Calculating Aerobraking/Captures with Math

[LexiSilva]

I am doing my best to learn the math of orbital mechanics, to help me do better in KSP (not having bad rockets when playing stock).

One thing I need to learn to do is calculate the delta-V required to capture oneself when coming in a hyperbolic orbit.

No, I do not want to use delta-V maps.

How would I be able to calculate the needed delta-V?

Related question: How would I calculate how much velocity an aerobraking/aerocapture maneuver would bleed off?

[K^2]

The only reliable ways I've seen involve integrating forces over trajectory. There might be some tools for doing this for KSP on the net already. There is no simple formula you can just plug numbers into.

Edit: Here, I'm talking about computing the actual aerobraking maneuver only. For computing how much delta-V you need to shed, see PakledHostage's answer bellow. There are simple formulae for periapsis/apoapsis velocities of hyperbolic and elliptic orbits.

Edited February 25, 2014 by K^2

[PakledHostage]

One thing I need to learn to do is calculate the delta-V required to capture oneself when coming in a hyperbolic orbit.

No, I do not want to use delta-V maps.

How would I be able to calculate the needed delta-V?

The simplest method is to patch together two orbits at their periapsis or apoapsis points. You can then estimate the delta-V required to transfer between them by calculateing the difference in speed between the two orbits at the point where they intersect.

For example, if your hyperbolic trajectory has an orbital speed at periapsis of 3500 m/s and your elliptical orbit has an orbital speed of 2250 m/s at a periapsis that matches the hyperbolic trajectory's periapsis, then you will need about 1250 m/s to enter the elliptical orbit from the hyperbolic trajectory.

In reality you will need a bit more delta-V than what you calculate with this method because you aren't executing your manoeuver instantaneously, but the estimate will be close.