# Are J2 and J4 cases of EGM2008 and other spherical harmonics models?

EGM84, EGM96, EGM2008... are Earth gravity models that contain coefficients "up to" a number of degrees. When using GMAT, STK, MATLAB Aerospace, this parameter is the "spherical harmonics degree" that is usually configured along with the desired EGM. For example, one may use EGM2008 with 60 degrees, or 120 degrees, etc.

I wonder if the simpler gravity model of order J2 or J4 are a low-order instance of EGM2008, and can therefore be expressd in terms of spherical harmonics degrees of such EGMs.

Is there a correspondence between J2/J4 and the degrees of these EGM?

$$J_2$$, $$J_3$$, $$J_4$$, ... correspond exactly to specific terms in a spherical harmonics expansion. For example, $$J_2 = -\sqrt{5}\,C_{2,0}$$. Similarly, higher order $$J_n$$ values are equal to the product of a fixed scale factor and the $$C_{n,0}$$ term in a fully normalized spherical harmonics expansion such as EGM2008 or GGM05C.
• @PaekSe Yes. One advantage of considering only $J_2$ etc. is that you don't have worry about the Earth's exact orientation. That is of course also a key disadvantage. The $J_3$ term is about the same magnitude as are the $C_{2,2}$ and $S_{2,2}$ terms. Except for $J_2$, there is no point in ignoring the Earth's daily rotation just so one can use just the zonal harmonics terms (i.e., the $C_{n,0}$ terms). The $J_3$ term is about the same order of magnitude as are the sectorial harmonic $C_{2,2}$ and $S_{2,2}$ terms. Jul 8, 2022 at 0:34
• I am still not sure I understand. For example, looking at the coefficents of EGM2008 as described here, is there any combination of coefficients (such as setting all $C_{n,m}$, $S_{n,m}$, and $P_{n,m}$ with $n>2$ and $m>0$ to zero) that yields an expression that is mathematically equivalent to the gravity model up to J2? Jul 8, 2022 at 7:34
• @PaekSe I'm not sure I understand your question. If you want something equivalent to a complete spherical harmonics model you need to use something equivalent. Using only the zonal harmonics (i.e., $C_{n,0}$, or alternatively, $J_n$) is not equivalent. The reason using $J_2$ and $J_2$ alone works fairly well is because the $C_{2,0}$ term (which is equivalent to $J_2$) dominates over all higher order / higher degree terms, at least for objects as large or larger than the Earth. The same cannot be said with regard to $J_3$, $J_4$, etc. Jul 9, 2022 at 14:57