I am looking for a formulation between the _vector_ of the hyperbolic excess velocity and the corresponding hyperbolic orbit, especially the turning angle and the pericenter distance vector.
I would like to model a gravity assist maneuver but how do I derive the turning of the spacecraft vecolicty vector correctly without knowing the pericenter distance. The incoming hyperbolic excess velocity vector is given, but without a corresponding position vector I cannot derive an orbit. But per definition the position is at infinity. I would appreciate any help!
I am aware that there are a number of equations to derive the scalar relations of this parameters, but is not the vector, i.e. the direction of the incoming excess velocity already setting these parameters as well? I do not assume that the pericenter of such a trajectory is independent of the direction, right?
Thanks in advance!
The usual definition of "hyperbolic excess velocity" is a scalar speed so I'm unclear what extra information you have that lets you call it a vector. If you only have a direction, you get a family of asymptotes like the blue dashed lines on the diagram to the right in the link. If you are approaching say from the bottom left then you need some positional information to work out the eccentricity from which you get the turning angle. Just a direction on its own is not enough, obviously the closer you approach the planet, the greater the change of direction. If you can work out how close the asymptote passes to the planet, working out the eccentricity is just solving the trigonometry.
https://en.wikipedia.org/wiki/Hyperbola#Nomenclature_and_features
https://en.wikipedia.org/wiki/Hyperbolic_trajectory#Hyperbolic_excess_velocity
In my opinion the position of the asymptote does not matter; only direction is important; you can imagine that distance B of asymptote from body center is a free control that you use to obtain desired turning angle. (distance B is negligible compared to interplanetary distances)
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