Need some help on orbital periods

[Dr-Drunk]

Hey all!

So yea, I've just finished building up my KSP interstellar powerplant in orbit around Kerbin. Now I need a satellite relay network to go allong with it so I can beam power all over the Kerbin SOI. For this, I'm going to have to have a minimum of 3 satellites in roughly the same orbit so as to asure maximum coverage trough line of sight.

I've been looking online for calculators to find one that could tell me how much height I need for a flat orbital period of for example 4 hours, or 6 hours or whatever. This simply so I can do a quick 2/3 maneuver and end up dropping the satellite at each apoapsis. I've been really struggling in finding such a tool though.

I've found a few that did roughly what I'm looking for but they mostly require you fill in an already known apoapsis height. I need one instead that will tell me my exact needed apo and periapsis to go allong with a certain orbital period. Anyone have any math tips, calculator links or tried and testing techniques for deploying such a network of satellites?

Edited January 7, 2015 by Dr-Drunk

[AlexinTokyo]

The equation for orbital period is:

T = 2À × sqrt(a³/μ)

where T is the period in seconds, a is the semi-major axis (km), and μ is the standard gravitational parameter (km³/s²).

You can rearrange this and solve for a to get the SMA for any given period. μ is available for all bodies in KSP on the KSP wiki.

For a circular orbit, the SMA is equal to the radius of the orbit, measured from the center of the planet. Altitudes in KSP are given from the surface of the planet, so remember to subtract the planet's radius from the SMA to get your apsides.

[Superfluous J]

Do this:

Get into a roughly circular orbit. Note the time to periapsis and the time to apoapsis. Subtract the smaller from the larger and multiply that number by 2. That'll give you your orbital period.

If you're too high, burn retrograde a bit. If you're too low, burn prograde a bit. Stop and keep checking those two times. Eventually, their difference will be what you want (2 hours, if you want a 4 hour orbit).

Now, this orbit will be pretty elliptical and you want a circular one. So, add the height values for the apoapsis and periapsis and divide that number by 2 to get the average height. THAT is the height you want your circular, 4 hour orbit.

Aaaaaand my math-lite, time-heavy approach has been Ninja'd by a math-heavy, time-lite approach

Edited January 6, 2015 by 5thHorseman

[Dr-Drunk]

Both interresting aproaches to solve my problem here so kudos to you guys!

I'm not much of a math wizard but I think I get the jest of it so I'm going to be giving both ways a shot here, given how I have a few of these networks to set up so plenty of chances to try different ways.

[cantab]

Yet another option is to use KER and configure it to show orbital period.

[AlexinTokyo]

Now I'm at a real computer, I'll add some actual numbers to the theory above.

First, the rearranged equation for a in terms of T:

a = (μ.(T/2À)²)^{â…“}

For Kerbin:

μ = 3.5316000 × 10¹² m³/s² = 3,531.6 km³/s²

R = 600 km (Radius of Kerbin)

P = 21,600 s (Rotational Period of Kerbin)

For a synchronous orbit, T = P, so:

a = 3468.75 km

For a stationary (i.e. circular, equatorial, synchronous) orbit, apoapsis and periapsis should be equal, so:

A_apo = A_peri = a - R = 2868.75 km

So if you want your constellation of satellites to end up in stationary orbit, you'll need to launch them to an apoapsis of 2868.75 km.

The periapsis will depend on the number of satellites you are launching.

For 3 equally spaced satellites you need a 4-hour period (3:2) for 4 you need 4.5 hours (4:3)

For your 4-hour orbit:

T = 14,400 s

a = 2647.15 km

A_peri = A_apo - 2(a-R) = -1225.55 km (negative because it's on the other side of Kerbin - it will be a positive reading in game)

The periapsis altitude for a 4.5-hour orbit is left as an exercise for the reader