1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Gravitational potential using spherical harmonics (WGS84)

  1. Sep 20, 2012 #1

    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]


    [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]


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


  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,

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook