# Geocentric latitude and longitude of sub-satellite point

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:

1. geocentric latitude: $$\ \ \sin \phi = \sin i \sin(\omega + f)$$
2. 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.

I can answer the question of why Semimajor Axis and Eccentricity aren't included in the calculation, at least.

What you have here are the main Keplerian Orbital Elements that define the plane of your orbit around Earth. Every orbit with the same inclination and Longitude of the ascending node lies in exactly the same plane, which intersects the globe of the Earth on a great circle.

Every orbit in that plane, with the same above elements. with the same Argument of Perigee has its Periapsis lined up along the same line drawn from the center of the Earth.

And True Anomaly is an angle measured in the plane of the orbit, from the Perigee to the current position of the spacecraft, along the direction of travel around the orbit.

As a result with those four elements fixed, seen from the center of the Earth, every object in orbit with the same Longitude of the Ascending Node, Inclination, Argument of Perigee, and True Anomaly at the same time is on the same line drawn from the center of the Earth, regardless of Semimajor Axis or Eccentricity.

This results in them all having the same surface Longitude and Latitude.

• Thank you so much for the clarification. I was wondering how to then derive the above expressions for the geocentric latitude and longitude of the sub-satellite point. I believe the geocentric longitude is the longitude of the ascending node which thereby depends on GAST. Is that right? What about the geocentric latitude? – budding physicist Dec 5 '19 at 23:58
• @MahithM Longitude of the Ascending Node has nothing to do with Geographic Longitude. It's the angle in the Reference Orbital Plane (For earth-centered orbits, this is usually the equatorial plane, and for solar orbits, usually the ecliptic plane) between the Reference Direction, through the center of the body being orbited, to the Ascending Node of the orbit. – notovny Dec 6 '19 at 0:04
• I understood now. So can you help me with a lead in deriving the expressions for the geocentric latitude and longitude of the sub-satellite point using the Keplerian elements?? I was wondering what transformation equations to use. Do you have any links where I could look up for the derivation? – budding physicist Dec 6 '19 at 0:09