# Tesseral Part of the Geopotential

I am working on the design of a constellation for academic purposes. I try to describe a phenomenon that appears in the simulations by means of analytic expression and I can't really find a good reference for this issue. I will try to describe the problem and maybe some on here can point me to the right book or an article.

Let assume we have two satellites orbiting the Earth in a near-polar near-circular orbit. The satellites are identical and the two orbits differ only on the RAAN angle. Using a high fidelity simulation I notice a drift between the two satellites, a drift that over time change the node crossing time. Basically, the mean anomaly change rate is not equal between the two satellites. For example, in a 600 km orbit, an inclination of 91 deg and RAAN difference of 60 deg I get a mean anomaly difference of 0.4 deg after 10 days. The effect seems to be secular or maybe with a very long period.

After further analysis, I realize that the cause of this effect is the geopotential, mainly the tesseral part of the geopotential. The initial RAAN difference between the two planes creates an uneven geopotential perturbation.

I am looking of a book or an article that provide equations of the mean anomaly derivative influenced by the tesseral part of the geopotential. I am looking for a straight-forward approximation, something that can help me quantify the effect so I can hopefully design an efficient controller to deal with this drift.

Thank you

Eviatar

• Can you say roughly how large the effect is? One second per orbit, per week, per year? I'm just curious. You might consider a simple numerical test instead of an analytical one. A few dozen lines of Python may be all you need. See my answer to Brouwer-Lyddane mean semi major axis bias. The problem there was that they were starting at the same place in space, but different epochs, actually rotating the earth, moving different gravitational potentials to the starting location of the satellite. You are doing the similar/same thing.
– uhoh
Commented Nov 6, 2018 at 13:40
• I think if you try to add some tesseral harmonics to a small script like that, you can learn a lot more than you can from an analytical approximation. It may take more time in the beginning, but you can just keep testing new things and learning more. Then you'd still move to standard software for full-blown simulations, but with better insight into what those results mean.
– uhoh
Commented Nov 6, 2018 at 13:43
• Answers to What's a Brouwer-Lyddane mean semi major axis, or any other, for an orbit in a lumpy gravity field? may be helpful as well.
– uhoh
Commented Nov 6, 2018 at 15:18