Can someone explain to me the TEME reference frame in defining orbital coordinates?

Question

I am undertaking a project of developing a "surface station" for my desktop, which will track the position of different satellites as they fly overhead. I have zero background in astrophysics and the such, but I have learned a ton so far.

I understand the importance of defining orbits to an ECI (earth-centered inertial) reference frame. From this, I can either translate my position on Earth to the ECI reference frame, or translate the satellites position from the ECI, to my ECEF (earth-centered, earth-fixed) reference frame.

I followed this article here, where we determine my local sidereal time, by determining the Greenwich sidereal time, which from what I understand, is the conversion between the x-axis in the ECI reference frame, to the ECEF reference frame (with the prime meridian as the x-axis). This Greenwich sidereal time, we use (what I assume to be) an empirical formula to determine the angle, with some time from the J2000 epoch.

Where I am getting confused now, is that apparently TLE (two-line element) data from satellites use a TEME (true equator, mean equinox) ECI reference frame, which a given epoch time with the TLE data.

My question then is, how do I determine the Greenwich sidereal time from a given TEME reference time - it makes sense to me when we have a well defined J2000 epoch - but doesn't make sense to me if this epoch time is changing every time TLE data is published.

Sorry if some of this doesn't make sense, I am trying to learn and any help would be greatly appreciated.

Answer 1

WARNING: This is an attempt to answer. The reason is that I thought I understood the TEME concept until, trying to answer your question leads me to be more cautious. Especially, when I discovered that "TEME" may have different definitions, interpretations, in various implementations of SGP4 (cf [1]).

My answer will be based largely on this paper [2].

First, we all know that TEME is in the class of ECI (non-rotating) reference systems, with the prime direction (X) pointing along the Vernal Equinox and the Z direction is the CEP (Celestial Ephemeris Pole). That's good so far, provided that this Z direction doesn't move wrt to the stars (Celestial Sphere). Yet it does! Wrt to the Celestial Sphere, Earth orbital trajectories have a nutation and a precession. With this in mind, we must understand first the difference between "true" and "mean" used in TEME. Wiki (https://en.wikipedia.org/wiki/Equatorial_coordinate_system) gives a clear definition: "true" is when all variations are accounted for and "mean" is when oscillating variations are ignored (eg "small periodic oscillations of nutation").

What does "true Equator" mean? According to my reading, it means the plane perpendicular to the CEP at a given specified time. The mean Equator is the plane perpendicular to an axis computed from the CEP from which the nutation terms are excluded.

What does "mean Equinox" refer to? It means the Equinox located on the "mean Equator". This is better explained in [2], page 4, just before Fig.1, which I also reproduce here.

In this Fig. we can see that the "Mean Equinox", in the TEME sense, is on the "True Equator" plane, but it is not the intersection of this True Equator with the (true) Ecliptic (so not really an Equinox in its classical sense!). As implied by Fig. 1, you have to take the intersection between the Ecliptic and the "Mean Equator" (Equator computed with precession equation, but ignoring nutation), then project it to the "True Equator". The "of date" term means that the precession is taken into account. The precession is a slow motion of the CEP with a perod of 25700 years.

I feel that my answer is correct. But I apologize in advance if my interpretations are wrong!

[1] https://celestrak.org/publications/AIAA/2006-6753/AIAA-2006-6753-Rev2.pdf

[2] https://geodesy.geology.ohio-state.edu/course/refpapers/AIAA4025.pdf

Answer 2

how do I determine the Greenwich sidereal time from a given TEME reference time

There is a bit of a problem. Because the reference time used in the TLEs, thus TEME, has never been officially confirmed. The following is from "Revisiting Spacetrack Report #3".

D. Coordinate System

[...]

However, you should be aware of an additional nuance, specifically the ‘of date’ and ‘of epoch’ formulations.

• TEME of Date—With this option, the epoch of the TEME frame is always the same as the epoch of the associated ephemeris generation time. The transformation to ECEF is done by first finding the conversion from TEME to TOD (third equation in Eq. (1)). Next the standard transformation from TOD to ECF is computed. We could have gone directly to PEF without the TOD frame (second equation in Eq. (1)), but this implementation enables comparison with the TEME of Epoch approach. All transformations are found using the complete IAU-76/FK5 formulae, including nutation.

• TEME of Epoch—In this approach, the epoch of the TEME frame is held constant. Subsequent rotation matrices must therefore account for the change in precession and nutation from the epoch of the TEME frame to the epoch of the transformation. This is accomplished by finding a static transformation from TEME to J2000—this includes the equation of the equinoxes, the nutation, and the precession which are all calculated at the epoch of the TLE. This static transformation is applied at each time requested in an ephemeris generation. Once the J2000 vector is found, standard techniques can convert this to other coordinate systems, at the appropriate time. This is computationally intensive, and introduces error into the subsequent solutions. All transformations, after the initial static calculations, are computed using the complete IAU-76/FK5 formulae, including all terms of the nutation theory.

Researchers generally believe the ‘of date’ option is correct, but confirmation from official sources is uncertain, and others infer that the ‘of epoch’ is correct. To be complete, we provide the equations and an example problem of both in the Appendix.

Also,

E. Time System Issues

[...]

The time system is assumed here to be UTC, but no formal documentation exists and UTC, as currently defined, was only introduced in 1972. UT1 is needed to calculate GMST for the coordinate transformations discussed in the appendix

If you think you can use UT1, "GMST is linked directly to UT1 through the equation"

GMST (in seconds at UT1=0) = 24110.54841 + 8640184.812866 * T
               + 0.093104 * T^2 - 0.0000062 * T^3

where T is in Julian centuries from 2000 Jan. 1 12h UT1

    T = d / 36525
    d = JD - 2451545.0