. h rmwr fIU20/06resolutions 2000/2006 IAU of framework The 5.2 . Introduction 5.1 Refer- Terrestrial International the between Transformation 5 neSse n h ecnrcClsilRfrneSystem Reference Celestial Geocentric the and System ence n“oecauefrfnaetlatooy NA (Capitaine (NFA) the Group astronomy” Working with IAU fundamental associated 2003-2006 for the terminology by “Nomenclature made on on were that recommendations resolutions quoted 2000/2006 the are IAU follows that publications chapter subsequent (Capitaine The original 29 of sections. Note number following the Technical a the to IERS in in in corresponding as well found products as be IERS 2002), can and resolutions software 2000 concepts, IAU 2000/2006 relevant IAU GCRS the the to ITRS with about the comply at chapter for (provided this procedures in the resolutions provided and are ITRS and that GCRS transformation the equator of the on definition origin The and pole respec- celestial of motion pole. choice any that polar for valid of from is (5.1) and Eq. the that pole, of Note rotation the the with from associated system, reference axis tively. celestial the the around in pole Earth celestial the of tion hs eouin eeedre yteIG n20 n 07 respectively. 2007, and 2003 in IUGG the by endorsed refer- were terrestrial resolutions and Those celestial the (see between systems transformation ence the August concern Prague, that and 2000, 2006) August (Manchester, Union Assemblies Astronomical General International XXVIth and XXIVth the the of by adopted were resolutions Several the of stages provided various chart the a illustrate including to chapter, order this IAU in of the Group end Working with process. the NFA consistently IAU at sys- systems the out intermediate these by set in among also transform objects also is celestial to resolutions systems of process reference directions the terrestrial computing tems, and for chapter. celestial required routines the the Software being of between end precession. the transformation 2006 towards IAU The described the 2003 are with IERS transformations compliant the the from be implementing revised to been order have in chapter values this in respec- contained values (ICRF), Numerical Frame Reference that Celestial International clear tively. the be for- and should theoretical (ITRF) adopted systems, Frame it IAU/IUGG are reference the However, those chapter, involves of formulas this realization those “systems”. in of reference implementation formulas numerical following to the the refer as that well mulations as (5.1), 2006 Eq. “IAU also the chapter in This Group Working http://syrte.obspm.fr/iauWGnfa/). recommendations). this by the recommenda- at provided of Glossary”(available “NFA were NFA list that as the definitions following for the the uses A in Appendix recommendations (see those tions” to refer will We where ytm(TS oteGoeti eeta eeec ytm(CS ttedate the Reference at Terrestrial (GCRS) International System Reference the Celestial relate Geocentric t to the to used (ITRS) be System to transformation The fteosraincnb rte as: written be can observation the of GR]= [GCRS] Q ( t ), R ( t and ) Q 1 ( t n pedxBfrtecmlt eto hs resolutions). those of text complete the for B Appendix and ) R W ( t ( ) < t W r h rnfrainmtie rsn rmtemo- the from arising matrices transformation the are ) 1 > ( t [ITRS] ) n nApni ) oedtie explanations detailed More B). Appendix in and , i.e. h nentoa ersra Reference Terrestrial International the tal. et Technical 2007). , tal et IERS Note (5.1) ., 43 No. 36 44 No. 36 .. A 00resolutions 2000 IAU 5.2.1 Note Technical IERS 1 http://syrte.obspm.fr/IAU rnm SF)srie(eto . n alc,19)adtersltoswere resolutions the As- and Fundamental 1998) Of 2003. Wallace, operationally. Standards made and January implemented the 5.9 been 1 (Section had and on software service 2003 (SOFA) and Conventions force tronomy data IERS into procedures, the came models, by B1.8 required available the and time, B1.7 that B1.6, By Resolutions 2000 IAU following: the ( in resolutions 2000 IAU The resolutions/Resol-UAI.htm • • • • n htteIR otnet rvd sr ihdt n loihsfor algorithms and data with finally users was provide It 2003 to transformation. January posi- Angle. conventional continue 1 the Rotation by the IERS GCRS, this Earth the implement the the to that in and steps and take CIP ITRS, IERS the the the of that in recommended CIP position the the 2000 of by defined IAU tion specified is ITRS the ERA. be UT1 between the and GCRS transformation the TIO, and to that the recommends proportionality also and linear B1.8 CIO along Resolution adopted the measured conventionally between angle a the CIP by as the defined of is equator (ERA) the Angle” Rotation “Earth The below). (see B2 tion gin designated were terrestrial They the and “non-rotating and 1979). the celestial recommends (Guinot, the B1.8 origin” Resolution in 2000 equator in IAU CIP systems, the CEP the reference the of on of that origins that longitude of For with extension 2000). coincides an (Capitaine, and domain is domain frequency definition frequency low Its the high its ITRS. the through and in definition GCRS CEP its the implement in of to both as how (CEP) Pole specifies at Ephemeris and direction Celestial 2003, the January of Pole place Intermediate 1 in Celestial implemented the be that recommends (CIP) B1.7 Resolution 2000 nutation.IAU 2000A with IAU consistent with precession that for and B1.6, expressions theories Resolution new the dynamical of 2000 at corre- development encouraged IAU the not model, why time does precession-nutation same is hence 2000A This IAU and to the theory. precession, corrections recommended dynamical 1976 of a IAU only to the consists spond of model 2000A rates IAU precession the the J2000.0 of at part equinox precession and The pole mean the GCRS. between the this values and in bias mas published frame 1 are offsets, the at model pole Dehant only celestial See model associated a document. need their who with of those together for functions level, 2000B transfer IAU version the shorter on its precession- based the (MHB2000 by replaced 2000A Mathews be IAU 1982) model Seidelmann, nutation 2003, 1981; January (Wahr, 1 nutation (Lieske on of model beginning precession 1976 that, IAU recommends the B1.6 Resolution 2000 IAU at transformations coordinate level. defining post-Newtonian and first tensor frameworkthe and metric general the a the system provides named expressing also solar now for It are respectively. the Celestial (GCRS) Geocentric Relativity for System the coor- General and Reference A4 (BCRS) of space-time System Resolution Reference framework of Celestial the 1991 Barycentric systems within IAU the Earth by that the defined specifies as B1.3 dinates Resolution 2000 IAU and ersra peei Origin Ephemeris Terrestrial ersra nemdaeOrigin Intermediate Terrestrial tal. et J 2000 20) o hs h edamdla h . a ee,or level, mas 0.2 the at model a need who those for (2002)) . nteGR swl steraiaino t motion its of realization the as well as GCRS the in 0 < 1 tal. et > htaerlvn oti hpe r described are chapter this to relevant are that ) rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 19)frmr eal.I diint that to addition In details. more for (1999) u renamed but , tal. et epciey yIU20 Resolu- 2006 IAU by respectively, , 97 n h A 90theory 1980 IAU the and 1977) , eeta peei Origin Ephemeris Celestial eeta nemdaeOri- Intermediate Celestial 5.3 Implementation of IAU 2000 and IAU 2006 resolutions IERS Technical

Note No. 36

5.2.2 IAU 2006 resolutions The IAU 2006 resolutions (cf. Appendix B) that are relevant to this chapter are the following: • IAU 2006 Resolution B1, proposed by the 2003-2006 IAU Working Group on “Precession and the Ecliptic” (Hilton et al., 2006), recommends that, beginning on 1 January 2009, the precession component of the IAU 2000A precession-nutation model be replaced by the P03 precession theory of Cap- itaine et al. (2003c) in order to be consistent with both dynamical theories and the IAU 2000 nutation. We will refer to that model as “IAU 2006 precession”. That resolution also clarifies the definitions of the precession of the equator and the precession of the ecliptic. • IAU 2006 Resolution B2, which is a supplement to the IAU 2000 resolutions on reference systems, consists of two recommendations: 1. harmonizing “intermediate” in the names of the pole and the origin (i.e. celestial and terrestrial intermediate origins, CIO and TIO instead of CEO and TEO, respectively) and defining the celestial and terrestrial “intermediate” systems; and 2. fixing the default orientation of the BCRS and GCRS, which are, unless otherwise stated, assumed to be oriented according to the ICRS axes. The IAU precession-nutation model resulting from IAU 2000 Resolution B1.6 and IAU 2006 Resolution B1, will be denoted IAU 2006/2000 precession-nutation in the following. A description of that model is given in Section 5.6.1 and Sec- tion 5.6.2.

5.3 Implementation of IAU 2000 and IAU 2006 resolutions 5.3.1 The IAU 2000/2006 space-time reference systems In order to follow IAU 2000 Resolution B1.3, the celestial reference system to be considered in the terrestrial-to-celestial transformation expressed by Eq. (5.1) must correspond to the geocentric space coordinates of the GCRS. IAU 1991 Resolution A4 specified that the relative orientation of barycentric and geocentric spatial axes in BCRS and GCRS are without any time-dependent rotation. This requires that the geodesic precession and nutation be taken into account in the precession-nutation model. Moreover, IAU 2006 Resolution B2 specifies that the BCRS and GCRS are oriented according to the ICRS axes. Concerning the time coordinates, IAU 1991 Resolution A4 defined TCB and TCG of the BCRS and GCRS respectively, as well as another time coordinate in the GCRS, Terrestrial Time (TT), which is the theoretical counterpart of the realized time scale TAI + 32.184 s. The IAU 2000/2006 resolutions have clarified the definition of the two time scales TT and TDB. TT has been re-defined (IAU 2000 Resolution B1.9) as a time scale differing from TCG by a constant rate, which is a defining constant. In a very similar way, TDB has been re-defined (IAU 2006 Resolution B3) as a linear transformation of TCB, the coefficients of which are defining constants. See Chapter 10 for the relationships between these time scales. The parameter t, used in Eq. (5.1) as well as in the following expressions, is defined by t = (TT − 2000 January 1d 12h TT) in days/36525. (5.2) This definition is consistent with IAU 1994 Resolution C7 which recommends that the epoch J2000.0 be defined at the geocenter and at the date 2000 January 1.5 TT = Julian Date 2451545.0 TT.

5.3.2 Schematic representation of the motion of the Celestial Intermediate Pole (CIP) According to IAU 2000 Resolution B1.7, the CIP is an intermediate pole sepa- rating, by convention, the motion of the pole of the ITRS in the GCRS into a celestial part and a terrestrial part. The convention is such that:

45 46 No. 36 .. h A 0020 elzto fteClsilItreit oe(CIP) Pole Intermediate Celestial the of realization 2000/2006 IAU The 5.3.3 Note Technical IERS rqec nGCRS in frequency rqec nITRS in frequency w as(nteGR)a ela h ia aitosi oa motion. polar in than variations less tidal observations periods the by with as nutations well IERS as forced the GCRS) the by the to (in provided terrestrial corresponding days that the be two pole including requires model ITRS the a also by of the It specified motion part in predictable epoch. a CIP account at into the taking offset precession- of observed 2006/2000 observed IAU motion the its the conventional on the with monitor the based together GCRS to IERS the model respect in the nutation with CIP that the offsets”) of requires pole position celestial thus “celestial as CIP (reported the differences band. of diurnal realization retrograde the models The and outside observations frequencies in astro-geodetic with through CIP variations IERS the periods the including of by for provided motion is nutation The the ITRS by forced observations. the provided (see astro-geodetic and corrections realized appropriate precession time-dependent through is additional for IERS GCRS plus days model the two IAU in than the CIP greater 2006/2000A IAU the by be the of 5.3.2) to with consistent has motion Section manner J2000.0 The a at orien- in CIP model. GCRS Earth’s the the precession-nutation the of of in pole direction the variations the from resolution, frequency be offset precession- this higher must for to and model pole According conventional diurnal celestial the tation. as in realized well epoch the at as B1.7, offset nutation an Resolution requires 2000 This CIP. IAU the follow that to of order results the In on (2006). based Capitaine are and sections Wallace following in the paper. provided in provided been in procedures have consistent discussed The resolutions procedures been have precession-nutation 2006 framework the IAU matrix 2006 and with precession-nutation IAU (2006), the the Wallace in forming and procedure Capitaine of rigorous ways a Various on based the choice. that this stating expressly makes parameterization, 2006 user specific IAU a the required adopting stipulate the B1 not Resolution does on 2006 precession 2006/2000A IAU depending precession IAU 2000B respectively). 2006 denoted 2006/200B, IAU IAU IAU being and or the precession-nutation 2000A on corresponding IAU based the (the model be B1, precision matrix nutation 2009, Resolution January transformation the 2006 1 on the on IAU and in starting and must, used B1.6 (5.1) be Resolution Eq. to 2000 IAU quantities follow precession-nutation to order In tp,o h CS(otm,wt pdsitdet h oaino h ITRS the of rotation the ITRS to the due shift in GCRS. cpsd viewed the 1 either to a with motion, respect (bottom), with polar GCRS its the and or the (top), CIP between separation the frequency of conventional precession-nutation the illustrates chart following The • • h ersra oino h I plrmto) nldsaltetrsoutside terms ( than the ITRS greater all the includes or in motion), band (polar diurnal retrograde CIP the the of motion ( terrestrial +0 terms the GCRS and the the (cpsd) all day in sidereal includes days per 2 (precession-nutation), cycles than CIP greater the periods of with motion celestial the − − 2.5 3.5 oa motion polar − − − 1.5 2.5 0 . cpsd). 5 − − . 05+. 25+. (cpsd) +3.5 +2.5 +1.5 +0.5 0.5 1.5 precession -nutation rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 − . 05+. 25(cpsd) +2.5 +1.5 +0.5 0.5 . cpsd), 5 oa motion polar i.e. rqece oe than lower frequencies i.e. rqece between frequencies − 1 Q . cpsd 5 ( − t of ) 0 . 5 5.4 Coordinate transformation consistent with the IAU 2000/2006 resolutions IERS Technical

Note No. 36

5.3.4 Procedures to be used for the terrestrial-to-celestial transformation consistent with IAU 2000/2006 resolutions Two equivalent procedures have been given in the IERS Conventions 1996 and 2003 for the coordinate transformation from the ITRS to the GCRS expressed by Eq. (5.1). According to the NFA recommendations, these procedures, which differ by the origin that is adopted on the CIP equator (i.e. the equinox or the CIO), will be called in the following “equinox based” and “CIO based”, respectively. Each of these procedures is based on a specific representation of the transformation matrix components Q(t) and R(t) of Eq. (5.1), which depends on the corresponding origin on the CIP equator, while the representation of the transformation matrix component W (t) is common to the two procedures. IAU 2000 Resolutions B1.3, B1.6 and B1.7 as well as IAU 2006 B1 can be im- plemented in any of these two procedures if the requirements described in Sec- tions 5.3.1 and 5.3.3 are followed for the space-time coordinates in the geocentric celestial reference system, for the precession and nutation model on which are based the precession and nutation quantities used in the transformation matrix Q(t) and for the polar motion used in the transformation matrix W (t). On the other hand, only the CIO based procedure can be in agreement with IAU 2000 Resolution B1.8, which requires the use of the “non-rotating origin” in both the GCRS and the ITRS as well as the position of the CIP in the GCRS and in the ITRS. This is why this chapter of the IERS Conventions provides the expres- sions for the implementation of the IAU resolutions with more emphasis on the CIO based procedure. However, the IERS must also provide users with data and algorithms for the conventional transformation, which implies in particular that the expression of Greenwich Sidereal Time (GST) has to be consistent with the new procedure. Consequently, this chapter also provides the expressions which are necessary to be compatible with the resolutions when using the conventional transformation. The following sections give the details of the CIO based procedure and the stan- dard expressions necessary to obtain the numerical values of the relevant param- eters for both procedures at the date of observation.

5.4 Coordinate transformation consistent with the IAU 2000/2006 resolu- tions

The coordinate transformation (5.1) from the ITRS to the GCRS corresponding to the procedure consistent with IAU 2000 Resolution B1.8 is expressed in terms of the three fundamental components W (t), R(t) and Q(t), as described in the following. According to IAU 2006 Resolution B2, the system at date t as realized from the ITRS by applying the transformation W (t) in both procedures is the “Terrestrial Intermediate Reference System” (TIRS). It uses the CIP as its z-axis and the TIO as its x-axis. The CIO based procedure realizes an intermediate celestial reference system at date t that uses the CIP as its z-axis and the CIO as its x-axis. According to IAU 2006 Resolution B2, it is called the “Celestial Intermediate Reference System” (CIRS). It uses the “Earth Rotation Angle” in the transformation matrix R(t), and the two coordinates of the CIP in the GCRS (Capitaine, 1990) in the transformation matrix Q(t). The classical procedure realizes an intermediate celestial reference system at date t that uses the CIP as its z-axis and the equinox as its x-axis. It is called the “true equinox and equator of date system”. It uses apparent Greenwich Sidereal Time (GST) in the transformation matrix R(t) and the classical precession and nutation parameters in the transformation matrix Q(t). Each of the transformation matrix components W (t), R(t) and Q(t) of Eq. (5.1) is a series of rotations about the axes 1, 2 and 3 of the coordinate frame. In the following, R1, R2 and R3 denote rotation matrices with positive angle about the axes 1, 2 and 3. The position of the CIP both in the ITRS and GCRS is provided

47 48 No. 36 Note Technical .. xrsinfrteCObsdtasomto arxfrErhrotation Earth for matrix transformation based CIO the for Expression 5.4.2 motion polar for matrix transformation the for Expression 5.4.1 .. xrsinfrtetasomto arxfrteclsilmto fteCIP the of motion celestial the for matrix transformation the for Expression 5.4.4 rotation Earth for matrix transformation based equinox the for Expression 5.4.3 IERS qao fteGR.O qiaety ihn1mcorscn vroecentury: one over microarcsecond 1 within equivalently, Or GCRS. the of equator and respectively GCRS by multiplied are expressions numerical 1296000 their factor and the following, the in “coordinates” the by where between GCRS, the to respect with the date moving of the is definition and CIP kinematical epoch the the reference when to the GCRS corresponding CIP the the in of NRO equator the on CIO and the to respect with sines) their (strictly “angles” corresponding the of safnto ftecoordinates the of function a as E Earth the of rotation the ( from CIP the arising of matrix axis the transformation around based CIO The quantity the of use The ( motion as: polar expressed from be can arising TIRS) matrix transformation The system. reference the aur 03 sncsayt rvd neatraiaino h “instantaneous the of meridian”). realization “TIO exact by an (designated provide meridian” to prime necessary is 2003, January 1 h I ae rnfrainmti rsn rmtemto fteCPi the in CIP the of motion the from arising ( matrix GCRS transformation based CIO The Apparent uses It ERA. the system. of instead date rotation, of of angle equator Earth’s and the equinox represent Time, true Sidereal the Greenwich to TIRS the matrix transformation based equinox The date at TIO the and Earth. CIO the the of between rotation Angle Rotation Earth t the is ERA where ihrsett h TSdet oa motion. kinematical polar the to moving of is due to expression CIP ITRS The corresponding the the when CIP ITRS to the the respect in of (NRO) with equator origin “non-rotating” the the on of definition TIO the of position and ITRS the in x p nteeutro h I,wihpoie ioosdfiiino h sidereal the of definition rigorous a provides which CIP, the of equator the on and and s X W s R Q s en uniy ae COlctr,wihpoie h oiino the of position the provides which locator”, “CIO named quantity, a being σ d x 0 ( y ( ( 0 ( i.e. t ( p t t t sin = = ) en uhta h oriae fteCPi h CSare: GCRS the in CIP the of coordinates the that such being and t = ) = ) = ) n Σ and = ) en h plrcodnts fteClsilItreit oe(CIP) Pole Intermediate Celestial the of coordinates” “polar the being eaigCR n CS,cnb xrse as: expressed be can GCRS), and CIRS relating − R y R 2 1 R d Z 3 3 opnnso h I ntvco.Teecmoet r called are components These vector. unit CIP the of components 0 3 cos t ( ( Z 0 ( − t − t r h oiin fteCOa 20. n the and J2000.0 at CIO the of positions the are s − 00 s 0 t X 0 ERA E / 0 ,Y E, ( s en uniy ae TOlctr,wihpoie the provides which locator”, “TIO named quantity, a being x 2 0 safnto ftecoordinates the of function a as ( ) ) π p t · ) y · ˙ R Y nodrt ersn h prxmt ausi arcseconds in values approximate the represent to order in N p R ˙ ) 2 , + 1 ( − 0 2 s ( t ( 0 ) − i.e. steacnignd fteeutrat equator the of node ascending the is hc a elce ntecasclfr ro to prior form classical the in neglected was which , x x ˙ − i.e. d p p Z ) ) y Y sin = ( p · · h nl ewe h qio n h I,to TIO, the and equinox the between angle the t ) X eaigTR n IS,cnb xrse as: expressed be can CIRS), and TIRS relating ( ) R R rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 t t dt. ) 3 1 u opeeso n uain t expression Its nutation. and precession to due X and ( ( ˙ E y d ( p sin ) t ) ) , · Y dt R ,Z E, R s(Capitaine is 3 − ( ( s t ( o at oaintasom from transforms rotation Earth for ) ) σ , 0 N 0 − cos = x Σ p 0 tal. et and N i.e. d, 0 ) , y 2000) , p eaigIR and ITRS relating is: x J oii fthe of -origin 2000 z . nthe in 0 ai of -axis (5.3) (5.7) (5.8) (5.4) (5.6) (5.5) 5.5 Parameters to be used in the transformation IERS Technical

Note No. 36

1 Z t s(t) = − [X(t)Y (t) − X(t )Y (t )] + X˙ (t)Y (t)dt − (σ N − Σ N ). (5.9) 2 0 0 0 0 0 0 t0

The arbitrary constant σ0N0 −Σ0N0, which had been conventionally chosen to be zero in previous references (e.g. Capitaine et al., 2000), was afterwards chosen to ensure continuity with the classical procedure on 1 January 2003 (see expression (5.31)). Q(t) can be given in an equivalent form directly involving X and Y as   1 − aX2 −aXY X    2  Q(t) =  −aXY 1 − aY Y  · R3(s), (5.10)   −X −Y 1 − a(X2 + Y 2)

with a = 1/(1 + cos d), which can also be written, with an accuracy of 1 µas, as a = 1/2 + 1/8(X2 + Y 2). Such an expression of the transformation (5.1) leads to very simple expressions of the partial derivatives of observables with respect to the terrestrial coordinates of the CIP, UT1, and celestial coordinates of the CIP.

5.4.5 Expression for the equinox-based transformation matrix for precession-nutation The equinox based matrix Q(t) that transforms from the true equinox and equator of date system to the GCRS can be expressed in several ways corresponding to different parameterization choices. • One rigorous way is that recommended in the previous version of the IERS Conventions. It is composed of the classical nutation matrix (using the nutation angles ∆ψ, ∆ in longitude and obliquity referred to the ecliptic of date and the mean obliquity of date, A), the precession matrix including four rotations (R1(−0) · R3(ψA) · R1(ωA) · R3(−χA)), and a separate rotation matrix for the frame biases. The precession angles are those of Lieske et al. (1977), in which 0 is the obliquity of the ecliptic at J2000.0, ψA and ωA are the precession quantities in longitude and obliquity referred to the ecliptic of epoch and χA is the precession of the ecliptic along the equator. • Another rigorous way is that proposed by Fukushima (2003) as an extension to the GCRS of the method originally proposed by Williams (1994). This way is more concise than the previous one, as it can be referred directly to the GCRS pole and origin without requiring the frame bias to be applied separately, and there is no need for separate precession and nutation steps. It is composed of the four rotations: R1(−) · R3(−ψ) · R1(φ¯) · R3(¯γ)), where the angles  and ψ are each obtained by summing the contributions from the bias, precession and nutation in obliquity and ecliptic longitude, respectively, φ¯ is the obliquity of the ecliptic of date on the GCRS equator, andγ ¯ is the GCRS right ascension of the intersection of the ecliptic of date with the GCRS equator.

5.5 Parameters to be used in the transformation 5.5.1 Motion of the Celestial Intermediate Pole in the ITRS

The standard pole coordinates to be used for the parameters xp and yp appearing in Eq. (5.3) of Section 5.4.1, if not estimated from the observations, are those published by the IERS with additional components to account for the effect of ocean tides (∆x, ∆y)ocean tides and for forced terms (∆x, ∆y)libration with periods less than two days in space:

(xp, yp) = (x, y)IERS + (∆x, ∆y)ocean tides + (∆x, ∆y)libration, (5.11)

where (x, y)IERS are pole coordinates provided by the IERS, (∆x, ∆y)ocean tides are the diurnal and semi-diurnal variations in pole coordinates caused by ocean tides, and (∆x, ∆y)libration are the variations in pole coordinates corresponding to motions with periods less than two days in space that are not part of the IAU 2000 nutation model. These variations are described in detail below.

49 50 No. 36 Note Technical IERS 5.5.1.3 5.5.1.2 5.5.1.1 h ira opnnso (∆ of components diurnal The Variations 8). Chapter that (see variations Conventions “ORTHO semi-diurnal routine and Ray the diurnal from from for those derived arguments of and been 8, amplitudes constituents have the Chapter tidal provide by 71 8.2b in caused and the coordinates considered 8.2a pole Tables orientation in tides. variations Earth ocean semi-diurnal in and diurnal variations including tidal are These not need ( and values con- Variations motion reported libration polar the the observed to of the variation added in secular be tidal contained the the already as are (i) well tribution, as of terms, (∆ date long-period the this The of for components contribution urnal the IERS adds the (∆ then of terms of “INTERP.F” and series interpola- routine date interpolates after the chosen which added by Center, be done to to Product appropriately therefore reported EOP is values are This motion and polar IERS tion. the the of by part distributed coordinates not and pole are variations in effects subdaily libration The and tidal ocean of Account 00ad20)icuepord ira n rgaesm-ira terms to semi-diurnal up prograde amplitudes with and GCRS diurnal the to prograde respect include with 2001) and ITRS. (Bretagnon 2000 the nutation in Earth pole al. the rigid et of for motion models considered corresponding not be Recent the are to have for space therefore model in Ap- and days a model see two nutation using B1.7; than 2000 IAU Resolution less the 2000 periods in (IAU with included motions CIP forced the A7), of pendix definition the to According urn rcso ria nlsswudgv austa ol ge within 0.1 agree of would level that values cut-off give adopted would the analysis model orbital geopotential precision JGM-3 current the (Brzezi´nski and of core coefficients of Stokes the coefficients (Tapley on of The based triaxiality are communication). 5.1a private the Table 2002, in table Dehant, the uncertainty 2003; in large account Capitaine, into the taken been to not due has 2003), Bretagnon, and Mathews amplitudes with components All 2003). 0.5 Bretagnon, than the greater and of developments Mathews and has Capitaine, Brzezi´nski Brzezi´nski,2003; and Earth models 2001; (TGP; nonrigid Earth potential nonrigid a generating tide on for based motion 2003), polar Mathews, in variations an by these provided for been polar model in the variations Folgueira of diurnal example for prograde prograde (see the to motion and term, correspond linear nutations the including long- semi-diurnal motion retrograde polar and the in prograde variations in to period designated correspond motion. nutations been diurnal polar effect has prograde on This The and effect effect, “nutation” geopotential. gravitation” the the as “tidal of past the terms the called non-zonal of also the part is by non-axisymmetric of expressed the effect on direct as torque the Earth from luni-solar) Chao originate (mainly “libration”, external to as the according here which, designated nutations, are (1991), semi-diurnal and diurnal These Earth nonrigid 2003). and (Getino and rigid formalism 2002 for Hamiltonian both on provided based been models also have nutation in terms 20) h otiuinfo h railt ftecr otediurnal the (Escapa level to cut-off core adopted the the Getino of exceed by triaxiality can estimated the it those from while with contribution terms, agreement The good very (2001). in are terms 98 n ;Souchay b; and 1998a , tal. et x, tal. et (∆ (∆ ∆ y 96,btayo h eptnilmdl omnyue in used commonly models geopotential the of any but 1996), , x, x, ) ca tides ocean 19) htruiei vial ntewbieo h IERS the of website the on available is routine That (1994). ∆ ∆ µ saegvni al .a h mltdso h diurnal the of amplitudes The 5.1a. Table in given are as y y ) ) dhoc ad ca tides ocean libration eie rmTbe .aad82,ad(i h di- the (ii) and 8.2b, and 8.2a Tables from derived x, okn ru Bzz´si 02 Brzezi´nski (Brzezi´nski, Group 2002; and Working rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 nplrmotion polar in ∆ x, tal. et nplrmotion polar in y ∆ µ ) libration swt hs fTbe5.1a. Table of those with as y ,y x, ) tal. et 99 osek 99 Bizouard 1999; Roosbeek, 1999; , libration ) IERS 01.Atbefroeainluse operational for table A 2001). , a ecmue ihteroutine the with computed be can . em,drvdfo al 5.1a. Table from derived terms, O.”bsdo h model the on based EOP.F” ∼ tal. et 15 x IERS tal. et µ 01 Escapa 2001; , s h semi-diurnal The as. , y 97 Folgueira 1997; , IERS x tal. et p aust a to values and 2002; , tal. et tal. et tal. et y tal. et p , , 5.5 Parameters to be used in the transformation IERS Technical

Note No. 36

PMSDNUT2.F, available on the Conventions Center website at <2>. These di- urnal components (namely, the 10 terms listed in Table 5.1a with periods near 1 day) should be considered similarly to the diurnal and semi-diurnal variations due to ocean tides (see above). Note that the 10 diurnal terms of Table 5.1a due to the influence of tidal gravi- tation on the non-axisymmetric part of the Earth are a subset of the 41 diurnal terms of Table 8.2a with, for each common term, an amplitude about 10 times lower than the corresponding amplitude for the variation in UT1 (or LOD) caused by ocean tides provided in Table 8.2a.

Table 5.1a: Coefficients of sin(argument) and cos(argument) in (∆x, ∆y)libration due to tidal gravita- tion (of degree n) for a nonrigid Earth. Listed are all terms with amplitudes greater than 0.5 muas. Units are µas, γ denotes GMST+π (where GMST = ERA + precession in RA (see Eq. (5.32))). The expressions for the fundamental arguments (Delaunay arguments) are given by Eq. (5.43).

Argument Doodson Period xp yp n Tide γ l l0 FD Ω number (days) sin cos sin cos 4 0 0 0 0 0 −1 055.565 6798.3837 0.0 0.6 −0.1 −0.1 3 0 −1 0 1 0 2 055.645 6159.1355 1.5 0.0 −0.2 0.1 3 0 −1 0 1 0 1 055.655 3231.4956 −28.5 −0.2 3.4 −3.9 3 0 −1 0 1 0 0 055.665 2190.3501 −4.7 −0.1 0.6 −0.9 3 0 1 1 −1 0 0 056.444 438.35990 −0.7 0.2 −0.2 −0.7 3 0 1 1 −1 0 −1 056.454 411.80661 1.0 0.3 −0.3 1.0 3 0 0 0 1 −1 1 056.555 365.24219 1.2 0.2 −0.2 1.4 3 0 1 0 1 −2 1 057.455 193.55971 1.3 0.4 −0.2 2.9 3 0 0 0 1 0 2 065.545 27.431826 −0.1 −0.2 0.0 −1.7 3 0 0 0 1 0 1 065.555 27.321582 0.9 4.0 −0.1 32.4 3 0 0 0 1 0 0 065.565 27.212221 0.1 0.6 0.0 5.1 3 0 −1 0 1 2 1 073.655 14.698136 0.0 0.1 0.0 0.6 3 0 1 0 1 0 1 075.455 13.718786 −0.1 0.3 0.0 2.7 3 0 0 0 3 0 3 085.555 9.1071941 −0.1 0.1 0.0 0.9 3 0 0 0 3 0 2 085.565 9.0950103 −0.1 0.1 0.0 0.6

0 2 Q1 1 −1 0 −2 0 −1 135.645 1.1196992 −0.4 0.3 −0.3 −0.4 2 Q1 1 −1 0 −2 0 −2 135.655 1.1195149 −2.3 1.3 −1.3 −2.3 2 ρ1 1 1 0 −2 −2 −2 137.455 1.1134606 −0.4 0.3 −0.3 −0.4 0 2 O1 1 0 0 −2 0 −1 145.545 1.0759762 −2.1 1.2 −1.2 −2.1 2 O1 1 0 0 −2 0 −2 145.555 1.0758059 −11.4 6.5 −6.5 −11.4 2 M1 1 −1 0 0 0 0 155.655 1.0347187 0.8 −0.5 0.5 0.8 2 P1 1 0 0 −2 2 −2 163.555 1.0027454 −4.8 2.7 −2.7 −4.8 2 K1 1 0 0 0 0 0 165.555 0.9972696 14.3 −8.2 8.2 14.3 0 2 K1 1 0 0 0 0 −1 165.565 0.9971233 1.9 −1.1 1.1 1.9 2 J1 1 1 0 0 0 0 175.455 0.9624365 0.8 −0.4 0.4 0.8 Rate of secular polar motion (µas/y) due to the zero frequency tide 4 0 0 0 0 0 0 555.555 −3.8 −4.3

5.5.2 Position of the Terrestrial Intermediate Origin in the ITRS The quantity s0 (i.e. the TIO locator) appearing in Eq. (5.3) and expressed by Eq. (5.4) is sensitive only to the largest variations in polar motion. Some compo- nents of s0 have to be evaluated, in principle, from the measurements and can be

2ftp://tai.bipm.org/iers/conv2010/chapter5/

51 52 No. 36 .. at oainAngle Rotation Earth 5.5.3 Note Technical IERS 3 ftp://tai.bipm.org/iers/conv2010/chapter8/ 5.5.3.1 5.5.3.2 ffcso ca ie n irto.Teeeet r ecie ndti below. detail the in for account described to are LOD effects be These and UT1 should libration. for and components IERS tides additional the ocean by 5.5.1), of published effects Section values the (cf. to motion added polar to Similarly 0 for the 0.5. that, UT1, Note given a value. to IERS corresponding ERA the UT1 provides quantity also relationship microarcsecond observations. above offsets. of the The pole independent at as celestial considered insensitive observed quantity be the is the must data, to CIO observational and processing the in model Therefore, on precession-nutation the based to UT1 level of definition This Capitaine by given that is 5.4.2 Section of (5.5) Eq. (ERA) in Angle used Rotation Earth be the to from UT1 defining relationship conventional The 0 of order the of values reach wobbles annual and Chandlerian the for plitudes (Capitaine considered of period the in respectively, wobbles, a as: written be can component main Its data. IERS the using extrapolated lnl mdl 2 (modulo alently 2002): Bizouard, and (Lambert gives wobbles annual and 0 and where c s and 0 ilteeoeb esta . a hog h etcnuy vni h am- the if even century, next the through mas 0.4 than less be therefore will est fteIR ovnin at Conventions al. IERS et the of website ∆UT1 terms tidal the in- routine of date the after this by added of for effect series be bution interpolates first which to the Center, therefore reported Product for UT1 values EOP are done IERS LOD and the appropriately or of IERS is “INTERP.F” UT1 This the the by of terpolation. distributed part not LOD and and are to UT1 variations in subdaily effects libration The and tidal ocean of Account eie rmTbe .aad83.Tesm-ira opnnso h li- the of components semi-diurnal The 8.3b. ∆UT1 contribution and bration 8.3a Tables from derived o h 1tdlcnttet ftoedunladsm-ira variations semi-diurnal and arguments diurnal “ORTHO and routine those the amplitudes by of from the derived caused constituents been provide have LOD tidal 8, 8.3b that 71 or Chapter and the UT1 in 8.3a Tables for in considered variations orientation tides. semi-diurnal Earth ocean and in diurnal variations including tidal are These Variations routine. that in included be should . s ERA s T 1 0 0 ERA 00 u a = = epciey sn h urn enapiue o h Chandlerian the for amplitudes mean current the Using respectively. , (uinU1date UT1 =(Julian a IERS − − en h vrg mltds(na)o h hnlra n annual and Chandlerian the of as) (in amplitudes average the being (1994). ( T ( 0 47 − T u . 0015 2 = ) u n LOD and T ob sd(fntetmtdfo h bevtos en the being observations) the from estimated not (if used be to UTC µ 2 = ) ∆ as π UT1 ,i re ordc osberudn errors, rounding possible reduce to order in ), π t. a (0 c 2 / ca tides ocean . +0 1 707724 1 + 7790572732640 IERS π . + 2 U1Jla a fraction day Julian (UT1 . 707724 0 + 7790572732640 − h 414.) n UT1=UTC+(UT1 and 2451545.0), a aust hsndt n hnad h contri- the adds then and date chosen a to values a 2 T,teU1Jla a rcini q 51)is (5.15) Eq. in fraction day Julian UT1 the UT1,
libration and rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 t, ∆ LOD r∆LOD or , . < 00273781191135448 ca tides ocean 3 > . 00273781191135448 ae ntemdlfo Ray from model the on based ) libration ca tides ocean nU1adLOD and UT1 in O.”(vial nthe on (available EOP.F” s eie rmTbe5.1b Table from derived tal. et rvddb al 5.2d Table by provided r∆LOD or , 96.Tevalue The 1986). , T − u ) T) requiv- or UTC), T tal. et , u ) . ca tides ocean (2000): (5.12) (5.15) (5.13) (5.14) . 5 00 , 5.5 Parameters to be used in the transformation IERS Technical

Note No. 36

5.5.3.3 Variations ∆UT 1libration and ∆LODlibration in UT1 and LOD The axial component of Earth rotation contains small variations due to the direct effect of the external (mainly luni-solar) torque on the non-axisymme- tric part of the Earth as expressed by the non-zonal terms of the geopo- tential. This effect was theoretically predicted by Tisserand (1891) and Kinoshita (1977), but was neglected in current practical applications due to its very small size (maximum peak-to-peak variation of about 0.1 mas, i.e. 0.007 ms in UT1). More complete descriptions were published when the effect, also designated as “tidal gravitation”, became observationally de- tectable by, e.g. Chao et al. (1991), W¨unsch (1991), Chao et al. (1996) and Brzezi´nskiand Capitaine (2003; 2010). An analytical solution for the sub-diurnal libration in UT1 has been derived by Brzezi´nskiand Capitaine (2003) for the structural model of the Earth consisting of an elastic mantle and a liquid core which are not coupled to each other. The reference solution for the rigid Earth has been computed by using the satellite-determined coefficients of geopotential and the recent developments of the tide generating potential. Table 5.1b provides the am- plitudes and arguments for all components in UT1 and LOD with amplitudes greater than 0.5 µas (i.e. 0.033 µs in UT1.) It consists of 11 semi-diurnal har- monics due to the influence of the TGP term u22 on the equatorial flattening of the Earth expressed by the Stokes coefficients C22, S22. There is excellent agreement between the values for the rigid Earth and the amplitudes derived by W¨unsch (1991), except for the term with the tidal code ν2, which seems to have been overlooked in the latter model. The amplitudes computed for an elastic Earth with liquid core are in reasonable agreement with those derived by Chao et al. (1991), but the latter model was not complete.

Table 5.1b: Coefficients of sin(argument) and cos(argument) of semi-diurnal variations in UT1 and LOD due to libration for a non-rigid Earth. Listed are all terms with amplitudes of UT1 greater than 0.033 µs. Units are µs, γ denotes GMST+π. Expressions for the fundamental arguments are given by Eq. (5.43). Argument Doodson Period UT1 LOD Tide γ l l0 FD Ω number (days) sin cos sin cos

2N2 2 −2 0 −2 0 −2 235.755 0.5377239 0.05 −0.03 −0.3 −0.6

µ2 2 0 0 −2 −2 −2 237.555 0.5363232 0.06 −0.03 −0.4 −0.7 N2 2 −1 0 −2 0 −2 245.655 0.5274312 0.35 −0.20 −2.4 −4.2 ν2 2 1 0 −2 −2 −2 247.455 0.5260835 0.07 −0.04 −0.5 −0.8 0 M2 2 0 0 −2 0 −1 255.545 0.5175645 −0.07 0.04 0.5 0.8 M2 2 0 0 −2 0 −2 255.555 0.5175251 1.75 −1.01 −12.2 −21.3 L2 2 1 0 −2 0 −2 265.455 0.5079842 −0.05 0.03 0.3 0.6 T2 2 0 −1 −2 2 −2 272.556 0.5006854 0.05 −0.03 −0.3 −0.6 S2 2 0 0 −2 2 −2 273.555 0.5000000 0.76 −0.44 −5.5 −9.5 K2 2 0 0 0 0 0 275.555 0.4986348 0.21 −0.12 −1.5 −2.6 0 K2 2 0 0 0 0 −1 275.565 0.4985982 0.06 −0.04 −0.4 −0.8

Note that the 11 semi-diurnal terms of Table 5.1b due to the influence of tidal gravitation on the triaxiality of the Earth are a subset of the 30 semi-diurnal terms of Table 8.3b, with, for each common term, an amplitude about 10 times lower than the corresponding amplitude for the variation in UT1 (or LOD) caused by ocean tides provided in Table 8.3b. Nevertheless, the max- imum peak-to-peak size of the triaxiality effect on UT1 is about 0.105 mas, hence definitely above the current uncertainty of UT1 determinations. A comparison with the corresponding model of prograde diurnal polar motion associated with the Earth’s libration (Table 5.1a) shows that the two effects are of similar size and that there is consistency between the underlying dy-

53 54 No. 36 .. ocdmto fteClsilItreit oei h GCRS the in Pole Intermediate Celestial the of motion Forced 5.5.4 Note Technical IERS tion. t ( amplitudes The amplitudes series. the nutation and and J2000.0 precession at the offset of pole and obliquity ( and longitude in cession non-polynomial largest the ( for 5.2b and 5.2a Tables from in extract terms An (5.16). in full at The Center 5.2b) ones. Conventions tab5.2b.txt and planetary IERS the the 5.2a for on (Tables (5.44) electronically website available Eq. are and given series ones are 2000/2006 lunisolar expressions IAU the whose for theory, (5.43) nutation Eq. the by of arguments fundamental the of parameter the uaineet.Tecoordinates The effects. nutation and in terms cosine and of part polynomial the of coefficients the of values numerical The e em nec coordinate each in terms few h A 0620Advlpet r sfollows: as are developments 2006/2000A IAU precession- The Conventions. Capitaine for by 2003 model provided IERS 2000 been They the to had IAU 2006). in that respect the Wallace, biases on and with frame (Capitaine and based J2000.0 nutation computed developments at been previous offsets have the GCRS equinox replace the and of details) pole more pole for corresponding the 5.6 Section their (see nutation on on 2000A based and IAU level, and microarcsecond of precession the function 2006 at as IAU valid developments the Developments by given quantities. be those can of 5.4.4 time Section of (5.10) Eq. in appearing h oriae fteCPi h CSt eue o h parameters the for used be to GCRS the in CIP the of coordinates The 2 a cf. s,j sin, 51) r eie rmtedvlpeta ucino ieo h pre- the of time of function a as development the from derived are (5.16)) B ) i yA reink,aalbeo h ovnin etrwbieat website Center Conventions the in on Brzezi´nski, available libration A. sub-diurnal by The employed. ∆ parameters UT1, the and models namical Y X i ( , t 2 ftesre o uaini ogtd [ longitude in nutation for series the of = a o,..wihoiiaefo rse em ewe rcsinadnuta- and precession between terms crossed from originate which ... cos, = X c,j and ) − + + + + +0 + + + + − − i UT o the for ( , 0 0 0 P · · · P P t · · · P P P . . . . 006951 00190059 Y b 19861834 016617 en ie yepeso 52 and (5.2) expression by given being 1 i i i c,j i i i libration [( [( [( [( [( [( , , sgvnbelow. given is b b b ) a a a X c c c i s s s Y ( , , , , , , , 0 2 1 2 1 0 ) ) ) and 00 ) ) ) 00 b i i i oriae h oyoiltrsof terms polynomial The coordinate. i i i t t a ecmue ihteruieULB.,provided UTLIBR.F, routine the with computed be can , s,j − t t cos( 2004 + sin( 2 00 00 2 cos( sin( t cos( t 0 ) sin( Y 3 3 i . 025896 ARGUMENT 0 + X ARGUMENT 0 + for ARGUMENT ARGUMENT epcieyaeeult h amplitudes the to equal are respectively ARGUMENT ARGUMENT and . . 191898 . rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 001112526 000007578 j X < 0 = 00 Y t 2 and > − hc oti otiuinfo crossed- from contribution a contain which , 00 ( + ) ( + ) , 1 ( + ) 22 ( + ) t , ( + ) tab5.2a.txt ( + ) Y − ,..aedrvdfo h amplitudes the from derived are ... 2, . 00 4072747 00 0 b a t oti oso em in terms Poisson contain t b a s 4 . c 4 , 4297829 × s b a , c 0 , 0 + 0 s 0 + , 1 c ) 1 , ) sin , 2 ) i 2 ) i i ) sin( i ) cos( i . t . i t 0000001358 t 0000059285  t 00 cos( 2 sin( 0 2 t ARGUMENT ARGUMENT sin( n biut,ecp o a for except obliquity, and ] 2 cos( ARGUMENT 00 t o the for ARGUMENT ARGUMENT 2 a ARGUMENT ARGUMENT s, 0 ) i X ( , 00 00 X tal. et t t )] b 5 en function a being and 5 )] c, oriaeand coordinate )] )] 0 ) )] )] i 20a and (2003a) Y ftesine the of t A X < sin, r given are X i 2 × > and and (5.16) . sin t cos,  Y Y 0 5.5 Parameters to be used in the transformation IERS Technical

Note No. 36

Table 5.2a: Extract from Table 5.2a (available at < 2>) for the largest non-polynomial terms (i.e. co- efficients of the Fourier and Poisson terms, with t in Julian centuries) in the development (5.16) for X(t) compatible with the IAU 2006/2000A precession-nutation model (unit µas). The expressions for the fundamental arguments appearing in columns 4 to 17 are given by Eq. (5.43) and Eq. (5.44). (Because the largest terms are all luni-solar, columns 9-17 contain only zeros in the extract shown.) 0 i (as,0)i (ac,0)i l l FD Ω LMe LV e LE LMa LJ LSa LU LNe pA 1 −6844318.44 1328.67 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 −523908.04 −544.75 0 0 2 −22000000000 3 −90552.22 111.23 0 0 2 0 2 0 0 0 0 0 0 0 0 0 4 82168.76 −27.64 0 0 0 0 2 0 0 0 0 0 0 0 0 0 5 58707.02 470.05 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ..... 0 i (as,1)i (ac,1)i l l FD Ω LMe LV e LE LMa LJ LSa LU LNe pA 1307 −3309.73 205833.11 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1308 198.97 12814.01 0 0 2 −22000000000 1309 41.44 2187.91 0 0 2 0 2 0 0 0 0 0 0 0 0 0 .....

The contributions (in µas) to expression (5.16) from the frame biases are

dX = −16617.0 − 1.6 t2 + 0.7 cos Ω, (5.17) dY = −6819.2 − 141.9 t + 0.5 sin Ω,

the first term in each coordinate being the contribution from the celestial pole offset at J2000.0 and the following ones from the equinox offset at J2000.0, also called “frame bias in right ascension.” The polynomial changes in X and Y due to the change in the precession model from IAU 2000 to IAU 2006 can be written in µas as (Capitaine and Wallace, 2006, after correcting a typographical error in the t2 term of dY ):

dX = 155 t − 2564 t2 + 2 t3 + 54 t4, (5.18) dY = −514 t − 24 t2 + 58 t3 − 1 t4 − 1 t5.

In addition, there are slight changes in a few periodic terms of the IAU 2006 version of the X,Y series with respect to the IAU 2000 version, corresponding to additional Poisson terms in nutation caused by introducing the IAU 2006 J2 rate value (see Section 5.6.3). The largest changes (Capitaine and Wallace, 2006) in µas are:

dXJ2d = 18.8 t sin Ω + 1.4 t sin 2(F − D + Ω), (5.19) dYJ2d = −25.6 t cos Ω − 1.6 t cos 2(F − D + Ω).

The periodic terms expressed by (5.19) are included in the IAU 2006/2000A ver- sion of the X,Y series. The difference between the IAU 2006 and IAU 2000A expressions for X and Y is available electronically on the IERS Conventions Cen- ter website (Table 5.2f) at < 2> in file tab5.2f.txt. The relationships between the coordinates X and Y and the precession-nutation quantities are (Capitaine, 1990):

X = X¯ + ξ0 − dα0 Y,¯ (5.20) Y = Y¯ + η0 + dα0 X,¯

where ξ0 and η0 are the celestial pole offsets at the basic epoch J2000.0 and dα0 the right ascension of the mean equinox of J2000.0 in the GCRS (i.e. frame bias

55 56 No. 36 al .b xrc rmTbe52 aalbeat (available 5.2b Table from Extract 5.2b: Table Note Technical IERS 6 2024.68 965 6 5017 5.200001000000000 0 0 0 0 0 0 0 0 1 0 0 0 0 853.32 11714.49 964 153041.79 963 ...... yE.(.3 n q 54) Bcuetelrettrsaealln-oa,clms9-17 columns luni-solar, all given are are terms 17 shown.) largest extract to the the 4 (Because in columns zeros (5.44). in only Eq. contain appearing and arguments (5.43) fundamental Eq. the by for expressions The ffiinso h ore n oso em,with terms, Poisson and (5.16) Fourier the of efficients i i 5 3.19866 0 0 0 0 0 0 0 0 0 2 0 2 0 0 97846.69 137.41 4 3 581 253.600001000000000 0 0 0 0 0 0 0 0 1 0 0 0 0 9205236.26 1538.18 2 1 − − − 5.6533.2002 0 0 573033.42 458.66 ( ( Y 74 23.2012 1 0 22438.42 17.40 29.05 b b s, s, ( t 1 0 optbewt h A 0620Apeeso-uainmdl(unit model precession-nutation 2006/2000A IAU the with compatible ) ) ) i i − Quantities sdi xrsin(.0 is: (5.20) expression in used 20. fo Chapront (from J2000.0 more for (1981) Standish See plane.) for momentum orbital value angular numerical the the The of is details. rotation momenta the angular two with the computed associated between vector difference momentum (The angular the ter. the to referred to an velocity to perpendicular the from relative is barycenter which the sense, plane of tional the velocity is Earth- the the which system from of inertial sense, motion computed orbital inertial refer. as the the barycenter solutions of in vector Moon analytical momentum mean ecliptic angular or dynamical the the numerical to “inertial with recent perpendicular the associated dynamical the but is “rotational past, which latter the the to The not in J2000.0” is used of considered as equinox be J2000.0” to of equinox J2000.0 mean of these equinox for mean (5.33) and The (5.21) in below provided numbers quantities.) the (See ascension). right in where where these epoch; of ecliptic fixed the on date of equator that true such the are of quantities node the of epoch, nlniueadolqiyrfre oteelpi feoh Setenumerical the (See ∆ ∆ angles epoch. (5.39).) nutation of in the quantities ecliptic from precession the obtained the to for referred provided obliquity developments and longitude in al. et h citco ae h olwn omlto rmAk n ioht 18)is (1983) Kinoshita and Aoki from formulation following The date. of ecliptic the 91.400002000000000 0 0 0 0 0 0 0 0 2 0 0 0 0 89618.24 − − 9.1002 0 0 290.91 ( ( 12 0 0 0 0 0 0 0 0 0 2 0 2 0 0 51.26 b b 97 eerdt h citco pc n ∆ and epoch of ecliptic the to referred 1977) , c, c, dα ω ψ  1 0 X 0 Y ¯ A ¯ ) ) = 0 i i =84381 (= and ( = ω X ¯ sin = = A l l l l − and ∆ + h oainleunxi soitdwt h citci h rota- the in ecliptic the with associated is equinox rotational The . ω 0 A − . 0 0 01460 . r h rcsinqatte nlniueadolqiy(Lieske obliquity and longitude in quantities precession the are  sin Y ¯ 406 1 ω ; D F D F r ie by: given are  sin 00 0 ± steIU20 au o h biut fteelpi at ecliptic the of obliquity the for value 2006 IAU the is ) cos − − − tal. et ψ, < 0 0 0 0 0 0 0 0 0 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 . ψ ω 00050) 2 > cos + = 2002), , Ω Ω o h ags o-oyoiltrs( terms non-polynomial largest the for ) ψ rotating dα rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 A L L 00  0 Me Me ∆ + . 0 sdrvdfo Chapront from derived as t ψ sin ω ∆ , nJla etre)i h development the in centuries) Julian in and , ψ L L ria ln fteErhMo barycen- Earth-Moon the of plane orbital ω 1 e V e V  , cos nlniueadolqiyrfre to referred obliquity and longitude in ψ L L ψ E E stelniue nteelpi of ecliptic the on longitude, the is ψ L L 1 Ma Ma ∆ ,  L L 1 J J r h uainangles nutation the are L L Sa Sa tal. et ψ L L 1 U U ∆ , 20)t be to (2002) L L  1 Ne Ne i.e. a be can (5.21) (5.22) (5.23) µ p p as). co- A A 5.5 Parameters to be used in the transformation IERS Technical

Note No. 36

accurate to better than one microarcsecond after one century:

∆ψ1 sin ωA = ∆ψ sin A cos χA − ∆ sin χA, (5.24) ∆1 = ∆ψ sin A sin χA + ∆ cos χA,

A being the mean obliquity of date and χA the precession of the ecliptic along the equator (Lieske et al., 1977). As VLBI observations have shown that there are deficiencies in the IAU 2006/2000A precession-nutation model of the order of 0.2 mas, mainly due to the fact that the free core nutation (FCN) (see Section 5.5.5) is not part of the model, the IERS will continue to publish observed estimates of the corrections to the IAU precession- nutation model. The observed differences with respect to the conventional celestial pole position defined by the models are monitored 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 coordi- nates. These corrections can be related to the equinox based celestial pole offsets δψ (along the ecliptic of date) and δ (in the obliquity of date) using the relation- ships (5.22) between X and Y and the precession-nutation quantities and (5.24) for the transformation from ecliptic of date to ecliptic of epoch. The relationship (5.25) below, which is to first order in the quantities, ensures an accuracy of one microarcsecond for one century, for values of δψ and δ lower than 1 mas:

δX = δψ sin A + (ψA cos 0 − χA)δ, (5.25) δY = δ − (ψA cos 0 − χA)δψ sin A.

These observed offsets include the contribution of the FCN described in Sec- tion 5.5.5. Using these offsets, the corrected celestial position of the CIP is given by

X = X(IAU 2006/2000) + δX, Y = Y (IAU 2006/2000) + δY. (5.26)

This is practically equivalent to replacing the transformation matrix Q with the rotation   1 0 δX   ˜   Q =  0 1 δY  QIAU, (5.27)   −δX −δY 1

where QIAU represents the Q(t) matrix based on the IAU 2006/2000 precession- nutation model.

5.5.5 Free Core Nutation Free core nutation is a free retrograde diurnal motion of the Earth’s rotation axis with respect to the Earth caused by the interaction of the mantle and the fluid, ellipsoidal core as it rotates. Due to the definition of the CIP, this free motion appears as a motion of the CIP in the GCRS. Because this effect is a free motion with time-varying excitation and damping resulting in a variable amplitude and phase, a FCN model was not included in the IAU 2000A nutation model. As a result, a quasi-periodic un-modeled motion of the CIP in the GCRS at the 0.1–0.3 mas level still exists after the IAU 2006/2000A model has been taken into account.

Depending on accuracy requirements, the lack of a FCN model should cause neg- ligible problems. However, for the most stringent accuracy applications, a FCN model may be incorporated to account for the FCN contribution to the CIP mo- tion in the GCRS. The FCN model of Lambert (2007) can be found at <4> and a copy is kept at the IERS Conventions Center website < 2>. It is expected that the coefficients of the model will be updated regularly by the IERS to describe the most currently

4http://syrte.obspm.fr/∼lambert/fcn/

57 58 No. 36 al .c al o h offiinso h miia oe ftertord C uigteinterval the during FCN retrograde the of model empirical the of coefficients the for Table 5.2c: Table Note Technical IERS 9421 (unit 1984-2010 Y ro fteapiueetmts l mltdsaei nt fmicroarcseconds. of units for in values are amplitudes the All that estimates. Note amplitude the of error vrtepro o hc h oe a endtrie,temdlshould annual next model the until extrapolation the rms of mas determined, period provided. 0.1 the than is been better for update accuracies has while provide rms should model mas model 0.05 the the than which better for accuracies provide period the Over time. in defined piecewise are and respectively, terms, at sine found model and be the can for updates coefficients and the provides 5.2c Table J2000.0. since days information this constant provides a It assumes the that of effect. representation coordinates sinusoidal FCN Y time-varying the X, of a for the period of account to form to added the GCRS be in the to quantities in the CIP describes model in- The date a by ( 1 identified realization July current are on the versions name, Successive their CIP. in the cluded of motion FCN observed where S µ 94047004.5 46066.0 45700.0 1985.0 1984.0 96046431.0 1986.0 97046796.0 1987.0 98047161.0 1988.0 99047527.0 1989.0 90047892.0 1990.0 99051179.0 50814.0 1999.0 50449.0 1998.0 50083.0 1997.0 49718.0 1996.0 49353.0 1995.0 48988.0 1994.0 48622.0 1993.0 48257.0 1992.0 1991.0 00051708. 5. 5.6 4.5 152.9 4.3 147.0 4.5 81.8 98.4 4.2 61.2 2.9 143.4 55197.0 24.2 161.1 5.7 54832.0 2010.0 154.7 54466.0 2009.0 159.7 54101.0 2008.0 115.7 53736.0 2007.0 117.6 53371.0 2006.0 108.7 78.2 53005.0 2005.0 65.6 52640.0 2004.0 10.8 52275.0 2003.0 51910.0 2002.0 51544.0 2001.0 2000.0 = as). erMJD Year − σ X X Y steaglrfeunyo h C =2 (= FCN the of frequency angular the is CN F CN F C st P 2010. , and = − Y = 3.3dy.Teeutosfrtemdlare model the for equations The days. 430.23 = C X Y = S S X sin( sin( S − − − − − − − − − − − − . − − − σt σt X 141.8 246.6 281.9 255.0 210.5 187.8 2. 042.8 2.9 30.4 11.9 3.9 126.8 4.6 19.5 104.0 4.8 28.6 5.5 44.6 108.9 26.3 128.7 141.2 163.0 Y 192. 2.6 25.0 19.7 3.1 81.9 19.7 96.7 S C + ) + ) < and X Y 4 rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 > C C Y − − − − − . cos( cos( C − − − − − − − X 0. 11.1 105.3 7. 9.5 170.2 5. 8.6 159.2 2. 2.3 2.5 127.1 137.4 X − 6619.7 36.6 368.1 43.6 867.3 88.6 746.4 57.4 012.6 20.1 232.1 42.3 2.7 76.8 C a edtrie rmterelationships the from determined be can S . 2.2 1.4 σt σt and ) ) , , X fcnnut100701 S S X r h mltdso h cosine the of amplitudes the are fcnnut100701 π/P a/)and rad/d) a enestablished been has ) S le versions Older . X steformal the is t sgvnin given is (5.28) 5.5 Parameters to be used in the transformation IERS Technical

Note No. 36

Note that the unmodeled FCN motion of the CIP is included in the published IERS celestial pole offsets dX and dY. These offsets should not be applied when the FCN model is used.

5.5.6 Position of the Celestial Intermediate Origin in the GCRS The numerical development of the quantity s (i.e. the CIO locator) appearing in Eq. (5.10), compatible with the IAU 2006/2000A precession-nutation model as well as the corresponding celestial offset at J2000.0, has been derived in a way similar to that based on the IERS Conventions 2003 (Capitaine et al., 2003b). It results from expression (5.8) for s using the developments of X and Y as functions of time given by (5.16) (Capitaine et al., 2003a). The numerical development is provided for the quantity s + XY/2, which requires fewer terms to reach the same accuracy than a direct development for s.

Table 5.2d: Development of s(t) compatible with the IAU 2006/2000A precession-nutation model: all terms exceeding 0.5 µas during the interval 1975–2025 (unit µas). The expressions for the fundamental arguments appearing in column 1 are given by Eq. (5.43). 2 3 P s(t) = −XY/2 + 94 + 3808.65t − 122.68t − 72574.11t + k Ck sin αk +1.73t sin Ω + 3.57t cos 2Ω + 743.52t2 sin Ω + 56.91t2 sin(2F − 2D + 2Ω) +9.84t2 sin(2F + 2Ω) − 8.85t2 sin 2Ω

Argument αk Amplitude Ck Ω −2640.73 2Ω −63.53 2F − 2D + 3Ω −11.75 2F − 2D + Ω −11.21 2F − 2D + 2Ω +4.57 2F + 3Ω −2.02 2F + Ω −1.98 3Ω +1.72 l0 + Ω +1.41 l0 − Ω +1.26 l + Ω +0.63 l − Ω +0.63

The constant term for s, which was previously chosen so that s(J2000.0) = 0, was subsequently fitted (Capitaine et al., 2003b) so as to ensure continuity of UT1 at the date of change (1 January 2003) consistent with the Earth Rotation Angle (ERA) relationship and the current VLBI procedure for estimating UT1 (see (5.31)). The complete series for s + XY/2 with all terms larger than 0.1 µas is available electronically on the IERS Conventions Center website < 2> in file tab5.2d.txt and the terms larger than 0.5 µas over 25 years in the development of s are pro- vided in Table 5.2d with µas accuracy. (There is no term where the change from IAU 2000 precession to IAU 2006 precession causes a change in s larger than 1 µas after one century.)

5.5.7 ERA based expressions for Greenwich Sidereal Time Greenwich Sidereal Time (GST), which refers to the equinox, is related to the “Earth Rotation Angle” ERA, that refers to the Celestial Intermediate Origin (CIO), by the following relationship (Aoki and Kinoshita, 1983; Capitaine and Gontier, 1993) at the microarcsecond level:

Z t ˙ GST = dT0 + ERA + (ψA\+ ∆ψ1) cos(ωA + ∆1)dt − χA + ∆ψ cos A + ∆ψ1 cos ωA, (5.29) t0

59 60 No. 36 al .e eeomn fE optbewt A 0620Apeeso-uainmdl all model: precession-nutation 2006/2000A IAU with compatible EO of Development 5.2e: Table Note Technical IERS O= EO em xedn 0 exceeding terms O= EO 0 + − − 0 . . dT 00000044 014506 Argument Ω+63 +2640 2 2 2 2 2 2Ω Ω l l l l 3Ω 0 0 0 F F F F F − Ω + − Ω + 5yasi h eeomn fteE r rvddi al .e o00 mi- 0.01 to 5.2e. Table in provided are EO the of ( development accuracy Conven- croarcsecond the IERS in the years on 25 the available on website based are Center Time model tions Sidereal Greenwich precession-nutation for in 2006/2000A expression continuity IAU the as providing (Capitaine series well change full as The of transformations, date the the per- among at those UT1 level with to microarcsecond consistent similar the EO computations at using for obtained expression for was numerical Capitaine formed expression by A provided The was change. model (2003c). of precession-nutation 2006/2000A date IAU the ascension the term right at constant equator. in the UT1 moving nutation t; in the date and the along precession to equinox accumulated J2000.0 the the from of for ascension accounts right EO based The CIO the is which hr Oi h euto fteoiis,dfie by: defined origins”, the of “equation the is EO where hscnb rte as: written be can This ecliptic). the ( equator of the precession along the ecliptic the of precession eerdt h citco pc and epoch of ecliptic the to referred ∆ The sn xrsin 51)frEA 51)frteclsilcodntso h CIP the of coordinates celestial the for century, for (5.16) one 5.2d Table after ERA, level and for microarcsecond (5.14) consistency the expressions ensures at transformation transformation using based based equinox CIO the the in with used quantity GST the for in expression appearing accu- that microarcsecond the to similar in with the provided ERA in terms and introduced complementary ∆ were GST equinoxes,” These between the of relationship racy. “equation the classical provide the from to subtracted be to expression, terms EO the in term last for development the viding − +1 +2 Ω + 3Ω + − − − Ω ψ Ω Z 1 . 2 2 2 t 00 5 0 ∆ , C t D D D − µ ( S ERA(UT1) = GST 00 k 0 ψ sdrn h nevl17–05(unit 1975–2025 interval the during as t +11 +11 2Ω + Ω + 3Ω +  offiinsaesmlrt the to similar are coefficients A 3 4612 1 \ α ie y(.4,bigtentto nlsi ogtd n obliquity and longitude in angles nutation the being (5.24), by given , − ∆ + k s ˙ n olwn rcdr,dsrbdblw htesrdconsistency ensured that below, described procedure, a following and ∆ . 156534 ψ ψ 1 cos( ) cos s .  < 00 A ω t 2 A i.e. − > − ∆ + − nfile in 1 s P ihtodigits). two with EO . n r qa oteececet pt 1 to up coefficients these to equal are and , 3915817  k 1 , i.e. ) C dt rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 tab5.2e.txt k 0 + − sin χ ESCnetos2003 Conventions IERS A P χ C 00 tal. et α hs eeomn sgvni 54) the (5.40), in given is development whose , A t k ESCnetos1996 Conventions IERS k 2 Amplitude k − offiinsapaigi al .dpro- 5.2d Table in appearing coefficients C s k 0 ∆ 2003b). , sicue nepeso 53) This (5.31). expression in included is i.e. sin ψ dT h em agrta 0 than larger terms the ; cos α 0 h ih seso opnn of component ascension right the − − − − − − k µ a hsnt nuecontinuity ensure to chosen was opie h complementary the comprises ,  as). 0 0 1 1 1 4 A ...... C 63 63 41 26 72 52 96 98 02 55 21 75 ∆ + k 0 ψ 1 cos elcn h two the replacing , eua term secular A . ω A , . 5 µ ψ µ s The as. sover as cos (5.31) (5.30) tal. et  A , 5.6 Description of the IAU 2000/2006 precession-nutation model IERS Technical

Note No. 36

As an alternative to using the developments set out in Table 5.2e, the EO can be calculated from the quantity s and the classical (i.e. equinox based) nutation × precession × bias matrix: see Wallace and Capitaine (2006), Eq. (5.16). This is the method used by SOFA.

The numerical values chosen for the constant term dT0 in GST to achieve conti- nuity in UT1 at the transition date (1 January 2003), and for the corresponding constant term in s are dT0 = + 14506 µas and s0 = + 94 µas. The polynomial part of GST (i.e. of ERA(UT1) − EO(t)) provides the IAU 2006 expression for Greenwich Mean Sidereal Time, GMST (Capitaine et al., 2003c):

GMST = ERA(UT1) + 0.01450600 + 4612.15653400t + 1.391581700t2 − 0.0000004400t3 (5.32) − 0.00002995600t4 − 0.000000036800t5 .

This expression for GMST clearly distinguishes between ERA, which is expressed as a function of UT1, and the EO (i.e. mainly the accumulated precession-nutation in right ascension), which is expressed as a function of TDB (or, in practice, TT), in contrast to the GMST1982(UT1) expression (Aoki et al., 1982), which used only UT1. The difference between these two processes gives rise to a term in GMST of (TT−UT1) multiplied by the speed of precession in right ascension. Using TT−TAI = 32.184 s, this was: [47+1.5(TAI−UT1)] µas, where TAI−UT1 is in seconds. On 1 January 2003, this difference was about 94 µas (see Gontier in Capitaine et al., 2002), using the value of 32.3 s for TAI−UT1. This is included in the values for dT0 and s0.

5.6 Description of the IAU 2000/2006 precession-nutation model The following sections describe the main features of the IAU 2006/2000 precession- nutation. Comparisons of the IAU 2000/2006 precession-nutation with other mod- els and VLBI observations can be found in Capitaine et al. (2009).

5.6.1 The IAU 2000A and IAU 2000B nutation model The IAU 2000A model, developed by Mathews et al., (2002) and denoted MHB- 2000, is based on the REN2000 rigid Earth nutation series of Souchay et al. (1999) for the axis of figure (available at: ftp://syrte.obspm.fr/pub/REN2000/). The REN2000 solution is provided as a series of luni-solar and planetary nutations in longitude and obliquity, referred to the ecliptic of date, expressed as “in-phase” and “out-of-phase” components with their time variations. The sub-diurnal terms arising for the imperfect axial symmetry of the Earth are not part of this solution, so that the axis of reference of the nutation model be compliant with the definition of the CIP. The rigid Earth nutation was transformed to the non-rigid Earth nutation by applying the MHB2000 “transfer function” to the full REN2000 series of the cor- responding prograde and retrograde nutations and then converting back into ellip- tical nutations. This “transfer function” is based on the solution of the linearized dynamical equation of the wobble-nutation problem and makes use of estimated values of seven of the parameters appearing in the theory (called “Basic Earth Pa- rameters” (BEP)), obtained from a least-squares fit of the theory to an up-to-date precession-nutation VLBI data set (Herring et al., 2002). The estimation of the dynamical ellipticity of the Earth resulting in a value differing slightly from that of the reference rigid Earth series (i.e. by the multiplying factor 1.000012249) was a part of the estimation of the parameters needed for the non-rigid Earth series. The MHB2000 model improves the IAU 1980 theory of nutation by taking into account the effect of mantle anelasticity, 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 non- linear terms which have hitherto been ignored in this type of formulation. The axis of reference is the axis of maximum moment of inertia of the Earth in steady rotation (i.e. ignoring time-dependent deformations).

61 62 No. 36 Note Technical IERS 5 ftp://tai.bipm.org/iers/conv2010/chapter5/ ∆ ∆ ψ  = = h rgnlIU20Antto eisaeaalbeeetoial nteIERS the on electronically available at are website in series Center given nutation Conventions be 2000A will IAU series original these The of arguments and 5.7. coefficients Section the about details More ota fteIU20Amdldrn h eid19–00 h oe is model The 1995–2050. on period electronically at respect available the website with IAU2000B.f, Center during mas Conventions subroutine 1 model IERS Fortran than the 2000A greater the IAU difference in with the a motion implemented in of pole terms result celestial that planetary not the the fewerto provides does of It includes that effect It the consideration. accuracy Such for under an account (2003). level. period to Luzum mas time bias 1 the and a the in plus McCarthy at terms by only lunisolar designated developed 80 model model, than been a need abridged has who an model those a B1.6, for available Resolution is with 2000 2000B, changed IAU IAU have by below recommended caption As the (Note at as available below. (2003) well Conventions given as IERS is files the components those to nutation of respect largest headings the the for that 5.3b and ∆ 5.3a longitude Tables in at nutation electronically files for available the are by “R06”) provided subscript 2000A the “IAU of nutation nutation” definition to “total corresponding The series The respectively. components, components. planetary all and includes lunisolar the for rmeohJ2000.0: epoch from h A 00ntto oe sgvnb eisfrntto nlniue∆ longitude in nutation for series a by ∆ given is obliquity model nutation 2000 IAU The Moon. the of node where n CSaewtotaytm-eedn oain h hoeia geodesic (Mathews theoretical model MHB2000 The the al. in rotation. used et time-dependent 1991) (Fukushima, any contribution without nutation are GCRS and of accuracy an have to expected is 10 time model their about That with components expression(5.35)). terms “out-of-phase” (see planetary and variations 687 “in-phase” and as terms expressed lunisolar are 678 which includes series nutation resulting The ul einulad1.-ertrst ecnitn ihicuigtegeodesic the including an- with the consistent to be ( contributions to precession nutation terms 18.6-year geodesic and the semiannual includes nual, nutation 2000A IAU The GCRS: the of pole (originally the offset of following direction the the d with from in associated bias frame is as series mas. provided nutation 0.3 2000A about IAU to direction The GCRS computed the the in in accuracy pole the celestial limits which the a considered model, of not 2000A is IAU 5.5.5), the Section of (see part rigorously predicted be cannot which motion P P 02 s in is, 2002) , i N i N =1 =1 l ∆ η 0 ξ ∆ 0 0 ( ( stema nml fteSnadΩtelniueo h ascending the of longitude the Ω and Sun the of anomaly mean the is ψ µ B A  sfrms fistrs nteohrhn,teFN en free a being FCN, the hand, other the On terms. its of most for as g g i i  p ( = ( = eerdt h citco ae with date, of ecliptic the to referred , + + g o Ω cos 1 = = A B 1 = i 0 i 0 − − t t sin( ) cos( ) µ − . 0 0 92 s o h uain nlniue∆ longitude in nutations the for as, . . 5 sin 153 0068192 0166170 00 tab5.3a.txt / ARGUMENT ARGUMENT etr)i h rcsinmdlads htteBCRS the that so and model precession the in century) , ψ l bias ψ 0 ± ± − n biut ∆ obliquity and 0 0 < i 2 sin 2 n d and . . rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 0000100) 0000100) ( + ) 5 ( + ) > Tbe53)and 5.3a) (Table ntefiles the in  A B bias l 0 i 00 i 00 i Ω sin 3 + < + 00 00 + ftedrcino h I tJ2000.0 at CIP the of direction the of ) 5 . , > A B i 000 . i 000  t t epciey netatfrom extract An respectively. , cos( ) sin( ) tab5.3a.txt , R06 t esrdi uincenturies Julian in measured < ARGUMENT seScin563frthe for 5.6.3 Section (see ” tab5.3b.txt ARGUMENT 5 ψ > g .) n biut ∆ obliquity and and ) ) < , . 2 Tbe5.3b), (Table tab5.3b.txt > hyare They .  (5.35) (5.34) (5.33) ψ g and 5.6 Description of the IAU 2000/2006 precession-nutation model IERS Technical

Note No. 36

Table 5.3a: Extract from Table 5.3a (nutation in longitude) (available at < 2>) providing the largest components for the “in-phase” and “out-of-phase” terms of the “IAU 2000AR06” nutation. Units are µas and µas/century for the coefficients and their time variations, respectively. The expressions for the Delaunay arguments appearing in columns 1 to 5 are given by Eq. (5.43); those for the additional fundamental arguments of the planetary nutations appearing in columns 6 to 14 (bottom part) are given by Eq. (5.44). (Because the largest terms are all luni-solar, columns 9-17 contain only zeros in the extract shown.) 00 0 i Ai Ai l l FD Ω LMe LV e LE LMa LJ LSa LU LNe pA 1 −17206424.18 3338.60 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 −1317091.22 −1369.60 0 0 2 −22000000000 3 −227641.81 279.60 0 0 2 0 2 0 0 0 0 0 0 0 0 0 4 207455.50 −69.80 0 0 0 0 2 0 0 0 0 0 0 0 0 0 5 147587.77 1181.70 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ..... 0 000 0 i Ai Ai l l FD Ω LMe LV e LE LMa LJ LSa LU LNe pA 1321 − 17418.82 2.89 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1322 −363.71 -1.50 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1323 −163.84 1.20 0 0 2 −22000000000 .....

Table 5.3b: Extract from Table 5.3b (nutation in obliquity) (available at < 2>) providing the largest components for the “in-phase” and “out-of-phase” terms of the “IAU 2000AR06” nutation. Units are µas and µas/century for the coefficients and their time variations, respectively. The expressions for the Delaunay arguments appearing in columns 1 to 5 are given by Eq. (5.43); those for the additional fundamental arguments of the planetary nutations appearing in columns 6 to 14 (bottom part) are given by Eq. (5.44). (Because the largest terms are all luni-solar, columns 9-17 contain only zeros in the extract shown.) 0 0 i Bi Bi l l FD Ω LMe LV e LE LMa LJ LSa LU LNe pA 1 1537.70 9205233.10 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 −458.70 573033.60 0 0 2 −22000000000 3 137.40 97846.10 0 0 2 0 2 0 0 0 0 0 0 0 0 0 4 −29.10 −89749.20 0 0 0 0 2 0 0 0 0 0 0 0 0 0 5 −17.40 22438.60 0 1 2 −22000000000 ..... 0 000 0 i Bi Bi l l FD Ω LMe LV e LE LMa LJ LSa LU LNe pA 1038 0.20 883.03 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1039 −0.30 −303.09 0 0 2 −22000000000 1040 0.00 −67.76 0 1 2 −22000000000 .....

5.6.2 Description of the IAU 2006 precession The IAU 2006 precession (Capitaine et al., 2003c) provides improved polynomial expressions up to the 5th degree in time t, both for the precession of the ecliptic and the precession of the equator, the latter being consistent with dynamical theory while matching the IAU 2000A precession rate for continuity reasons. While the precession part of the IAU 2000A model consists only of corrections 00 00 (δψA = (−0.29965 ± 0.00040) /century, δωA = (−0.02524 ± 0.00010) /century) to the precession rates of the IAU 1976 precession, the IAU 2006 precession of the equator was derived from the dynamical equations expressing the motion of the 00 mean pole about the ecliptic pole, with 0 = 84381.406 for the mean obliquity at J2000.0 of the ecliptic (while the IAU 2000 value was 84381.44800). The IAU 2006

63 64 No. 36 .. rcsindvlpet optbewt h A 0020 model 2000/2006 IAU the with compatible developments Precession 5.6.4 .. A 06ajsmnst h A 00 nutation 2000A IAU the to adjustments 2006 IAU 5.6.3 Note Technical IERS Q P A A = +4 = − 46 . 199094 . 811015 dψ dε 00 00 J J t h ai xrsin o h rcsino h citcadteeutraepro- are equator the and with ecliptic respectively, the (5.39), of and precession (5.38) by the vided for expressions basic ob- The origins be the of model-dependent. can directly equation latter of is the The independent for is expression which 5.5.7). CIP, the Angle, ( and Section Rotation the model, Earth (see of precession-nutation the time for the coordinates sidereal expression quan- the GCRS for from derived the or tained and (5.16), for equations, the by dynamical those and given the ecliptic as of the such solutions for tities, quantities direct basic are the that for equator developments (Capitaine the developments separately polynomial vide precession 2006 IAU The n biut saalbeeetoial nteIR ovnin etrwebsite Center Conventions IERS the at on 5.2f) electronically difference (Table available The is IAU obliquity the 5.9). and with (Section implemen- use precession-nutation SOFA 2000A IAU for the 2006/2000A the in revised between IAU account been the into has taken of 2000A nutation are tation adjustments “IAU the The that notation precession. indicate the 2006 to included, used are be adjustments can small these Whenever at huhvr ml hne r eddi e fteIU20Anutation for values 2000A IAU 2006 IAU the the of with few compatibility precession a ensure the in the to in needed essentially order are lies in changes 2000 amplitudes small IAU very and though 2006 part, IAU between difference The the in effects perturbing precession the 2006 from for is IAU corrections precession quantities. include the geodesic observed also the to These and contributions MHB2000, (1991). other and Brumberg (1994) The Williams from 2000. are rates IAU Earth’s in 2000 the account IAU includes into the with precession consistent 2006 quite − IAU is flattening the dynamical value; Earth’s the for value 2 2 t 0 + i.e. d d 3 0 + = 47 + = • • × J . h itnebtenteCOadteeunxaogteCPeutr,which equator), CIP the along equinox the and CIO the between distance the 1939873 2 10 . h ags em ntecreto ple oteIU20Antto in nutation 2000A IAU the in to are, applied effect correction this the for in longitude terms of largest case The the To the in of needed amplitudes amplitudes. is sin the obliquity nutation adjustment multiply by 1980 to 2000A longitude necessary IAU in IAU is the nutation it the from change, estimating this different for when is for compensate value used obliquity 2006 2006 was IAU IAU that the the to that adjustment fact the of effect The nthe in to proportional are which of coefficients the nutation, in 2006 IAU the Introducing 10 0510283 rate. − − − 9 6 25 / / etr) otydet h otgailrbud hc a o taken not was which rebound, post-glacial the to due mostly century), etr) h ags hne ( changes largest The century). . . ,Y X, 6 8 d 00 t t 00  t ψ 2 o Ω cos 3 + Ω sin < t 2 − = eis in series) 2 0 + > 0 . 00022466 − − . nfile in 00052413 8 1 R06 . . . i Ω sin 1 6 7 t t n A 00epesosfrtentto nlongitude in nutation the for expressions 2000 IAU and µ tab5.2f.txt o 2( cos i 2( sin sae(aian n alc,2006): Wallace, and (Capitaine are as 00 00 J t 3 − 2 t rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 F F 3 −  aevlegvsrs oadtoa oso terms Poisson additional to rise gives value rate 0 IAU2000 − − − . 0 µ i 2( sin 6 0 . as: D 000000912 D . 000000646 )+0 + Ω) + Ω) + . ,Y X, / cf. sin F 51)frtecrepnigchanges corresponding the for (5.19) .  .) − 0  IAU2006 84381 = 00 D . 00 6 t 4 t Ω) + t 4 0 + i 2( sin − 1 = J 0 . . 0000000120 . 2 . 406 0000000172 F aeeet( effect rate . 0007.(osuch (No 000000470. J Ω) + 00 ˙ 2 :  /J ,Y X, 0 tal. et eut rmthe from results 2 − ( i.e. hc were which , 00 0 03)pro- 2003c) , 00 . t 6 5 t − i.e. 5 t 2 i 2Ω sin .  7774 (5.38) (5.37) (5.36) 0 J ˙ R06 2 and , = × ” 5.6 Description of the IAU 2000/2006 precession-nutation model IERS Technical

Note No. 36

00 00 2 3 00 4 00 5 ψA = 5038.481507 t − 1.0790069 t − 0.00114045 t + 0.000132851 t − 0.0000000951 t (5.39) 00 00 2 00 3 00 4 00 5 ωA = 0 − 0.025754 t + 0.0512623 t − 0.00772503 t − 0.000000467 t + 0.0000003337 t

The precession quantities PA = sin πA sin ΠA and QA = sin πA cos ΠA are the secular parts of the developments of the quantities P = sin π sin Π and Q = sin π cos Π, π and Π being the osculating elements (i.e. the inclination and lon- gitude of the ascending node, respectively) of the Earth-Moon barycenter orbit referred to the fixed ecliptic for J2000.0. The precession quantities ψA and ωA, defined in Section 5.4.5, are solutions of the dynamical equations expressing the motion of the mean pole about the ecliptic pole. Derived expressions for other precession quantities are given below:

00 00 2 00 3 χA = 10.556403 t − 2.3814292 t − 0.00121197 t +0.00017066300 t4 − 0.000000056000 t5,

00 00 2 00 3 A = 0 − 46.836769 t − 0.0001831 t + 0.00200340 t −0.00000057600 t4 − 0.000000043400 t5,

γ¯ = −0.05292800 + 10.55637800 t + 0.493204400 t2 − 0.0003123800 t3 (5.40) −0.00000278800 t4 + 0.000000026000 t5,

φ¯ = +84381.41281900 − 46.81101600 t + 0.051126800 t2 + 0.0005328900 t3 −0.00000044000 t4 − 0.000000017600 t5,

ψ¯ = −0.04177500 + 5038.48148400 t + 1.558417500 t2 − 0.0001852200 t3 −0.00002645200 t4 − 0.000000014800 t5,

where χA is the precession of the ecliptic along the equator and A is the mean obliquity of date;γ ¯, φ¯ and ψ¯ are the angles (Williams 1994, Fukushima 2003) referred to the GCRS.γ ¯ is the GCRS right ascension of the intersection of the ecliptic of date with the GCRS equator, φ¯ is the obliquity of the ecliptic of date on the GCRS equator and ψ¯ is the precession angle plus bias in longitude along the ecliptic of date. The last three series are from Table 1 of Hilton et al. (2006). Due to their theoretical bases, the original development of the precession quanti- ties as functions of time can be considered as being expressed in TDB. However, in practice, 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 (where l0 is the mean anomaly of the Sun), 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 below the required microarcsecond accuracy. This applies to Eq. (5.38)-(5.40), as well as to the polynomial part of Eq. (5.16) (i.e. the expression for the CIP’s GCRS X,Y coordinates).

5.6.5 Summary of the different ways of implementing the IAU 2006/2000A precession- nutation There are several ways to implement the precession-nutation model, and the pre- cession developments to be used should be consistent with the procedure being used. Using the CIO based paradigm, the complete procedure to transform between the ITRS and the GCRS compatible with the IAU 2006/2000A precession-nutation is based on the IAU 2000 expression (i.e. Eq. (5.15)) for the ERA and on expressions provided by (5.16) and Tables 5.2a–5.2d for the positions of the CIP and the CIO in the GCRS. These already contain the proper expressions for the new precession- nutation model and the frame biases.

65 66 No. 36 .. eeomn fteagmnso uioa nutation lunisolar of arguments the of Development 5.7.2 .. h utpir ftefnaetlagmnso uaintheory nutation of arguments fundamental the of multipliers theory The nutation of 5.7.1 arguments fundamental The 5.7 Note Technical IERS hne ntentto mltdsrsligfo h contributions, the from resulting amplitudes nutation time, the of in changes functions as nutations planetary and ∆ ( longitude each in components for “out-of-phase” provide, and phase” nutation “in 2000 the IAU of Tables h xrsin o h udmna ruet fntto r ie ytefol- the by given are nutation of arguments where developments fundamental lowing the for expressions The expres- precession coordinates. the for CIP for expressions GCRS case the the the in for is TDB it of ( place as sions in arguments, practice nutation in fundamental used the be can TT Consequently, 0 than ( arguments TDB difference the of arguments fundamental the for expressions original the replacement In ∆ the for that series fact the the as- in from necessarily sin(ARGUMENT) arise not assignments the multipliers do in ( obliquity of differences The and set ries. longitude same in the nutations. nutations luni-solar sign of the of tables case Different the in variations time their plus the for write, frequency may the one studies, nutation and in independent, involved scales time Over L 94 als34(.)ad35() ae nIR 92cntnsadWilliams and constants 1992 IERS on based (b)) 3.5 and (b.3) 3.4 Tables 1994: ( Earth the including planets the of longitudes mean the are h rqec hsdfie spstv o ottrs n eaiefrsome. for negative fact the and in only terms, terms most nutation luni-solar for the that from positive differ is terms nutation defined Planetary thus frequency The five of set a by characterized is Arguments Fundamental series five the nutation the in terms integers lunisolar the of Each precession 2006 IAU Sidereal Greenwich bias, for expression 2006 2000 IAU ( the IAU Time with rigor- conjunction the in the used upon be routines of should SOFA based by one are supported following They are ( requires that 5.4.5 GCRS 5.9). Section transfor- the (Section in based mentioned and equinox procedures ITRS the ous the using between model 2006/2000A/B mation IAU the Implementing h uaindsrbdb h emi ie by given is term the by described nutation the The term. the characterize = ARGUMENT N cf. J j , =1 q 53) n h ajse)IU20Antto.Teetransformations These nutation. 2000A IAU (adjusted) the and (5.39)) Eq. L ARGUMENT Sa , ARGUMENT ω cf. 14 . cf. 01 , N ≡ ) eto ..) hsas ple otennplnma ato q (5.16) Eq. of part non-polynomial the to applies also This 5.6.4). Section L − → q 53)adTbe5.2e). Table and (5.31) Eq. µ j d U s hc ssgicnl eo h eurdmcorscn accuracy. microarcsecond required the below significantly is which as, hc eemnsteagmn o h ema iercmiainof combination linear a as term the for argument the determines which ω ( , ARGUMENT k l t Ne ( + N = β and ) = j P k =1 P r epnil o ieec nteCPlcto hti less is that location CIP the in difference a for responsible are ) ω j 5 , =1 j 14 14 k − =1 t copne yrvra ftesg ftececetof coefficient the of sign the of reversal by accompanied ) F T(hs ags emi . ms 1.7 is term largest (whose TT ) + N /dt. 14 N j β F j 0 F t k stegnrlpeeso nlongitude in precession general the is F j . smaue nJla etre fTB(Simon TDB of centuries Julian in measured is j , j 0 , r ucin ftm,adteaglrfeunyof frequency angular the and time, of functions are hr h aus( values the where F F rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 6 j N to aeyteDlua aibe ( variables Delaunay the namely , j ψ F oapriua emi h uainse- nutation the in term particular a to 13 n ∆ and k sntdi als52-.bad5.3b, and 5.2a-5.2b Tables in noted as , htr ntentto series, nutation the in term th t  smaue nTB oee,the However, TDB. in measured is evsteesre unchanged. series these leaves ARGUMENT N 1 , · · · N , × sin L 5 ,tececet of coefficients the ), ω F ftemultipliers the of ) ψ Me 1 seetvl time- effectively is l 0 - n biut ∆ obliquity and otenutation the to ) F , p 14 L A ω e V . ,l l, k fluni-solar of (TDB , 0 ,D, F, , L E , (5.42) (5.41) tal. et − L Ma TT), Ω):  , , , 5.7 The fundamental arguments of nutation theory IERS Technical

Note No. 36

et al. (1991), for precession.

F1 ≡ l = Mean Anomaly of the Moon

= 134.96340251◦ + 1717915923.217800t + 31.879200t2

+0.05163500t3 − 0.0002447000t4,

0 F2 ≡ l = Mean Anomaly of the Sun

= 357.52910918◦ + 129596581.048100t − 0.553200t2

+0.00013600t3 − 0.0000114900t4,

F3 ≡ F = L − Ω

= 93.27209062◦ + 1739527262.847800t − 12.751200t2 (5.43)

−0.00103700t3 + 0.0000041700t4,

F4 ≡ D = Mean Elongation of the Moon from the Sun

= 297.85019547◦ + 1602961601.209000t − 6.370600t2

+0.00659300t3 − 0.0000316900t4,

F5 ≡ Ω = Mean Longitude of the Ascending Node of the Moon

= 125.04455501◦ − 6962890.543100t + 7.472200t2

+0.00770200t3 − 0.0000593900t4

where L is the Mean Longitude of the Moon.

Note that the SOFA implementation of the IAU 2000A nutation takes the MHB2000 code (T. Herring 2002) as its definition of the IAU 2000A model. As part of this strict compliance, SOFA uses the original MHB2000 expressions for the Delaunay variables l0 and D, that differ from Eq. (5.43) in that the fixed term is rounded to five digits (i.e. 1287104.7930500 instead of 1287104.79304800 for the Eq. (5.43) value in the l expression converted into arcseconds and 1072260.7036900 instead of 1072260.70369200 for the Eq. (5.43) value in the l expression converted into arc- seconds), respectively. The CIP location is insensitive to this difference of 2 µas in the nutation arguments at a level better than 10−9 arcsec accuracy.

It should also be noted that the SOFA equinox based implementation of the IAU 2000A nutation follows the MHB2000 Fortran code in neglecting time varia- 000 000 tions of the out of phase components, i.e. the Ai and Bi columns of Table 5.3a (see Section 5.6.1). The difference in the CIP location is just over 2 µas after one century.

5.7.3 Development of the arguments for the planetary nutation

The mean longitudes of the planets used in the arguments for the planetary nu- tations are essentially those provided by Souchay et al. (1999), based on theories and constants of VSOP82 (Bretagnon, 1982) and ELP 2000 (Chapront-Touz´eand Chapront, 1983) and developments of Simon et al. (1994: Tables 5.8.1-5.8.8). Their developments are given in Eq. (5.44) in radians with t in Julian centuries.

67 68 No. 36 Note Technical . rgaeadrtord uainamplitudes nutation retrograde and Prograde 5.8 IERS (∆ (∆ ψ ψ op ip ( ( aeu fi-hs n u-fpaeparts out-of-phase and in-phase of up ∆ of made positive parts the periodic with purely system, The equatorial mean the X in vector two-dimensional moving L n writing and ∆ quantities The 0 than from less different is (slightly CIP location Neptune CIP of (5.44), the longitude Eq. to the in makes for that this expression MHB2000 difference that inal maximum is variables The code 0 Delaunay MHB2000 is the the case. for with nutation SOFA used tary of are compliance expressions strict simplified the of part Another precession, general The n oaigi h poiesne h eopsto sfclttdb factoring by facilitated is decomposition The sign sense. the opposite out the sense same in the rotating in one (rotating positive nutation the circular from rotating prograde as uniformly two a into constituting decomposed one be may vectors, vectors these of Each respectively. u-fpaevcosi em ftee n obtains one these, of terms in vectors out-of-phase bandfo hs ytereplacement the by these from obtained n (cos and (sin Because with t t sin ) sin ) Ne and . χ 013 χ 5 = k   ω q 0 k , 0 Y k F F F F F (∆ , k − , (∆ F F F F ω nraiglnal ihtm.Tepi fvcosaoete becomes then above vectors of pair The time. with linearly increasing t µ 14 13 12 11 10 χ χ ∆ . xspitn ln h ietoso nraig∆ increasing of directions the along pointing axes ∆ k 6 9 8 7 cos 1868 3 + 311886287 + ψ satroecnuy h OAipeetto loue h orig- the uses also implementation SOFA The century. one after as k k ψ   , | ≡ op op ip ip β − nrae ierywt ie h uulyotooa ntvectors unit orthogonal mutually the time, with linearly increases q χ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ k k ( ( ( ( ω sin t k t t t = sin ) of sin ) )=(∆ = )) )=(∆ = )) n (cos and ) k | L p L L L L L L L . ψ q χ ω 01 A k Me Ne U Sa J Ma E e V k ( k X  (  t r nrtord oain nrsligtei-hs and in-phase the resolving On rotation. retrograde in are ) i.e. 0 | sin ) 0 µ ω rmteargument, the from , , xstwrstepositive the towards axis satroecentury. one after as k ψ ψ ∆ ∆ F | 0 = 5 = 5 = 0 = 0 = 6 = 1 = 3 = 4 = t L k k ip op 14    + 0 op Ne χ . ip sin 8133035638 sfo ioht n oca (1990). Souchay and Kinoshita from is , sin k q ( ( n ∆ and , k t t ...... sin )=( = )) )=(∆ = )) 5 = ψ  β 02438175 3 + 311886287 7 + 481293872 21 + 874016757 52 + 599546497 334 + 203480913 628 + 753470314 1021 + 176146697 2608 + 402608842  0 0 k ( sin( t rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 ) cos( χ sin ) ≡ . k 2190 3 + 321159000  oaei rgaesneadtevectors the and sense prograde a in rotate ) q ( ω q ω k t k k a evee stecmoet fa of components the as viewed be may ) ψ  ∆ k χ t 0 k t op ψ × + k × n ∆ and + χ , k ip q sin t β k k β sin 0 + k t k en uhthat such being . . .Temxmmdffrnei the in difference maximum The ). − → )  8133035638 4781598567 ) , . . 0 , 3299104960 9690962641   . . . Y cos ∆ 0612426700 3075849991 0 00000538691 ( ∆ . . t 3285546211 7903141574 sin  o emo frequency of term a for ) xs n h te retrograde a other the and axis) χ  k ip χ k op k . k aey( namely , χ cos( 812777400 sin( q , k , k ω ∆ ω ∆ × × k ψ × × k  t  t t, t, × × k ip k op n ∆ and + F t, t, × × + × cos t, t, 1 sin β − t, t, - β t F k 2 k sin × . )) χ 5 )) χ  k , k respectively. , , nteplane- the in ) χ ) , t . k nta of instead , − cos ω (5.47) (5.46) (5.44) (5.45) (5.48) k χ are k ) 5.9 Algorithms for transformations between ITRS and GCRS IERS Technical

Note No. 36

ip ip pro ip ret ip (∆ψ (t) sin 0, ∆ (t)) = Ak (sin χk, − cos χk) + Ak (− sin χk, − cos χk), (5.49) 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 ), (5.50) pro op 1 op op Ak = 2 (∆ψk sin 0 + qk∆k ),

ret op 1 op op Ak = 2 (∆ψk sin 0 − qk∆k ). The expressions providing the corresponding nutation in longitude and in obliquity from circular terms are

∆ψip = qk Apro ip − Aret ip
, k sin 0 k k

∆ψop = 1 Apro op + Aret op
, k sin 0 k k (5.51) ip pro ip ret ip
∆k = − Ak + Ak ,

op pro op ret op
∆k = qk Ak − Ak .

The contribution of the k-term of the nutation to the position of the Celestial In- termediate Pole (CIP) in the mean equatorial system is thus given by the complex coordinate

  pro iχk ret −iχk ∆ψ(t) sin 0 + i∆(t) = −i Ak e + Ak e , (5.52)

pro ret where Ak and Ak are the amplitudes of the prograde and retrograde compo- nents, respectively, and are given by

pro pro ip pro op ret ret ip ret op Ak = Ak + iAk ,Ak = Ak + iAk . (5.53)

The decomposition into prograde and retrograde components is important for studying the role of resonance in nutation because any resonance (especially in pro ret the case of the nonrigid Earth) affects Ak and Ak unequally. In the literature (Wahr, 1981) one finds an alternative notation, frequently fol- lowed in analytic formulations of nutation theory, that is:   pro − −iχk ret − iχk ∆(t) + i∆ψ(t) sin 0 = −i Ak e + Ak e , (5.54)

with

pro − pro ip pro op ret − ret ip ret op Ak = Ak − iAk ,Ak = Ak − iAk . (5.55)

Further details can be found in Defraigne et al. (1995) and Bizouard et al. (1998).

5.9 Algorithms for transformations between ITRS and GCRS

Software routines to implement the IAU 2006/2000A transformations are provided by the IAU Standards Of Fundamental Astronomy service.6 The routines vary in complexity from simple modules to complete transformations, allowing the application developer to trade off simplicity of use against computational efficiency and flexibility. Implementations in Fortran77 and C are available. The SOFA software supports two equivalent ways of implementing the IAU reso- lutions in the transformation from ITRS to GCRS provided by expression (5.1),

6The SOFA software collection may be found at the website http://iau-sofa.hmnao.com/.

69 70 No. 36 Note Technical IERS OAruie htspotteIU20/00 oesicuetefollowing the include models 2006/2000A IAU the others): (among support that routines SOFA quantity the is required coordinates matrix, the motion polar into polar components the these cases, all combine In to then and matrix. forms, terrestrial-to-celestial classical complete or the based for CIO choosing the (5.1), expression of components various two the the ERA links equivalently with that origins”, 5.4.5, quantity the The of and censions, “equation angles). 5.4.3 classi- the nutation 5.4.1, the is ( and systems Sections Time (b) precession classical Sidereal in and Greenwich of described 5.5) use and Section procedure equinox in based the equinox on described based parameters transformation using cal 5.4.4, the and and Origin 5.4.2 Intermediate ( Celestial Angle the Rotation on Earth based transformation the (a) namely POM00 steGenih(paet ieelTm.Ti a eotie ycligthe calling by obtained be can SOFA This the calling Time. by Sidereal obtained (apparent) Greenwich be the can is which ERA, matrix Angle, defines routine Rotation that Earth rotation the Earth is for angle the routine R is the component calling intermediate by The with obtained transpose be the can taking matrix and true-to-celestial the known, otn,wihipeet h A 00 rnae oe.Oc ∆ Once SOFA model. the truncated of 2000B IAU means the by implements which precession, routine, 2006 IAU match routine to adjustments with model, ∆ components nutation matrix to routine 2006/2000A SOFA routine IAU the The CIO. intermediate- using the the transpose) of position the is (as this obtained transformation, be C2IXYS, based is can CIO bias it frame the matrix, and For to-celestial precession nutation, (5.1). of expression effects in combined the for matrix The BP06 XY06 SP00 POM00 PNM06A NUT06A NUM06A GST06A GMST06 ERA00 EORS C2TCIO C2T06A C2IXYS C2I06A XYS06A ( t nepeso 51.FrteCObsdtasomto,teagei question in angle the transformation, based CIO the For (5.1). expression in ) n hntasoigtersl ( result the transposing then and NUT06A XYS06A ERA00 trigfo h I position CIP the from starting α Q CIO eeta-otu arx( matrix celestial-to-true X quantity the matrix motion polar 2006/2000A IAU matrix, celestial-to-true 2006/2000A IAU components, nutation 2006/2000A IAU matrix, nutation 2006/2000A IAU Time, Sidereal (apparent) Greenwich 2006 IAU Time, Sidereal Mean Greenwich Angle and Rotation matrix Earth celestial-to-true given origins, the of equation matrix celestial-to-terrestrial based CIO 2006/2000A IAU matrix, celestial-to-terrestrial given matrix, celestial-to-intermediate 2006/2000A IAU matrix, celestial-to-intermediate X ( t , , h onepr ntecs fteeunxbsdtransformation based equinox the of case the in counterpart The . stetu-oclsilmti.T banti arxrqie the requires matrix this obtain To matrix. true-to-celestial the is ) Y Y atr u esacrt,peitosaeaalbefo the from available are predictions accurate, less but Faster, . ntecs fteeunxbsdtasomto,tecounterpart the transformation, based equinox the of case the In . and , s rmsm-nltclsre,IU2006/2000A IAU series, semi-analytical from A 2006/2000A IAU , − x S.Frbt rnfrain,tepoeuei ofr the form to is procedure the transformations, both For GST. p α , y Υ p eerdt h I n h qio,rsetvl) or respectively), equinox, the and CIO the to referred , hscnb copihdb aln h OAroutine SOFA the calling by accomplished be can This . i.e. ψ s s 0 0 oee yteroutine the by modeled , n ∆ and h I ae rcdr ecie nScin 5.4.1, Sections in described procedure based CIO the TR rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5  . hs a epeitduigteIU2000A IAU the using predicted be can these ; etc. W ( ,gvn∆ given ), t X e.g. nepeso 51,i edd sn the using needed, is (5.1), expression in ) α ,Y s Y, X, CIO , Y sn h upr routine support the using n h quantity the and − r vial ycligteSOFA the calling by available are α ψ X Υ SP00 n ∆ and , dffrnebtenrgtas- right between (difference Y Q . and (  t and ) s s htdfie the defines that R ( ψ t areither pair ) s n ∆ and TR i.e. NUT00B .Also ). PN06  Q the are ( t ) 5.10 Notes on the new procedure to transform from ICRS to ITRS IERS Technical

Note No. 36

SOFA routine GST06, given the celestial-to-true matrix that was obtained earlier. The three components – the precession-nutation matrix, the Earth rotation quan- tity and the polar motion matrix – are then assembled into the final terrestrial- to-celestial matrix by means of the SOFA routine C2TCIO (CIO based) or C2TEQX (equinox based), followed by TR as required. Two methods to generate the terrestrial-to-celestial (i.e. ITRS-to-GCRS) matrix Q(t)R(t)W (t), given TT and UT1, are set out below. In each case it is assumed that observed small corrections to the IAU 2006/2000A model, either as ∆X, ∆Y or as d∆ψ, d∆, are available and need to be included.

Method (1): the CIO based transformation The CIO based transformation is a function of the CIP coordinates X,Y and the quantity s. For the given TT, call the SOFA routine XY06 to obtain the IAU 2006/2000A X,Y from series (see Section 5.5.4) and then the routine S06 to obtain s. Any CIP corrections ∆X, ∆Y can now be applied, and the corrected X, Y, s can be used to call the routine C2IXYS, giving the GCRS-to-CIRS matrix. Next call the routine ERA00 to obtain the ERA corresponding to the current UT1, and apply it as an R3 rotation using the routine RZ, to form the CIRS-to-TIRS matrix. Given 0 xp, yp, and obtaining s by calling the routine SP00, the polar motion matrix (i.e. TIRS-to-ITRS) is then produced by the routine POM00. The product of the two matrices (GCRS-to-TIRS and TIRS-to-ITRS), obtained by calling the routine RXR, is the GCRS-to-ITRS matrix, which can be inverted by calling the routine TR to give the final result.

Method (2): the equinox based transformation The classical transformation, based on angles and using sidereal time is also avail- able. Given TT, the IAU 2006/2000A nutation components ∆ψ, ∆ are obtained by calling the SOFA routine NUT06A. Any corrections d∆ψ, d∆ can now be applied. Next, the GCRS-to-true matrix is obtained using the routine PN06 (which employs the 4-rotation Fukushima-Williams method described in Section 5.3.4, final para- graph). The classical GCRS-to-true matrix can also be generated by combining separate frame bias, precession and nutation matrices. The SOFA routine BI00 can be called to obtain the frame bias components, the routine P06E to obtain various precession angles, and the routine NUM06A to generate the nutation matrix. The product N×P×B is formed by using the routine RXR. Next call the routine GST06 to obtain the GST corresponding to the current UT1, and apply it as an R3 rotation using the routine RZ to form the true matrix-to-TIRS. Given xp, yp, and obtaining s0 with the routine SP00, the polar motion matrix (i.e. TIRS-to- ITRS) is then obtained using the routine POM00. The product of the two matrices (GCRS-to-TIRS and TIRS-to-ITRS), obtained by calling the routine RXR, is the GCRS-to-ITRS matrix, which can be inverted by calling the routine TR to give the final result.

Methods (1) and (2) agree to microarcsecond precision. Both methods can be abridged to trade off speed and accuracy (see Capitaine and Wallace, 2008). The abridged nutation model IAU 2000B (see Section 5.5.1) can be substituted in Method (2) by calling NUT00B instead of NUT06A. Depending on the application, the best compromise between speed and accuracy may be to evaluate the full series to obtain sample values for interpolation.

5.10 Notes on the new procedure to transform from ICRS to ITRS The transformation between the GCRS and the ITRS, which is provided in detail in this chapter for use in the IERS Conventions, is also part of the more gen- eral transformation for computing directions of celestial objects in intermediate systems or terrestrial systems. The procedure to be followed in transforming from the celestial (ICRS) to the terrestrial (ITRS) systems has been clarified to be consistent with the improving

71 72 No. 36 Note Technical IERS n elnto n saaoost h rvosdsgaino aprn right “apparent of ascension designation right previous intermediate the the to declination.” called analogous and is ascension is system and reference declination this and in 2006/2000A position IAU IntermediateThe the Conventional equinox. by the the and realized replaces equator is Origin its Ko- that defines and (CIP) model Seidelmann Pole precession-nutation Intermediate also Celestial (See The system systems reference reference intermediate terrestrial (2002).) terrestrial a valevsky and to celestial transforming of use CIO in make B1.8). parallel we Resolution the before, 2000 and As (IAU B1.3) processes Resolution based trans- 2000 equinox ICRS-to-BCRS-to-GCRS-to-ITRS (IAU and the relativity show general to in is chart formation docu- this of Group purpose Working The NFA followed. IAU be the to procedures in 5.1 (Capitaine Figure ments See accuracy. observational tal. et 07 o iga fteCObsdadeunxbased equinox and based CIO the of diagram a for 2007) , rnfrainbtenteIR n h GCRS the and ITRS the between Transformation 5 5.10 Notes on the new procedure to transform from ICRS to ITRS IERS Technical

Note No. 36

Figure 5.1: Process to transform from celestial to terrestrial reference systems (Chart from the IAU Working Group on Nomenclature for Fundamental Astronomy (2006)). The chart summarizes the system, and the elements that are associated with that system, i.e. the name for the positions (place), the processes/corrections, the origin to which the coordinates are referred, and the time scale to use. In particular the blue type in the box in the “Process” column is the operation/correction to be applied, and the purple type indicates the quantities required for that process. CIO and equinox based processes are indicated using grey and yellow shading, respectively.

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