<p><em>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 &quot;TEME&quot; may have different definitions, interpretations, in various implementations of SGP4 (cf [1]).</em></p>
<p>My answer will be based largely on this paper [2].</p>
<p>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 &quot;true&quot; and &quot;mean&quot; used in TEME. Wiki (<a href="https://en.wikipedia.org/wiki/Equatorial_coordinate_system" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Equatorial_coordinate_system</a>) gives a clear definition: &quot;true&quot; is when all variations are accounted for and &quot;mean&quot; is when oscillating variations are ignored (eg &quot;small  periodic oscillations of nutation&quot;).</p>
<p>What does &quot;true Equator&quot; mean? According to my reading, it means the plane perpendicular to the CEP at a given specified time. <strong>The mean Equator is the plane perpendicular to an axis computed from the CEP from which the nutation terms are excluded.</strong></p>
<p>What does &quot;mean Equinox&quot; refer to? <strong>It means the Equinox located on the &quot;mean Equator&quot;</strong>. This is better explained in [2], page 4, just before Fig.1, which I also reproduce here.</p>
<p><a href="https://i.sstatic.net/qs3z1.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/qs3z1.png" alt="Fig 1 from Ref 2" /></a></p>
<p>In this Fig. we can see that the &quot;Mean Equinox&quot;, <strong>in the TEME sense</strong>, is on the &quot;True Equator&quot; 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 &quot;Mean Equator&quot; (Equator computed with precession equation, but ignoring nutation), then project it to the &quot;True Equator&quot;. <strong>The &quot;of date&quot; term means that the precession is taken into account. The precession is a slow motion of the CEP with a perod of 25700 years.</strong></p>
<p>I feel that my answer is correct. But I apologize in advance if my interpretations are wrong!</p>
<p>[1] <a href="https://celestrak.org/publications/AIAA/2006-6753/AIAA-2006-6753-Rev2.pdf" rel="nofollow noreferrer">https://celestrak.org/publications/AIAA/2006-6753/AIAA-2006-6753-Rev2.pdf</a></p>
<p>[2] <a href="https://geodesy.geology.ohio-state.edu/course/refpapers/AIAA4025.pdf" rel="nofollow noreferrer">https://geodesy.geology.ohio-state.edu/course/refpapers/AIAA4025.pdf</a></p>

CC BY-SA 4.0
2021-06-03T17:43:09.597Z
2021-10-04T21:44:29.990Z
2022-09-27T22:27:30.220Z
Ng Ph
<blockquote>
<p>how do I determine the Greenwich sidereal time from a given TEME
reference time</p>
</blockquote>
<p>There is a bit of a problem.  Because the reference time used in the TLEs, thus TEME, <a href="https://celestrak.org/publications/AIAA/2006-6753/AIAA-2006-6753.pdf" rel="nofollow noreferrer">has never been officially confirmed</a>.  The following is from &quot;<a href="https://celestrak.org/publications/AIAA/2006-6753/AIAA-2006-6753.pdf" rel="nofollow noreferrer">Revisiting Spacetrack Report #3</a>&quot;.</p>
<blockquote>
<p><strong>D. Coordinate System</strong></p>
<p>[...]</p>
<p>However, you should be aware of an additional nuance, specifically the
‘of date’ and ‘of epoch’ formulations.</p>
<ul>
<li><p>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.</p>
</li>
<li><p>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.</p>
</li>
</ul>
<p>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.</p>
</blockquote>
<p>Also,</p>
<blockquote>
<p><strong>E. Time System Issues</strong></p>
<p>[...]</p>
<p>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</p>
</blockquote>
<p>If you think you can use UT1, &quot;<a href="https://lweb.cfa.harvard.edu/%7Ejzhao/times.html#GMST" rel="nofollow noreferrer">GMST is linked directly to UT1 through the equation</a>&quot;</p>
<pre><code>GMST (in seconds at UT1=0) = 24110.54841 + 8640184.812866 * T
               + 0.093104 * T^2 - 0.0000062 * T^3
</code></pre>
<p>where T is in Julian centuries from 2000 Jan. 1 12h UT1</p>
<pre><code>    T = d / 36525
    d = JD - 2451545.0
</code></pre>

CC BY-SA 4.0
2022-03-01T08:33:40.380Z
2022-03-01T08:33:40.380Z
2022-09-27T22:27:30.220Z
Yasushi Shoji
<p>I am undertaking a project of developing a &quot;surface station&quot; 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.</p>
<p>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.</p>
<p>I followed this article <a href="https://celestrak.org/columns/v02n02/" rel="nofollow noreferrer">here</a>, 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.</p>
<p>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.</p>
<p>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.</p>
<p>Sorry if some of this doesn't make sense, I am trying to learn and any help would be greatly appreciated.</p>

CC BY-SA 4.0
2021-06-02T22:05:54.470Z
Welcome to Astronomy SE! Could you please add some details regarding what some of your acronyms are? Also, congrats on posting our 11,000th question! (as of 6/2/2021)
WarpPrime
CC BY-SA 4.0
2021-06-02T23:34:13.120Z
Thanks, I think I have now defined all acronyms.
werdnerd
CC BY-SA 4.0
2022-03-02T22:04:59.463Z
TEME isn't well defined, but the website that article is on has at least "an" answer.  The link below has source code in several languages, and most will contain a teme2eci and teme2ecef function. https://celestrak.com/software/vallado-sw.php
Greg Miller
<orbit><coordinate>
Can someone explain to me the TEME reference frame in defining orbital coordinates?
werdnerd
# Can someone explain to me the TEME reference frame in defining orbital coordinates?

(Asked by: username_0 on 2021-06-02)

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](https://celestrak.org/columns/v02n02/), 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.

**Tags:** orbit, coordinate

> **Comments:**
> **username_1**: Welcome to Astronomy SE! Could you please add some details regarding what some of your acronyms are? Also, congrats on posting our 11,000th question! (as of 6/2/2021)  
> **username_0**: Thanks, I think I have now defined all acronyms.  
> **username_2**: TEME isn't well defined, but the website that article is on has at least "an" answer.  The link below has source code in several languages, and most will contain a teme2eci and teme2ecef function. https://celestrak.com/software/vallado-sw.php  

---

## 2 answers

### Answer by username_3 on 2021-06-03. Score: 5

*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.

[![Fig 1 from Ref 2](https://i.sstatic.net/qs3z1.png)](https://i.sstatic.net/qs3z1.png)

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 by username_4 on 2022-03-01. Score: 3

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


---

