8 Polar motion and nutation coefficients: Expressions and numerical results

8.1 Circular motions

According to Eq. (20),

$$\tilde p(t) = Q^{\pm} {\rm e}^{{\rm\pm i} \Theta_\omega(t)},$$ (71)

$$Q^+=\tilde p(\sigma_{\rm p}){\rm e}^{{\rm -i}\zeta_{n,m}},\qquad Q^-=\tilde p(\sigma_{\rm r}){\rm e}^{{\rm i}\zeta_{n,m}}.$$ (72)

Therefore, if we write $Q^\pm = Q^\pm_{\rm R}+{\rm i}Q^\pm_{\rm I}$, it follows from Eq. (19) that the coefficients of  $\sin\Theta_\omega(t)$ and  $\cos\Theta_\omega(t)$ (identified by superscripts s and c, respectively) in the polar motion variables are,

$$x_{\rm p}^{\rm s} = \mp Q^\pm_I, ~ x_{\rm p}^{\rm c}=Q^\pm_{\... ...}, ~ y_{\rm p}^{\rm c} =\pm x_{\rm p}^{\rm s}= -Q^\pm_{\rm I}.$$ (73)

As usual, the upper and lower signs are for prograde and retrograde circular polar motions (frequencies $\sigma_{\rm p}$ and $\sigma_{\rm r}=-\sigma_{\rm p}$, respectively), and $\omega$ and $\Theta_\omega(t)$ pertain to the tidal potential which excites these motions. The combined polar motion, which is elliptical, in general, is obtained by adding the two parts. The explicit expressions for $\tilde p(\sigma)$ which have to be used for computing the above coefficients follow from Eq. (37) taken together with (26a), (26b), or (26c), as the case may be:

$$Q^{\pm} F_{n,m}^{(\pm)} \left[~(-1)^{m+1}~{\rm e}^{\mp {\rm i}\zeta_{n,m}}~(\pm C_{n,m\pm 1}+{\rm i}S_{n,m\pm 1})\right],$$ = (74a)

$$F_{n,m}^{(\pm)} {A\over A_{\rm m}}{1\over (1\pm\sigma_{\rm p}) (\pm\sigma_{\rm p}-\sigma_1)}\ G^{(\pm)}_{n,m}H_\omega^{(n,m)},$$ = (74b)

with the exception that for the retrograde case with m=0,

$$F_{n,0}^{(-)}\left[-{\rm e}^{{\rm i}\zeta_{n,0}}(C_{n,1}+{\rm i}S_{n,1})\right],$$                                Q^- = (75a)

$${A\over A_{\rm m}}{1\over (1-\sigma_{\rm p}) (-\sigma_{\rm p}-\sigma_1)}\ G^{(+)}_{n,0}H_\omega^{(n,0)}.$$ F_n,0^(-) = (75b)

The real part of the square bracketed factor is, in general, one of the pair of geopotential coefficients (C or S) of the relevant degree and order, and the imaginary part is the other member of the pair, both to within a sign; what sign goes with each, is determined by the parities (odd/even) of m and (n-m). In view of Eqs. (73), the coefficients of polar motion are now directly given by these quantities, apart from the factor  $F_{n,m}^{(\pm)}$, as shown in Table 3, provided that  $F_{n,m}^{(\pm)}$ can be taken to be real. The reality assumption is not strictly valid because of the presence of an imaginary part in the Chandler frequency $\sigma_1$, but its effect is quite negligible except for excitation frequencies close to $\sigma_1$, which exist only in the case of (n,0) potentials. Close to the resonance, the sine and cosine coefficients contain admixtures of C_n,1 and S_n,1, weighted by the real and imaginary parts of  $F^{(\pm)}_{n,0}$.

 

 

Table 3: Coefficients of $F_{n,m}^{(\pm)}\sin\Theta_\omega(t)$ and $F_{n,m}^{(\pm)}\cos\Theta_\omega(t)$ in polar motions excited by various potentials.

Forcing	Coefficients in $x_{\rm p}(t)$		Coefficients in $y_{\rm p}(t)$
potential	$\sin\Theta_\omega(t)$	$\cos\Theta_\omega(t)$	$\sin\Theta_\omega(t)$	$\cos\Theta_\omega(t)$
Prograde polar motions
(2, 1)	C2,2	S2,2	-S2,2	C2,2
(3, 0)	-C3,1	-S3,1	S3,1	-C3,1
(3, 1)	-S3,2	C3,2	-C3,2	-S3,2
(3, 2)	-C3,3	-S3,3	S3,3	-C3,3
(4, 0)	S4,1	-C4,1	C4,1	S4,1
(4, 1)	C4,2	S4,2	-S4,2	C4,2
Retrograde polar motions
(3, 0)	-C3,1	S3,1	S3,1	C3,1
(3, 2)	C3,1	S3,1	S3,1	-C3,1
(3, 3)	S3,2	-C3,2	-C3,2	-S3,2
(4, 0)	-S4,1	-C4,1	-C4,1	S4,1

We have carried out the numerical evaluation of the coefficients of polar motions due to tidal potentials of degrees up to 4, starting from the tidal amplitudes defined according to the conventions of Cartwright & Tayler (1971). Actually, we used the RATGP series of Roosbeek (1996) and converted the amplitudes from this series to their Cartwright-Tayler equivalents through multiplication by the appropriate factors f_n,m taken from Table 6.5 of the IERS Conventions 1996. (The sign of the factor f_3,1 given there has to be reversed; the value f_4,0= 0.317600 not listed there was needed to compute polar motions with coefficients down to 0.05 $\mu$as, there being a few that are excited by (4, 0) potentials.) For the prograde diurnal polar motions, we have also used the alternative approach explained in Sect. 7, using the known amplitudes of the long period nutations as inputs instead of the tidal amplitudes. The JGM3 values listed in Table 2 were used for the geopotential coefficients in computations for the nonrigid Earth; the IERS92 values were used for the rigid Earth case, to facilitate comparisons with the results of earlier workers.

We present in Table 4 the periodic polar motions having amplitudes exceeding 0.5 $\mu$as. Only the low frequency polar motions due to (3, 0) potentials and the prograde diurnals excited by the (2, 1) potentials attain these magnitudes. The secular polar motion due to the constant term in the (4, 0) potential is also shown. The argument of the polar motion, denoted by (arg) in the Table, is $\Theta_\omega(t)$ if the motion is prograde, and $-\Theta_\omega(t)$ if retrograde. It is expressed here as a linear combination,

$${\rm (arg)}=n_1(\Phi+\pi)+n_2l+n_3l'+n_4F+n_5D+n_6\Omega,$$ (76)

where $\Phi=GMST$ and $l,l',F,D,\Omega$, (which are themselves linear combinations of the last five of Doodson's tidal arguments) are DeLaunay's fundamental arguments for nutations. For polar motions excited by potentials of type (n,m), $n_1=\pm m$ according as the motion is prograde/retrograde. Retrograde motions do not appear in Table 4, as their amplitudes are below the cut-off, except for those excited by (2,1) potentials which, however, are treated as nutations rather than polar motions.

The argument of the HF nutation equivalent to the polar motion of (76) is

$$(n_1+1)\Phi+n_2l+n_3l'+n_4F+n_5D+n_6\Omega.$$ (77)

The periods of these circular nutations are shown in the last column of the table for convenience. They are all prograde.

 

 

Table 4: Coefficients, in $\mu$as, of $\sin({\rm arg})$ and $\cos({\rm arg})$, arg = $\pm \Theta _\omega (t)$, in polar motions ( $x_{\rm p}(t),y_{\rm p}(t)$) excited by potentials of different degrees n, and periods, in solar days, of equivalent high frequency nutations.

n	Multipliers of						Period	$x_{\rm p}(t)$		$y_{\rm p}(t)$		Nutation
	$\Phi+\pi$	l	l'	F	D	$\Omega$	of PM	$\sin$	$\cos$	$\sin$	$\cos$	Period
3	0	-1	0	-1	0	-1	-13.719	1.39	.17	-.17	1.39	1.07545
3	0	0	0	-1	0	0	-27.212	2.48	.30	-.30	2.48	1.03521
3	0	0	0	-1	0	-1	-27.322	15.75	1.93	-1.93	15.75	1.03505
3	0	0	0	-1	0	-2	-27.432	-.82	-.10	.10	-.82	1.03489
3	0	-1	0	-1	2	-1	-193.560	.81	.10	-.10	.81	1.00243
3	0	1	0	-1	0	0	-2190.35	1.86	.24	-.24	1.86	.99772
3	0	1	0	-1	0	-1	-3231.50	12.32	1.59	-1.59	12.32	.99758
3	0	1	0	-1	0	-2	-6159.14	-.68	-.09	.09	-.68	.99743
3	0	-1	0	1	0	2	6159.14	.78	.09	-.09	.78	.99711
3	0	-1	0	1	0	1	3231.50	-16.16	-1.83	1.83	-16.16	.99696
3	0	-1	0	1	0	0	2190.35	-2.78	-.31	.31	-2.78	.99682
3	0	1	1	-1	0	0	438.360	-.63	.12	-.12	-.63	.99501
3	0	1	1	-1	0	-1	411.807	1.05	.27	-.27	1.05	.99486
3	0	0	0	1	-1	1	365.242	1.31	.20	-.20	1.31	.99455
3	0	1	0	1	-2	1	193.560	2.10	.27	-.27	2.10	.99216
3	0	0	0	1	0	2	27.432	-.87	-.11	.11	-.87	.96229
3	0	0	0	1	0	1	27.322	16.64	2.04	-2.04	16.64	.96215
3	0	0	0	1	0	0	27.212	2.62	.32	-.32	2.62	.96201
3	0	1	0	1	0	1	13.719	1.28	.16	-.16	1.28	.92969
2	1	-1	0	-2	0	-1	1.11970	-.44	.25	-.25	-.44	.52747
2	1	-1	0	-2	0	-2	1.11951	-2.31	1.32	-1.32	-2.31	.52743
2	1	1	0	-2	-2	-2	1.11346	-.44	.25	-.25	-.44	.52608
2	1	0	0	-2	0	-1	1.07598	-2.14	1.23	-1.23	-2.14	.51756
2	1	0	0	-2	0	-2	1.07581	-11.36	6.52	-6.52	-11.36	.51753
2	1	-1	0	0	0	0	1.03472	.84	-.48	.48	.84	.50782
2	1	0	0	-2	2	-2	1.00275	-4.76	2.73	-2.73	-4.76	.50000
2	1	0	0	0	0	0	.99727	14.27	-8.19	8.19	14.27	.49863
2	1	0	0	0	0	-1	.99712	1.93	-1.11	1.11	1.93	.49860
2	1	1	0	0	0	0	.96244	.76	-.43	.43	.76	.48977
Rate of secular polar motion ($\mu$as/yr) due to the zero frequency tide, arg = 0
4	0	0	0	0	0	0			-3.80		-4.31	.99727

8.2 Coefficents of circular high frequency nutations

The coefficients of the HF nutations may be inferred from those of the equivalent polar motions from the following considerations.

Beginning with the fact that $\tilde\eta(t)=-\tilde p(t)~{\rm e}^{{\rm i}GMST}$, which produces an overall sign difference between the coefficients of  $\sin\Theta_\omega(t)$ and  $\cos\Theta_\omega(t)$ in  $x_{\rm p}(t)$ and  $y_{\rm p}(t)$, on the one hand, and those of $\sin\Xi_\nu(t)$ and  $\cos\Xi_\nu(t)$ in $\Delta\psi(t)\sin\epsilon$ and $\Delta\epsilon$ on the other hand, one takes note of the other sign differences that occur: a sign (-1)^m depending on the order m of the tidal potential giving rise to the motions, which arises from the relations (15) between $\Theta_\omega(t)$ and $\Xi_\nu(t)$, and a further minus sign that arises between the coefficients in $y_{\rm p}(t)$on the one hand and those in $\Delta\epsilon(t)$ on the other because of the fact that $\tilde p(t)=x_{\rm p}(t)-{\rm i}y_{\rm p}(t)$ while $\tilde\eta(t)=\Delta\psi\sin\epsilon +{\rm i}\Delta\epsilon$. Thus, with superscripts s and c identifying coefficients of the sine and cosine functions, respectively, of (arg) or $\Xi_\nu$ as the case may be, we have

$$\Delta\psi_\nu^{\rm s}\sin\epsilon = (-1)^{m+1}x_{\rm p}^{\rm s}(... ...\quad \Delta\psi_\nu^{\rm c}\sin\epsilon = (-1)^{m+1}x_{\rm p}^{\rm c}(\sigma),$$ (78a)

$$\Delta\epsilon_\nu^{\rm s} = (-1)^m y_{\rm p}^{\rm s}(\sigma)\quad \Delta\epsilon_\nu^{\rm c} = (-1)^m y_{\rm p}^{\rm c}(\sigma).$$ (78b)

The coefficients thus obtained are for circular nutations.

The above relations, when combined with (73), show that the coeffients of any circular polar motion and of the corresponding nutation can all be obtained from just two of them, say $x_{\rm p}^{\rm s}$ and  $x_{\rm p}^{\rm c}$. They supersede Eqs. (24) of Mathews & Bretagnon (2002) which fail to be valid in general as the sign factors referred to below Eq. (17) were overlooked.

Our results for a few of the leading terms in the semidiurnal nutations due to degree 2 potentials, which are strictly prograde and hence circular, are compared with the results from earlier works in Table 5. The coefficients $\Delta\epsilon^{\rm s}$ and $\Delta\epsilon^{\rm c}$ are shown both for the rigid and the nonrigid Earth; $\Delta\psi^{\rm s}= -\Delta\epsilon^{\rm c}/\sin\epsilon_0$ and $\Delta\psi^{\rm c}=\Delta\epsilon^{\rm s}/\sin\epsilon_0$ in the present case. The values shown against BCpc were obtained by conversion from recent polar motion coefficients of Brzezinski & Capitaine (private communication, 2002). Elliptical nutations (including semidiurnal ones) that are induced by higher degree potentials are considered below.

   

Table 5: Coefficients (in $\mu$as) of prograde semidiurnal nutations: Comparisons with earlier works^a.

Period	Rigid Earth			Nonrigid Earth
(days)	Authors	$\Delta\epsilon^{\rm s}$	$\Delta\epsilon^{\rm c}$	Authors	$\Delta\epsilon^{\rm s}$	$\Delta\epsilon^{\rm c}$
0.51753	BRS97	5.79	10.09	GFE01	6.54	11.39
	FBS01	5.8	10.0	FBS01	6.5	11.3
	BCpc	5.83	10.16	BCpc	6.57	11.45
	Present	5.87	10.22	Present	6.52	11.36
0.50000	BRS97	2.43	4.23	GFE01	2.74	4.77
	FBS01	2.37	4.13	FBS01	2.7	4.7
	BCpc	2.44	4.25	BCpc	2.75	4.79
	Present	2.46	4.28	Present	2.73	4.76
0.49863	BRS97	-7.27	-12.67	GFE01	-8.21	-14.30
	FBS01	-7.12	-12.40	FBS01	-8.0	-14.0
	BCpc	-7.32	-12.75	BCpc	-8.25	-14.37
	Present	-7.27	-12.67	Present	-8.19	-14.27

^a BRS97: Bretagnon et al. (1997); FBS01: Folgueira et al. (2001);
GFE01: Getino et al. (2001); and BCpc: Brzezinski & Capitaine
(private communication, 2002).

8.3 Elliptical motions

 

 

Table 6: Coefficients (in $\mu$as) of elliptical nutations of the rigid Earth; Comparisons with Bretagnon et al. (1997) and Folgueira et al. (2001) (abbreviated as BRS97 and FBS01).

Type of	Period (days) of		Coefficients
Tide	Wobble	Nutation	$\Delta\psi^{\rm s}$	$\Delta\psi^{\rm c}$	$\Delta\epsilon^{\rm s}$	$\Delta\epsilon^{\rm c}$
Diurnal nutations
(3,0)	-27.322	1.03505	-34.201	-4.204	-1.672	13.604
(3,2)	-.50790	-1.03505	.604	-.074	-.030	-.240
Total			-34.805	-4.278	-1.642	13.364
BRS97			-34.821	-4.271	-1.640	13.371
FBS01			-35.404	-4.351	-1.587	12.911
(3,0)	-3231.5	.99758	-19.881	-2.444	-.972	7.908
(3,2)	-.49871	-.99758	.031	-.004	-.002	-.012
Total			-19.912	-2.448	-.970	7.896
BRS97			-19.854	-2.491	-.988	7.873
FBS01			-19.940	-2.451	-.972	7.906
(3,0)	27.322	.96215	-38.080	-4.680	-1.862	15.147
(3,2)	-.48970	-.96215	.050	-.006	-.002	-.020
Total			-38,130	-4.686	-1.860	15.127
BRS97			-38.128	-4.695	-1.863	15.127
FBS01			-38.231	-4.699	-1.857	15.106
Semidiurnal nutations
(3,1)	.89050	.527517	-.074	-.108	-.043	.029
(3,3)	-2.89050	-.527517	.106	-.154	-.061	-.042
Total			-.180	-.262	.018	-.013
BRS97			-.178	-.258	.020	-.013
FBS01			-.109	-.388	.234	-.092
(3,1)	.963499	.507904	-.206	-.301	-.120	.082
(3,3)	-2.963499	-.507904	.013	-.019	-.008	-.005
Total			-.219	-.320	-.112	.077
BRS97			-.219	-.321	-.113	.077
FBS01			-.244	-.356	-.097	.067

The combination of two circular motions differing only in the sign of the frequency describes an elliptical motion. In Table 4, such prograde-retrograde pairs of terms appear only among the low frequency polar motions. The argument  $\Theta_\omega(t)$ of the prograde part is assigned to the elliptical polar motion. Since $({\rm arg}) = -\Theta_\omega(t)$ for the retrograde part, the signs of the coefficients in the sine columns have to be reversed in the row pertaining to any retrograde PM before adding to the coefficients in the row pertaining to the corresponding prograde PM to obtain the coefficients for the elliptical motion. For the elliptical PM with the 27.322 day period, for instance, one finds the coefficients (in the same order as in the table) to be ( 0.89, 3.99, -0.11, 32.35) $\mu$as; they are ( -28.49, -0.24, 3.44, -3.85) $\mu$as for the 3231.496 day polar motion. Note the predominance of the cosine part of $y_{\rm p}$ in the former case and of the sine part of $x_{\rm p}$ in the latter. The difference in behaviour is due to the presence of the Chandler mode in between these periods.

For higher frequency PM, (e.g., the semidiurnals), the prograde and retrograde parts originate in the action of the same potential on different geopotential coefficients, e.g., by the action of (3, 2) potentials on C_3,3 and S_3,3 for prograde semidiurnals, and on C_3,1 and S_3,1 for the retrograde ones. The largest of these, with periods of +0.52752and -0.52752 have coefficients ( -0.330, -.0041, -0.041, 0.330) and ( -0.028,-0.055,0.055,-0.028) $\mu$as, respectively; both sets are below the cuf-off for inclusion in Table 4. The elliptical motion from their combination has coefficients ( -0.358, -0.096, 0.014, 0.302) $\mu$as.

In contrast to elliptical polar motions, elliptical HF nutations result from the combination of a pair of prograde and retrograde nutations produced by different potentials acting on the same C and S coefficients. It must be noted that the semidiurnal nutations arising from the action of degree 2 potentials on C_2,2 and S_2,2 are strictly prograde and circular: there exists no (2, 3) potential to generate retrograde components. Thus the elliptical nutations are generated only by higher degree potentials. Table 6 shows a few examples from our computations for the rigid Earth, and comparisons with the results of Bretagnon et al. (1997) and Folgueira et al. (2001) - both of which are for elliptical nutations only. Since both these works have employed the values listed under IERS92 in Table 2 for C_n,m and S_n,m, our numbers used for the comparison are based on the same values. The ratios $(-\Delta\psi^{\rm s}/\Delta\psi^{\rm c})$ and $(\Delta\epsilon^{\rm c}/ \Delta\epsilon^{\rm s})$ should be equal to (-C_3,1/S_3,1) for the diurnal nutations and  (S_3,2/C_3,2) for the semidiurnals, as may be seen from our theory. This requirement is satisfied rather well by all the sets of coefficients shown, except those of Folgueira et al. (2001) for the 0.527517 day nutation which are inconsistent with the above requirement. In fact, the fractional differences of their numbers from ours are not really small for the other listed semidiurnals too. Our sets of values are very close to those of Bretagnon et al.; and for the diurnal nutations, they are quite close to Folgueira et al. too.

8.4 Effect of triaxiality of the core

The coefficients shown in Table 4 for the prograde diurnal polar motions do not take account of possible triaxiality of the core. How much of a difference could core triaxiality make? To answer this question, we have made computations based on Eqs. (64) with nonzero $Z_{\rm f}$ as well as with $Z_{\rm f}=0$, and taken the difference. To facilitate comparison with the results of Escapa et al. (2002), we present in Table 7 our results for the contributions from $Z_{\rm f}$ to the equivalent semidiurnal nutations when  $Z_{\rm f}=0.8112~Z$, together with numbers from the IT columns of Table 1 of their paper which pertain to the same ratio for $Z_{\rm f}/Z$ which is, in their notation, $d_{\rm c}/d$. Only the coefficients of the increment $\delta\Delta\epsilon$ due to $Z_{\rm f}$ are shown. It is evident that the Escapa et al. values are 2.4 to 3 times as large as ours, except for the .51753 day nutation for which the factor is nearly 8. We have not been able to discern the reason for the discrepancies; and we find no scope for modifying our expressions to bridge the gap, our derivations being entirely transparent.

   

Table 7: Contributions (in $\mu$as) from triaxiality of the core (with $Z_{\rm f}=0.8112~Z$) to coefficients of semidiurnal nutations; comparison with Escapa et al. (2002).

Nutation perioda		Present work		Escapa et al.
PSD	LF	$\delta\Delta\epsilon^{\rm s}$	$\delta\Delta\epsilon^{\rm c}$	$\delta\Delta\epsilon^{\rm s}$	$\delta\Delta\epsilon^{\rm c}$
.49863	$\infty$	.468	.815	1.211	2.110
.49860	-6798.38	.068	.118	.175	.304
.50000	182.62	-.045	-.079	-.134	-.233
.49795	-365.26	-.021	-.036	-.049	-.085
.51753	13.66	-.010	-.017	-.076	-.132

^a Periods, in solar days, of the prograde semidiurnal (PSD)
  and low frequency (LF) nutations produced by the same
  retrograde diurnal potential are shown in each row.