I was trying to derive the expressions for the geocentric latitude and longitude of the sub-satellite point using Keplerian elements. The final equations connecting the combination of the Keplerian elements with the geocentric latitude and longitude doesn't contain the eccentricity and semi-major axis, and I was wondering why? I was also wondering what transformation equations to use to derive the expressions for the geocentric latitude and longitude. Any leads would be highly appreciated.
The starting variables that I used are the classical orbital elements, and I would like to arrive at:
- geocentric latitude: $ \ \ \sin \phi = \sin i \sin(\omega + f)$
- geocentric longitude: $ \ \ \tan(\lambda - \Omega) = \cos i \tan(\omega + f) $
Here $i$ is inclination, $\omega$ is the argument of perigee, $\Omega$ is the longitude of ascending node and $f$ is the true anomaly.