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:
(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.