The Best Relay Orbit | KSP Let's Do The Math

amazing i love it

May I point out the comment on line 2 🙂 I've been using it since your com network video, thank you.

https://docs.google.com/spreadsheets/d/13qE-oKEdiMUFLx4UuB5cgZ3GOWbQW8Wp6XRfdWlUMjk/edit?usp=sharing

This is the best KSP videos I've ever found. Thank You Mike so much for making these videos

I needed that thanks ! Very pedagogic explanation.

Hi Mike, first I love your videos! they're amazing. Second, I have a question. In my career game I'm trying to put a Relay Comm on Kerbin (because I only have connection with KSP), so I follow your videos and put on each relay sat two HG-5 and the orbit period are 12h (so the orbit altitude is 4.906.298m) but I have one problem, the relay satellites don't communicate with each other. Trying to know the reason, I found the distance between each other is 8.497km (please correct me if I'm wrong, I found it because in a right triangle the short side (orbit altitude) is 4906km and the ang is 60° so tang 60 x 4906 = 8497km x 2 -> 16994km). So, what i'm doing wrong? How can I know if the power of the antennas are enough to communicate between each other?.
Thank you for this class of videos, they are amazing.

There's another method I just discovered for getting equidistant relays around ANY body, and its actually simplifies things a lot. (TLDR at the bottom)

So I was number crunching seeing if I could come up with a simplified formula that didn't require you to target specific altitudes or orbital periods. With the idea that I don't have to memorize/lookup a handful of data and tables to make the right relay orbit, something that would be easy to remember…which would make things a lot more flexible. I started with the formula that finds a semi-major axis, "x" with the given Orbital period: (apologies for the excessive parenthesis upcoming in the formulas, writing formulas in Youtube is not the best)
x = (GM * (T^2)/4pi^2)^1/3, where x is the semi-major axis of the phasing orbit

I then substituted T with the Orbital Period of the phasing orbit which is the 2/3rds the duration of the larger relay orbit:
T = 2/3 * (2pi/GM * y^3)^(1/2), where "y" is the semi-major axis of the larger, circular orbit
giving the formula:
x = (GM * (2/3 * 2pi*(y^3/GM)^(1/2))^2 / 4pi^2) ^(1/3)

Most of the variables like the gravitational standard parameter mu and pi and the exponents cancel out. heres how it looks simplified:
x = (2/3)^(2/3) * y
Moving the leftover (2/3)^(2/3) into variable "o" (which I initially called the phasing orbit modifier, this simplifies things later).
o = ((n-1)/n)^(2/3), where "n" is the number of relays you want in the orbit

I then substitute "x" with the semi-major axis formula (P+A)/2 , and y as A (since its both the phasing orbit's apoapsis and is also a circular orbit). then simplify further
(P+A)/2 = 2*o * A – A =>
P = A* (2o – 1), where "P" is the phasing orbit's periapsis, and"A" is apoapsis of the phasing orbit

Great so if we know the apoapsis of the relay orbit we can also easily find the target periapsis needed for the phasing orbit using this phasing orbit modifier.

BUT WAIT THERE'S MORE!!
While crunching for a "simplified formula" I had a hunch that eccentricity may be the thing I needed.
I didn't recall seeing the formula for Eccentricity ever posted in the video series so I had to look it up. The formula is as follows:
E = 1-2/(A/P+1), where A is the orbits apoapsis and P periapsis. (this formula doesn't apply to escaping orbits, i.e. parabolic/hyperbolic)

I had these two numbers up next to each other and quickly noticed that this also holds true:
E = o-1
The phasing orbit mod is directly related to the phasing orbit's eccentricity!

This means that since KSP shows you the eccentricity, you don't need to plan for an explicit period or attitude(within reason, apoapsis still needs to be within the min/max bounds as mentioned in the video), you just burn for an explicit eccentricity and you'll get the perfect phasing orbit, which works on ALL CELESTIAL BODIES.

Its also possible to burn from periapsis, but only if the apoapsis is in the correct range. If the periapsis is very low the apoapsis can end up under the minimum bounds and thus get occluded from other relays by the body it's orbiting. Also like normal, after the relays have been deployed and circularized you still may need to do minor correction burns so that the sibling relays sync their orbital periods.

TLDR;
Keep your apoapsis anywhere above 2x the body's radius but below the SOI (or for me, my upper bound is also the relays range / 25 for strong ground signal). Then burn for an eccentricity of 0.310371 (for 3 equidistant relays). If your eccentricity is less than this, burn retrograde. if its greater, burn prograde.

Math went over my head :/

That's precisely the video I needed to watch. Thank you!

Me at 0:10 – seems alright.
Me at 1:10 – wft.
Great videos @mikeaben keep it up! Do you have a background in Maths to be able to do all these calculations? I am the reason for the extra 20 views in the last 3hrs lol. Now to fight with excel to get the equations onto my KSP helper spreadsheet

This is a great tool i found to easely plan your network if your too lazy to do math xD
https://meyerweb.com/eric/ksp/resonant-orbits/

thought the thumbnail was the orbits trying to form a pentagram

So, the antenna rating (500k, 5M, 2G, etc.) is just the max range?

Any special reason why we want a signal strength of 80%?

This is the first time math has been fun.Thank you so much for the well paced explanation! I wish you were my math teacher back then 😉

I wanted funni space game.
Instead I get math

Thanks for the video, it helped a lot

Someone may have asked this already, but why did you chose 80% as your minimum effective strength? Is that the minimum signal strength it takes to get your +40% bonus when transmitting science?

Great video! Even a few years later, this video still rocks and is still useful. I was wondering if there is any benefit to using more than three satellites in a relay network. Am I correct in thinking that using more satellites allows you to have a lower minimum orbit to maintain connectivity? If I'm thinking about the geometry correctly, using 6 sats instead of 3 would lower your minimum altitude by half.

That would give you a bigger range in which you can pick your relay orbit. So instead of adding antenna (or upgrading to more powerful antenna) to increase the min-max orbital range, you could add satellites.

However, it's probably cheaper (certainly easier) to just add more antenna (or upgrade antenna) than it is to add more satellites. And as you say in the video, this only really matters for the Mun.

Mmmmmmmmm
Maph