Math question: How to calculate hexagon, and octagon formation in orbit

[Sokar408]

How do you calculate even positions of say 6 or 8 satellites in a circular orbit?

[Greenspan]

360 degrees in a circle / 8 = 45 degrees separation

360 degrees in a circle / 6 = 60 degrees separation

[Vanamonde]

Quick question quickly moved to gameplay questions.

[Starstrider42]

If you mean how do you get the separations between adjacent ships (which IMO is the easiest way to set up the formation, just use the built-in rendezvous planner), the distance is 2r sin (θ/2), where r is the orbital radius (including Kerbin's radius) and θ is the angle Greenspan mentioned.

[Astronut25]

the rotation of orbital planet divided by the number in formation, that is the time between launches.

For example Kerbin has a 6 hour day, so you launch your rockets 1 hour apart (for 6 formation). If you can't launch them that fast, then you take your current time in hours MODULO 6 and track which ones you have launched. Assuming you're on a 0 inclination. I have a more advanced procedure for non-equatorial ones if you need them.

For more advanced formations with some satellites offset, just place their positions on a circle, then place a 6 hour clock (for Kerbin only) over it, then that is your launch times MODULO 6. Again assuming 0 inclination, and the aforementioned formula would still work if needed.

between .17 and .21 I have set up many GPS constellations around Kerbin, Mun, and Jool, with extreme accuracy.

Edited March 14, 2014 by Astronut25

[ScottyDoesKnow]

An easy way is to place them all in one launch. First thing you need to know is the period of the desired final orbit. To make it easy, let's say you want to place 8 satellites and the final orbit period is 4 hours. You'll launch, get your apoapsis to the desired orbit, then change your periapsis until your orbital period is 7/8 of the final (3 1/2 hours in this case). Each time you hit apoapsis, release a satellite and have it circularize itself (give it a bit of fuel and a weak engine). Something like kerbal engineer or mechjeb is very useful for this, as is a very weak engine or RCS to get it very exact. Each orbit will stagger the next satellite by your fraction (30 minutes in this case) and after 8 orbits you'll have 8 satellites each 1/8 of an orbit apart.

However exact your orbital periods are will determine how long your satellites stay evenly distributed. How circular your orbit is doesn't matter here (though they'll already be close), just the orbital period.

[WaveFunctionP]

An easy way is to place them all in one launch. First thing you need to know is the period of the desired final orbit. To make it easy, let's say you want to place 8 satellites and the final orbit period is 4 hours. You'll launch, get your apoapsis to the desired orbit, then change your periapsis until your orbital period is 7/8 of the final (3 1/2 hours in this case). Each time you hit apoapsis, release a satellite and have it circularize itself (give it a bit of fuel and a weak engine). Something like kerbal engineer or mechjeb is very useful for this, as is a very weak engine or RCS to get it very exact. Each orbit will stagger the next satellite by your fraction (30 minutes in this case) and after 8 orbits you'll have 8 satellites each 1/8 of an orbit apart.

However exact your orbital periods are will determine how long your satellites stay evenly distributed. How circular your orbit is doesn't matter here (though they'll already be close), just the orbital period.

Then hyperedit their orbits to simulate the satellites setting up orbital corrections. And never, ever, fly them again.

[Starstrider42]

For example Kerbin has a 6 hour day, so you launch your rockets 1 hour apart (for 6 formation). If you can't launch them that fast, then you take your current time in hours MODULO 6 and track which ones you have launched. Assuming you're on a 0 inclination. I have a more advanced procedure for non-equatorial ones if you need them.

Umm, doesn't that require that every launch follow the exact same trajectory (exact same gravity turn, exact same timing of orbital burns, etc.)? Or am I missing something?

[Taki117]

Umm, doesn't that require that every launch follow the exact same trajectory (exact same gravity turn, exact same timing of orbital burns, etc.)? Or am I missing something?

Technically yes, but if you are putting sats in KSO, or you know the orbital period of your first Sat, then it doesn't matter as much. All you have to do at that point is match orbital periods. They will naturally form an analemma relative to each other, but they won't drift out of orbit too far.

[Astronut25]

Umm, doesn't that require that every launch follow the exact same trajectory (exact same gravity turn, exact same timing of orbital burns, etc.)? Or am I missing something?

Right, the more exact you are, the closer they will be to target, but I've done it without Mechjeb without too much trouble. Something that can show orbital and/or target information would be very useful if you want a lot of precision.

and it looks like Scotty has described my other technique in setting up constellations, but I'll see if I can upload my math file.

Its not looking very good for the upload, so what kind of orbital parameters are you planning (altitude, period etc)

assuming final orbit is circular.

1a) If you got the altitude you want then run the following formula:

R = body radius

r = desired altitude apsis

P = √( [(4À^2)*([R+r]^3)] /μ)

1b) if you want a specific period then:

p = desired orbital period

r = third√( [p^2 μ] / [4À^2] )-R

2) now you got what you need next you want to divide it by the number of satellites in the orbit then run the next formula:

r2 = other apsis

p = P / number of sats in orbit

r2 = 2*[third√( p^2 μ / 4À^2)] - (r+R)

now you should have the needed Apsides. Now you'll have to launch all of the satellites on a single rocket and deploy them as Scotty described.

in case something doesn't work with the fonts

√ = root

μ = mu (in this case is gravitational constant times mass of planet)

À = pi

Everything I know about Orbital Mechanics I learned from:

http://www.braeunig.us/space/orbmech.htm

Edited March 14, 2014 by Astronut25
formula clarification