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Calculate periapsis velocity based on known values on apoapsis, perapsis and apoapsis velocity *without* the standard gravitational parameter


Instresu
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I'd like kOS to calculate the velocity at periapsis for me with the apoapsis height, periapsis height and apoapsis velocity as variables. However, if one is using the specific orbital energy v2/2 - µ/r = constant, you must know the standard gravitational parameter. I could hardcode the values into my script for every celestial body, but I want it to be as general as possible (if you decide to alter the default masses with mods etc). How do I get rid of the dependency of µ in my formula?

Edited by Instresu
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If you know other parameters of your orbit, you could calculate using other orbital equations. For example knowing your orbital period and SMA using Kepler's 3rd law, or the vis-viva equation using your current velocity, position and SMA...

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Angular momentum is conserved.

mass * velocity * radius

If you know the apoapsis velocity and the apoapsis, as well as the periapsis, you can put them into an equation. Mass cancels out( if you haven't lost any) so we get:

apoapsis velocity * apoapsis radius == periapsis velocity * periapsis radius

Solve for periapsis velocity 

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