How to calculate the maneuver vector

[RocketSon4]

I have been interested in orbital physics for a long time. One thing I have recently been trying to figure out is how to calculate the burn vector at any point in your orbit. So, the question is: if you have the orbital elements of your initial orbit, and you have the orbital elements of your next orbit, how do you calculate the 3D maneuver vector if you know how long the maneuver lasts? I can find literally no answers on this on the internet. Thank you.

[UmbralRaptor]

With finite TWR / non-zero burn time? You're deep in numeric solutions land, so the various equations you can find won't be useful as anything beyond a starting point, outside of some very specific situations.

[RocketSon4]

With finite thrust, not TWR. Also, yes, non-zero burn time. Does anyone know how KSP calculates it. I understand that we, the players, get to change how much of normal, radial, and prograde we use. However, what orbital elements do those each individually change? Thanks

[UmbralRaptor]

IIRC, the stock maneuver nodes hand-wave some stuff. Namely, will it does give a burn time, it pretends that the duration is either 0 or so much shorter than the period etc as not not matter.

If you dig through various listings of how to get the trajectory vector components out of orbital elements (eg: http://braeunig.us/space/orbmech.htm), you can get the differences between your current and target orbit -> Δv -> propellant expenditure (via the rocket equation) -> burn time via engine propellant consumption. You can alternatively do Δv (in each direction) -> new orbit, and overall Δv -> burn time.

 

Principia does things differently, with numeric integration across burns.