ISS (ZARYA) TLE to Latitude Longitude Conversion

Question

I am trying to take TLE of ISS (ZARYA) and get the latitude longitude and height. I am aware the TLE is in TEME~(True Equator Mean Equinox) frame. This means that the TLE was generated considering the True Equator as of TLE generation time and X axis pointing to mean equinox. Thus, if I were to calculate instantaneous latitude, longitude and altitude at the TLE generation point I need not worry about the precession and nutation complexity. I am also aware that LLH of ISS is expressed in geodetic coordinates. Now getting the exact sidereal time at the TLE generation time seems to be a little involved process involving UT1 and GMST equations (and UTC!), but I am able to bypass that problem by considering a UT1 time and sidereal time at 1 Jan 2023 12:00PM and then assume them to linearly related for short time. I am assuming within an year deviation will be small enough.

Anyway, I am first trying to attack the latitude problem, which is indepedent of sidereal angle and must come very close to actual number.

I am considering this TLE. Which is for date 2023-05-25 07:49:28.500 I derived the date using tle to date online utility

ISS (ZARYA)             
1 25544U 98067A   23145.32602431  .00014206  00000+0  25493-3 0  9994
2 25544  51.6413  81.8570 0005396  18.1967 120.0030 15.50160974398246

Now using historical ISS data I am able to pull the latitude longitude and height at this time ISS Hitorical Data

Lat: 31.581 deg
Lon: -127.093 deg
Ht : 419.88 km

The inertial vectors I derived using the mathematical equations as defined in kep2cart doc are

-3498.170819 -4621.733107  3548.392785     2.785672    -5.551759    -4.477993

Now using this ECEF to LLH online utility I can convert to geodetic lat lon ht:

Latitude  : 31.63493   deg N
Longitude : 232.87807   deg E (forget this for a while)
Height    : 423939.5   m

The two latitudes are off by 0.05 degrees. Why there is this difference? Or I am assuming something about precession and nutation that I cannot?

Another issue is the altitude, using WGS84 system altitude should also come exact. Why is there difference in altitude?

Answer 1

The problem is TLEs do not mean what you think they mean, so you can't use them that way. The equations in the kep2cart doc you linked are simply wrong when applied to TLEs. If you want to successfully convert a TLE to inertial vectors or LLH, you must use SGP4, which exists solely for that purpose, and does far more math than you can possibly hope to code successfully on your own. Nothing about TLEs is exact, because they describe the average state, not the instantaneous one. Errors of several kilometers and several meters per second are routine at epoch, and grow quadratically from there. Finally, the TLE model is so old that they actually use WGS72, not WGS84.