Geostationary Transfer Math Help

[MAFman]

How do I calculate the surface longitude where I start my transfer burn to geostationary altitude, given the surface longitude where I want my satellite to end up, my current altitude and orbital period, and my current longitude? I'm trying to write a kOS program that puts a satellite into a precise-ish slot in geostationary orbit.

 

My probe is powered by a single ion engine and has a wet mass of 1.726 tonnes.

Edited April 23, 2020 by MAFman
added a detail

[Raptor42]

I have not done the entire calculation myself but here's something that might help you:

1. Suppose that your vessel is in circular orbit and can do the entire transfer burn at once.
2. Calculate the semi-major axis of your transfer orbit, that is -> [(your current altitude in km + 600* km) + geostaionary orbit altitude in km + 600* km] / 2
*Note: the 600 km is the radius of Kerbin
3. Using Kepler's third law calculate the orbital period of your transfer orbit   (μ is the standard gravitational parameter (for Kerbin it is about 3.531× 10³  km³/s² ), a is the semi-major axis), then divide it by 2 (to get your time from Pe to Ap)
4. This is your time to Ap (in seconds), which means that when you reach Ap, the point you're trying to reach is now (that is, when you do your transfer burn) this amount of time "behind" compared to when it will be when you reach Ap.
     For example if you depart from about 300km orbit going to the geostationary orbit (2863 km-ish) the point that will be directly under you when you reach Ap is the same point you are perfoming your burn above -> this means that the time it takes you to from Pe to Ap is exactly the same as the time it takes Kerbin to rotate 180°
5. SOME NOTES: Kerbin rotates at the rate of 1° per minute, so you can take the time to Ap you calculated, divide it by 60 to get how many degrees will Kerbin rotate. You then want to have your desired point this many degrees behind where the Ap will be.

6. You really don't need to do this calculation when you do this manually, you can just create the maneuver node in-flight to see how much time to Ap it will take and just perform step 5.

I hope it helps you in some way, good luck trying to translate this into kOS program, and also I hope that this explanation makes at least some sense. You really don't need the mass of your probe as it is almost none compared to Kerbin and also the planets are on rails, which means that your ship will NOT pull Kerbin at all.

[MAFman]

I can get the orbital period of the current orbit and the transfer directly from the game. What I need help with is how to calculate what longitude I need to start the burn at.

[Rhomphaia]

Should it not just be 

Desired longitude + 180 + transfer time in minutes

[Raptor42]

15 hours ago, MAFman said:

I can get the orbital period of the current orbit and the transfer directly from the game. What I need help with is how to calculate what longitude I need to start the burn at.

What you need to calculate is the orbital period for the transfer orbit, then you can do the rest.

The calculation for your burn longtitude should be:

Desired long. + 180 + how many deg. will Kerbin rotate when you get to Ap*

for kerbin it is transfer time in minutes

Edited April 25, 2020 by Raptor42