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Gravitational potential using spherical harmonics (WGS84)

  1. Sep 20, 2012 #1
    Hi,

    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
     
  2. jcsd
  3. Sep 20, 2012 #2
    It means derivative. As in d/dt, d^2/dt^2, etc
     
  4. Sep 20, 2012 #3
    Ah right, now I feel stupid, lol.

    Thanks for that,

    Ryan
     
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