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IIRC, above something like geostationary they tend to decay upwards? Though the old orbital ring white paper wasn't suggesting anything like that, this was an alternative to needing to go so high in the first place.

(I may be misremembering or getting confused with a thing specific to tidal locking?)

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That is indeed what you get with tidal forces - bodies closer than geostationary orbit lose angular momentum and decay inward, bodies further out steal angular momentum from Earth and move outward.

I suppose the same effect is there with satellites much smaller than the moon, but it would be tiny.

I wonder if they could have an orbit high enough to move away from earth with some kind of drag cables dangling from them into low orbit to counter the outward movement. Would that work?

You mean something like a skyhook?

Yes but no hook or anchor needed I'd assume. The cable alone in lower orbit would weigh enough to be pulled by gravity.

These orbits have vastly different speeds though. Consider a high geosynchronous orbit vs. something like ISS which goes around the Earth in an hour or two.

but you are towing the cable and you'd only get so far down for the counteraction of force from gravity to pull on the cable. Would speed or friction on a cable be a problem.

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