- #1

ryan88

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I am looking to use the definition from WGS84 to calculate Earth's gravitational potential using spherical harmonics, however I am having some difficulty finding the definition of one of the variables. Gravitational potential is given as the following:

[itex]V = \frac{GM}{r}\left [ 1 + \sum_{n=2}^{n_{max}} \sum_{m=0}^{n} \left( \frac{a}{r} \right )^n \bar{P}_{nm} \left( \sin{\phi} \right ) \left( \bar{C}_{nm} \cos{m\lambda} + \bar{S}_{nm} \sin{m\lambda} \right ) \right ][/itex]

where:

[itex]V[/itex] is the gravitational potential function

[itex]GM[/itex] is the Earth's gravitational constant

[itex]r[/itex] is the distance from the Earth's centre of mass

[itex]a[/itex] is the semi-major axis of the WGS84 ellipsoid

[itex]n,m[/itex] are the degree and order respectively

[itex]\phi[/itex] is the geocentric latitude

[itex]\lambda[/itex] is the longitude

[itex]\bar{C}_{nm},\bar{S}_{nm}[/itex] are normalised gravitational coefficients

[itex]\bar{P}_{nm}\left( \sin \phi \right) = \left[ \frac{(n-m)!(2n+1)k}{(n+m)!} \right] P_{nm}(\sin\phi)[/itex]

[itex]P_{nm}(\sin\phi) = (\cos\phi)^m \frac{d^m}{d(\sin\phi)^m}[P_n(\sin\phi)][/itex]

[itex]P_n(\sin\phi) = \frac{1}{2^n n!} \frac{d^n}{d(\sin\phi)^n}\left( \sin^2\phi -1 \right )^n[/itex]

[itex]m=0,k=1[/itex]

[itex]m\ne0,k=2[/itex]

However, I can't find what the definition of [itex]d[/itex] is. Can anyone offer any help?

Thanks,

Ryan