P03-BASED PRECESSION-NUTATION TRANSFORMATIONS

1 P03-BASED PRECESSION-NUTATION WALLACE1, N. CAPITAINE21 HMNAO, CCLRC / Rutherford Appleton Laboratory,Chilton, Didcot, OX11 0QX, United Kingdome-mail: Observatoire de Paris61, avenue de l Observatoire, 75014 - Paris, Francee-mail: IAU WG on precession and the ecliptic has recommended theadoption ofthe P03 models of Capitaine et al. (2003). We discuss methodsfor generating the rotationmatrices that transform celestial to terrestrial coordinates, taking into account frame bias (B),P03 precession (P), P03-adjusted IAU 2000A nutation (N) andEarth rotation.

2 The NPB portioncan refer either to the equinox or the celestial intermediate origin (CIO), requiring either theGreenwich sidereal time (GST) or the Earth rotation angle (ERA) as the measure of Earthrotation. The equinox based NPB transformation can be formed using various sequences ofrotations, while the CIO based transformation can be formedusing series for theX,Ycoordinatesof the celestial intermediate pole (CIP) and for the CIO locators; also, either matrix can becomputing using series for thex,y,zcomponents of the rotation vector.

3 Common to bothmethods is the CIP, which forms the bottom row of the transformation matrix. In the caseof the CIO based transformation , the CIO is the top row of the NPB matrix, whereas in theequinox based case it enters via the GST formulation in the form of the equation of the origins(EO). The EO is the difference between ERA and GST and equivalently the distance betweenthe CIO and equinox . The choice of method is dictated by considerations of internal consistency,flexibility and ease of use; the different ways agree at the level of a few microarcseconds overseveral centuries, and consume similar computing INTRODUCTIONAt the 2003 IAU General Assembly, a working group was formed to select models for theprecession of the ecliptic and equator that are consistent with dynamical theories to replace thesimple rate corrections adopted in 2000 (see Hilton et al.)

4 2005). The new precession will be usedwith the existing IAU 2000A nutation , and in order to be consistent with the new precession ,this requires small ( 10 as) corrections for the effects of (i) the change in obliquityfrom theIAU 1980 ecliptic to the P03 ecliptic and (ii) the secular variation in the Earth s dynamicalflattening, not taken into account in the IAU 2000A model. In this paper, based on the recentstudy of Capitaine & Wallace (2005), we review methods for using the new precession -nutationin practical following matrix:Mclass=NPB(1)is needed in two forms, namely the classical form based on theequinox and the new form based onthe celestial intermediate origin (CIO).

5 The matricesB,PandNare the successive contributionsof the frame bias, precession and nutation . In order to predict terrestrial coordinates, or hourangles, formulations for both Greenwich sidereal time and Earth rotation angle are end-to-end transformation for both forms is between celestial and terrestrial coordinates,represented by the matrixRin:R=R3(ERA) MCIO(2)=R3(ERA) R3( EO) M =R3(GST) M .(3)The link between the two methods is the equation of the origins, EO, a quantity somewhatakin to the equation of the equinoxes but including precession as well as nutation .

6 Note that inthe equinox based version, (3), theR3rotation is a function both of Earth rotation and time,whereas in the CIO based version, (2), the rotation-relatedand time-related components arekept precession -NUTATIONThe two types of NPB matrices,MCIOandM , can be generated in a number of ways,including semi-analytical expressions for the CIP locationX(t),Y(t)and CIO locators(t), clas-sical methods using precession and nutation angles, and models for the Euler axis and angle (the r-vector ). The matricesMcan both be expressed in terms of three Euler anglesE,d,E+ ,whereE,dare the GCRS polar coordinates of the CIP and the angle selects the origin of rightascension:M=R3( E ) R2(d) R3(E),(4)or:M =R3( ) M ,(5)where the matrixM is a function of the CIPX,Y,Z:M =R3( E) R2(d) R3(E)= 1 aX2 aXY X aXY1 aY2 YXY1 a(X2+Y2) ,(6)with:a= 1/(1 + cosd) = 1/(1 +Z) = 1/[1 + (1 X2 Y2)1/2],(7)For the CIO- based matrixMCIO, =s.

7 For the equinox - based matrixM , = EO+ convenient way of writing theMmatrices is as three unit vectorsv:MCIO vCIOvCIP vCIOvCIP ,M v vCIP v vCIP .(8)98In each case the top row (vCIOandv ) is the the RA origin of date, namely the CIO or theequinox respectively. The bottom row is the GCRS position ofthe CIP, which of course iscommon to both most conservative method of forming the equinox based matrixM is to provide indi-vidual rotation matrices for each of the three stages, delivering successively mean place of epochand mean place of date.

8 In this scheme, the precession stage can use four angles that comedirectly from P03, or alternatively the traditionalz, and , giving a total of either ten or ninerotations respectively. The Fukushima-Williams method (Fukushima 2003) instead condensesthese into only four rotations, different uses of which deliver the full transformation or stop shortat one of the earlier the CIO based matrix, the starting-point is the CIP position and the CIO (8), any of the above classical methods can be used to obtainX,Ysimply by evaluatingonly the matrix elementsM(3,1)andM(3,2).

9 However, an efficient and foolproof alternative issemi-analytical series for the coordinates themselves, that deal with frame bias, precession andnutation in a single radically different approach, capable of generating bothM andMCIO, is to use semi-analytical series to generate thex,yandzcoordinates of the rotation vector . This is the Euleraxis unit vector scaled by the amount of rotation, from whichthe more familiar rotation matrixcan be EARTH ROTATION AND THE ORIGIN OF RIGHT ASCENSIONThe choice of equinox or CIO as the longitude zero affects how Earth rotation is expressed,namely as sidereal time or Earth rotation angle.

10 These two measures are related through theequation of the origins (EO), which is the distance between the CIO and the equinox , so thatGST=ERA CIO is located by the quantitys, through (5) and =s. It can be obtained quitereadily by numerical integration, but for much faster results in practical applications a series isalways used. Series forsitself exist, but a much more concise result is obtained by evaluating theexpressions+XY/2: see Table 1. Even fewer terms are needed to computes+XY/2+D, whereD= Y2t2(X1t/3+Xnut), but this involves intermediate results from evaluating theX,Yseries,a complication that probably outweighs the small performance gains.