More orbital math?

[ChainsDD]

I'm sure that there's a simple answer to this that I just haven't found yet...

Given the apoapsis and period of a desired orbit, how would you go about calculating that the periapsis should be?

I realize I could just fiddle with it using KER until I got what I wanted, but I'd like to know how to do it on paper...

Edited February 13, 2014 by ChainsDD
Answered

[EtherDragon]

HINT:

The period of an orbit is (essentially*) determined only by it's semi-major axis (SMA). How do you calculate the SMA?

See Kepler's Laws of Planetary Motion, specifically, the third law.

*There are other factors, like the gravitational force of the central mass, etc... that really don't matter, in this case. You can figure out your ratio constant (for the 3rd Law of Planetary Motion) quite simply and use that in your calculation since you are only worried about orbits within the same sphere of influence.

[alexmun]

Period = 2 * pi * sqrt( a^3 / mu )

Apoapsis radius = a * (1 + e)

Periapsis radius = a * (1 - e)

Given the period and gravitational parameter of the central body (mu), you can solve for the semi-major axis (a). Given the semi-major axis and the apoapsis you can solve for the eccentricity (e). Given the semi-major axis and eccentricity the periapsis is trivial to calculate.

[cpottinger]

T = 2 * pi * ( a^3 / mu )^.5

a = ( ra + rp ) / 2

where

T = Period

a = semi-major axis

mu = gravitational parameter

ra = apoapsis measured from the centre of the body

rp = periapsis measured from the centre of the body

Therefore

a = ( ( T / ( 2 * pi ) ^2 * mu )^(1/3)

so

rp = ( 2 * ( ( T / ( 2 * pi ) ^2 * mu )^(1/3) ) - ra

[ChainsDD]

Thanks guys. Exactly what I was looking for.

One day I'll learn all that fancy math...

[MrFlibble1978]

I'm a math idiot but to try and learn more I've been playing with the Orbital calculations in KSP and made some basic calculators.

I made one to work out the periapse from orbital period & apoapse.

Calculator it's built in HTML5.

It'll also tell you the what the circular orbit will be for a given orbital period by clicking the circle between the apoapse & periapse text boxes. Also be sure to click the "total seconds" button before the calculate button.

Hope it's of help to someone.

Ps only works for Kerbin at the moment