Compatibility of the IERS earth rotation representation and its relation to the NRO conditions

Transcription

1 Journées 005 Systèmes de Référence Spatio-emporels Earth dynamics and reference systems: Five years after the adoption of the IAU 000 Resolutions Warsaw, 9- September 005 Compatibility of the IERS earth rotation representation and its relation to the NRO conditions Athanasios Dermanis Department of Geodesy and Surveying he Aristotle University of hessaloniki

2 Summary he mathematical representation of a time-dependent rotation matrix relating the terrestrial reference system to the celestial one, requires the determination of three functions of time and is directly related to a mathematically compatible rotation vector (vector of instantaneous angular velocity). Representations involving more than three time functions, as the one provided by the IERS, must be accompanied by a number of conditions equal to the number of superfluous functions. In a general separation of the total rotation in three parts, one for precession-nutation with three functions X () t, Yt (), s() t, one for diurnal rotation with one function θ () t and one for polar motion with three functions x() t, yt (), s () t, three conditions can be identified as follows: wo directional conditions are introduced by imposing that at any instant the direction of the mathematically implied rotation vector is the same as the direction of the axis of diurnal rotation by an angle θ () t. One magnitude condition is introduced by imposing that at any instant the magnitude of the rotation vector is equal to the time derivative of the diurnal rotation angle θ () t. A fourth condition for reducing the seven rotation parameters (functions) to three is missing, reflecting the fact that adding an arbitrary function to s() t while subtracting it from s () t, gives the same rotation matrix and rotation vector. It is shown that the NRO conditions imply the magnitude condition only when combined with the direction conditions. hus the direction conditions together with the NRO conditions constitute the required set of 4 independent conditions that reduces the original number of 7 parameters to the required minimal number of. hese four conditions can be used to produce an IERS-like representation with new compatible functions X () t, Y () t, s () t, θ () t, x () t, y () t, s () t, which might be more appropriate for the analysis of high-precision observations, sufficiently dense in time to be sensitive to the direction and magnitude of the compatible rotation vector. Relations are given for the computation of the position of the Compatible Celestial Pole (), which should replace the smoothed Celestial Intermediate Pole, as well as for the compatible Universal ime related to the compatible angle of diurnal rotation θ () t.

3 . Introduction One of the main function of the International Earth Rotation and Reference Systems Service (IERS) is the provision of the necessary parameter series, which allow, for any epoch, the computation of an orthogonal matrix R transforming coordinates x in the errestrial Reference System into coordinates x C in the Celestial Reference System, according to x = Rx () C If we denote the elements e, e, e of any geocentric orthonormal basis by a row-matrix e = [ ee e ], the matrix R, which represents the rotation of the earth, transforms the celestial reference system e C into the terrestrial one e according to e C = e R. () he orthogonal matrix R () t can be completely described by only parameters (functions of time), for example Euler angles of elementary rotations around the axes. he IERS representation, following the IAU 000 resolutions, as well as the previous classical astronomic representation, have two main characteristics: (a) an overparameterization, where more than parameters are used for the descripton of R. (b) a separation of R into parts R = QDW, the precession-nutation part Q, the diurnal rotation part D and the polar motion part W. he classical representation involves 9 parameters, for precession, for nutation, for diurnal rotation and for polar motion. he IERS representation starts with 7 parameters ( for precession-nutation, for diurnal rotation and for polar motion), which by the ingenious introduction of the so called Non Rotating Origin (NRO) conditions are reduced to 5 only parameters ( for precession-nutation, for diurnal rotation and for polar motion). In general, the diurnal rotation part D consists of a rotation around a direction (unit vector) p, called the celestial pole, which serves as the common rd axis ( IC I p = e = e ) of two intermediate reference systems, the Intermediate Celestial (IC) and the Intermediate errestrial (I). hus the precession-nutation matrix Q transforms the celestial system to the intermediate celestial, the diurnal rotation matrix D transforms the intermediate celestial to the intermediate terrestrial, and the polar motion matrix W transforms the intermediate terrestrial to the terrestrial system according to e IC C = e Q, I IC e = e D, I e = e W. () he most direct way to represent a rotation matrix involving a rotation around a celestial pole p is by 5 parameters R = Q( ΞΗ, ) R ( ψ ) W ( ξη, ) (4) where Ξ, Η are any parameters giving the direction of p in the celestial system, while ξ, η are any parameters giving the direction of p in the terrestrial system, as already proposed by Dermanis (977) and used for studying the accuracy of earth rotation parameters determined from VLBI observations. However this representation fails to associate the diurnal rotation angle ψ, with a proper measure of the earth s rate of rotation which serves for the definition Universal ime (U), that is time defined by using the rotating earth as a clock. he IERS representation solves this problem by separating the original rotation angle into parts, ψ = s + θ s, with s passed over to precession-nutation and s to polar motion, according to R = Q ( Ξ, Η) R ( s) R ( θ ) R ( s ) W ( ξη, ) = Q( Ξ, Η, s) R ( θ) W ( s, ξη, ). (5) 0 0 he NRO conditions define the origin-direction of the angle θ (Celestial Ephemeris Origin, CEO) and its end- direction (errestrial Ephemeris Origin, EO), on the equatorial plane ( p ), in such a way that the earth rotation angle θ properly represents the rotation of the earth around the celestial pole p. he NRO conditions take the analytical form s = s( Ξ, Η ), s = s ( ξ, η) and the earth rotation is ultimately represented by 5 parameters only ( s ) ( s ) R( ΞΗ,, θ, ξη, ) = Q ΞΗ,, ( ΞΗ, ) R ( θ) W ξη,, ( ξη, ). (6)

4 he essential feature of this type of representation is that the position of CEO and EO and hence the angle θ are independent of the parameterizations used for the direction of the celestial pole p in either the celestial ( Ξ, Η ), or the terrestrial ( ξ, η ) system. On the contrary the angle ψ of the (also 5-parameter) straight-forward representation (4) strongly depends on such parameterization choices. In the classical astronomical representation the origin of the diurnal rotation was defined with the help of the direction of the vernal equinox, while its end-direction followed from the use of a particular parameterization of the polar motion, namely W = R( xp ) R ( yp ). he remaining matter is the choice of the direction of the celestial pole p and its relation to the rotation vector of the earth ω. Precession and nutation is determined primarily from theory, by solving the differential equation that govern the rotation of the earth, with observation restricted to the provision of the necessary integration constants. On the contrary information about polar motion as well the angular velocity of rotation ω = ω (length-of-the-day variation) are primarily determined from observations. In the classical astronomical approach p was identified with the theoretically best available estimate of the direction of the rotation axis n = ω ω and the corresponding direction is characterized as the Celestial Ephemeris Pole (CEP). On the contrary in the new IERS representation, implementing the IAU000 resolutions, p is identified with a smoothed version of the best estimate of n and the corresponding direction is characterized as the Celestial Intermediate Pole (CIP). High frequency terms of periods smaller than days with respect to the celestial system are removed from the precession-nutation direction so that the CEP is replaced by the CIP. he removed terms are included in the polar motion part (Note to IAU 000 Resolution B.7). More details can be found in the IERS conventions 00 (McCarthy and Petit, 004), as well as in a series of papers preceding and following the adoption of the resolutions (Capitaine, N., 986, Capitaine & Gontier, 99, Capitaine et al. 000, 00, IERS, 00, Johnston et al., 000, Lambert & Bizouard, 00, McCarthy, 996, Seidelmann & Kovalevsky, 00). On the other hand the representation offers a total rotation matrix R, which is the best available estimate of the corresponding true rotation matrix, from which a compatible rotation vector ω = e ω can be derived according to de the requirement that = e [ ω ] from which follows that dr [ ω ] = R (7). Here [ a ] denotes the antisymmetric matrix having the column a as its axial vector, i.e. a a = a a 0 a a [ a ] = a 0 a (8) a a 0 Generalizing and extending previous work (Dermanis, 00), we will examine here the conditions under which the direction IC I p = e = e of diurnal rotation corresponds to the direction n = ω ω of the rotation vector ω, which is compatible with the model representation of the rotation matrix R. Furthermore the compatibility conditions will be related to the NRO conditions. he end result is a model for the rotation matrix R = QDW, with separation into three parts (precession-nutation Q, diurnal rotation D = R ( θ ) and polar motion W ) where the 4 compatibility conditions reduce the original 7 parameters into the required number of independent parameters. In addition the magnitude ω = ω of the compatible rotation vector provides a definition of Universal ime which is compatible with R, through a relation of the form U A B ω = +, with A and B being appropriate constants. We end this introduction by emphasizing the fact that R depends on both the choice of the celestial and the terrestrial reference system. he requirement that the celestial system is quasi inertial makes the choice of the celestial system of no particular importance, since all possible choices are connected by time-independent rotation vectors. he same would be true for the terrestrial system if it was chosen to be one of the equivalent isserand systems of the earth, i.e. systems with axes such that the relative angular momentum of the earth is vanishing. However the current representations use the IRF as a terrestrial system which does not coincide with a isserand system of the earth. In Dermanis (00) it is shown how convert the IRF to an estimate of a isserand system by utilizing the best available information about the shape of tectonic plates, their internal mass density distribution and their motion with respect to the IRF. 4

5 . Decomposition of the rotation vector I IC C he components of any vector v = e v = e vi = e vic = e vc, v C in the celestial system e C, v IC in the intermediate celestial system e = e Q, v I in the intermediate terrestrial system e = e D and v in the terres- IC C IC I I C trial system e = e W = e R, are related to each other through vc = QvIC = QDvI = QDWv = Rv Q vc = vic = DvI = DWv C = IC = I = DQv Dv v Wv C = C = IC = I = R v W D Q v W D v W v v (9) In addition to the compatible rotation vector ω = e ω of the terrestrial system with respect to the quasi-inertial celestial system, having terrestrial components ω provided by equation (7) it is possible to define by similar means relative rotation vectors between consecutive systems. he relative rotation vector ω IC Q = e ( ωq) IC of e IC C with respect to e, the relative rotation vector I I ω D = e ( ωd) I of e IC with respect to e and the relative rotation vector ω W = e ( ωw) of e with respect to e I, are uniquely defined by their components determined respectively from dq dd dw [( ωq) IC ] = Q, [( ωd) I ] = D, [( ωw) ] = W. (0) dq dw he explicit form of the components [( ωq) IC ] = Q and [( ωw) ] = W depend on the particular parametrizations chosen for Q and W. From D = R ( θ ) follows directly that ( ω ) = ( ω ) = i () D I D IC θ da where we use the general notation a = for derivatives, while i k denotes the kth column of the identity matrix I = [ iii ]. Replacing R = QDW into (7) and carrying out the necessary computations we arrive at Q IC D I W [ ω ] = W D [( ω ) ] DW + W [( ω ) ] W + [( ω ) ] () which upon using the property [( Ma) ] = M[ a ] M holding for any orthogonal matrix M, yields Q IC D I W ω = W D ( ω ) + W ( ω ) + ( ω ) () Utilizing the relations ω IC = DWω, ( ωd ) IC = D( ω D ) I and ( ωw) IC = DW( ω W) from equations (9) we may compute the components in the intermediate celestial system as ω = DWω = ( ω ) + D( ω ) + DW( ω ) = ( ω ) + ( ω ) + ( ω ) (4) IC Q IC D I W Q IC D IC W IC Multiplying with IC e from the left we arrive at ω = ω + ω + ω Q D W (5) which is the decomposition of the total rotation vector into three relative rotation vectors. 5

6 . he compatibility conditions In order to secure that the direction IC I p = e = e coincides with the direction n = ω ω of the compatible rotation vector ω, it suffices to ensure that IC ω has no equatorial components in the directions of e IC and e, in which case we arrive at the two direction conditions IC ω = 0, IC ω = 0. (6) In order to secure that the angle θ of diurnal rotation corresponds to the compatible Universal ime linked to the IC magnitude ω = ωicω IC of the compatible rotation vector ω = e ω IC, it suffices to ensure that the temporal rate of θ equals the angular velocity ω. hus we arrive at the magnitude condition IC IC IC IC IC [ IC ] [ IC ] [ IC ] ( ωd) IC θ = ω. (7) Using the fact that ω = ω ω = [ ω ] + [ ω ] + [ ω ] we can express the magnitude condition in the form ω = ω + ω + ω = θ. aking into account that according to (4) ω = ( ω ) + ( ω ) + ( ω ) while according to () = θ the magnitude condition takes the form IC Q IC D IC W IC [ ω ] [ ω ] [( ω ) θ ( ω ) ] θ. (8) IC + IC + Q IC + + W IC = he last form considers the magnitude condition independently of the direction conditions. If the direction conditions are already imposed the magnitude condition takes the simpler form Q IC ωw IC ( ω ) + ( ) = 0. (9) In a representation with Q = Q0( ΞΗ, ) R ( s) and W = R( s ) W 0( ξη, ), the rotation matrix R = Q ( Ξ, Η) R ( s) R ( θ) R ( s ) W ( ξ, η) (0) 0 0 depends on 7 parameters which the above compatibility conditions reduce to 4 parameters, one more than the minimum required number of. hus there must be one condition missing. Indeed if a representation of the form R = Q0R( s θ s ) W 0 of equation (0) is used and s() t is an arbitrary function, then the representation R = Q R ( s θ s ) W with s = s+ s and s = s s, also satisfies the compatibility conditions and provides 0 0 the same rotation matrix R. he compatibility conditions determine the magnitude of the angle θ () t but not its starting and ending directions on the equatorial plane. hus the missing condition is no other than the arbitrary choice of the starting direction (or equivalently the ending one) of the diurnal rotation angle, which corresponds to an arbitrary choice of s() t (or equivalently of s () t ). he missing information is contained in the two NRO conditions, which we will present in a somewhat unusual form. hus the CEO condition reads: he relative rotation vector ωq of the intermediate celestial system with respect to the celestial system has no component along the direction of diurnal rotation e = e. IC I Correspondingly the EO condition reads: he relative rotation vector ωw of the intermediate terrestrial system with respect to the terrestrial system has no IC I component along the direction of diurnal rotation e = e. Expressed in equations the two NRO conditions are Q IC ( ω ) = 0, W IC ( ω ) = 0. () For representation of the form (0) which satisfies the NRO conditions (e.g. the current IERS representation) but not the direction condition, the magnitude condition takes the form IC ωic [ ω ] + [ ] = 0 () 6

7 which is not satisfied in general. If the NRO conditions and the direction conditions are satisfied then the magnitude condition is trivially satisfied. hus the direction conditions and the NRO conditions provide the 4 independent conditions required for the reduction of the 7 original parameters into the minimal number of independent ones. 4. Celestial and terrestrial direction parameters of the Compatible Celestial Pole () When a representation of the earth rotation matrix R does not satisfy the direction compatibility conditions then IC I n ω = ω p = e = e, it is possible to compute the direction parameters of the Compatible Celestial Pole () C n = e n = e n in both the celestial and terrestrial system by means of C X X n = Y = Y = ω = ω C C C ω Z ω CωC X Y, () ξ ξ n = η = η = ω = ω ω ζ ω ω ξ η (4) he terrestrial components ω are given by equation (), which with D= R ( θ ) and ( ωd) I = ( ωd) IC = θ i becomes ω = W R ω + W i + ω = ( θ)( Q) IC ( θ ) ( W) = W { R ( θ)( ω ) + θi + W( ω ) ] (5) Q IC W and can be also be written in the form ω W [ R ( ) a i a ]. (6) = θ Q + θ W Here we have introduced the auxiliary vectors a = ( ω ), a = W( ω ) = WW D ( ω ) = R( θ )( ω ) (7) Q Q IC W W W IC W IC which according to the first and last of equations (0) are determined by dq dw [ aq ] = Q, [ aw ] = W. (8) From equations (9) we have that ω = QDWω = QR ( θ ) Wω which applied to (6) gives C ω = Qa [ + θi R ( θ) a ]. (9) C Q W With Q = Q 0 R () s, W = R( s ) W 0 the above relations become ω = Q [ b + θi R ( s θ s ) b ] (0) C 0 Q W ω W [ R ( ) b i b ] () = 0 s+ θ + s Q + θ W where we have set 7

8 bq = R () s a Q, bw = R ( s ) a W. () C he magnitude ω of the rotation vector can be computed from either ω = ω ω or ω = ω ω both yielding C ω = θ + b b + b b b R θ b. () Q Q W W Q ( s s ) W With b Q and b W from () we may compute ω C and ω, respectively, which divided with ω from () provide the components of the, either in the celestial system ( X, Y, Z ) or in the terrestrial one ( ξ, η, ζ ), using equation () or (4), respectively. hese relations are general enough to hold for any particular representation of the matrices Q 0 and W 0. However the computation of the initial vectors a Q and a W from (8) depend on the particular representations used. 5. he compatibility conditions for the IERS representation of earth rotation he IERS representation has the particular form R = QR ( θ ) W where Q = Q R () s = R ( E) R ( d) R ( E) R () s = R ( E) R ( d) R ( S) (4) 0 W = R ( s ) W = R ( s ) R ( F) R ( g) R ( F) = R ( S ) R ( g) R ( F) (5) 0 where we have set S = s + E, S = s + F. (6) From the first of the relations (0) we have after performing the required computations dq [( ωq) IC ] = Q = E R( S) R( d)[ i ] R( d) R( S) + d R( S)[ i ] R( S) S [ i ] (7) from which it follows that E sin d ( ωq) IC = E R( S) R( d) i + d R( S) i S i = = R( S) d. (8) E cos d S Noting the similarity of W = R( F) R( g) R ( S ) with Q = R( E) R( d) R ( S) it follows that the vector a determined by (8) will have the same form as a = ( ω ), with EdS,, replaced by F, gs,, respectively W Q Q IC aw = R( θ )( ωw) IC = F R( S ) R( g) i + g R( S ) i S i (9) On the other hand W = ( W) = ( W) IC = ( θ )( W) IC a W ω WW D ω R ω (40) and thus ( ωw) IC = F R( θ S ) R( g) i g R( θ S ) i + S i = F sin g ( θ ). (4) F cos g S = R S g 8

9 he components ω = ( ω ) + ( ω ) + ( ω ) of the rotation vector in the intermediate celestial system are IC Q IC D IC W IC thus given by Esin d Fsin g ωic = R( S) d + θi R( θ S ) g. (4) Ecos d S Fcos g S he NRO conditions become ( ω Q) IC = E cosd S = 0, ( ω W) IC = F cosg S = 0 (4) so that S = s + E = Ecos d, S = s + F = Fcos g,leading to the more familiar forms s = E (cos d ), s = F (cos g ). (44) With the implementation of the NRO conditions the intermediate celestial components become Obviously ωic Stan d S tan g ωic = R( S) d + θi R( θ S ) g. (45) 0 0 = θ, while projection on the equatorial plane provides the direction conditions in the form ω IC Stan d S tan g = R( S) ( θ S ) = R 0, (46) ω d g IC or equivalently tan tan ( S ) S g ( θ S) S R = R d. (47) g d Explicitly ωic = Stan dcos S d sin S + S tan gcos( θ + S ) + g sin( θ + S ) (48) ωic = Stan dsin S + d cos S + S tan gcos( θ + S ) g sin( θ + S ) (49) and the direction conditions, which are not satisfied by the IERS representation, become S tan gcos S + g sin S = Stan dcos( θ S) + d sin( θ S) (50) S tan gsin S + g cos S = Stan dsin( θ S) + d cos( θ S) (5) he above relation states that once θ, E, d and thus S = S( E, d) are determined, then S and g are uniquely determined from a system of two differential equations, and finally F is also detemined from the EO condition F = S /cosg. From (46) it follows that [ ωic ] + [ ω IC ] = S tan d + d + S tan g+ g cos( S θ S )( Stand S tan g+ dg ) + sin( S θ S )( Stand g ds tan g). (5) 9

10 or with the implementation of the NRO conditions S = E cos d, S = F cos g IC ωic [ ω ] + [ ] = = E sin d + d + F sin g+ g cos( S θ S )( EF sin dsin g+ dg ) + sin( S θ S )( Eg sin d df sin g). (5) Under the NRO conditions Q IC ( ω ) = 0 W IC IC Q IC D IC W IC ( ω ) = 0 it holds that ω = ( ω ) + ( ω ) + ( ω ) = θ, the angular IC IC IC velocity of rotation ω = [ ω ] + [ ω ] + [ ω ] is determined from ω = θ + E sin d + d + F sin g+ g cos( E+ s θ s F)( EF sindsin g+ dg ) + + sin( E+ s θ s F)( Eg sin d df sin g) θ (54) and thus the magnitude condition ω = θ is not satisfied by the IERS representation. 6. he celestial and terrestrial directions of the compatible celestial pole expressed in terms of the corresponding directions of the Celestial Intermediate Pole he direction p = e IC = e components are given by I of the CIP has obviously components pic = pi = i, while its celestial and terrestrial X cos Esin d pc = Y IC 0 () s ( E) ( d) sinesind = Qp = Q R i = R R i = X Y cos d (55) ξ cos F sin g p = η = W pic = W0 R( s ) i = R( F) R( g) i = sinfsing ξ η cos g (56) Direct differentiation gives X = E sin Esin d + d cos Ecos d (57) Y = E cos Esin d + d sin Ecos d (58) For the directions of the compatible celestial pole () we need the already computed vector Esin d Esin d aq = ( ωq) IC = R( S) d = R( s) R( E) d = R( s) bq 0 0 (59) Esin d Ecos Esin d d sin E bq = R() s aq = R( E) d = E sinesind + dcose 0 0 (60) while due to the similarity between W and Q 0

11 b W F cos Fsin g g sin F = E sin Fsin g+ g cos F. (6) 0 Calculating = sin, and sin cos XY YX d E and Y are the relations XX + YY = d dd, the basic relations for expressing b Q in terms of X sin d = X + Y, cos d = X Y. (6) = sin, sin cos XY YX d E Introducing these values into(6) we obtain b Q XX + YY = d dd X Y, cos E =, sin E =. (6) sin d sin d Y( XX + YY) XYX ( XY ) b X Y Q X( XX + YY) = b Q = Y( YX XY ) + (64) X + Y X Y 0 0 he above relation can not be used in the calculations because of the presence of the very small term Y X the denominator. Instead we may use expressions involving one of the ratios L =, M =, namely X Y X + Y, in b Q X + LY ( LX Y ) L X Y X + LY = LLX ( Y ) +, + L X Y 0 b Q MX + Y M( X MY ) X Y MX + Y = ( X MY ) + M (65) + M X Y 0 the first being preferable when Y X and the second when X Y. We need also to compute the contribution of b Q to ω, which according to (64) is X + Y Q Q Q Q b b = ( b ) + ( b ) = [( Y ) X + ( X ) Y + XYXY ] (66) X Y he matrix Q 0 is already provided by the IERS resolutions (McCarthy & Petit, 004) ax axy X Q 0 = axy ay Y, X Y a( X + Y ) a = + cosd = + X Y. (67) Noting the similarity of W 0 to Q and of b W to b Q, we have directly

12 a ξ a ξη ξ W 0 = a ξη a η η, ξ η a ( ξ + η ) a = = + cosg + ξ η (68) where l η =, m = ξ and ξ η ηξξ ( ηη) ξηξ ( + ξy ) b ξ η W ξξξ ( + ηη) bw = bw = ηηξ ( ξy ) + = ξ + η 0 ξ η 0 ξ + lη m ( l ξ + η ξ η) l m( ξ m η) ξ η ξ η ξ + lη m ξ + η = ll ( ξ η) + = ( ξ m η) + m (69) + l ξ η + m ξ η 0 0 ξ + η W W bw bw b b = ( ) + ( ) = [( η ) ξ + ( ξ ) η + ξηξη ]. (70) ξ η he remaining term for the calculation of ω is Q s θ s W= s θ s bqbw + bqbw s θ s bqbw bqbw b R ( ) b cos( )( ) sin( )( ). (7) Applying equations (0), () and (), (4) we obtain the components n C of the unit vector of the in the celestial system, as wells as its components n in the terrestrial system X ax axy X bq cos( s θ s ) bw sin( s θ s ) w b Y axy ay Y bq sin( s θ s ) bw cos( s θ s ) b = + w ω θ Z X Y a( X + Y ), (7) ξ aξ aξη ξ cos( s θ s ) bq sin( s θ s ) Q w b b η a ξη a η η sin( s θ s ) bq cos( s θ s ) bq b = w ω ζ θ ξ η a ( ξ + η ) (7) with X + LY MX + Y bq = ( LX Y) L M( X MY) = (74) + L X Y + M X Y X + LY MX + Y bq = L( LX Y) ( X MY) M + = + (75) + L X Y + M X Y

13 ξ + lη m ξ + η bw = ( l ξ η) l = ( ξ m η) + m (76) + l ξ η + m ξ η ξ + lη m ξ + η bw = l( l ξ η) + = ( ξ m η) + m (77) + l ξ η + m ξ η where Y L =, X M X η =, l =, m = ξ, (78) Y ξ η X + Y θ ω = + [( Y ) X + ( X ) Y + XYXY ] + X Y ξ + η + [( η ) ξ + ( ξ ) η + ξηξη ] + ξ η Q W Q W θ Q W Q W + cos( s θ s )( b b + b b ) sin( s s )( b b b b ) (79) According to the IERS Conventions (McCarthy & Petit, 004) the polar motion matrix is given by W = R ( s ) W = R ( s ) R ( x ) R ( y ). (80) 0 P If this taken as the exact definition then the components of the CIP in the terrestrial system will be given by P ξ = sin x P, η = sin yp cos xp (8) and W 0 will have the exact value W0 = R( yp ) R ( xp ). aking into account that x P, y P and g have very small values, we may neglect terms of rd order and higher to obtain the approximations ξ = xp, η = yp. he same approximations can be obtained if the model W0 = R( F) R( g) R( F) is used as the exact model, in which case W0 R( xp ) R ( yp ) is only an approximation, while ξ = cos F sin g gcos F = xp, η = sin F sin g gsin F = yp (8) here are three choices for representing the precession-nutation matrix Q and the polar motion matrix W, namely W R ( s ) R ( F) R ( g) R ( F) (8) Q = R( E) R( d) R( E) R ( s), = Q = R( Ξ) R( Η) R ( s), Q = R( B) R( A) R ( s), W = R ( s ) R ( Π) R ( Σ) (84) W = R ( s ) R ( C) R ( D) (85) In addition to the above symmetric combinations, its is possible to use all the non-symmetric combinations, leading to 9 different possibilities. Each model is accompanied by its own particular form of the NRO conditions (Dermanis, 005), as well of different forms for the matrices a Q and a W, as determined from equations (8), which lead to different equations for the computation of the direction parameters X, Y, ξ, η of the Compatible Celestial Pole. From these it is easy to derive the corresponding angles E = arctan( Y / X ), d = arctan( X + Y / Z ), F = arctan( η / ξ ), g = arctan( ξ + η / ζ ), while the implementation of the NRO conditions yields s = s ( E, d ) and s = s ( F, g ). he remaining compatible angle of diurnal rotation θ can be derived by either integrating the compatible magnitude θ = ω() t or by utilizing the IERS provided rotation matrix R, and the new matrices of precession-nutation

14 Q = Q ( E, d ) and polar motion W = W ( F, g ), in which case the relation R = QR ( θ ) W yields ( ) R = Q RW. θ 7. Implementation of observational corrections to the IERS representation In order to accommodate for discrepancies between the IERS representation of the earth rotation matrix R = QDW and observational evidence, a corrected rotation matrix R = δqr is used, with an observation-provided additional infinitesimal rotation matrix (McCarthy & Petit, 004) 0 δ X δq = 0 δy (86) δx δy a procedure which is equivalent with replacing the original precession-nutation matrix Q with a corrected one Q = δqq. he terrestrial components ω of the corrected rotation vector are defined by d dδ d [ ω ] = R R == R δq Q R+ R R = R [ ωδ Q ] R + [ ω ] (**87) where we have introduced the auxiliary vector ω δq defined by dδq [ ωδq ] =δq ω δ Q δy = δ X. (88) 0 he celestial components of the corrected rotation vector are ω C = Rω =δqrω and from (**) it follows that = δq + = 0 ( s + θ s) 0 δq + ω R ω ω W R Q ω ω (89) ω = δq ( ω + ω ). (90) C δq C he corresponding corrected magnitude can be derived from = ωcωc = ω + ωδ QωδQ + ωcωδq ω. (9) he corrected direction parameters are X X ω nc = Y C δ ( δq ω C ) δ δq δ Y = ω = Q ω + n = Qω + Q ω ω ω ω Z Z (9) ξ ω n = η = ω = R ωδq + ω = R ωδq + n = ω ω ω ω ω ζ ξ ω = W0 R( s + θ s) Q0ω δq + η ω ω. (9) ζ With already determined values of δq, ω δq, W 0 and Q 0 we arrive after performing the relevant computation at the desired observationally corrected values 4

15 ω = ω + δx + δy + ω( Y δx X δy) (94) X X δy ω X Y δx + Y ω = Y + δx ω X Y δy + ω ω Z Z δ X( δ Y + ω X ) δ Y( δ X + ω Y ) ξ η a ( ξ + η ) ξ ( ) ξ a ξ a ξη ξ cosψ sinψ 0 axyδ X ax δy ω η = η + a ξη a η η sinψ cosψ 0 ( ay ) δx + axyδy ω ω ζ ζ 0 0 YδX XδY (95) (96) where the abbreviation ψ = s + θ s has been used. We close by outlining how a compatible representation can be derived directly from observations only, in order to compare with the combined theoretical-observational approach of the IERS. Observations sensitive to the earth rotation (mainly VLBI observations) can be analyzed using a -parameter model for the orthogonal rotation matrix of the form R = R ( θ ) R ( θ ) R ( θ ) (97) k k m m n n where k, m, n are integers from the set {,, } with the only restriction that k m and m n. he obtained parameter estimates θ k( ti), θ m( ti), θ n( ti) at discrete epochs or time intervals t i, must then be interpolated in order to derive the functions θ k () t, θ m () t, θ n () t and their time derivatives θ k () t, θ m () t, θ n () t. he terrestrial and celestial components of the rotation vector can be then estimated from (Dermanis, 005) ω = θ i + θ R ( θ ) i + θ R ( θ ) R ( θ ) i (98) k k m k k m n k k m m n ω = θ R ( θ ) R ( θ ) i + θ R ( θ ) i + θ i (99) C k n n m m k m n n m n n C C Dividing by the magnitude ω = ω ω = ω ω we obtain the direction components and n C = [ ξηζ] = ω ω and the direction angles C n = [ XYZ] = ω ω E = arctan Y, X X + Y η d = arctan, F = arctan, Z ξ ξ + η g = arctan, (00) ζ he angles s and s can be derived by integrating the NRO conditions s = E (cos d ) and s = F (cos g ). Finally the diurnal rotation angle θ can be calculated from = = k m n R ( θ) Q RW R ( s) R ( E) R ( d) R ( E) R( θ, θ, θ ) R ( F) R ( g) R ( F) R ( s ). (0) Finally a primary unsmoothed compatible Universal ime can be computed using U = A + Bθ. (0) where the constants A and B are selected to bring the origin and scale of Universal ime in best agreement with a desired time system, e.g. a system of atomic time. he obtained representation will not contain all the frequency components of precession-nutation predicted by theories of earth rotation, but only those up to a certain resolution associated with the temporal density of the analyzed observations. 5

16 8. Conclusions We have examined representations of the orthogonal rotation matrix R transforming coordinates from the terrestrial to the celestial system, which involve a separation R = QR ( θ ) W into three consecutive parts: polar motion W, diurnal rotation R ( θ ) around a direction (unit vector) p called the celestial pole, and precession-nutation Q. Since any orthogonal matrix is mathematically connected to a rotation vector ω (vector of angular velocity), this vector supplies a direction n = ω ω which we have called the compatible celestial pole, while its magnitude ω = ω provides a compatible angular rate of rotation which can be associated with a compatible Universal ime. Including terms of rotations around p in both precession-nutation Q = Q0( Ed, ) R ( s) and polar motion W = R( s ) W 0( F, g), 4 conditions are required to hold between the 7 parameters E, d, s, θ, s, F, g, to the minimum of parameters representing any orthogonal matrix R. hree natural choices are the direction conditions following from setting p = n, i.e. identifying the celestial pole with the compatible one, and the magni- tude condition following from setting ω = θ dθ /. o arrive at the required 4 conditions we have decomposed the compatible rotation vector ω = ωq + ωd + ωw into three parts, ωq for precession-nutation, ωd = θ p for diurnal rotation and ωw for polar motion. By requiring that both ωq and ωw have no component in the direction of p, the Non Rotating Origin (NRO) conditions follow: the Celestial Ephemeris Origin (CEO) condition ω Q p = 0 and the errestrial Ephemeris Origin (EO) condition ω p = 0. It comes out that the sought 4 conditions are the W combination of the NRO conditions and the direction conditions, which combined they also secure the fulfillment of the magnitude condition. Since the current IERS representation satisfies only the NRO conditions, it follows that the implemented celestial pole, which is the Celestial Intermediate Pole (CIP), does not coincide with the Compatible Celestial Pole (), a fact which is fundamentally due to the non-fulfillment of the direction conditions and not primarily to the removal of higher frequency terms from the CIP. We have provided equations for the computation of the components of the, namely the components X,, in the terrestrial system. he more general equations Q and W 0, have been specialized to the particular IERS representations, in order to express the components as functions of the first celestial components X, Y of the CIP and the polar motion parameters x P and y P. In addition the relevant results have been extended to the case where an infinitesimal correction matrix factor is incorporated into the IERS representation to account for discrepancies with observational data. Finally we have demonstrated how the observational data can be analyzed to provide a compatible representation having the same form as the IERS representation, where the diurnal rotation takes place around the compatible celestial pole (direction of the compatible rotation vector), with an angular rate equal to the compatible angular velocity (magnitude of the compatible rotation vector). Y, Z in the celestial system and ξ, η, ζ applying to any choice of representation of the matrices 0 References Capitaine, N. (986): he earth rotation parameters: conceptual and conventional definitions. Astron. Astrophys., 6 (986), -9. Capitaine, N. and A.-M. Gontier (99): Accurate procedure for deriving U at a submilliarcsecond accuracy from Greenwich Sidereal ime or from the stellar angle. Astron. Astrophys., 75 (99), Capitaine, N., B. Guinod, D.D. McCarthy (000): Definition of the Celestial Ephemeris Origin and of U in the International Celestial Reference Frame. Astron. Astrophys., 55 (000), Capitaine, N., D. Gambis, D.D. MacCarthy, G. Petit, J. Ray, B. Richter, M. Rotacher, E. Myles Standish, J. Vondrak (eds.) (00): IERS echnical Note No. 9: Proceedings of the IERS Workshop on the Implementation of the New IAU Resolutions. Observatoire de Paris, 8-9, April 00. Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt am Main 00. Dermanis, A. (977): Design of Experiment for Earth Rotation and Baseline Parameter Determination from Very Long Baseline Interferometry. Report No. 45, Department of Geodetic Science, he Ohio State University, Columbus, Ohio. Dermanis, A. (00): Global Reference Frames: Connecting Observation to heory and Geodesy to Geophysics. IAG 00 Scientific Assembly Vistas for Geodesy in the New Millenium -8 Sept. 00, Budapest, Hungary. Available at: 6

17 Dermanis, A. (00): Some remarks on the description of earth rotation according to the IAU 000 resolutions. From Stars to Earth and Culture. In honor of the memory of Professor Alexandros sioumis, pp School of Rural & Surveying Engineering, he Aristotle University of hessaloniki. Dermanis, A. (005): Coordinates and Reference Systems (in Greek), 480 pp. Ziti Publications, hessaloniki. IERS (00): IERS Annual Report 00. Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt am Main 00. Johnston, K.J., D.D. MacCarthy, B.J. Luzum, G.H. Kaplan (eds.) (000): Proceedings of IAU Colloquium 80: owards Models and Constants for Sub-Microarcsecond Astrometry. Washington, DC, 7-0 March 000. U.S. Naval Observatory, Washington, DC, 000. Lambert, S. and C. Bizouard (00): Positioning the errestrial Ephemeris Origin in the International errestrial Reference Frame. Astron. Astrophys., 94 (00), 7-. McCarthy, D.D. (ed.) (996): IERS Conventions 996. IERS echnical Note, Central Bureau of IERS, Observatoire de Paris. McCarthy D.D. and G. Petit (eds.), 004: IERS Conventions (00). IERS echnical Note No., Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt am Main 004. Seidelmann P.K. and J. Kovalevsky (00): Application of the new concepts and definitions (ICRS, CIP and CEO) in fundamental astronomy. Astron. Astrophys., 9 (00) 4-5. Author s address: Prof. Athanasios Dermanis Department of Geodesy and Surveying he Aristotle University of hessaloniki University Box hessaloniki Greece Phone: Fax: dermanis@topo.auth.gr 7