Orbital Parameters for a Tetrahedral Satellite Constellation

Zyx Abacab · January 18, 2017

Introduced in 1.2, the CommNet feature made communications-satellite constellations relevant in the base game. A relatively easy way to provide pretty good communications coverage is a three-satellite constellation in an equatorial orbit. However, such a constellation does not provide full coverage over the entire surface of a celestial body.

But a four-satellite constellation, if placed in a tetrahedral configuration, does!

With that in mind: I've compiled a list of orbital parameters for this kind of tetrahedral constellation for every celestial body in the Kerbol system.

Legend

INC.
- Inclination
ECC.
- Eccentricity
SMA.
- Semi-Major Axis
LAN.
- Longitude of the Ascending Node
LPE.
- Longitude of Periapsis (a.k.a. Argument of Periapsis)
MNA.
- Mean Anomaly at Epoch
EPH.
- Epoch

Moho

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	1812500m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	1812500m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	1812500m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	1812500m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

Eve

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	5075000m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	5075000m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	5075000m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	5075000m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

Gilly

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	94250m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	94250m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	94250m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	94250m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

Kerbin

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	4350000m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	4350000m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	4350000m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	4350000m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

Mun

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	1450000m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	1450000m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	1450000m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	1450000m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

Minmus

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	435000m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	435000m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	435000m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	435000m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

Duna

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	2080000m^*
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	2080000m^*
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	2080000m^*
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	2080000m^*
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

* Note: The correct semi-major axis of 2320000 metres causes satellites to eventually be pulled away by Ike's gravity. The listed SMA corresponds to the highest stable orbit I could find.

Ike

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	942500m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	942500m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	942500m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	942500m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

Dres

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	1000500m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	1000500m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	1000500m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	1000500m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

Jool

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	43500000m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	43500000m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	43500000m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	43500000m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

Laythe

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	2909000m^*
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	2909000m^*
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	2909000m^*
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	2909000m^*
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

* Note: As pointed out by boccelounge: given these orbital parameters, the correct SMA of 3625000 metres is outside of Laythe's sphere of influence. The listed SMA is the closest possible value that is still within Laythe's sphere of influence.

Vall

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	2175000m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	2175000m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	2175000m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	2175000m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

Tylo

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	4350000m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	4350000m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	4350000m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	4350000m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

Bop

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	471250m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	471250m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	471250m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	471250m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

Pol

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	319000m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	319000m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	319000m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	319000m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

Eeloo

Satellite 1	INC.	33°
	ECC.	0.28
	SMA.	1522500m
	LAN.	0°
	LPE.	270°
	MNA.	0
	EPH.	0
Satellite 2	INC.	33°
	ECC.	0.28
	SMA.	1522500m
	LAN.	90°
	LPE.	90°
	MNA.	-1.57078
	EPH.	0
Satellite 3	INC.	33°
	ECC.	0.28
	SMA.	1522500m
	LAN.	180°
	LPE.	270°
	MNA.	3.14159
	EPH.	0
Satellite 4	INC.	33°
	ECC.	0.28
	SMA.	1522500m
	LAN.	270°
	LPE.	90°
	MNA.	1.57078
	EPH.	0

The information in this post is based on John Draim's paper, Three- and Four-Satellite Continuous-Coverage Constellations, and an excellent thread by maltesh. The maths are all his; I only did the number-crunching and table-setting.

Edited March 21, 2021 by Zyx Abacab
Updated wording; removed broken image.

Sharpy · January 18, 2017

As an extra advice to keep the constellation stable: once you get the satellites into desired orbits, perform tiny prograde/retrograde burns that set their orbital periods to identical values to under a second of difference and the constellation will remain stable for years to come. KER can provide orbital period, you need to add the parameter to its GUI though, it's not visible by default. (it's good to set RCS thrust limiter to 1%, press caps lock for fine controls and perform the final adjustment with H/N keys).

Edited January 19, 2017 by Sharpy

tseitsei89 · January 18, 2017

Yeah nice work. BUT

If you just put 3 evenly spaced satellites at equator and 3 evenly spaced at polar orbit you get full coverage. Yes you use 2 more satellites BUT all of them can be very conveniently place to their orbits with only 2 launches (1 equatorial launch and 1 polar orbit launch).

In this tetrahedral case the orbits are so inclined compared to each other that you need more launches or more dv in orbit to constantly change your inclination and place the satellites --> more money (and time/effort) spent.

But that said I really like how this constellation actually works It is beautiful

Zhetaan · January 18, 2017

Indeed, you're looking at the prospect of four launches, at least for Kerbin. On the other hand, the inclinations and apsides are the same, so they're four identical launches, just at different times of day to account for the different longitudes of ascending node. That means that you'll have to phase three of the orbits to get the satellites into the correct relative position, but that's no different from having to phase the orbits of a single-launch trigonal-planar constellation.

Perhaps this should be in Tutorials?

Sharpy · January 18, 2017

1 hour ago, tseitsei89 said:

Yeah nice work. BUT

If you just put 3 evenly spaced satellites at equator and 3 evenly spaced at polar orbit you get full coverage. Yes you use 2 more satellites BUT all of them can be very conveniently place to their orbits with only 2 launches (1 equatorial launch and 1 polar orbit launch).

In this tetrahedral case the orbits are so inclined compared to each other that you need more launches or more dv in orbit to constantly change your inclination and place the satellites --> more money (and time/effort) spent.

Only for Kerbin network. I wish this was available when I was making my network for Gilly. Construction of a craft is vastly more costly than deployment in such case.

For "arrival from the outside" getting the right orbital parameters is cheap in terms of delta-V but tricky in execution. But I believe I know what my Jool network will look like

Zyx Abacab · January 18, 2017

7 hours ago, tseitsei89 said:

If you just put 3 evenly spaced satellites at equator and 3 evenly spaced at polar orbit you get full coverage.

Sure, that's certainly true: the technique isn't nearly as straightforward as what you describe...but this way works too!

Previously, there was only one thread about it—and that was a scenario thread! I really felt that this information, expanded for the Kerbol system, needed to be put down in writing somewhere.

5 hours ago, Sharpy said:

I believe I know what my Jool network will look like

I'm glad I could help.

Edited January 18, 2017 by Zyx Abacab

Spricigo · January 18, 2017

22 minutes ago, Zyx Abacab said:

Previously, there was only one thread about it—and that was a scenario thread! I really felt that this information, expanded for the Kerbol system, needed to be put down in writing somewhere.

Personally I think somewhere should be the tutorial section . My suggestion its to ask a Moderator to move it.

Also, good job.

ajburges · January 19, 2017

I use the same constellation for my career save. Injector releases from high, equitorial, circular orbit. Sats were placed in three burns each: inclination, phase correction, final orbit. I used a Google sheet to calculate parameters (I wanted the tetrahedron orbit to have even resonance with a polar sacansat for science so I used various SMA to radius ratios).

Wonder if I should share it.

diomedea · January 19, 2017

As suggested above, this thread is best for the tutorial section. Moved.

boccelounge · December 5, 2017

I hope no one objects to this reply to an older thread-- if so, apologies.

But I've been fascinated by this idea since it first came to my attention, and have finally gotten around to implementing it. Mun and Duna are looking good with their new constellations.

My problem is Laythe-- the SMA listed above puts my relays in an escape trajectory. Thankfully, I was able to salvage the mission and set the sats around Jool instead (which may actually be better, as they now cover Laythe as well). But I wonder if anyone can help me resolve this issue?

And thanks much to the OP for putting this info out there.

ajburges · December 5, 2017

24 minutes ago, boccelounge said:

I hope no one objects to this reply to an older thread-- if so, apologies.

But I've been fascinated by this idea since it first came to my attention, and have finally gotten around to implementing it. Mun and Duna are looking good with their new constellations.

My problem is Laythe-- the SMA listed above puts my relays in an escape trajectory. Thankfully, I was able to salvage the mission and set the sats around Jool instead (which may actually be better, as they now cover Laythe as well). But I wonder if anyone can help me resolve this issue?

And thanks much to the OP for putting this info out there.

That is an issue. A Draim constellation needs a SMA axis of at least 7.25 times the objects radius or the geometry fails (they can loose LoS to each other and are too close at Pe). There is no Laythe orbit that will satisfy both this constraint and the eccentricity constraint that can be contained in the sphere of influence.

boccelounge · December 5, 2017

3 hours ago, ajburges said:

That is an issue. A Draim constellation needs a SMA axis of at least 7.25 times the objects radius or the geometry fails (they can loose LoS to each other and are too close at Pe). There is no Laythe orbit that will satisfy both this constraint and the eccentricity constraint that can be contained in the sphere of influence.

That was my assumption-- I certainly couldn't find an altitude that worked. Just wanted to see if anyone could confirm that-- thank you.

But, as I mentioned, I actually think the Jool constellation is optimal, as it covers Laythe quite nicely...

If anyone's interested, I'm at this moment burning for Tylo; I'm very curious to see if I can set things up there. Should know later this evening.

EDIT: Follow-up: Yep, this formation works just fine at Tylo (and is very fun to set up).

Edited December 6, 2017 by boccelounge
Follow-up

Orbital Parameters for a Tetrahedral Satellite Constellation

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