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Deriving Apoapsis and periapsis formulas

  1. Jul 18, 2011 #1
    Hello, I'm wondering if anyone can tell me where these formulas for RA and RP (apoapsis and periapsis) come from.

    RA = [itex]\frac{a(1-e^{2})}{(1-e^{2}sin^{2}(LatB))^{3/2}}[/itex]

    RP = [itex]\frac{a}{(1-e^{2}sin^{2}(LatB))^{1/2}}[/itex]

    If you multiply RA by [itex]\frac{1-e^{2}sin^{2}(LatB)}{1-e^{2}}[/itex], you can get RP. Seems to be a clue, but I cant figure it out.

    I am trying to go from Latitude/Longitude to XYZ co-ordinates with A as the origin somewhere on the Earth, and B as the point I want to place with respect to A. Right now I am using an Excell spreadsheet which has these formulas. The spreadsheet uses the WGS84 Earth model for a,b and e.

    a = semi-major axis
    b = semi-minor axis
    e = ellipsoid eccentricity
    LatB is the latitude of the second point

    I found these two on the internet, but I dont get how they relate to the other two.
    RA = a(1+e)
    RP = a(1-e)

    It then does
    R = [itex]\sqrt[]{RA\times RP}[/itex]
    which is the geometric mean to find a spherical model radius for the AB area.

    Any help is appreciated. Thank you.
     
  2. jcsd
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