Math question for sat network

[eberkain]

I used to use the Visual Remote Tech Planner which is a great source of info.  I tried looking through the source of that to find my answer but didn't get anywhere with it. 

i want to be able to deploy a synchronized sattelite network of either 3 or 4 relays depending on the body.  I've got the deployment mastered, but I need a way to calculate the proper height for using a resealed system.  Take these for example.  

If I know the length of r1, what is the formula to solve for r2 in each situation?  I have tried google searching also and not had any luck either.  

[Jas0n]

Some quick literal back of the envelope math tells me that it r2=2*r1. 

Because if you make r1 the radius that goes to the point where the triangle and inner circle intersect and r2 the radius from center to vertex of the triangle, you can form a 30-60-90 triangle. 

[Kryxal]

More generally, for a constellation of n satellites:

r1/r2 = cos (360/2n) -> r2 = r1/cos(360/2n)

For 3 satellites, that's cos 60 = 1/2, and for 4 it's cos 45 = .7071... (1/sqrt(2))

Incidentally, the second diagram makes this easy to see, given the alignment of r1 and r2.  The angle you measure is for the right triangle from the center to one of the satellites, to a point tangent to the body's surface, and back to the center.  Also, you might want to add a bit extra to r2 for practical use, allows for drift.

Edited February 5, 2018 by Kryxal

[Spricigo]

What @Kryxal said.

BTW you could find it searching for circumscribed circle and inscribed circle.

[eberkain]

34 minutes ago, Kryxal said:

More generally, for a constellation of n satellites:

r1/r2 = cos (360/2n) -> r2 = r1/cos(360/2n)

For 3 satellites, that's cos 60 = 1/2, and for 4 it's cos 45 = .7071... (1/sqrt(2))

Incidentally, the second diagram makes this easy to see, given the alignment of r1 and r2.  The angle you measure is for the right triangle from the center to one of the satellites, to a point tangent to the body's surface, and back to the center.  Also, you might want to add a bit extra to r2 for practical use, allows for drift.

Thanks, I was able to get the formulas in my spreadsheet working with this. 

[Human Person]

You can do the calculation based on your orbital period, using this:

 

Orbital Period = some constant * semi major axis ^ (3/2)

 

don‘t worry about some constant, it will cut out in your calculation.

 

the semi major axis is the sum of your apoapsis, periapsis and kerbins diameter.

 

The orbital period of your deployment orbit has to be 2/3 the orbital period of your desired orbit (for 3 satelites), 3/4 for 4 satelites, 4/5 for 5 satelites and so on.

 

 

Edited February 5, 2018 by Human Person