5 Solution of the dynamical equations

The formal solution of the pair of linear Eqs. (32) is nearly trivial. Denoting the secular determinant of this system by $D(\sigma)$ and the coefficient $[1+e_{\rm f}+(1+\beta)\sigma]$ of  $\tilde m_{\rm f}(\sigma)$ in the second equation by $X(\sigma)$, we have

$$\tilde m(\sigma)= {X(\sigma)\over D(\sigma)}~{\tilde\Gamma(\sigma)\over {\rm i}\Omega_0^2 A},$$ (33)

$$D(\sigma) = (A_{\rm m}/A) (\sigma-\sigma_1)(\sigma-\sigma_2),$$ (34)

where $A_{\rm m}$ is the mean equatorial moment of inertia of the mantle, and $\sigma_1$ and $\sigma_2$ are the resonance frequencies associated, respectively, with the Chandler wobble (CW) and the nearly diurnal free wobble corresponding to the FCN. They are given by

$$\sigma_1={A\over A_{\rm m}}(e-\kappa)\quad {\rm and}\quad \sigma_2=-1-{A\over A_{\rm m}}(e_{\rm f}-\beta)$$ (35)

to the first order in the ellipticity and compliance parameters, none of which exceeds 1/300 in magnitude. Quantities of this order of magnitude will be referred to as of  $O(\epsilon)$. The retrograde diurnal wobble frequency band centered at $\sigma= -1$ corresponds to the low frequency nutations which we have expressly excluded from consideration here. The frequencies of the wobbles corresponding to high frequency nutations are all far from -1 cpsd. Therefore the small parameters referred to above may be neglected in comparison with $(\sigma+1)$; they may also be dropped from X for similar reasons. Then $(\sigma-\sigma_2)\approx (\sigma+1)$, and $X\approx (1+\sigma)$. Hence, the approximation

$$\tilde m(\sigma)= {A\over A_{\rm m}} {1\over (\sigma-\sigma_1)}~{\tilde\Gamma(\sigma)\over {\rm i}\Omega_0^2 A}$$ (36)

is good for all wobbles outside the retrograde diurnal band. The amplitude of the associated polar motions is then, in view of (21),

$$\tilde p(\sigma)= {A\over A_{\rm m}} {1\over (1+\sigma)(\sigm... ... \over {\rm i}\Omega_0^2 A}, \qquad (\vert 1+\sigma\vert>1/2).$$ (37)

If the Earth were rigid, $A/A_{\rm m}$ would reduce to unity, and $\sigma_1$ would be just e. With these substitutions, the expression (36) reduces to the wobble amplitude  $\tilde m_{\rm R}(\sigma)$ of the rigid Earth. So the transfer function from the rigid to the nonrigid Earth becomes

$$T(\sigma) \equiv \tilde m(\sigma)/\tilde m_{\rm R}(\sigma) = {A(\sigma-e)\over A_{\rm m}(\sigma-\sigma_1)}\cdot$$ (38)

When the excitation frequencies $\sigma$ are outside the low frequency band too, i.e., when $\vert\sigma\vert>1/2$ (besides $\vert 1+\sigma\vert>1/2$), further simplification of (36) and (38) is possible since e and $\sigma_1$ are then negligible relative to $\vert\sigma\vert$ which dominates by factors of the order of 300. Thus the transfer function becomes very nearly constant, equal to $(A/A_{\rm m})$; this is the case for all nutations except the low frequency and prograde diurnal ones.

Long period wobbles and polar motions (or the associated prograde diurnal nutations) have to be handled with care, because of the resonance associated with the Chandler wobble whose frequency appears in the low frequency band. These are dealt with in Sect. 6. The zero frequency term in the torque, which pertains to this band, merits special consideration, for reasons to be outlined there.

For numerical computations, which are done to an accuracy of 0.1 $\mu$as and without making use of any of the approximations made above or later, we take $A/A_{\rm m}=1.1284$, e=0.00328455, and, for an elastic Earth, $\kappa=0.0010505$; the last two are the estimates due to Mathews et al. (2002). The other parameters involved are $e_{\rm f}=0.0026490$, $\xi=0.0002248$, $\gamma=0.0019825$, $\beta=0.0006227$, but they contribute only marginally, if at all, to our numerical results.