Could two satellites in the same orbit, but phased differently, be tethered in a sustainable manner?

Was reading this, and I immediately saw the problem with the angular momenta not lining up. Unfortunately, my mind likes to keep turning scenarios over and asking what would happen if x? So my question is best described by the following diagram: Assumptions would be:

• These two satellites are in the exact same orbit, and are tethered together.
• They are phased by t seconds in this orbit, and weigh the exact same amount (w kg).
• The drag on these two satellites is non-existent (Earth's atmosphere left).
• The tether is of an unbreakable material (does it need to be?).
• The satellites will not be so far apart as to drag the rope across your roof.
• I shouldn't have to say the satellites/tether not to scale, but I will.
• Edit: In accordance with kbelder's comment, the tether may or may not be a straight line.

Would this system be sustainable, or would the tether being closer to Earth create a pull on the central part of the tether after x separation between the two? What would the separation be before this system becomes unstable? I currently am under the assumption that if the tether was massless, there would be no problems; you can assume any rope material if you're doing calculations.

KBelder pointed out that the tether would not look like that in certain scenarios, I suppose the scenario I want to consider is as I've mentioned in the comments:

I was assuming the two objects would be connected by connecting one end, then performing a phasing maneuver (allowing slack during the transfer) to get to the other one. This would mean that it would not be following the same orbital path (though I do not know what the path it would follow is).

I've been playing what-if in my head. If all of this is stupid, that's a completely valid answer.

• I think your diagram is wrong, in that I don't think the tether would be straight. Assuming everything has a constant velocity, I think it would follow the orbital curve. It's moving at orbital velocity. – kbelder May 13 at 21:11
• @kbelder that is most likely correct, depending on how they were tethered. I'm starting to think the key point here may be that it's impossible to tether two objects like this easily. I was assuming the two objects would be connected by connecting one end, then performing a phasing maneuver (allowing slack during the transfer) to get to the other one. This would mean that it would not be following the same orbital path (though I do not know what the path it would follow is). – Magic Octopus Urn May 13 at 21:15
• @Kbelder if the orbital transfer method would result in a tether in the orbital plane then I'm just making some dumb assumptions in my head that weren't ever true (and that wouldn't make a bad answer). – Magic Octopus Urn May 13 at 21:20
• I think it depends a lot on the specifics of the tether - for example, if it's massless and slack, seems like it wouldn't exert any force on the satellites. – Organic Marble May 13 at 22:32
• I've removed the hypothetical tag per this answer. – uhoh Sep 28 at 10:26

Nice question; it's gedankenexperiment time!

Let's assume the Earth's gravitational field is spherical (no oblateness or other lumps) and stick with the OP's draglessness. Let's also assume the cable has some finite mass.

If we place just the two satellites in the same circular orbit but phased differently, then they would continue in that orbit spaced by the same amount, indefinitely.

If we then started laying out some little 1 cm long segments of cable, unconnected, also along this orbit in between the satellites, they would also stay "in place" relative to the two satellites in their own orbit. We could keep doing this until they were connected, and nothing would change.

So if the two satellites were connected by a cable and the cable perfectly followed the path of the circular orbit, then you would have a stationary state. It would stay that way.

However that doesn't mean that the stationary state would be stable.

For example, the five Lagrange points in the circular restricted three body problem (CR3BP) are all stationary states. Some are conditionally stable, some are unstable to small perturbations.

What would happen if this system were slightly perturbed?

I don't know! It's a dynamical system with large degrees of freedom since as soon as there is some slack the cable can potentially assume an infinite number of different possible shapes.

• Displacements transverse to the orbital plane may simply oscillate with the period of the orbit, as if parts of the cable were simply at different inclinations.

• Displacements in the plane of the orbit are more problematic as they would tend to "orbit" faster or slower than the two satellites.

A definitive answer about this situation probably needs to be addressed numerically. It is certainly possible and probable that this has been done already. It may take some further searching to find it, stay tuned... Others are encouraged to find it first!

• Can I say “Es ist gedankenexperimentzeit!”? – uhoh May 14 at 4:07
• "Es ist Gedankenexperimentzeit!" is valid but so complex of a word that it is unlikely to be used. Nevertheless it is completely usable and also interesting as an eye catcher/rhetorical instrument – GittingGud May 14 at 11:14
• @GittingGud please feel free to leave another answer there if it seems appropriate. – uhoh May 14 at 12:02
• If you replace the tether with a rod, then in the static state there is neither tension nor compression on it. The rod would insure the distance / phase remains always the same. With the tether, it would eventually go slack and the two move closer. Then the motion is complex and the tether serves only to limit the maximum distance. – bitchaser May 16 at 16:49