cdn.discordapp.com/attachments/241626066693783554/… – Alexander Ivanov Jun 18 at 20:57