How is stability of Geosynchronous Orbits (GSO) affected by Earth’s mass?

Question

I'm researching stable orbits slightly beyond GSO and I need to find out how close to GSO I could place an object without risking collision. If the Earth was homogeneous the altitude should be the same for all but as it is not (based on the Geoid measured by the ESA satellite GEOC) is there a maximum and minimum altitude and how would the geoid affect/limit the orbit of an object just above GSO?

Answer 1

tl;dr: A satellite just above GEO would pass by all longitudes, and so the question asks if Earth's gravity deviations would lead to the GEO band being higher or lower in some places. Of course the non-synchronous orbit would do the same thing, but regardless, the effect is of order 0.1 part per million and the scatter of orbits of even active GEO satellites will probably be far larger than this.

The spatial variation gravitational potential of nearly spherical bodies such as the Earth is closely approximated by the first few coefficients of the spherical harmonic expansion. This is especially true out at GEO so many Earth-radii away.

You can read about his in many places, including @DavidHammen's excellent answer, Wikipedia's article on Geopotential model and many papers including this one (click "print this article" to see the pdf).

The dominant variation in latitude (pole to equator) is $J_2$ or $C_2$, and in longitude (around the equator) is $J_{22}$ or $C_{22}$. The ratio of these modulations to Earth's main monopole field are of the order of these coefficients times $(R_E/r)^2$ where $R_E$ is a normalization to the Earth's average equatorial radius (about 6378 km) and $r$ would in this case be the geostationary distance to the center of the Earth, roughly 42,164 km, so that term is about 0.02.

Earth's $J_2$ is about 1.1E-03, and $J_{22}$ is about 600 times lower at 1.8E-06. There are some coefficients in the polynomials of order 3.

Putting it all together, at a distance of about 42,000 kilometers from Earth's center, the gravitational potential varies by +/-7E-05 pole to equator (~70 ppm), but only +/-1E-07 (~0.1 ppm) around the equator.

If the potential varies by 1ppm, so will the force, and the period, roughly speaking. So it's a period difference of only a few tens of milliseconds, or for synchronous orbit, tens of meters. In terms of velocity, less than a millimeter per second.

This variation is likely to be small compared to the existing variation in period, altitude, and velocity of even those GEO satellites with good, active stationkeeping. In other words, this effect is probably in the noise.