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I am currently trying to understand the definition of the True Equator Mean Equinox (TEME) frame. I found a very nice explanation in @NgPh's answer to Can someone explain to me the TEME reference frame in defining orbital coordinates? where it is specified that the Z axis is defined as the direction between the center of mass of Earth and the Celestial Ephemeris Pole (CEP).

However, after doing some research, I have not been able to find a clear definition of the CEP.

  • Could anybody explain or point me to an appropriate source regarding the definition of the CEP?
  • How does it differ from the instantaneous axis of rotation of Earth?
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    $\begingroup$ I adjusted the format a bit, please feel free to roll back or adjust further. $\endgroup$
    – uhoh
    Commented Sep 29, 2021 at 10:06

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Thanks for bringing this up.

I am not an expert in this field (Geodesy), and to be honest, this term has given me headaches. Loosely speaking it is closely related to the "instantaneous" axis of Earth's rotation, depending on how you understand "instantanenous"... This seems to be the key source of ambiguity (IMHO).

Earth's rotation is complex, with cycles expressed in tens of thousands of years, but also down to a day (yes, it spins unpredictably from one day to another, i.e. from one "revolution" to the other!). Mathematically speaking, how can one define an "axis" of rotations then?

For an introductory reading of this fascinating science to model Earth's rotation, consider this link.

For more advanced discussions, but definitely very involved, is this 1985 paper by Nicole Capitaine: Clarification concerning the definition and determination of the Celestial Ephemeris Pole

Recall that the axis of rotation of the Earth moves with respect to the Celestial Sphere. The first major component of this motion is the Precession, with a period of 25700 years (about). It also has shorter-period movements that astronomers call Nutation. Here, “shorter than precession period” has a broad meaning in modern astronomy, a few thousands years down to a day, and below as astronomy science progresses.

Complicating things further, the axis also moves with respect to the Earth’s crust. This is termed “polar motion” by astronomers. The causes behind are the elasticities of the core, the oceans and the atmosphere (Earth as a non-rigid body).

Capitaine's 1985 paper introduction told us that the IAU replaced their model for the nutation in 1984, the first difference with respect to the previous model being the modelling of a non-rigid Earth (emphasis and [clarifications/interpretations] added by me) :

The difference between these two nutation series are due to the introduction of the non-rigid model of the Earth, in all the coefficients [of nutation], and to the pole [motion] for which the coefficients are computed.

Secondly, the IAU decided in 1984 to change the reference axis from the “instantaneous” to the new concept of Celestial Ephemeris Pole.

The newly adopted reference … was selected to have no diurnal or quasi-diurnal motions, with respect either to a space-fixed [i.e. non-rotating] or Earth-fixed coordinate systems.

Hence, if the motion of the instantaneous axis rotation, with respect to an inertial coordinate system, could be decomposed into a Fourier series, then the main term has a period of 25700 years and other terms have shorter periods, down to whichever smaller value that can be either predicted by a model and/or confirmed by observations. The CEP (IAU 1984) is the truncation of the series by excluding all terms (harmonics) with period equal to, or shorter than, a sidereal day. An equivalent way to state this is that the difference between the instantaneous axis and the CEP is the summation of all harmonics with period equal to or shorter than one sidereal day.

In the Section on “Practical realization of the CEP”, Capitaine stated:

Since the instantaneous axis is unobservable, there is no need to refer the obliquity to this axis.

We have to interpret this sentence as: Since the exact position of the instantaneous axis cannot be derived from observations of the stars, it is not logical to use it as a reference (as the IAU did prior to the 1984 decision). Capitaine further reminded that:

Thus old observations which were not generally corrected for diurnal polar motion, actually refer to the CEP… instead, the failure to correct for the diurnal motion resulted in an increase noise of the data with respect to the CEP.

Nicole Capitaine is definitely an authoritative author on this subject and you can easily find her other papers, if you wish to explore further.

To summarize:

  • The « mean Equator » is the plane perpendicular to the axis of rotation that takes into account only the precession motion (period ~25700 years).
  • The « true Equator » is the plane perpendicular to the axis of rotation that takes into account the precession and all the nutations, except for diurnal and sub-diurnal ones. This axis is also termed the Celestial Ephemeris Pole (CEP).
  • To obtain an « instantaneous pole », all known/computable nutations, but previously neglected, have to be added to the CEP. The first component of these fast and known nutations has a period of one sidereal day and an amplitude of 0.0087 arcsecond. An instantaneous pole with just that first term rotates in a cone of 0.0087 arcsecond around the CEP, in one sidereal day.

(I also think this subject would have a better home in Astronomy SE).

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    $\begingroup$ @Rafa, actually I am inclined to interpret the CEP as the (short-term) mean value of observations by definition (averaged over several days, at least 2 days). As we can read from Capitaine's 1985 paper, "the instantaneous axis is unobservable". Hence, the delta between the CEP and the instantaneous pole can only be derived from a given model, e.g., the "IAU 1980 nutation theory". I admit this is confusing. I will try to improve my answer when I have enough seld-confidence! $\endgroup$
    – Ng Ph
    Commented Oct 2, 2021 at 20:58
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    $\begingroup$ @Rafa, done. Tell me if you agree. $\endgroup$
    – Ng Ph
    Commented Oct 4, 2021 at 13:17
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    $\begingroup$ @Rafa, I have also made necessary corrections to my original interpretation of TEME. Thanks again for your questionings. $\endgroup$
    – Ng Ph
    Commented Oct 4, 2021 at 21:05
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    $\begingroup$ Thank you so much for providing your interpretation and the extended clarification in the reply! I completely agree with your interpretation. This seems to fit what is described by Capitaine's 1985 paper. Thanks a lot again for the very insightful discussion and for bringing light to this subject! $\endgroup$
    – Rafa
    Commented Oct 6, 2021 at 3:03
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    $\begingroup$ @Rafa, still one minor point that I hesitated to include in my answer. It concerns the 0.0087 arcsecond. My interpretation is that it's not included in the CEP (its period is exactly one sidereal day). It's the largest harmonic of the difference between the 2 axis and it's independent of the non-rigid Earth model (any such model has this component). That's the meaning of "constant" in Capitaine's paper (cf Section 2.2). If you agree, I will include this. $\endgroup$
    – Ng Ph
    Commented Oct 6, 2021 at 9:26
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The CEP has been superseded by the CIP.

From the glossary on the International Earth Rotation and Reference Systems Service (IERS) site:

Celestial Ephemeris Pole (CEP)

used from 1984 to 2003 with the IAU 1980 Theory of Nutation as the reference pole for nutation and polar motion; the axis of figure for the mean surface of a model Earth in which the free motion has zero amplitude. This pole was originally defined as having no nearly-diurnal nutation with respect to a space-fixed or Earth-fixed coordinate system and being realized by the IAU 1980 nutation. It was afterwards determined by using VLBI observations of celestial pole offsets. It is now replaced by the CIP, which is defined by IAU 2000 Resolution B1.7.

and

Celestial Intermediate Pole

Acronym: CIP

The CIP is an intermediate pole separating the motion of the pole of the ITRS in the GCRS (Geocentric Celestial Reference System) by dividing nutation from polar motion explicitly at the 2-day nutation period.

Nutations with periods less than two days are modeled by their equivalent polar motion.

The reason for this change is given here:

How does the new CIP (Celestial Intermediate Pole) differ from the CEP (Celestial Ephemeris Pole)?

The CEP did not allow for modeling of diurnal and subdiurnal nutation periods. The CIP is an intermediate pole separating, by convention, the motion of the pole of the ITRS in the GCRS by dividing nutation from polar motion explicitly at the 2-day nutation period. Nutations with periods less than two days are modeled by their equivalent polar motion.

Further related information may be found on their
The new IAU resolutions page.

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  • $\begingroup$ For astronomers, the CEP is outdated. But to my understanding, NORAD still use the CEP in its computation of TLE data. Is it your understanding as well? $\endgroup$
    – Ng Ph
    Commented Oct 5, 2021 at 10:12
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    $\begingroup$ @NgPh Sorry, I don't know. I just had a brief look on CelesTrak, but I couldn't find any recent info, just Revisiting Spacetrack Report #3AIAA 2006-6753, from 2006. I guess it makes sense to use CEP with TLE data to maintain consistency. OTOH, it should be easy to explicitly specify which Earth rotation model is being used in the new OMM format. $\endgroup$
    – PM 2Ring
    Commented Oct 5, 2021 at 11:06
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    $\begingroup$ Or may be it is not worth making the update, considering other inaccuracies "baked into" the model released by NORAD, not to mention the lose definition of the TEME frame as noted by Vallado in AIAA 2006-6753. $\endgroup$
    – Ng Ph
    Commented Oct 6, 2021 at 13:02
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    $\begingroup$ The IERS has a comprehensive collection of Earth Orientation data here: iers.org/IERS/EN/DataProducts/EarthOrientationData/eop.html Figuring out how to use that data is left as an exercise for the reader. ;) $\endgroup$
    – PM 2Ring
    Commented Oct 12, 2021 at 2:35
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    $\begingroup$ @Rafa The difference is very slight. The CEP definition excludes nutation components with a period less than 1 (sidereal) day, the CIP has its cut-off at 2 days. The longer period makes the CIP more predictable. $\endgroup$
    – PM 2Ring
    Commented Oct 12, 2021 at 9:20

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