# Conversion of satellite position coordinates to local horizon reference frame (lat & long)

I'm currently working on a bit of code to calculate satellite passes for a ground station. I've found a variety of references that discuss the conversion form other coordinate reference frames to the local horizon (lat, long, radius to target) frame. Aside from other issues it would be really beneficial to have a worked step-by-step example of how to get from Earth-centered inertial (ECI - as output by SGP propagators) to the local horizon frame. Either a worked-example in an answer or a web link to a worked example would be great!

I'm currently using this reference: Transformation to local horizontal coordinates, I'm stuck on step 3 - conversion matrix.

• This is otherwise called the Cartesian to Geodetic conversion. It should help you find good references. For a worked-example, I'd recommend CelesTrak / Dr. T.S. Kelso's Orbital Coordinate Systems, parts 1-3. Dec 5, 2014 at 8:41

I couldn't comment on the question due to low reputation. So, I'd first like to clarify that Local Horizon frame and Long, Lat, Radial distance are two entirely different frames.

Local Horizon frame would be Earth-surface-centered (at Ground station) and principle axes being one towards local east (reason for the name 'Local horizon) , Other towards zenith and third one completing the right hand coordinate system and towards local North. While lat, long ,radial distance while specifically used for ECEF (Earth centered Earth Fixed) , can be generally used for any spherical coordinate frame.

Trying to identify your task, I'd assume you'll need local horizon frame to record observed spacecraft position, ECEF or ECR (Non-inertial geo-centric and Earth fixed) as intermediate frame to convert from/to ECI w.r.t the GS coordinates and local time of observation.

For ECI-ECEF rotation matrix, you'll need single rotation matrix which is referenced with greenwich time and rotation angle being [A(s)+(We)(Tu)] in Z axis rotation matrix. (Look up in wikipedia- Rotation matrix -under Basic 3x3 rotations)

Here, A(s) is rotation angle from Vernal Equinox to the sun given by time/day of the year i.e. position of earth in orbit around sun. We- Angular velocity of Earth's rotation Tu- Universal time

Further, for subsequent change from ECEF to Local horizon frame you'd just need Ground station coordinates in spherical form i.e. latitude, longitude and distance for rotation matrix.

The rotation matrix would be R(ECEF_LH)= R(0.5pi)R(lat-0.5pi) R(-long) R being the rotation matrix as usual. You can refer to sign conventions of elementary rotation and order of elementary rotation in a composite rotation to self verify the matrix.

So, with these conversions you can go from ECI to ECEF to LH while reverse conversion can be done by using inverse of the composite rotation matrix. Referring to rules of combining simpler rotations might aid the understanding.

P.S I'd really like to help if you could clarify the question more. You can however, use the answer to choose out particular transformation for your problem. This is my first answer. So, Sorry for the lack of perfection in presentation.