. h rmwr fIU20 Resolutions 2000 IAU of Framework The 5.1 Terrestrial and Celestial the Between Transformation 5 Systems iey h rm sraie rmte[R]b pligtetransfor- the applying respec- by motion [TRS] the from from and realized pole, as the mations the frame of from axis system, The celestial the tively. the around in pole the celestial of the of motion the where 03adta h ESwl otnet rvd sr ihdt and data with users provide to continue transformation. will conventional January is the IERS 1 It by for GCRS, the this Angle. algorithms the implement Rotation that in to Earth and steps the CIP takes and 2003 the IERS ITRS, of the the the position in that between CIP recommended the transformation the by of the proportional specified position that the linearly be and GCRS be the Angle and and UT1 Rotation ITRS CEO Earth that the the between recommends CIP to resolution the the as of This are defined TEO. origins is the Angle” along these Terrestrial Rotation measured the and “Earth angle and ITRS The (CEO) (TEO). the Origin Origin and Ephemeris Celestial GCRS the the as (NRO) in designated origin” both “non-rotating the CEP 1979) of the (Guinot, use of the recommends that 2000). the B1.8 with (Capitaine, of Resolution that domain coincides of frequency and extension low domain motion an the its frequency is in of high definition realization the Its the 1 ITRS. its in as and CEP through on well GCRS definition as (CEP) the GCRS its in Pole the implement both in Ephemeris to J2000.0 Celestial how at direction the specifies and of 2003 place (CIP) January Pole in Intermediate implemented celestial Celestial the associated be that their recommends with B1.7 document. need together Resolution this who level, those in mas for published 1 2000B offsets, the IAU version pole at shorter only its model or the level, a on mas replaced based 0.2 be the 2000 at (MHB Mathews of 2000A of functions IAU Theory transfer model 1980 -nutation IAU the and IAU by the Model 2003, January Precession 1 on 1976 beginning that, recommends gen- coordinate B1.6 level. Resolution defining a post-Newtonian and first provides tensor the also metric at the It transformations expressing the the for respectively. named framework Ce- and (GCRS) eral Geocentric now system System the are solar and Reference Relativity (BCRS) the lestial General System coordinates for Reference of space-time (1991) Celestial framework Barycentric A4 of the Resolution systems within IAU Earth the by that defined as specifies B1.3 the at Resolution provided also chapter. is this resolutions of of IAU among directions the end transform with procedures. computing to consistent process IERS for systems the these the systems, required intermediate in also in implemented objects being celestial be transformation to a therefore Such reference are terrestrial that and and of 2000) celestial systems Assembly the August between General transformation (Manchester, the XXIVth Union concern the Astronomical by International adopted the were resolutions Several of frame at (CRS) system reference terres- celestial the the from epoch to transform the (TRS) to system used reference be trial to transformation coordinate The Q ( W t ), t ( fteosraincnb rte as: written be can observation the of t R n then and ) ( t t and ) .” CS = [CRS] W ( R t r h rnfrainmtie rsn from arising matrices transformation the are ) ( t ilb ald“h nemdaereference intermediate “the called be will ) Q tal. et ( t ) R 20) o hs h edamodel a need who those for (2002)) , ( t ) W ( t [TRS] ) , Technical IERS Note (1) 33 No. 32 34 No. 32 . mlmnaino A 00Resolutions 2000 IAU of Implementation 5.2 IERS Note Technical i h CS swl stetdlvrain nplrmto.Two motion. polar in variations tidal days the pole two as the than well be less of TRS as motion periods the GCRS) with terrestrial in the the CIP forced including (in the the predictable model a to of a account motion corresponding into by the taking specified observations that by part offset requires IERS observed the also the by its It provided on the with based together GCRS monitor epoch. the Model at IERS in Precession-Nutation CIP the to 2000 the respect that of IAU with position offsets”) requires celestial pole conventional thus “celestial the as CIP (reported the differences observed of realization outside astro-geodetic The frequencies through with IERS band. the variations diurnal by retrograde including provided the is models ITRS and motion the observations The in by CIP observations. provided greater the astro-geodetic corrections periods of appropriate the time-dependent for through by nutation additional IERS realized Precession- forced plus the is and 2000A days GCRS the precession IAU two in for the than pole CIP model with the the 2000 from consistent of offset IAU motion manner be The a the to Model. has in in reso- Nutation J2000.0 GCRS this at epoch the to CIP at the According of of offset orientation. direction an higher the Earth’s and lution, the requires diurnal ce- in as This realized variations the well CIP. frequency the as on precession-nutation the B1.7, for based Resolution be model conventional the follow be must on to must pole depending order Q(t) lestial 2000B In matrix IAU precision. or quantities transformation required 2000A precession-nutation the IAU the in model B1.6, precession-nutation used Resolution be TT. follow the to 2451545.0 to at Date and order Julian geocenter = In the rec- TT at which 1.5 defined (1994) January be 2000 C7 J2000.0 date Resolution epoch IAU the with that consistent ommends is definition This the for 10 Chapter parameter See The scales. an- (2000). time as B1.9 these well between resolution 32 relationships IAU + the as TAI by is scale respectively, which re-defined time (TT), realized GCRS been Time the defined Terrestrial and of GCRS, (1991) counterpart BCRS the theoretical A4 in the coordinate Resolution of time other IAU TCG and coordinates, TCB time the Concerning precession- the the in that account requires into model. taken This nutation be and rotation. nutation BCRS and dependent in precession rela- time axes geodesic the any spatial that without coor- geocentric specified are and space (1991) GCRS barycentric geocentric A4 of Resolution the orientation IAU to tive which GCRS. correspond system, the must reference of celestial CRS, dinates the here B1.3, designated Resolution is follow to order In found be explanations can (Capitaine detailed products 29 IERS Note More Technical and IERS software chapter. in are concepts, the relevant transformations of the the end IERS about the the implementing of towards slightly subroutines continuity described been Fortran ensure have be to chapter values to products. transfor- provisional this necessary conventional earlier in are from the contained which revised values using expressions Numerical when the resolutions Resolu- mation. IAU provides the in also the with described It is of compatible which implementation B1.8. transformation Con- the new tion IERS for the the using expressions of resolutions chapter the this provides above, ventions recommendations the Following t (TT = t sdi h olwn xrsin,i endby defined is expressions, following the in used , rnfrainBtenteClsiladTretilSystems Terrestrial and Celestial the Between Transformation 5 − 00Jnay1 2 T ndays in TT) 12h 1d January 2000 tal et ,2002). ., / 36525 . 8 n has and s 184 . (2) 5.3 Coordinate Transformation consistent with the IAU 2000 Resolutions IERS Technical

Note No. 32

equivalent procedures were given in the IERS Conventions (McCarthy, 1996) for the coordinate transformation from the TRS to the CRS. The classical procedure, which was described in detail as option 1, makes use of the for realizing the intermediate reference frame of date t. It uses apparent Greenwich (GST) in the transformation matrix R(t) and the classical precession and nutation parameters in the transformation matrix Q(t). The procedure, which was described in detail as option 2, makes use of the “non-rotating origin” to realize the intermediate reference frame of date t. It uses the “Earth Rotation Angle,” originally referred to as “stellar angle” in the transformation matrix R(t), and the two coordinates of the celestial pole in the CRS (Capitaine, 1990) in the transformation matrix Q(t). Resolutions B1.3, B1.6 and B1.7 can be implemented in any of these pro- cedures if the requirements described above are followed for the space- time coordinates in the geocentric celestial system, for the precession and nutation model on which are based the precession and nutation quan- tities used in the transformation matrix Q(t) and for the used in the transformation matrix W (t). On the other hand, only the second procedure can be in agreement with Resolution B1.8, which requires the use of the “non-rotating origin” both in the CRS and the TRS as well as the position of the CIP in the GCRS and in the ITRS. However, the IERS must also provide users with data and algorithms for the conventional transformation; this implies that the expression of Greenwich Sidereal Time (GST) has to be consistent with the new procedure. The following sections give the details of this procedure and the stan- dard expressions necessary to obtain the numerical values of the relevant parameters at the date of the observation.

5.3 Coordinate Transformation consistent with the IAU 2000 Resolutions

In the following, R1, R2 and R3 denote rotation matrices with positive angle about the axes 1, 2 and 3 of the coordinate frame. The posi- tion of the CIP both in the TRS and CRS is provided by the x and y components of the CIP unit vector. These components are called “coor- dinates” in the following and their numerical expressions are multiplied by the factor 129600000/2π in order to provide in arcseconds the value of the corresponding “angles” with respect to the polar axis of the reference system. The coordinate transformation (1) from the TRS to the CRS corre- sponding to the procedure consistent with Resolution B1.8 is expressed in terms of the three fundamental components as given below (Capitaine, 1990) 0 W (t) = R3(−s ) · R2(xp) · R1(yp), (3)

xp and yp being the “polar coordinates” of the Celestial Intermediate Pole (CIP) in the TRS and s0 being a quantity which provides the po- sition of the TEO on the equator of the CIP corresponding to the kine- matical definition of the NRO in the ITRS when the CIP is moving with respect to the ITRS due to polar motion. The expression of s0 as a function of the coordinates xp and yp is: Z t 0 s (t) = (1/2) (xpy˙p − x˙ pyp) dt. (4) t0 The use of the quantity s0, which was neglected in the classical form prior to 1 January 2003, is necessary to provide an exact realization of the “instantaneous prime meridian.”

35 36 No. 32 .. ceai ersnaino h oino h CIP the of Motion the Transformation of the Representation Schematic in used be 5.4.1 to Parameters 5.4 IERS Note Technical s ( t = ) bevbe ihrsett h ersra oriae fteCP UT1, CIP, CIP. the of the of derivatives of partial coordinates coordinates the terrestrial celestial the of and to expressions respect simple very with observables to leads (1) mation 1 with 1 on procedure classical the with Q (36)). continuity expression ( ensure (see references 2003 to January previous chosen in now zero is be to chosen ain t xrsina ucino h coordinates the GCRS, of the function to a respect itaine as the epoch with expression the of moving Its and definition is tation. epoch CIP kinematical reference the the the when between to GCRS the corresponding in CIP NRO the of equator and E n,b ovnin h oino h oeo h R nteCSinto CRS the in TRS the of pole separat- the pole of intermediate parts: motion an two the is convention, CIP by the ing, B1.7, Resolution to According constant arbitrary The within equivalently, Or GCRS. century the one of over equator microarcsecond the 1 and in J2000.0 respectively at GCRS equator the of origin where Earth. the of rotation sidereal the date θ • • ( µ en h at oainAgebtenteCOadteTOat TEO the and CEO the between Angle Rotation Earth the being − and h ersra oino h I plrmto) nldn l the all including ( motion), TRS the (polar in band CIP diurnal the retrograde the of outside ( terms motion CRS terrestrial the in the days 2 than greater between the all periods including with (precession/nutation), CIP terms the of motion celestial the oe than lower t s as as, a egvni neuvln omdrcl involving directly form equivalent an in given be can ) 2 1 s Q t a [ tal. et en uniywihpoie h oiino h E nthe on CEO the of position the provides which quantity a being ( X nteeutro h I,wihpoie ioosdfiiinof definition rigorous a provides which CIP, the of equator the on σ d 1 = t 0 = ) s ( en uhta h oriae fteCPi h R are: CRS the in CIP the of coordinates the that such being X ( t a n Σ and ) t / = ) Y − 2000) , 1 = sin = 1+cos + (1   ( 0 − t . ) onsprsdra a cs)ad+0 and (cpsd) day sidereal per counts 5 − 1 / Q 1 − − 0 1 + 2 rnfrainBtenteClsiladTretilSystems Terrestrial and Celestial the Between Transformation 5 . − ( Z pdo rae than greater or cpsd 5 − d aXY t r h oiin fteCOa 20. n the and J2000.0 at CEO the of positions the are X t = ) aX cos X 0 t d ( X t / ,wihcnas ewitn iha cuayof accuracy an with written, be also can which ), 0 2 ,Y E, 8( R ( ) t Y σ 3 X ) ( Y 0 1 ( ˙ − − N t 2 + 1 − ( 0 − R X X aXY E t + 0 ]+ )] ) aY Y ( ) − t − Y Z = ) · 2 R 2 Σ Y Z ( sin = .Sc nepeso ftetransfor- the of expression an Such ). t t 2 0 ( ) 0 t R ( N t 1 − ) X 3 0 X ˙ − d ( ˙ hc a enconventionally been had which , N ( d − ) ( t a ) 0 t − sin · θ ) Y ( R ) X 0 dt steacnignd fthe of node ascending the is e.g. , Y ( . 3 ,Z E, t cpsd). 5 t 2 ( ) − E dt + u opeeso n nu- and precession to due ( ) Capitaine Y σ − · 0 2 R N ( ) σ 3 0 ( 0   s N − X cos = ) . · , 0 cpsd), 5 Σ i.e. i.e. R and − 0 3 tal. et X N ( Σ s d, 0 frequencies frequencies 0 Y ) and ) N , , s(Cap- is 2000), , 0 ) . Y (10) as (7) (6) (8) (5) (9) x - 5.4 Parameters to be used in the Transformation IERS Technical

Note No. 32

frequency in TRS -3.5 -2.5 -1.5 -0.5 +0.5 +1.5 +2.5 (cpsd) polar motion polar motion

frequency in CRS -2.5 -1.5 -0.5 +0.5 +1.5 +2.5 +3.5 (cpsd) nutation 5.4.2 Motion of the CIP in the ITRS

The standard pole coordinates to be used for the parameters xp and yp, if not estimated from the observations, are those published by the IERS with additional components to account for the effects of ocean and for nutation terms with periods less than two days.

(xp, yp) = (x, y)IERS + (∆x, ∆y)tidal + (∆x, ∆y)nutation, where (x, y)IERS are pole coordinates provided by the IERS, (∆x, ∆y)tidal are the tidal components, and (∆x, ∆y)nutation are the nutation components. The corrections for these variations are described below.

Corrections (∆x, ∆y)tidal for the diurnal and sub-diurnal variations in polar motion caused by ocean tides can be computed using a routine available on the website of the IERS Conventions (see Chapter 8). Ta- ble 8.2 (from Ch. Bizouard), based on this routine, provides the am- plitudes and arguments of these variations for the 71 tidal constituents considered in the model. These subdaily variations are not part of the polar motion values reported to and distributed by the IERS and are therefore to be added after interpolation. Recent models for rigid Earth nutation (Souchay and Kinoshita, 1997; Bretagnon et al., 1997; Folgueira et al., 1998a; Folgueira et al., 1998b; Souchay et al., 1999; Roosbeek, 1999; Bizouard et al., 2000; Bizouard et al., 2001) include prograde diurnal and prograde semidiurnal terms with respect to the GCRS with amplitudes up to ∼ 15 µas in ∆ψ sin 0 and ∆. The semidiurnal terms in nutation have also been provided both for rigid and nonrigid Earth models based on Hamiltonian formalism (Getino et al., 2001, Escapa et al., 2002a and b). In order to realize the CIP as recommended by Resolution B1.7, nutations with periods less than two days are to be considered using a model for the corresponding motion of the pole in the ITRS. The prograde diurnal nutations correspond to prograde and retrograde long periodic variations in polar motion, and the prograde semidiurnal nutations correspond to prograde diurnal variations in polar motion (see for example Folgueira et al. 2001). A table for operational use of the model for these variations (∆x, ∆y)nutation in polar motion for a nonrigid Earth has been provided by an ad hoc Working Group (Brzezi´nski,2002) based on nonrigid Earth models and developments of the tidal potential (Brzezi´nski,2001; Brzezi´nski and Capitaine, 2002; Mathews and Bretagnon, 2002). The amplitudes of the diurnal terms are in very good agreement with those estimated by Getino et al. (2001). Components with amplitudes greater than 0.5 µas are given in Table 5.1. The contribution from the triaxiality of the core to the diurnal waves, while it can exceed the adopted cut-off level (Escapa et al., 2002b; Mathews and Bretagnon, 2002), has not been taken into account in the table due to the large uncertainty in the triaxiality of the core (Dehant, 2002, private communication). The Stokes coefficients of the geopotential are from the model JGM-3. The diurnal components of these variations should be considered simi- larly to the diurnal and semidiurnal variations due to ocean tides. They are not part of the polar motion values reported to the IERS and dis- tributed by the IERS and should therefore be added after interpolation. The long-periodic terms, as well as the secular variation, are already contained in the observed polar motion and need not be added to the reported values.

37 38 No. 32 .. oiino h E nteITRS the in TEO the of Position 5.4.3 .. at oainAngle Rotation Earth 5.4.4 IERS Note Technical al . offiinso i(ruet n o(ruet n(∆ in cos(argument) and sin(argument) of Coefficients 5.1 Table l l χ n 8.6 9.0950103 085.565 2 0 3 0 0 0 3 8.5 9.1071941 085.555 3 0 3 0 0 0 3 6.6 7222 0 13.718786 075.455 27.212221 1 065.565 0 0 1 0 0 1 1 0 0 0 0 3 0 3 3 6.5 7318 0 27.321582 065.555 1 0 1 0 0 0 3 7.5 .646 0 555.555 0.9624365 14 0 175.455 0.9972696 0 0 165.555 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 0 1 0 0 1 2 0 1 2 0 1 2 0 0 1 2 0 0 1 2 1 27.431826 1 2 065.545 1 2 1 2 2 1 2 0 2 1 1 0 1 0 0 0 1 0 1 0 0 3 1 1 0 0 3 1 0 3 0 0 3 0 3 0 0 3 0 0 3 0 3 4 − − − − − − − 7.5 14.698136 073.655 1 2 1 0 1 5.5 .378 0 1.0347187 155.655 0 0 0 0 1 0 0 1 1 1 2190.3501 055.665 3231.4956 055.655 6159.1355 0 055.645 1 0 2 0 1 0 1 0 1 0 1 0 1 1 n h xrsin o h udmna ruet Dlua ruet)aegiven are arguments) (Delaunay arguments (40). fundamental by the for expressions the and rvtto o eren o orgdErh nt are Units Earth. nonrigid a for n) degree (of gravitation aeo eua oa oin( motion polar secular of Rate ruetDosnPeriod Doodson argument 0 − − − − − − − − D F 2 2 0 0 2 2 0 2 0 2 2 438.35990 056.444 0 0 0 1 1 eainhpwt T sgvnb Capitaine by given as UT1 with relationship Angle, Rotation Earth The quantity The a main Its data. IERS as: the written using be extrapolated can be component can and measurements of components Some tion. h urn enapiue o h hnlra n nulwobbles annual and Chandlerian 2002): Bizouard, the and for (Lambert amplitudes gives mean current the (Capitaine considered period al. the et in respectively wobbles, annual and rian obe ec auso h re f0 of annual order and Chandlerian the the of for values amplitudes reach the wobbles if even century, next the − − − c 2 1 1 193.55971 057.455 365.24219 056.555 1 1 2 1 and 96.Tevleof value The 1986). , θ ( − − − − − − − − − a T ubr(as i o i cos sin cos sin (days) number Ω a 6.6 .913 1 0.9971233 165.565 1.0027454 1 163.555 1.0758059 2 1.0759762 145.555 1.1134606 145.545 2 1.1195149 137.455 1 1.1196992 135.655 2 135.645 2 1 0 411.80661 056.454 1 6798.3837 055.565 1 u 2 = ) en h vrg mltds(nacscns fteChandle- the of ) arc (in amplitudes average the being s π 0 (0 rnfrainBtenteClsiladTretilSystems Terrestrial and Celestial the Between Transformation 5 sol estv otelretvrain nplrmo- polar in variations largest the to sensitive only is . 707724 1 + 7790572732640 µ sy u otezr rqec frequency zero the to due as/y) s 0 = s s − 0 0 θ s 0 aet eeautd npicpe rmthe from principle, in evaluated, be to have ilteeoeb esta . a through mas 0.4 than less be therefore will sotie yteueo t conventional its of use the by obtained is , 0 . 0015( = − 47 a − − c 2 − − − − − − − − − − − − − µ . / 28 11 00273781191135448 . as 1 0 0 0 0 0 0 4 4 2 0 2 0 0 5 ...... 00 + 2 50 05 80 08 10 11 20 0 02 14 93 89 05 76 27 0 69 65 53 93 2 76 6 84 1 36 0 14 1 44 0 31 44 0 0 30 0 19 99 0 0 46 03 t. n 0 and x p x, a tal. et µ − − − − − − − − a 2 as; ∆ ) 3 0 0 8 0 0 1 0 t, . y ...... 1 70 07 10 11 30 33 70 07 62 97 80 0 21 30 43 8 19 0 15 3 08 23 11 11 0 73 48 52 23 25 32 25 37 21 26 00 63 ) 00 χ nutation (2000), epciey Using respectively. eoe GMST+ denotes − − − − − − − − − − − − − − 0 0 0 2 6 1 0 1 0 0 0 0 0 0 ...... T 10 01 10 01 12 01 00 5 00 02 132 11 01 30 43 14 19 15 55 42 11 11 0 73 48 52 23 25 32 25 2 1 1 17 19 25 0 18 05 u otidal to due u ) y , p − − − − − − − − − − − − 11 4 1 0 0 3 4 2 0 2 0 0 ...... (13) (11) (12) 55 88 66 56 09 39 31 68 76 27 68 92 86 93 84 36 91 40 04 11 76 14 44 31 44 55 π 5.4 Parameters to be used in the Transformation IERS Technical

Note No. 32

where Tu=(Julian UT1 date−2451545.0), and UT1=UTC+(UT1−UTC), or equivalently

θ(T ) = 2π(UT1 number elapsed since 2451545.0 u (14) +0.7790572732640 + 0.00273781191135448Tu),

the quantity UT1−UTC to be used (if not estimated from the observa- tions) being the IERS value. This definition of UT1 based on the CEO is insensitive at the microarc- second level to the precession-nutation model and to the observed celes- tial pole offsets. Therefore, in the processing of observational data, the quantity s provided by Table 5.2c must be considered as independent of observations.

5.4.5 Motion of the CIP in the GCRS Developments of the coordinates X and Y of the CIP in the GCRS, valid at the microarcsecond level, based on the IERS 1996 model for precession, nutation and pole offset at J2000.0 with respect to the pole of the GCRS, have been provided by Capitaine et al. (2000). New developments of X and Y based on the IAU 2000A or IAU 2000B model (see the following section for more details) for precession-nutation and on their corresponding pole offset at J2000.0 with respect to the pole of the GCRS have been computed at the same accuracy (Capitaine et al., 2003a). These developments have the following form:

X = −0.0166169900 + 2004.1917428800t − 0.4272190500t2 −0.1986205400t3 − 0.0000460500t4 + 0.0000059800t5 P + i[(as,0)i sin(ARGUMENT) + (ac,0)i cos(ARGUMENT)] P (15) + i[(as,1)it sin(ARGUMENT) + (ac,1)it cos(ARGUMENT)] P 2 2 + i[(as,2)it sin(ARGUMENT) + (ac,2)it cos(ARGUMENT)] + ··· ,

Y = −0.0069507800 − 0.0253819900t − 22.4072509900t2 +0.0018422800t3 + 0.0011130600t4 + 0.0000009900t5 P + i[(bc,0)i cos(ARGUMENT) + (bs,0)i sin(ARGUMENT)] P (16) + i[(bc,1)it cos(ARGUMENT) + (bs,1)i t sin(ARGUMENT)] P 2 2 + i[(bc,2)it cos(ARGUMENT) + (bs,2)it sin(ARGUMENT)] + ··· ,

the parameter t being given by expression (2) and ARGUMENT being a function of the fundamental arguments of the nutation theory whose expressions are given by (40) for the lunisolar ones and (41) for the planetary ones. These series are available electronically on the IERS Convention Cen- ter website (Tables 5.2a and 5.2b) at <4>. tab5.2a.txt for the X coordinate and at tab5.2b.txt for the Y coordinate. An extract from Tables 5.2a and 5.2b for the largest non-polynomial terms in X and Y is given hereafter. The numerical values of the coefficients of the polynomial part of X and Y are derived from the development as a function of time of the precession in longitude and obliquity and pole offset at J2000.0 and the amplitudes (as,j)i,(ac,j)i,(bc,j)i,(bs,j)i for j = 0, 1, 2, ... are derived from the amplitudes of the precession and nutation series. The ampli- tudes (as,0)i,(bc,0)i of the sine and cosine terms in X and Y respectively are equal to the amplitudes Ai × sin 0 and Bi of the series for nutation

4ftp://.usno.navy.mil/conv2000/chapter5/

39 40 No. 32 394.928.100202000000000 0 0 0 0 0 0 0 0 2 0 2 0 2 0 0 2187.91 0 41.19 12814.01 197.53 1309 1308 1307 IERS xrc rmTbe .aad52 aalbeat (available for 5.2b (15) and development 5.2a Tables from Extract oun o1 r ie y(0 n (41). and (40) by given are 17 to 4 columns (unit Model Precession-Nutation Note Technical 6 5018 7.900001000000000 0 0 0 0 0 0 0 0 1 0 0 0 0 2024.68 878.89 11714.49 965 153041.82 964 963 ...... i i i i 2 3 1 80.2400 0 0 0 0 0 0 0 0 0 0 0 0 1 0 470.05 58707.02 82168.76 5 4 581 253.600001000000000 0 0 0 0 0 0 0 0 1 0 0 0 0 9205236.26 1538.18 2 1 3.19866 0 0 0 0 0 0 0 0 0 2 0 2 0 0 97846.69 137.41 5 4 3 − − 841.412.700001000000000 0 0 0 0 0 0 0 0 1 0 0 0 0 1328.67 6844318.44 − 523908.04 − 05.2112 0 0 0 0 0 0 0 0 0 2 0 2 0 0 111.23 90552.22 − 384 0831 0 0 0 0 0 0 0 0 0 1 0 0 0 0 205833.15 3328.48 − − 5.6533.2002 0 0 573033.42 458.66 ( ( ( ( a a 74 23.2012 1 0 22438.42 17.40 29.05 b b s, s, s, s, 0 0 1 1 ) ) ) ) i i i i − X 91.400002000000000 0 0 0 0 0 0 0 0 2 0 0 0 0 89618.24 − − − − ( 4.6002 0 0 544.76 8.2002 0 0 289.32 ( ( t ( ( a a tppr)ad(6 for (16) and part) ( ) 76 0 0 0 0 0 0 0 0 0 2 0 0 0 0 27.64 09 0 0 0 0 0 0 0 0 0 2 0 2 0 0 50.99 b b c, c, c, c, Setenmespoie eo n(9 n 2)frteequantities.) these for (28) and (19) in below provided numbers the (See and where ffcs h coordinates The effects. ordinate longitude in issare biases (in contributions The nutation. and uainqatte r Cptie 1990): (Capitaine, are a quantities coordinates and nutation the section re- between following angles relationships the computation. nutation The in the the described for are for available date and is of quantities subroutine developments precession The the the to Model. for ferred Precession-Nutation used 2000 be IAU to the by vided CIP, at the time offset each of equinox ascension.” coordinates the right celestial from in The ones bias following “frame the called celestial the and also from J2000.0 J2000.0 contribution at the being offset coordinate pole each in term first the t 0 0 1 1 2 ) ) ) ) i i i i sin, dα µ l l l l l l l l t ξ s.Teepesosfrtefnaetlagmnsapaigin appearing arguments fundamental the for expressions The as). 0 2 0 o,..wihoiiaefo rse em ewe precession between terms crossed from originate which ... cos, h ih seso ftema qio fJ000i h CRS. the in J2000.0 of equinox mean the of ascension right the X and 0 0 0 0 t safnto ftepeeso n uainqatte pro- quantities nutation and precession the of function a as and × D F D F D F D F η 0 sin dX dY rnfrainBtenteClsiladTretilSystems Terrestrial and Celestial the Between Transformation 5 − − − − − Y r h eeta oeost ttebsceohJ2000.0 epoch basic the at offsets pole celestial the are 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 2 2 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 2 2  0 hc oti otiuinfo crossed-nutation from contribution a contain which < n biut,ecp o e em nec co- each in terms few a for except obliquity, and = = µ Ω Ω Ω Ω X 4 Y Y s oepesos(5 n 1)fo h frame the from (16) and (15) expressions to as) > ( X L L L L − − o h ags o-oyoiltrsi the in terms non-polynomial largest the for ) t bto at optbewt A 2000A IAU with compatible part) (bottom ) Me Me Me Me = = 6819 16617 and L L L L . Y X ¯ 2 ¯ Y . e V e V e V e V 0 − + + − oti oso em in terms Poisson contain 141 η ξ L L L L X 1 0 0 E E E E . + 6 − . and 9 t L L L L dα dα 2 t X Ma Ma Ma Ma 0 + 0 + 0 0 Y and X, , Y ¯ a lob bandat obtained be also can , ¯ . . L L L L o Ω cos 7 i Ω sin 5 J J J J Y L L L L n h precession- the and Sa Sa Sa Sa , , L L L L U U U U t L L L L sin, Ne Ne Ne Ne t (18) (17) p p p p cos, A A A A 5.4 Parameters to be used in the Transformation IERS Technical

Note No. 32

The mean equinox of J2000.0 to be considered is not the “rotational dy- namical mean equinox of J2000.0” as used in the past, but the “inertial dynamical mean equinox of J2000.0” to which the recent numerical or analytical solutions refer. The latter is associated with the ecliptic in the inertial sense, which is the plane perpendicular to the vector of the orbital motion of the Earth- barycenter as com- puted from the velocity of the barycenter relative to an inertial frame. The rotational equinox is associated with the ecliptic in the rotational sense, which is perpendicular to the vector angular momentum computed from the velocity referred to the rotating orbital plane of the Earth-Moon barycenter. (The difference between the two angular momenta is the an- gular momentum associated with the rotation of the orbital plane.) See Standish (1981) for more details. The numerical value for dα0 as derived from Chapront et al. (2002) to be used in expression (18) is

00 dα0 = (−0.01460 ± 0.00050) . (19)

Quantities X¯ and Y¯ are given by:

X¯ = sin ω sin ψ, (20) Y¯ = − sin 0 cos ω + cos 0 sin ω cos ψ

00 where 0 (= 84381.448 ) is the obliquity of the ecliptic at J2000.0, ω is the inclination of the true equator of date on the fixed ecliptic of epoch and ψ is the longitude, on the ecliptic of epoch, of the node of the true equator of date on the fixed ecliptic of epoch; these quantities are such that ω = ωA + ∆1; ψ = ψA + ∆ψ1, (21)

where ψA and ωA are the precession quantities in longitude and obliquity (Lieske et al., 1977) referred to the ecliptic of epoch and ∆ψ1, ∆1 are the nutation angles in longitude and obliquity referred to the ecliptic of epoch. (See the numerical developments provided for the precession quantities in (30) and (31).) ∆ψ1, ∆1 can be obtained from the nutation angles ∆ψ, ∆ in longitude and obliquity referred to the ecliptic of date. The following formulation from Aoki and Kinoshita (1983) has been verified to provide an accuracy better than one microarcsecond after one century:

∆ψ sin ω = ∆ψ sin  cos χ − ∆ sin χ , 1 A A A A (22) ∆1 = ∆ψ sin A sin χA + ∆ cos χA,

ωA and A being the precession quantities in obliquity referred to the ecliptic of epoch and the ecliptic of date respectively and χA the plane- tary precession along the equator (Lieske et al., 1977). As VLBI observations have shown that there are deficiencies in the IAU 2000A of the order of 0.2 mas (Mathews et al., 2002), the IERS will con- tinue to publish observed estimates of the corrections to the IAU 2000 Precession-Nutation Model. The observed differences with respect to the conventional celestial pole position defined by the models are mon- itored and reported by the IERS as “celestial pole offsets.” Such time dependent offsets from the direction of the pole of the GCRS must be provided as corrections δX and δY to the X and Y coordinates. These corrections can be related to the current celestial pole offsets δψ and δ using the relationship (20) between X and Y and the precession-nutation quantities and (22) for the transformation from ecliptic of date to eclip- tic of epoch. The relationship can be written with one microarcsecond accuracy for one century:

δX = δψ sin  + (ψ cos  − χ )δ, A A 0 A (23) δY = δ − (ψA cos 0 − χA)δψ sin A.

41 42 No. 32 .. oiino h E nteGCRS the in CEO the of Position 5.4.6 IERS Note Technical al .cDvlpetof Development 5.2c Table s ( t = ) +9 − +1 oe:altrseceig0.5 exceeding terms all Model: XY/ . . 84 71 saalbeeetoial nteIR ovninCne est at website Center Convention of IERS velopment the on tab5.2c.txt electronically available that for is series so complete (36)). current The chosen (see the consis- UT1 and 2003) relationship estimating previously January (ERA) for (1 Angle procedure was change Rotation VLBI of Earth date which the the with at tent s, UT1 of continuity for ensure term s for constant development direct The a than accuracy same quantity the the reach IERS for the provided (Capitaine on is (16) based and at that (15) offset by to similar celestial way corresponding for a in the (Capitaine 1996 derived as Conventions been well has as J2000.0 Model sion-Nutation of development numerical The where matrix transformation the rotation replacing the to with equivalent practically is This of position celestial Precession- corrected by 2000 given the IAU is offsets, the CIP Nuta- these on the Core Using 5.5.1 Free sub-section Model. the in Nutation of contribution described the (FCN) include tion offsets observed These inNtto Model. sion-Nutation Ω+1 l l l l 3Ω 2 2 2 2 2 2Ω Ω Argument ( 0 0 F F F F F +0 − Ω + t t J +1 − Ω + 4+3808 + 94 + 2 2 i 3 + Ω sin 2000 s Ω + 3Ω + − − − sin(2 +0 Ω +1 Ω 8 sn h eeomnsof developments the using (8) 2 2 2 Q D D D . X IAU )=0 a o enfi (Capitaine fit been now has 0, = 0) F Ω+4 2Ω + Ω + 3Ω + = 2Ω) + α . 57 ersnsthe represents k s X n h em agrta 0.5 than larger terms the and s t r rvddi al .cwt iraceodaccuracy. microarcsecond with 5.2c Table in provided are IU20)+ 2000) (IAU ( o Ω+743 + 2Ω cos . t rnfrainBtenteClsiladTretilSystems Terrestrial and Celestial the Between Transformation 5 35 optbewt A 00 Precession-Nutation 2000A IAU with compatible ) − Q t ˜ 8 − = . 85 119

t 2 s . i 2Ω sin − 94 + µ Q δX 1 0 0 1 . sdrn h nevl17–05(unit 1975–2025 interval the during as tal. et 53 t tal. et XY/ ( s X Y δX, 2 t s arxbsdo h A 00Preces- 2000 IAU the on based matrix ) + t − 2 optbewt h A 00 Preces- 2000A IAU the with compatible i 56 + Ω sin XY/ − 72574 03) h ueia development numerical The 2003a). , ihaltrslre hn0.1 than larger terms all with 2 00.I eut rmteexpression the from results It 2000). , δY X ,wihrqie ee em to terms fewer requires which 2, and . − − Amplitude − − − − 09 δX δY 1 = 63 2640 1 2 11 11 t ...... 3 Y . 63 63 41 72 26 98 02 57 . . . µ Y 91 53 21 75 + ! soe 5yasi h de- the in 25 over as sfntoso iegiven time of functions as IU20)+ 2000) (IAU tal. et . t 73 P Q 2 sin(2 IAU k C C 03)i re to order in 2003b) , , k k F sin s − . α 2 k D δY. 2Ω) + µ (24) (25) as). µ as Q 5.5 IAU 2000A and IAU 2000B Precession-Nutation Model IERS Technical

Note No. 32

5.5 IAU 2000A and IAU 2000B Precession-Nutation Model 5.5.1 Description of the Model The IAU 2000A Precession-Nutation Model has been adopted by the IAU (Resolution B1.6) to replace the IAU 1976 Precession Model (Lieske et al., 1977) and the IAU 1980 Theory of Nutation (Wahr, 1981; Seidel- mann, 1982). See Dehant et al. (1999) for more details. This model, developed by Mathews et al. (2002), is based on the solution of the lin- earized dynamical equation of the wobble-nutation problem and makes use of estimated values of seven of the parameters appearing in the the- ory, obtained from a least-squares fit of the theory to an up-to-date precession-nutation VLBI data set (Herring et al., 2002). The nutation series relies on the Souchay et al. (1999) Rigid Earth nutation series, rescaled by 1.000012249 to account for the change in the dynamical el- lipticity of the Earth implied by the observed correction to the lunisolar precession of the equator. The nonrigid Earth transformation is the MHB2000 model of Mathews et al. (2002) which improves the IAU 1980 Theory of Nutation by taking into account the effect of mantle anelas- ticity, ocean tides, electromagnetic couplings produced between the fluid outer core and the mantle as well as between the solid inner core and fluid outer core (Buffett et al., 2002) and the consideration of nonlinear terms which have hitherto been ignored in this type of formulation. The resulting nutation series includes 678 lunisolar terms and 687 plan- etary terms which are expressed as “in-phase” and “out-of-phase” com- ponents with their time variations (see expression (29)). It provides the direction of the celestial pole in the GCRS with an accuracy of 0.2 mas. It includes the geodesic nutation contributions to the annual, semiannual and 18.6- terms to be consistent with including the geodesic preces- sion pg in the precession model and so that the BCRS and GCRS are without any time-dependent rotation. The IAU 1976 Precession Model 00 uses pg = 1.92 /c and the theoretical geodesic nutation contribution (Fukushima, 1991) used in the MHB model (Mathews et al., 2002) is, in µas, for the nutations in longitude ∆ψg and obliquity ∆g ∆ψ = −153 sin l0 − 2 sin 2l0 + 3 sin Ω, g (26) ∆g = 1 cos Ω, where l0 is the of the and Ω the longitude of the ascending node of the Moon. On the other hand, the FCN, being a free motion which cannot be predicted rigorously, is not considered a part of the IAU 2000A model. The IAU 2000 nutation series is associated with improved numerical values for the precession rate of the equator in longitude and obliquity, which correspond to the following correction to the IAU 1976 precession:

00 δψA = (−0.29965 ± 0.00040) /c, 00 (27) δωA = (−0.02524 ± 0.00010) /c, as well as with the following offset (originally provided as frame bias in dψbias and dbias) of the direction of the CIP at J2000.0 from the direction of the pole of the GCRS:

00 ξ0 = (−0.0166170 ± 0.0000100) , 00 (28) η0 = (−0.0068192 ± 0.0000100) . The IAU 2000 Nutation Model is given by a series for nutation in longi- tude ∆ψ and obliquity ∆, referred to the mean ecliptic of date, with t measured in Julian centuries from epoch J2000.0:

PN 0 00 000 ∆ψ = (Ai + Ait) sin(ARGUMENT) + (Ai + Ai t) cos(ARGUMENT), i=1 (29) PN 0 00 000 ∆ = i=1(Bi + Bit) cos(ARGUMENT) + (Bi + Bi t) sin(ARGUMENT).

43 44 No. 32 nt r a n a/ o h offiinsadtertm aitosrsetvl.Prosaein are Periods respectively. variations time their and coefficients the obliquity. and for longitude mas/c in terms days. and “out-of-phase” mas and at “in-phase” are (available the Units nutations) for (planetary components 5.3b largest and the nutations) viding (lunisolar 5.3a Tables from Extract 12- 6.2 152 009 952 .29001 .00003 -0.0001 0.0000 0.0132 0.0367 0.0000 0.0000 0.0000 0.0000 0.0318 0.0111 0.0816 0.0358 0.0001 0.0299 -0.0063 0.0000 0.0380 -9.5929 12.9025 -0.0872 0.0000 0.0018 0.0000 0.0005 -0.0174 -0.0036 20.0728 -0.0494 -0.6750 0.0002 -0.1924 -30.1461 21.5829 -0.0015 -0.0524 -0.0367 0.0000 1.1817 0.0073 -0.0001 -0.0677 -38.7298 0.1374 22.4386 -0.0291 -0.0184 9.133 0.0000 71.1159 365.225 7.3871 0.0002 0.1226 -0.0698 0.2796 13.633 2 0.0470 -2 2 -0.0485 -0.3633 -0.0003 2 -51.6821 27.555 0 -89.7492 97.8459 -0.4587 -1 0 2 1 147.5877 0.0012 0 0 0.0207 0.0002 1 2 0 -0.0234 121.749 -1.3696 0 0 1.5377 -0.3015 365.260 207.4554 0 0 -227.6413 573.0336 2 0.0029 0 -2 1 2 0 -0.1675 3.3386 13.661 0 1 -3399.192 0 0 0.9086 2 -1317.0906 0 1 9205.2331 0 0 2 182.621 -17.4666 0 0 -17206.4161 0 2 2 0 -6798.383 -2 0 2 1 0 0 0 0 0 0 IERS Note Technical l l 10000388 012 002 .000.0000 0.0246 0.0000 -0.0232 0.0000 -0.0435 -0.0026 -0.0460 -0.1223 0.0000 37883.60 398.88 0.0000 0.0000 0.0000 0.0505 0.1485 0.0000 0 0.0000 0.0000 583.92 0.0000 0 0 0.1440 0.0000 0.1604 0.0932 0 34075700.82 -0.1166 0 -0.0641 -0.0462 0 2165.30 0.0000 0.0319 -651391.30 0 0 0 0 0.0000 0.0598 0 0 0 -0.0771 -0.2150 0 0.1200 0 -0.1286 2 0 2957.35 0 2 0.1647 -88082.01 0.2409 -1 5 0 0 0.2735 0 -0.1444 0 311927.52 -1 0 4 0.5123 0 2 0 0 0 1 0 -16 -0.3084 0 0 1 0 311921.26 3 0 8 0 0 0 2 0 -8 0 0 0 -1 0 0 0 0 1 0 1 4 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 5 0 0 0 0 0 12 0 0 5 0 0 -2 -8 0 0 0 0 0 -3 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 1 0 0 l l 6 5 ftp://maia.usno.navy.mil/conv2000/chapter5/IAU2000B.f ftp://maia.usno.navy.mil/conv2000/chapter5/IAU2000A.f 0 0 D F D F Ω Period Ω L Me L e V L E L Ma e est,frtelnslradpaeaycmoet,rsetvl at respectively Cen- components, Convention planetary IERS and 5.8. lunisolar section the tab5.3a.txt the in on given (see for electronically website, be series available ter will these are below) of 5.3b series and arguments These 5.3a and Tables coefficients the of the extract about details More vial lcrnclyo h ESCneto etrwbieat IAU not website the Center is of Convention does subroutine that IERS 2000B that to the IAU The respect on accuracy consideration. with electronically 1995–2050. an available mas period under the 1 with during period than the model motion greater 2000A time for difference pole account the a to in celestial in result bias the mas terms a provides 1 plus planetary the It terms the at lunisolar (2003). of Luzum only 80 and effect model than McCarthy by fewer a developed designated includes need been has It model, who model those abridged a Such for an available level. B1.6, is 2000B, Resolution IAU to by obser- used astronomical recommended be recent As also the can software on The based FCN vations. (FCN). nutation expected excep- core the the free model with components the 1976 all of IAU includes tion d the nutation” bias “total longitude to The frame Precession- in corrections obliquity. the nutation the 2000A plus of model: rates, IAU contribution 2000 precession the the MHB plus the implement obliquity, on and to based quantities elec- Model available Nutation the is at produces website Herring, Center It T. Convention IERS by the provided on subroutine, tronically 2000A IAU The A i L J L Sa L A U i 0 and L Ne rnfrainBtenteClsiladTretilSystems Terrestrial and Celestial the Between Transformation 5 tab5.3b.txt p A B i eidLniueObliquity Longitude Period . B i 0 ψ bias A A i i 00 n d and A A  i 00 bias i 000 < nlniueand longitude in 5 > B B . i i 00 < 4 > B B pro- ) < i i 00 000 6 > . 5.6 Procedure to be used for the Transformation consistent with IAU 2000 Resolutions IERS Technical

Note No. 32

5.5.2 Precession Developments compatible with the IAU2000 Model The numerical values for the precession quantities compatible with the IAU 2000 Precession-Nutation Model can be provided by using the devel- opments (30) of Lieske et al. (1977) to which the estimated corrections (27) δψA and δωA to the IAU 1976 precession have to be added. The expressions of Lieske et al. (1977) are

00 00 2 00 3 ψA = 5038.7784 t − 1.07259 t − 0.001147 t , 00 2 00 3 ωA = 0 + 0.05127 t − 0.007726 t , 00 00 2 00 3 (30) A = 0 − 46.8150 t − 0.00059 t + 0.001813 t , 00 00 2 00 3 χA = 10.5526 t − 2.38064 t − 0.001125 t , and 00 00 2 00 3 ζA = 2306.2181 t + 0.30188 t + 0.017998 t , 00 00 2 00 3 θA = 2004.3109 t − 0.42665 t − 0.041833 t , (31) 00 00 2 00 3 zA = 2306.2181 t + 1.09468 t + 0.018203 t , 00 with 0 = 84381.448 . Due to their theoretical bases, the original development of the precession quantities as function of time can be considered as being expressed in TDB. The expressions compatible with the IAU 2000A precession and nutation are: 00 00 2 00 3 ψA = 5038.47875 t − 1.07259 t − 0.001147 t , 00 00 2 00 3 ωA = 0 − 0.02524 t + 0.05127 t − 0.007726 t , 00 00 2 00 3 (32) A = 0 − 46.84024 t − 0.00059 t + 0.001813 t , 00 00 2 00 3 χA = 10.5526 t − 2.38064 t − 0.001125 t ,

and the following series has been developed (Capitaine et al., 2003c) in order to match the 4-rotation series for precession R1(−0) · R3(ψA) · R1(ωA) · R3(−χA), called the “canonical 4-rotation method,” to sub- microarcsecond accuracy over 4 centuries: 00 00 00 2 00 3 ζA = 2.5976176 + 2306.0809506 t + 0.3019015 t + 0.0179663 t −0.000032700t4 − 0.000000200t5, 00 00 2 00 3 θA = 2004.1917476 t − 0.4269353 t − 0.0418251 t −0.000060100t4 − 0.000000100t5, 00 00 00 2 00 3 zA = −2.5976176 + 2306.0803226 t + 1.0947790 t + 0.0182273 t +0.000047000t4 − 0.000000300t5. (33)

Note that the new expression for the quantities ζA and zA include a con- stant term (with opposite signs) which originates from the ratio between the precession rate in ωA and in ψA sin 0. TT is used in the above expressions in place of TDB. The largest term in the difference TDB−TT being 1.7 ms × sin l0, the resulting error in the precession quantity ψA is periodic, with an annual period and an amplitude of 2.700 × 10−9, which is significantly under the required mi- croarcsecond accuracy.

5.6 Procedure to be used for the Transformation consistent with IAU 2000 Resolutions There are several ways to implement the IAU 2000 Precession-Nutation Model, and the precession developments to be used should be consis- tent with the procedure being used. The subroutines available for the different procedures are described below.

45 46 No. 32 . xrsino rewc ieelTm eerdt h CEO the to referred Time Sidereal Greenwich of Expression 5.7 IERS Note Technical S = GST dT 0 + θ + biut eerdt h citco pc and epoch of ecliptic the to referred obliquity eern otenttossmlrt hs sdi al .c h nu- the 5.2c, Table in used those to is similar expression notations merical the IERS to the Referring on available based is at Time website Model Sidereal Center Precession-Nutation Convention Greenwich IAU2000A for as the (Capitaine expression transformation change on the of new providing date the the series at The with UT1 in level continuity as microarcsecond well a at performed The consistency those to 2003). similar January computations Precession-Nutation for using (1 obtained, 2000 change been IAU the of has the Model date along with the consistent equinox expression at the numerical UT1 from in continuity measured ensure CEO and and the equator, GST precession of moving t. accumulated ascension epoch the the right for to J2000.0 the account from (34) ascension right of in nutation parts equator. four the along last precession The planetary the (32), in given ∆ level: microarcsecond a at 1993) Gontier, and Capitaine Angle” 1983; Rotation “Earth the to related θ is (GST) Time Sidereal Greenwich four Capitaine including matrix (in precession the Wallace ( is series, by 2000 GCRS MHB the described the using procedure and model TRS rigorous 2000A/B al. the a IAU between follow the transformation to classical implementing the for using option recommended taking but The corrections. angles 2000 same for IAU the rates the give precession account (33) into 1976 Expressions contri- IAU (31)). the the expressions including to (see correction model the precession-nutation provides of subroutine the butions IAU2000A of the rates. paradigm, components precession classical the the 1976 of IAU support In the to quantities (32).) precession corrections series, expressions on the nutation (See as 2000 account MHB well into the as on taking J2000.0 based at be computa- must offsets the This on for (22). used to be (18) and sions also coordinates model can con- the precession-nutation already procedure of new These Another tion GCRS. the biases. the for for frame in 5.2 expressions the CEO Tables and the proper (16) and the and CIP tain (15) the by of precession-nutation provided positions 2000A expressions the IAU the the the from on with transform based compatible to is ITRS procedure the complete to the GCRS paradigm, new the Using etos19 MCrh,19)adte pl h orcin othe to software. Con- 2000A/B corrections IERS IAU the the appropriate apply in the Nu- then by described of provided as and Theory past model expressions 1996) 1980 the (McCarthy, classical IAU in 1996 and the as ventions Model using proceed Precession should continue one 1976 to tation, IAU elects frame the one on the when based for case matrix the rotation In separate a and quantities bias. these for (32) Z R t eerdt h E ytefloigrltosi Ak n Kinoshita, and (Aoki relationship following the by CEO the to referred ψ 0 02.Ti rcdr scmoe ftecasclntto matrix nutation classical the of composed is procedure This 2002). , 1 t 1 ( s ( ∆ , − ψ n olwn rcdr,wihi ecie eo,t ensure to below, described is which procedure, a following and  A 0 \  ) ∆ + 1 · ˙ ie y(2,bigtentto nlsi ogtd and longitude in angles nutation the being (22), by given , R 3 ψ ( 1 ψ cos( ) A rnfrainBtenteClsiladTretilSystems Terrestrial and Celestial the Between Transformation 5 ) · R ω 1 A ( dT ω X ∆ + A 0 ) and sacntn emt efitdi re to order in fitted be to term constant a is ·  R 1 ) 3 tab5.4.txt dt Y ( − − fteCPi h CSuigexpres- using GCRS the in CIP the of χ A χ )uigteudtddevelopments updated the using )) A ∆ + . ψ cos χ A  hs eeomn is development whose , A − ∆ ψ ψ 1 tal. et A cos ω , − θ ζ ω A 2003b). , A provides A  , θ , , A Z χ , (34) z , A et A , 5.8 The Fundamental Arguments of Nutation Theory IERS Technical

Note No. 32

GST = 0.01450600 + θ + 4612.1573996600t + 1.3966772100t2 00 3 00 4 −0.00009344 t + 0.00001882 t + ∆ψ cos A (35) P 0 00 − k Ck sin αk − 0.00000087 t sin Ω. P 0 The last two terms of GST, − k Ck sin αk − 0.87 µas t sin Ω, are the complementary terms to be added to the current “equation of the ,” ∆ψ cos A, to provide the relationship between GST and θ with microarcsecond accuracy. This replaces the two complementary terms provided in the IERS Conventions 1996. A secular term similar to that appearing in the quantity s is included in expression (35). This expression for GST used in the classical transformation based on the IAU 2000A precession-nutation ensures consistency at the microarcsec- ond level after one century with the new transformation using expres- sions (14) for θ, (15) and (16) for the celestial coordinates of the CIP and Table 5.2c for s. The numerical values for the constant term dT0 in GST which ensures continuity in UT1 at the date of change (1 January 2003) and for the corresponding constant term in s have been found to be dT = +14506µas, 0 (36) s0 = +94µas. The change in the polynomial part of GST due to the correction in the precession rates (27) corresponds to a change dGMST (see also Williams, 1994) in the current relationship between GMST and UT1 (Aoki et al., 1982). Its numerical expression derived from expressions (35) for GST and (13) for θ(UT1), minus the expression for GMST1982(UT1), can be written in microarcseconds as

dGMST = 14506 − 274950.12t + 117.21t2 − 0.44t3 + 18.82t4. (37)

The new expression for GST clearly distinguishes between θ, which is expressed as a function of UT1, and the accumulated precession-nutation in , which is expressed in TDB (or, in practice, TT), whereas the GMST1982(UT1) expression used only UT1. This gives rise to an additional difference in dGMST of (TT − UT1) multiplied by the speed of precession in right ascension. Using TT−TAI=32.184 s, this can be expressed as: 47 µas +1.5µas (TAI−UT1), where TAI−UT1 is in seconds. On 1 January 2003, this difference will be about 94 µas (see Gontier in Capitaine et al., 2002), using an estimated value of 32.3 s for TAI−UT1. This contribution for the effect of time scales is included in the value for dT0 and s0.

5.8 The Fundamental Arguments of Nutation Theory

5.8.1 The Multipliers of the Fundamental Arguments of Nutation Theory Each of the lunisolar terms in the nutation series is characterized by a set of five integers Nj which determines the ARGUMENT for the term as a linear combination of the five Fundamental Arguments Fj, namely the 0 P5 Delaunay variables (`, ` ,F,D, Ω): ARGUMENT = j=1 NjFj, where the values (N1, ··· ,N5) of the multipliers characterize the term. The Fj are functions of time, and the angular frequency of the nutation described by the term is given by

ω ≡ d(ARGUMENT)/dt. (38)

The frequency thus defined is positive for most terms, and negative for some. Planetary nutation terms differ from the above only in that P14 0 0 ARGUMENT = j=1 NjFj, F6 to F13, as noted in Table 5.3, are the mean longitudes of the to including the Earth

47 48 No. 32 .. eeomn fteAgmnso uioa Nutation Lunisolar of Arguments the of Development 5.8.2 .. eeomn fteAgmnsfrtePaeayNutation Planetary the for Arguments the of Development 5.8.3 IERS Note Technical F F F F F orsodn mltdso nutation. of amplitudes corresponding o h ags ffc ntentto ruet,wihpoue neg- a produces which 10 arguments, than nutation (less the difference in TDB effect ligible to due largest difference the the as for TDB of place in used expressions, original the with In radians 2000 in below ELP given and are centuries. developments 1982) Their (Bretagnon, Simon (1994). of VSOP82 developments plan- and of the 1983) Souchay(Chapront-Touz´e for Chapront, constants arguments by and and the provided in those theories used are planets nutations the etary of longitudes mean The oeta nteMB20 oe ipie xrsin r sdfrthe for amplitudes 0 nutation used than the are less in expressions difference is simplified maximum code, The 2000 nutation. MHB planetary the in that Note Williams and constants 1992 by given (Simon where are TDB nutation developments of following arguments fundamental the the for expressions The ∆ for the series unchanged. that the series fact in these sin(ARGUMENT) the of from coefficient the arises of assignments the ( in replacement differences The multipliers necessarily series. of not set do obliquity same and the longitude assign in nutations of tables Different the for write, may frequency the series, one nutation studies, and nutation time-independent, in involved tively scales time Over longitude in ( where 5 4 3 2 1 l Me ≡ ≡ ≡ ≡ ≡ , = Ω D F l l 0 l L e V 125 = 297 = = 93 = = 357 = = 134 = = steMa ogtd fteMoon. the of Longitude Mean the is , enLniueo h sedn oeo h Moon the of Node Ascending the of Longitude Mean +0 Sun +0 the from Moon the of Elongation Mean − L +0 enAoayo h Sun the of Anomaly Mean +0 enAoayo h Moon the of Anomaly Mean l E 0 − . . . . 27209062 . . p , . . . . . 007702 006593 001037 000136 051635 N 1 52910918 04455501 85019547 96340251 tal. et a Ω l µ . Ma j as. =1 rnfrainBtenteClsiladTretilSystems Terrestrial and Celestial the Between Transformation 5 , , 94 als34(.)ad35() ae nIERS on based (b)) 3.5 and (b.3) 3.4 Tables 1994: , 14 l 00 00 00 00 00 Ju ) t t t t t ◦ 3 3 3 3 3 − → , RUET= ARGUMENT ◦ ◦ ◦ ◦ 1739527262 + − − − − 0 + l − 1602961601 + 129596581 + 1717915923 + Sa 0 0 0 0 6962890 . . . . . ( , 00005939 00003169 00000417 00001149 00024470 t N l Ur smaue nTB oee,T a be can TT However, TDB. in measured is tal. et j − =1 , 2 , µ N l 14 Ne swt eido n er nthe in year) one of period a with as . j t 5431 copne yrvra ftesign the of reversal by accompanied ) 19)frprecession. for (1991) and ) oapriua emi h nutation the in term particular a to 00 00 00 00 00 . . smaue nJla etre of centuries Julian in measured is 8478 0481 t t t t t . . 2178 2090 4 4 4 4 4 ω , , , , 00 k t t 00 00 7 + F 00 00 t t + 14 t t − − α 31 + . − 4722 12 0 stegnrlprecession general the is k 6 . tal. et . 5532 − . . 3706 7512 . Ti . mas 0.9 is TT 00 8792 t 2 00 00 00 t 19) ae on based (1999), k 2 00 t t ψ htr nthe in term th 2 2 t 2 n ∆ and t ω nJulian in seffec- is  × leaves tal. et sin (40) (39) l 0 5.9 Prograde and Retrograde Nutation Amplitudes IERS Technical

Note No. 32

F6 ≡ lMe = 4.402608842 + 2608.7903141574 × t, F7 ≡ lV e = 3.176146697 + 1021.3285546211 × t, F8 ≡ lE = 1.753470314 + 628.3075849991 × t, F9 ≡ lMa = 6.203480913 + 334.0612426700 × t, F10 ≡ lJu = 0.599546497 + 52.9690962641 × t, (41) F11 ≡ lSa = 0.874016757 + 21.3299104960 × t, F12 ≡ lUr = 5.481293872 + 7.4781598567 × t, F13 ≡ lNe = 5.311886287 + 3.8133035638 × t, 2 F14 ≡ pa = 0.024381750 × t + 0.00000538691 × t .

5.9 Prograde and Retrograde Nutation Amplitudes

The quantities ∆ψ(t) sin 0 and ∆(t) may be viewed as the components of a moving two-dimensional vector in the mean equatorial frame, with the positive X and Y axes pointing along the directions of increasing ∆ψ and ∆, respectively. The purely periodic parts of ∆ψ(t) sin 0 and ∆(t) for a term of frequency ωk are made up of in-phase and out-of- phase parts

ip ip ip ip (∆ψ (t) sin 0, ∆ (t)) = (∆ψ sin 0 sin(ωkt + αk), ∆ cos(ωkt + αk)), k k (42) op op op op (∆ψ (t) sin 0, ∆ (t)) = (∆ψk sin 0 cos(ωkt + αk), ∆k sin(ωkt + αk)),

respectively. Each of these vectors may be decomposed into two uni- formly rotating vectors, one constituting a prograde circular nutation (rotating in the same sense as from the positive X axis towards the pos- itive Y axis) and the other a retrograde one rotating in the opposite sense. The decomposition is facilitated by factoring out the sign qk of ωk from the argument, qk being such that

qkωk ≡ |ωk|. (43)

and writing ωkt + αk = qk(|ωk|t + qkαk) ≡ qkχk, (44)

with χk increasing linearly with time. The pair of vectors above then becomes

ip ip ip ip (∆ψ (t) sin 0, ∆ (t)) = (qk∆ψ sin 0 sin χk, ∆ cos χk), k k (45) op op op op (∆ψ (t) sin 0, ∆ (t)) = (∆ψk sin 0 cos χk, qk∆k sin χk).

Because χk increases linearly with time, the mutually orthogonal unit vectors (sin χk, − cos χk) and (cos χk, sin χk) rotate in a prograde sense and the vectors obtained from these by the replacement χk → −χk, namely (− sin χk, − cos χk) and (cos χk, − sin χk) are in retrograde ro- tation. On resolving the in-phase and out-of-phase vectors in terms of these, one obtains

ip ip pro ip ret ip (∆ψ (t) sin 0, ∆ (t)) = A (sin χk, − cos χk) + A (− sin χk, − cos χk), k k (46) op op pro op ret op (∆ψ (t) sin 0, ∆ (t)) = Ak (cos χk, sin χk) + Ak (cos χk, − sin χk),

where pro ip 1 ip ip Ak = 2 (qk∆ψk sin 0 − ∆k ), ret ip 1 ip ip Ak = − 2 (qk∆ψk sin 0 + ∆k ), pro op 1 op op (47) Ak = 2 (∆ψk sin 0 + qk∆k ), ret op 1 op op Ak = 2 (∆ψk sin 0 − qk∆k ).

49 50 No. 32 .0Poeue n ESRuie o rnfrain rmIR to ITRS from Transformations for Routines IERS and Procedures 5.10 IERS Note Technical 7 ftp://maia.usno.navy.mil/conv2000/chapter5 GCRS ie nteIR ovnin e ae hc sat provided: is are which routines page, following pro- web The Conventions are IERS transformations the 2000 IAU on the vided implement that routines Fortran ute ealcnenn hstpccnb on nDefraigne in found be can Bizouard topic and this (1995) concerning detail Further fre- notation, is: alternative that theory, an nutation finds of formulations one analytic 1981) in (Wahr, followed quently literature the affects In Earth) nonrigid the resonance of any because case equally. nutation the impor- in resonance in is of (especially components role the retrograde studying and for tant prograde into decomposition The and are longitude terms in circular nutation from obliquity corresponding in the providing expressions The opnns epciey n r ie by given are and respectively, components, with thus is frame equatorial coordinate mean complex the the in by given (CIP) Pole Intermediate the Celestial of contribution The where XYS2000A T2C2000 SP2000 POM2000 NU2000B NU2000A GST2000 GMST2000 ERA2000 EECT2000 EE2000 CBPN2000 BPN2000 A k pro A A k pro ∆ − k pro  ∆ ( = and t ψ + ) = A ( t k ip pro sin ) A A rnfrainBtenteClsiladTretilSystems Terrestrial and Celestial the Between Transformation 5 i ∆ ∆ k ip pro ∆ k ret ∆ ∆ ψ ψ ψ   ,Y s matrix Y, X, celestial to terrestrial form quantity the matrix polar-motion form 2000B IAU nutation, 2000A IAU nutation, Time Sidereal (apparent) Greenwich Time Sidereal Mean Greenwich Angle Rotation Earth terms complementary (EE) EE equinoxes the of equation matrix true-to-celestial equinox-based matrix intermediate-to-celestial CEO-based  k ( k op op 0 − k ip k ip tal. et r h mltdso h rgaeadretrograde and prograde the of amplitudes the are t + sin ) + iA i k = = = = iA ∆ k op pro tr ftentto otepsto fthe of position the to nutation the of -term  (1998).  k op pro 0 ( q − t sin sin = = ) k q A , 1  k   − A A 0 0 A , s i − 0  k ip pro k op pro A A i A k op pro k ip pro k pro A k ret k pro + − k ret − − A A e e + − = = − iχ k ip ret k op ret iχ A A k A A k k op ret k ip ret + k ip ret k ip ret +   A , . A k ret   k ret − + A , , e < − k pro iA iA − iχ 7 e > k op ret k op ret k iχ and  . k ,  , . . A k ret tal., et (52) (50) (51) (48) (49) un- 5.10 Procedures and IERS Routines for Transformations from ITRS to GCRS IERS Technical

Note No. 32

The above routines are to a large extent self-contained, but in some cases use simple utility routines from the IAU Standards of Fundamental software collection. This may be found at <8>. The SOFA collection includes its own implementations of the IAU 2000 models, together with tools to facilitate their rigorous use. Two equivalent ways to implement the IAU Resolutions in the transfor- mation from ITRS to GCRS provided by expression (1) can be used, namely (a) the new transformation based on the Celestial Ephemeris Origin and the Earth Rotation Angle and (b) the classical transforma- tion based on the equinox and Greenwich Sidereal Time. They are called respectively “CEO-based” and “equinox-based” transformations in the following. For both transformations, the procedure is to form the various com- ponents of expression (1), or their classical counterparts, and then to combine these components into the complete terrestrial-to-celestial ma- trix. Common to all cases is generating the polar-motion matrix, W (t) in expression (1), by calling POM2000. This requires the polar coordinates 0 xp, yp and the quantity s ; the latter can be estimated using SP2000. The matrix for the combined effects of nutation, precession and frame bias is Q(t) in expression (1). For the CEO-based transformation, this is the intermediate-to-celestial matrix and can be obtained using the routine BPN2000, given the CIP position X,Y and the quantity s that defines the position of the CEO. The IAU 2000A X, Y, s are available by calling the routine XYS2000A. In the case of the equinox-based transfor- mation, the counterpart to matrix Q(t) is the true-to-celestial matrix. To obtain this matrix requires the nutation components ∆ψ and ∆; these can be predicted using the IAU 2000A model by means of the rou- tine NU2000A. Faster but lower-accuracy predictions are available from the NU2000B routine, which implements the IAU 2000B truncated model. Once ∆ψ and ∆ are known, the true-to-celestial matrix can be obtained by calling the routine CBPN2000. The intermediate component is the angle for Earth rotation that de- fines matrix R(t) in expression (1). For the CEO-based transformation, the angle in question is the Earth Rotation Angle, θ, which can be ob- tained by calling the routine ERA2000. The counterpart in the case of the equinox-based transformation is the Greenwich (apparent) Sidereal Time. This can be obtained by calling the routine GST2000, given the nutation in longitude, ∆ψ, that was obtained earlier. The three components are then assembled into the final terrestrial-to- celestial matrix by means of the routine T2C2000. Three methods of applying the above scheme are set out below.

Method (1): CEO-based transformation consistent with IAU 2000A precession-nutation

This uses the new (X, Y, s, θ) transformation, which is consistent with IAU 2000A Precession-Nutation. Having called SP2000 to obtain the quantity s0, and knowing the polar motion xp, yp, the matrix W (t) can be obtained by calling POM2000. The Earth Rotation Angle provided by expression (13) can be predicted with ERA2000, as a function of UT1. The X, Y, s series, based on expressions (15) and (16) for X and Y , the coordinates of the CIP, and on Table 5.2c for the quantity s, that defines the position of the CEO, can be gener- ated using the XYS2000A routine. (Note that this routine computes the full series for s rather than the summary model in Table 5.2c.) The

8http://www.iau-sofa.rl.ac.uk

51 52 No. 32 .1Ntso h e rcdr oTasomfo CSt ITRS to ICRS from Transform to Procedure new the on Notes 5.11 IERS Note Technical h oiini hsrfrnesse scle h nemdaeright designation intermediate previous .” the the and to called ascension analogous right is is “apparent system and of reference declination and this ascension in position the The replaces Origin model Ephemeris Precession-Nutation Conventional the IAU2000A/B Poleequinox. and the Intermediate equator by its Celestial realized defines also The is (See that we intermediate (2002).) System. case (CIP) an Reference Kovalevsky this Celestial of In and Intermediate use system. Seidelmann the make terrestrial system a we that to before, call transforming As in system and followed. reference (McCarthy be 5.1 to Figure procedures with See Capitaine consistent be (in accuracy. to (ICRS) Capitaine clarified observational celestial been the improving has from systems the transforming (ITRS) in terrestrial followed the to be in to more procedure objects the The celestial of of part also directions is systems. Conventions, computing intermediate IERS for detail the in transformation in provided use is general for which chapter ITRS, this to GCRS in the from transformation The use to is step first matrix the the 1, Method for IAU-2000-compa- As new GST. the for and the expression Model using tible transformation, Precession-Nutation equinox-based, 2000A classical, IAU the is alternative An using transformation, equinox-based the (2A): Method the and Angle Rotation Earth the matrix. matrix, intermediate-to-celestial polar-motion the of the calling means by specifying by obtained is generated matrix be terrestrial-to-celestial then can coordinates matrix qaino h qioe opeetr em n vnngetn the neglecting even and motion. terms complementary polar setting equinoxes the including of equation possible, are substituting but optimizations used, is (2A) Method for the in as but procedure same mas, IAUThe 1 full the about using to when model. accuracy than 2000A onerous the for less limits significantly as are 2000B Precession-Nutation transformation computations 2000B IAU classical IAU Using truncated the the out on Model. based carry but to 2A, is Method possibility third The using transformation, classical the (2B): Method matrix. intermediate-to-celestial the and Time Sidereal Greenwich of and means the equator by calling generated true by be the then from can transforms coordinates that CBPN2000 GCRS to matrix date The of equinox UT1. as well calling as by predicted is calling Time by Sidereal planetary) + (lunisolar NU2000A Q ( ial,tefiihdtretilt-eeta arxi obtained is matrix terrestrial-to-celestial finished the Finally, . t httasom rmteitreit ytmt h GCRS the to system intermediate the from transforms that ) W eedn nteacrc eurmns ute efficiency further requirements, accuracy the on Depending . ( T2C2000 t A 00 precession-nutation 2000A IAU A 00 precession-nutation 2000B IAU ,given ), rnfrainBtenteClsiladTretilSystems Terrestrial and Celestial the Between Transformation 5 otn,seiyn h oa-oinmti,the matrix, polar-motion the specifying routine, x tal. et p y , p et opt h uaincomponents nutation the compute Next, . 02)fradarmo h e n old and new the of diagram a for 2002)) , NU2000A GST2000 SP2000 h rewc (apparent) Greenwich The . hsrqie ∆ requires This . s 0 and BPN2000 ozr,oitn the omitting zero, to POM2000 T2C2000 h finished The . ψ oobtain to NU2000B routine, n TT and 5.11 Notes on the new Procedure to Transform from ICRS to ITRS IERS Technical

Note No. 32

Fig. 5.1 Process to transform from celestial to terrestrial systems. Differences with the past process are shown on the right of the diagram.

References Cartwright, D. E. and Tayler, R. J., 1971, “New Computations of the Tide-Generating Potential,” Geophys. J. Roy. astr. Soc., 23, pp. 45–74. Aoki, S., Guinot, B., Kaplan, G. H., Kinoshita, H., McCarthy, D. D., and Seidelmann, P. K., 1982, “The New Definition of ,” . Astrophys., 105, pp. 359–361. Aoki, S. and Kinoshita, H., 1983, “Note on the relation between the equinox and Guinot’s non-rotating origin,” Celest. Mech., 29, pp. 335–360.

53 54 No. 32 IERS Note Technical Capitaine, Capitaine, Capitaine, Capitaine, Capitaine, Capitaine, Capitaine, B. Buffett, Brzezi´nski, Brzezi´nski, Brzezi´nski, Bretagnon, Bretagnon, Bizouard, Bizouard, Bizouard, ei rgncnitn ihteIU20Aprecession-nutation 2000A Ephe- IAU Celestial the and with model,” Pole consistent Intermediate Origin Celestial meris the for pressions 2002, Resolutions, Main. IAU am Geod¨asie, Frankfurt New und f¨ur Kartographie the Bundesamts of Implementation http://www.iers.org/iers/publications/tn/tn29/ the on (eds.), Vondr´ak,shop J., and M., Standish, 2002, M., Rothacher, B., Richter, Ref- International the in UT1 Frame,” of and erence origin Ephemeris Celestial the Sidereal Greenwich angle,” from stellar accuracy from submilliarcsecond or Time a at UT1 ing Use,” and Properties Mech. Definition, Celest. Equator: Instantaneous the on Observatory, gin Naval U.S. (eds.), H. G. Kaplan, 153–163. and pp. J., B. Luzum, AstrometrySub-Microarcsecond in Origin,” Ephemeris Celestial Astr. Dyn. coupling,” Res. electromagnetic phys. of Effects nutation-precession: geophysi- Earth: 51–58. the the of pp. of Proc. figure the Spatio-temporels triaxial in the aspects,” to cal due rotation Earth publication. for prepared the being sion-nutation,” of 243–251. Proc. pp. the Paris, de in Observatoire Earth,” (ed.), nonrigid taine a R´ef´erence for Journ´ees Syst`emes de model Spatio-temporels theoretical ple Earth,” rigid the of tion VSOP82,” solution tes. mo- polar on the Temporels influence of and Proc. in series S., tion,” between Dick, comparison Ser., nutations: Conf. 613–617. pp. Pac. (eds.), B. Soc. Luzum, and Astron. Colloquium IAU D., the McCarthy, Proc. of in Publications series,” nutation 178, Earth rigid period short amplitu- nutation the of variability temporal des,” and forcing spheric .Geod. J. h,Fluia . n oca,J,20,“hr periodic “Short 2001, J., Souchay, and M., Folgueira, Ch., the of “Comparison 2000, J., Souchay, and M., Folgueira, Ch., atmo- “Diurnal 1998, S., Petrov, and A., Brzezi´nski, Ch., . hpot . abr,S,adWlae . 2003 P., Wallace, and S., Lambert, J., Chapront, N., J., Ray, G., Petit, D., D. McCarthy, D., Gambis, of N., “Definition 2000, D., D. McCarthy, and B., Guinot, N., deriv- for procedure “Accurate 1993, A.-M., Gontier and N. Ori- Non-rotating “A 1986, J., Souchay, and B., Guinot, N., the and Pole Ephemeris Celestial the of “Definition 2000, N., Coordinates,” Pole Celestial “The 1990, N., ESTcnclNote Technical IERS . ahw,P . n ern,T,20,“oeigof “Modeling 2002, T., Herring, and M., P. Mathews, A., . n aian . 02 Lnslrprubtosin perturbations “Lunisolar 2002, N., Capitaine and “Preces- A., WG 19 Commission IAU 2, Circular 2002, July A., sim- a nutation: of terms subdiurnal and “Diurnal 2001, A., . ohr . n io,J-. 97 Ter fterota- the of “Theory 1997. J.-L., Simon, and P., Rocher, P., plan`e- des l’ensemble de mouvement Th´eorie du “ 1982, P., srn Astrophys. Astron. aian,N e.,Osraor ePrs p 260–265. pp. Paris, de Observatoire (ed.), N. Capitaine, , , , rnfrainBtenteClsiladTretilSystems Terrestrial and Celestial the Between Transformation 5 0,B4 107, 48 , , p 127–143. pp. , 39 72 srn Astrophys. Astron. 01 .Cptie(d) bevtied Paris, de Observatoire (ed.), Capitaine N. 2001, p 283–307. pp. , p 561–577. pp. , 10.1029/2001JB000056. , or´ e 00Ss`msd R´ef´erence Syst`emesJourn´ees de 2000 Spatio- srn Astrophys. Astron. srn Astrophys. Astron. , , srn Astrophys. Astron. 400 29 ontn .J,MCrh,D D., D. McCarthy, J., K. Johnston, , oad oesadCntnsfor Constants and Models Towards p 1145–1154. pp. , rceig fteIR Work- IERS the of Proceedings or´ e ytee eR´ef´erenceJourn´ees Syst`emes de , 355 p 398–405. pp. , , , 114 319 , 275 p 278–288. pp. , p 305–317. pp. , 00 .Capi- N. 2000, eet Mech. Celest. p 645–650. pp. , elgdes Verlag , .Geo- J. a “Ex- , References IERS Technical

Note No. 32

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