How do stable equilibrium points work in GEO? If all geosynchronous spacecraft suddenly lost stationkeeping, would most "fall into" one or the other?

Question

The (currently unanswered) question Quantitatively, how deep are the stable equilibrium points in GEO? How much delta-v to move from one to the other? (also see comments at Delta-v to move from GEO to GEO) is premised on their existence, and I have the impression that unstationkept satellites starting from geostationary or at least geosynchronous orbit would eventually "collect near" or tend to migrate towards certain longitudes. I suppose you can think of that as a kind of phasing; the period may not change but the phase would.

But now I'm not sure if that's what would happen, and if it is, why it would. Without stationkeeping the spacecraft would certainly start to move from it's original longitude in earth-fixed coordinates due to Earth's lumpy gravity and the Sun and Moon and other things, but would it have to exchange angular momentum with the Earth to move to its new orbit and settle into its new phase but keep its original period, since it might be now at a different gravitational potential and semimajor axis.

As an extension of this, if all the satellites currently stationkept in GEO were to suddenly loose propulsion, after a decade or so (baring collisions) if we looked at a map of all their ground tracks, would we see two groups of Analemmae clustered near the two equilibrium longitudes? I'm thinking of a map like that below but with hundreds of them now tall and clustered in two groups.

Question: How do the stable equilibrium points work in GEO? If all geosynchronous spacecraft suddenly lost stationkeeping, would most "fall into" one or the other?

Possibly helpful: NASA TN D-3557 The Earth's Longitude Gravity Field as Sensed by the Drift of Three Synchronous Satellites

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"Family portrait" from this answer to Are there any satellites in geosynchronous but not geostationary orbits?

Answer 1

There is a very nice article about the synchronized variation of the debris motion at GEO, available here.

Authors give an example of the 'well-known GEO stable plane, which is a fixed point of the doubly-averaged differential equations (governing inclination and right ascension of the ascending node). Objects iniitialized at this equilibrium configuration with i=7.4 and RAAN=0 exhibit dramatically-reduced inclination andRAAN variations over the 53-year cycle'. See attached figure:

This article is also referenced in the Wikipedia page about the Geostationary orbit: link (yeah, I know, wiki is not the best source, sorry). There, it is stated that 'there are two stable equilibrium points (at 75.3°E and 252°E) and two corresponding unstable points (at 165.3°E and 14.7°W). Any geostationary object placed between the equilibrium points would (without any action) be slowly accelerated towards the stable equilibrium position, causing a periodic longitude variation', which kinda answers your question. So, according to this statement, your assumptions about two clusters of Analemmae, which would form in a veeery long time, sounds legit.

However, after quick look at the referenced article, I wasn't able to find the same conclusion there (or at least formulated in the same way). Thus, I would take the wiki statement with a pinch of salt and read the referenced article carefully.

Answer 2

Only have time for a quick rundown without numbers.

Short explanation of equlibrium points

Even at the GEO layer, the earth has gravitational differences and it not completely round. Land masses have a greater graviational pull than oceans.

This means that, except for the stable equilibrium points (and unstable, on paper), a satellite in GEO will experience a gravitational pull from these land masses.

Imagine a 2D model: High gravity points (stable equilibrium) as valleys and low gravity points (unstable equilibrium) as mountains. Put a marble somewhere on a slope and it will roll down towards the valley, reach the opposite point of it and roll back - repeating this infinitely if there is no friction.

A satellite that is not on a point of equilibrium will do roughly the same.

Satellites between equilibrium points

Usually the satellites will use small bursts from their engines to stay in their window (or constant burns from ion engines). Seen from "above" (north pole), all satellites in GEO move counter-clockwise (prograde).

But if the satellite wouldn't counteract the gravitational pull, the satellite would accelerate along his orbit in the direction of the pull. This would mean that the orbit's distance from earth will increase. It would pass behind stable satellites and keep increasing height until reaching the equilibrium point, where it will start being pulled back by the mountain range and continue until it reaches the opposite point of the starting point, where its speed will be equal to the one it bad before. Then it will move back towards the equilibrium point (orbit height decreasing) until it passes it, returning to the starting point.

If the equilibrium is on the other (retrograde) side, it would do the same thing but the other way around: Slow down, decrease orbit height, pass in front of other satellites on a lower orbit and so on.

From the point of view of a stable equilibrium point, an uncontrolled satellite would always move counter-clockwise around it (if earth's north pole is considered "up", that is).

Answering the question

So if all satellites lost stationkeeping, they would all do this. Chaos would ensue, and while they wouldn't immediately crash into each other it would only be a matter of time.