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I've implemented the Earth harmonics calculation in JGM-3 model, using the coefficients from here

2    0   -0.10826360229840e-02     0.0
2    1   -0.24140000522221e-09     0.15430999737844e-08
2    2    0.15745360427672e-05    -0.90386807301869e-06
3    0    0.25324353457544e-05     0.0

Now, I want to switch to EGM2008, the coefficients are taken from here (Tide-free).

2    0   -0.484165143790815e-03    0.000000000000000e+00
2    1   -0.206615509074176e-09    0.138441389137979e-08
2    2    0.243938357328313e-05   -0.140027370385934e-05
3    0    0.957161207093473e-06    0.000000000000000e+00

There is a major difference. Probably, I should make some operations on the coefficients? For example, multiply $C_{20}$ by $\sqrt{5}$?

Multiplied all coefficients of EGM2008 by $\sqrt{2*degree+1}$, got

2   0   -1.082626173852220E-03  0.0000000000000E+00
2   1   -4.6200632349558E-10    3.0956435701202E-09
2   2   5.4546274930574E-06 -3.1311071889349E-06
3   0   2.5324105185677E-06 0.0000000000000E+00

Especially for $C_{21}$ and $C_{22}$ the difference is still major.

Extra

To calculate the Earth gravity potential in JGM-3, the equation was used: $U_{har}=\frac{\mu}{r}[1+\sum_{i=2}^d\sum_{j=0}^o (\frac{R_{eq}}{r})^iP_{ij}(\sin\phi)(S_{ij}\sin{j\lambda}+C_{ij}\cos{j\lambda})]$,

As you see, the argument of Lejendre polynom $P_{ij}$ is $sin\phi$.

However, here for EGM2008 it's said to use $cos\phi$. Is it specific for EGM2008 model?

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    $\begingroup$ sqrt(5) is for C20, but you also need to normalize/unnormalize the other coefficients: sqrt(degree * 2 + 1). $\endgroup$ – Cristiano Aug 3 '18 at 10:41
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    $\begingroup$ Both C and S. It depends only on the degree. No link, sorry. $\endgroup$ – Cristiano Aug 3 '18 at 10:58
  • $\begingroup$ @Cristiano I multiplied all coefficients by $\sqrt{degree*2+1}$, but the results are still strange. Probably, there is something specific with EGM2008 $\endgroup$ – Leeloo Aug 3 '18 at 16:33
  • $\begingroup$ Then probably it's an overflow problem; what does "the results are still strange." mean? $\endgroup$ – Cristiano Aug 3 '18 at 16:46
  • $\begingroup$ @Cristiano I don't undertand why for $C_{22}$ in JGM-3 the value is 0.15e-5, but in EGM-2008 it's 0.54E-05. So major difference? May be it depends on order also? Added the coefficients to the question. $\endgroup$ – Leeloo Aug 3 '18 at 17:24
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Spherical harmonics coefficients for gravity models can span several orders of magnitude. To avoid underflow problems and loss of accuracy, they are commonly normalized with a factor that depends on both $n$ and $m$. It looks like the coefficients reported by JGM-3 are the unnormalized $C_{n,m}, S_{n,m}$, while the ones in EGM2008 are the normalized $\overline{C}_{n,m}, \overline{S}_{n,m}$.

From Montenbruck and Gill (2000): Screenshot from Montenbruck and Gill describing normalized / unnormalized coefficients

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