# Reducing degree of ICGEM's Gravity Field Models for orbit propagation

I would like to use one of the static gravity field models published at ICGEM for satellite orbit propagation. For example, EGM2008. Its degree is 2190. I would like to use only 360 (may be even less) coefficients of this model to get precession that is good enough while keeping algorythm peformance acceptable.

Accoding to .gfc file the coefficients are "fully_normalized".

Does that mean that I can't just peform 360 iterations instead of 2190? And I need to convert 2190 nomalized coefficients to unnormalized then limit their number to 360 and renormalize them? If so, am I right that I can use eq. 1-2 from this article for that?

• Note that the computational cost of a spherical harmonics gravitation model is $O(N^3)$, where $N$ is the degree of the model. It's much more expensive than extra iterations (which implies $O(N)$, at least kinda-sorta) to add complexity to such a model. Commented May 10, 2023 at 17:05

Regarding EGM2008 versus other models

Your goal is orbit propagation. You do not need a 2190x2159 Earth gravitation model for orbit propagation. It's worse than pointless unless your field of study is improving models of the Earth's gravitational field, and you'll need access to a supercomputer plus measurements from the Earth's surface and from space for that endeavor. Keep in mind that most spherical harmonics expansion algorithms are $$O(N^3)$$ algorithms, where $$N$$ is the degree of the gravitational model. You'll need access to a supercomputer to compute large degree and order spherical harmonics expansions for orbit propagation.

Moreover, those higher degree terms quickly drop off with increased distance from the Earth. Point mass gravitation drops off as an inverse square ($$1/R^2$$). The second order spherical harmonics terms drop off as $$1/R^4$$, and in general the higher order terms drop off as $$1/R^{N+2}$$. Even JPL, which does have access to supercomputers, only uses a fifth degree model for the Earth's gravitational field and a sixth degree model for the Moon's gravitational field.

In addition, the uncertainties in non-gravitational perturbations, the Earth's orientation, and thrusting (if the vehicle is active) quickly overwhelm the contributions of higher degree/order terms. I recommend using a degree / order approximation that is consonant with the perturbations you are modeling (or ignoring), and the uncertainties in them.

Regarding normalized versus unnormalized coefficients

It doesn't matter whether you use unnormalized or normalized coefficients in the sense that you are still going to be hit to some extent by the spherical harmonics equivalent of Gibbs phenomenon. There are techniques for mitigating this problem; see The resolution of the Gibbs phenomenon for spherical harmonics for details. Those techniques are a mitigation, not a removal. Truncating results in some loss of fidelity. "It is what it is."

It does matter whether you use unnormalized or normalized coefficients in the sense that you will be hammered massively (or even worse, hit with infinities) if you use an algorithm that uses finite precision arithmetic (e.g., double precision arithmetic) and that algorithm uses unnormalized coefficients. Normalized coefficients and algorithms that use finite precision arithmetic and use normalized coefficients (without having to denormalize them) fare much, much better than do algorithms that use finite precision arithmetic and use unnormalized coefficients. There's a reason spherical harmonics gravitation models have universally gravitated toward normalized coefficients. Use the normalized coefficients and use algorithms that take advantage of those normalizations.

Which algorithm, which model, and which degree/order?

Three widely used spherical harmonics expansion algorithms are the Lear, Pines, and Gottlieb algorithms. You will not be fired (or have your paper or dissertation rejected) if you use any one of these three algorithms. Be careful as bugs have been uncovered in these algorithms. Use an implementation that overcomes these bugs. Regarding the linked reference, using a recently published algorithm that has not undergone near as much scrutiny as those three widely used algorithms and that has only been cited a dozen times? You might well get fired or have your paper or dissertation rejected for doing that.

Regarding degree and order, there's no reason to go beyond degree 32 or so for most applications for Earth orbit propagation, and that's probably overkill. There are exceptions, but if you are working on one of those exceptionally challenging applications (e.g., modeling LAGEOS) you would already have been educated on the challenges -- and you wouldn't be using EGM2008.