How much more unstable are satellites in retrograde then a pro-grade orbit? Does the stability of a retrograde orbit increase with altitude?
This answer said this orbit is unstable and made me think, is it? https://astronomy.stackexchange.com/a/28696/15415
How would propagation differ between a GEO and a Retrograde GEO?
I can give the start of an answer. Somebody with a simulator can do better quantitatively, or I will in a few (several) days if there are no other takers.
By rotating the opposite direction as the orbited body (retrograde), there may be a cancelation of the tangential force integrated over one orbit. I'll look for more about that, and hopefully someone will address it numerically in a separate answer.
In the mean time I'll summarize other situations where retrograde motion tends to cancel the effects of orbital perturbations with periodicities related to the orbital period:
Retrograde orbits, those going the opposite direction as some other more dominant sense of rotation in the system, can offer stability under some situations.
@MarkAdler's comment
A distant retrograde orbit around the Moon might be close. It is claimed to be stable over at least hundreds of years, and it would only rarely be eclipsed by the Earth or the Moon. However I don't know if anyone has run it out for 10,000 years.
@Hobbes' answer which quotes Jason Davis' Planetary Society 2014 blogpost The latest on NASA’s Asteroid Redirect Mission
In both cases, the spacecraft and captured asteroid are inserted into what’s called a Distant Retrograde Orbit, or DRO, around the moon. DROs are very stable over long periods of time. “The orbit that we currently have in mind is about 75,000 kilometers [47,000 miles] above the surface of the moon,” said ARM Program Director Michelle Gates during the June 19 briefing. She added the spacecraft and asteroid would be stable for more than a hundred years.
@BrianLynch's answer
The reason is that a lunar orbit (specifically, a distant retrograde orbit or DRO) is much more stable compared to a low Earth orbit so you can leave it there for a long time and study it in the future. You also don't have to worry about it decaying and colliding with the Earth -- although the size of the target asteroid (actually only a piece of an asteroid) probably makes it nothing to worry about.
...which after repair might end up linking to Collin J. Bezrouk's UC Boulder Ph.D. Thesis Ballistic Capture into Lunar and Martian Distant Retrograde Orbits
Once a spacecraft is in a stable DRO, the long duration evolution of that orbit is of interest. Using a high fidelity dynamical model and numerical precision techniques, the evolution of several DROs in the Earth-Moon system is studied over a period of 30,000 years. The perturbing forces that cause a DRO to transition into an unstable orbit are identified and analyzed. DROs larger than 60,000~km grow in amplitude due to solar gravity until they depart the Moon after several centuries. DROs smaller than 45,000~km remain stable for 25,000 years or more, but decay in size due to the Moon's solid tide bulge, which eventually causes the DRO to depart the Moon. The DROs evolve chaotically and occasionally experience periods of relatively fast amplitude growth when the period of the DRO is in resonance with the frequency of particular perturbing forces.
In this example the orbit is extremely high and the perturbing force is distant, but the perturbing stimulus has a period equal to the orbit period, while the orbit you are describing would receive gravitational perturbations from the Earth's field with both period=1 and higher. Still, the basic idea that retrograde might offer some stability benefits is discussed.
@TildalWave's answer also discusses the advantages of distant retrograde orbits.
For more on the Asteroid Redirect Mission see @OrganicMarble's thorough answer to What ever happened to the Asteroid Redirect Mission?
In this question I discuss this paper that addresses the stability of clumps of matter in a retrograde ring around a star. For that see also NPR's Spin To Survive: How 'Saturn On Steroids' Keeps From Self-Destructing or watch this cool Vimeo video (then this one).
Unfortunately, the question Distant retrograde Orbit in CR3BP Question remains without a sufficiently helpful answer.
See also the answers to the Quora question Why are retrograde orbits more stable than prograde ones? which link to the classic On The Stability of Direct and Retrograde Orbits. See also the @DavidHammen’s insightful answer to the question Are retrograde capture orbits “easier” than prograde capture orbits? as well as this paywalled but probably very helpful paper On retrograde orbits, resonances and stability
