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This answer discusses a spacecraft in GEO using a two-impulse (circular → elliptical → circular) raise & lower maneuver to move (phase) itself by 180 degrees in longitude then return to synchrony. (synchronicity?) over a period of weeks or months.

Since the Geopotential is lumpy and the tesseral part is dominated by the quadrupole J₂₂ term, there are two maxima and two minima in the gravitational potential along a circular path around the Earth's equator.

This results in two stable equilibrium points (valleys) and two unstable equilibrium points (hills) that would tend to "attract" and "repel" bodies at GEO altitudes, making East-West station-keeping something that depends on longitude.

The answer estimates that the amount of delta-v necessary to get up out of one of these stable equilibrium points (and presumable also to drop down into one) to be small, and cites another estimate (of about 2 m/s per year) of the penalty to East-West station keeping at unstable points as evidence of this.

Question: But I'm curious about how deep these stable points are in terms of delta-v quantitatively. For a slow transfer from one stable point to the other, how much delta-v is needed to pick ones self up out of one "gravity well" and then to settle back down in the other? Can a quantitative, verifiable source be found for this, or simply calculated approximately using Earth's J₂₂?

Let's assume the initial state is zero inclination and station-keeping as been done recently so that the East-West drift has been zeroed.

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    $\begingroup$ You've shown some detailed orbital simulation work in the past IIRC; do you have a decent geopotential model you could just slap this problem into and see if it works? $\endgroup$ – Russell Borogove Dec 30 '18 at 1:30
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    $\begingroup$ @RussellBorogove all I know how to do is the way I apply J₂ here and here. I could probably add a J₂₂ term with a cos(longitude) using this stuff but I wouldn't be confident I was doing it right without a check by someone else's calculation. So far I am baffled how to implement gravity models into my scripts correctly. fyi I've edited the question, is that better? $\endgroup$ – uhoh Dec 30 '18 at 1:45
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    $\begingroup$ Yeah, no issues now. $\endgroup$ – Russell Borogove Dec 30 '18 at 2:23

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