Appendix C: TEME Coordinate System

Overview

This appendix describes the equations necessary to implement the nutation equations for the TEME (True Equator, Mean Equinox) approach. There are two approaches:

Using the GMST (Greenwich Mean Sidereal Time)
Using the equation of the equinoxes

Sidereal Time (GMST)

GMST is found using UT1. From McCarthy (1992:30):

$\theta_{GMST1982} = 67{,}310.548\,41^s + (876{,}600^h + 8{,}640{,}184.812\,866^s) T_{UT1} + 0.093\,104 \, T_{UT1}^2 - 6.2 \times 10^{-6} \, T_{UT1}^3 \tag{C-1}$

Transformation to ITRF

The transformation to ITRF is found using the polar motion ( $x_p$ , $y_p$ ) values and the GMST. Note that “PEF” implies the pseudo-Earth-fixed frame, where polar motion has not yet been applied (Vallado, 2007:214).

$[\mathbf{W}]_{ITRF-PEF} = ROT1(y_p) \, ROT2(x_p) \tag{C-2a}$

$\vec{r}_{ITRF} = [\mathbf{W}]_1^T \left[ ROT3(\theta_{GMST1982}) \right]^T \vec{r}_{TEME} \tag{C-2b}$

$\vec{v}_{ITRF} = [\mathbf{W}]_1^T \left\{ ROT3(\theta_{GMST1982}) \right\}^T \left[ \dot{\vec{v}}_{TEME} + \vec{\omega}_{\oplus} \times \vec{r}_{PEF} \right\} \tag{C-2c}$

IAU-80 Nutation: Delaunay Variables

If the equation of the equinox approach is taken, you must find the nutation parameters. The IAU-80 nutation uses so-called Delaunay variables and coefficients to calculate nutation in longitude ( $\Delta\psi_{1980}$ ) and nutation in the obliquity of the ecliptic ( $\Delta\varepsilon_{1980}$ ). From McCarthy (1992:32):

$M_\ell = 134.962\,981\,39^\circ + 1{,}717{,}915{,}922.6330^\circ T_{TT} + 31.31 T_{TT}^2 + 0.064 T_{TT}^3 \tag{C-3a}$

$M_O = 357.527\,723\,33^\circ + 129{,}596{,}581.2240^\circ T_{TT} - 0.577 T_{TT}^2 + 0.012 T_{TT}^3 \tag{C-3b}$

$\mu_\ell = 93.271\,910\,28^\circ + 1{,}739{,}527{,}263.1370^\circ T_{TT} - 13.257 T_{TT}^2 - 0.011 T_{TT}^3 \tag{C-3c}$

$D_O = 297.850\,363\,06^\circ + 1{,}602{,}961{,}601.3280^\circ T_{TT} - 6.891 T_{TT}^2 + 0.019 T_{TT}^3 \tag{C-3d}$

$\Omega_\ell = 125.044\,522\,22^\circ - 6{,}962{,}890.5390^\circ T_{TT} + 7.455 T_{TT}^2 + 0.008 T_{TT}^3 \tag{C-3e}$

Where:

$M_\ell$ — Mean anomaly of the Moon (l)
$M_O$ — Mean anomaly of the Sun (l’)
$\mu_\ell$ — Mean argument of latitude of the Moon (F)
$D_O$ — Mean elongation of the Moon from the Sun (D)
$\Omega_\ell$ — Mean longitude of the ascending node of the Moon ( $\Omega$ )

Nutation Parameters

The nutation parameters are then found using (McCarthy, 1992:33):

$a_{p_i} = a_{e1_i} M_\ell + a_{e2_i} M_O + a_{e3_i} \mu_\ell + a_{e4_i} D_O + a_{e5_i} \Omega_\ell \tag{C-4a}$

$\Delta\psi = \sum_{i=1}^{106} (A_{p_i} + A_{p1_i} T_{TDB}) \sin(a_{p_i}) \tag{C-4b}$

$\Delta\varepsilon = \sum_{i=1}^{106} (A_{e_i} + A_{e1_i} T_{TDB}) \cos(a_{p_i}) \tag{C-4c}$

Nutation Corrections and True Obliquity

Corrections to the nutation parameters ( $\delta\Delta\psi_{1980}$ and $\delta\Delta\varepsilon_{1980}$ ) supplied as Earth Orientation Parameters (EOP) from the IERS are simply added to the resulting values in Eq. C-4 to provide compatibility with the newer IAU 2000 Resolutions (Kaplan, 2005). These corrections also include effects from Free Core Nutation (FCN) that correct errors in the IAU-76 precession and IAU-80 nutation. However for TEME, these corrections do not appear to be used.

The nutation parameters let us find the true obliquity of the ecliptic, $\varepsilon$ . From McCarthy (1992:29—31):

$\Delta\psi_{1980} = \Delta\psi + \delta\Delta\psi_{1980} \tag{C-5a}$

$\Delta\varepsilon_{1980} = \Delta\varepsilon + \delta\Delta\varepsilon_{1980} \tag{C-5b}$

$\bar{\varepsilon} = 84{,}381.448^{\prime\prime} - 46.8150 T_{TT} - 0.000\,59 T_{TT}^2 + 0.001\,813 T_{TT}^3 \tag{C-5c}$

$\varepsilon = \bar{\varepsilon} + \Delta\varepsilon_{1980} \tag{C-5d}$

Equation of the Equinoxes

The equation of the equinoxes ( $EQ_{eqe1980}$ ) can then be found. The last two terms in the $EQ_{eqe1980}$ are probably not included in AFSPC formulations. From McCarthy (1992:30):

$EQ_{eqe1980} = \Delta\psi_{1980} \cos(\bar{\varepsilon}) + 0.002\,64^{\prime\prime} \sin(\Omega_\ell) + 0.000\,063^{\prime\prime} \sin(2\Omega_\ell) \tag{C-6a}$

$\theta_{GMST1982} = 67{,}310.548\,41^s + (876{,}600^h + 8{,}640{,}184.812\,866^s) T_{UT1} + 0.093\,104 \, T_{UT1}^2 - 6.2 \times 10^{-6} \, T_{UT1}^3 \tag{C-6b}$

$\theta_{GAST1982} = \theta_{GMST1982} + EQ_{eqe1980} \tag{C-6c}$

Transformation Equations

These relations let us form the transformation equations:

Precession and Nutation Matrices

$[\mathbf{P}]_{MOD \to J2000} = ROT3(\zeta) \, ROT2(-\Theta) \, ROT3(z) \tag{C-7a}$

$[\mathbf{N}]_{TOD \to MOD} = ROT1(-\bar{\varepsilon}) \, ROT3(\Delta\Psi) \, ROT1(\varepsilon) \tag{C-7b}$

TEME to J2000 Transformation

$\vec{r}_{J2000} = [\mathbf{P}][\mathbf{N}]\left[ROT3(-EQ_{eqe1980})\right] \vec{r}_{TEME} \tag{C-7c}$

$\vec{v}_{J2000} = [\mathbf{P}][\mathbf{N}]\left[ROT3(-EQ_{eqe1980})\right] \vec{v}_{TEME} \tag{C-7d}$

Worked Example

An example is useful to show the various options and their effect on the resulting vectors. Consider an initial ECI (J2000.0, IAU76/FK5) state vector.

Initial Conditions

June 28, 2000, 15:8:51.655 000 UTC
dUT1 = 0.162 360 s,  dAT = 21 s,  x_p = 0.098 700",  y_p = 0.286 000"

r_J2000 =  3961.744 260 3    6010.215 610 9    4619.362 575 8  km
v_J2000 = -5.314 643 386      3.964 357 585      1.752 939 153  km/s

Standard TOD, PEF (IAU 76/FK5)

Converting without the nutation corrections ( $\delta\Delta\psi_{1980}$ and $\delta\Delta\varepsilon_{1980}$ ), but using the two additional terms with $EQ_{eqe1980}$ :

JD_UT1 = 2,451,724.131 155 293 40
T_TT   = 0.004 904 360 547

r_TOD  =  3961.421 498 5    6010.475 268 8    4619.301 531 0  km
v_TOD  = -5.314 833 569      3.964 181 915      1.752 759 802  km/s

r_PEF  =   298.803 632 8   -7192.314 622 9    4619.301 531 0  km
v_PEF  =  6.105 014 271     -0.131 824 177      1.752 759 802  km/s

TEME Transformation

Using equations (C-3) to (C-6) to find the approximate parameters (with 4 nutation terms in Eq (C-4), no additional two terms in Eq (C-6), and no small angle approximations), then transforming TOD or PEF to TEME using Eq (C-7) or Eq (C-2) respectively:

r_TEME = 3961.003 549 8    6010.751 174 0    4619.300 930 1  km
v_TEME = -5.315 109 069     3.963 813 071      1.752 758 562  km/s

Notice this vector is “in between” TOD and PEF, but much closer to the TOD value — a reason it is sometimes (imprecisely) considered “inertial”. The authors consider these numbers are within a few mm of CMOC results.

TEME “of date” vs. “of epoch” Demonstration

Using the following TLE data at epoch and at 3 days into the future:

1 00005U 58002B   00179.78495062  .00000023  00000-0  28098-4 0  4753
2 00005  34.2682 348.7242 1859667 331.7664  19.3264 10.82419157413667

TEME vector at a time 3 days in the future (day = 182.784 950 62) from TLE epoch:

r_TEME = -9060.473 735 69    4658.709 525 02     813.686 731 53  km
v_TEME =  -2.232 832 783    -4.110 453 490      -3.157 345 433  km/s

Nutation Transformation Matrix (“of date”)

$[\mathbf{R}]_{TEME} = \begin{bmatrix} 0.999\,999\,999\,56 & 0.000\,000\,000\,00 & 0.000\,029\,505\,95 \\ -0.000\,000\,000\,65 & 0.999\,999\,999\,76 & 0.000\,022\,007\,05 \\ -0.000\,029\,505\,95 & 0.000\,022\,007\,05 & 0.999\,999\,999\,32 \end{bmatrix} \tag{C-8}$

Using Eq (7), the inertial (“J2000”) vector at the future time:

r_J2000 = -9059.941 378 6    4659.697 200 0     813.958 887 5  km
v_J2000 =  -2.233 348 094   -4.110 136 162      -3.157 394 074  km/s

Nutation Transformation Matrix (“of epoch”)

For the future time using the “of epoch” option, the nutation matrix is found at the epoch time as:

$[\mathbf{R}]_{TEME} = \begin{bmatrix} 0.999\,999\,999\,55 & 0.000\,000\,000\,00 & 0.000\,030\,111\,90 \\ -0.000\,000\,000\,66 & 0.999\,999\,999\,76 & 0.000\,021\,766\,37 \\ -0.000\,030\,111\,90 & 0.000\,021\,766\,37 & 0.999\,999\,999\,31 \end{bmatrix} \tag{C-9}$

And the resulting vector is:

r_J2000 = -9059.951 079 9    4659.680 755 6     813.945 045 1  km
v_J2000 =  -2.233 336 111   -4.110 141 024      -3.157 396 220  km/s

This is a difference of 23.6 m in just 3 days.

Summary of Coordinate Frames

Frame	Full Name	Description
J2000	J2000.0 ECI	Mean equator and equinox of J2000.0 epoch (inertial)
MOD	Mean of Date	Precessed from J2000 to date (mean equator/equinox of date)
TOD	True of Date	Nutated from MOD (true equator/equinox of date)
TEME	True Equator, Mean Equinox	SGP4 output frame — between TOD and PEF
PEF	Pseudo Earth Fixed	Rotated from TOD/TEME by sidereal time (no polar motion)
ITRF	International Terrestrial Reference Frame	Earth-fixed, includes polar motion

Transformation Chain

J2000  --[P]--->  MOD  --[N]--->  TOD  --[R3(EQ_eqe)]--->  TEME
                                   |
                            [R3(GAST)]
                                   |
                                   v
                                  PEF  --[W]--->  ITRF

Where:

$[\mathbf{P}]$ is the IAU-76 precession matrix
$[\mathbf{N}]$ is the IAU-80 nutation matrix
$[\mathbf{W}]$ is the polar motion matrix
GMST/GAST provides the sidereal rotation