Earth’s rotation: observations and theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 302

the mean radius of the Earth d = 6371.012 km (Lambeck, 1980, p. 27) enter the

discussion.
Neglecting dissipation and rotational deformation,

C − A
A Ω .

σ 0 = (4.74)

Since

C − A

A =

C − A 1

1/ H

1
304.456 , C = 1 =

(4.75)

C + A − −

σ 0 is 1/304.456 of the angular frequency of Earth's rotation. The period of free
wobble of a rigid Earth is then 304.456 sidereal days or 305.290 solar days. This
is called the Euler wobble period. In the real, deformable Earth, the resonant fre-
quency and Q fitted by expression (4.15) to the maximum entropy spectrum of the
VLBI pole path are, respectively,

f 0 = 0.83813 cpy and Q = 228. (4.76)

The resonant frequency is not a ected by dissipation, which is reflected only in

the value of Q . The observed resonant frequency corresponds to a period of 435.80
solar days or 434.61 sidereal days. Then, from (4.61), the value of the Love num-
ber, k 2 , required to account for the increase in the observed period of the Chandler
wobble by rotational deformation, is 0.2820. This value of k 2 is close to the val-
ues computed from seismically determined Earth models (Farrell, 1972), but the
agreement is more apparent than real, because the liquid core actually shortens the
wobble period by roughly 32 days, while the oceans lengthen the period by roughly
28 days (Lambeck, 1980, p. 202).
The value of Q , determined from the maximum entropy spectrum, is consist-
ent with values for the mantle and crust found from seismic wave attenuation
(Lambeck, 1988, p. 63), which are in the range 100 to 600. Conventional spectral
analysis has usually given much lower values. This is likely due to the limitations
on frequency resolution due to finite record e

ff

ff

ects, discussed in Section 2.4.2. Even
for a 'good' window, such as the Parzen window, frequency resolution is limited
by half-power points separated by 1.82/ T for a record length T . The half-power
points of the resonance determined by (4.15) are separated in frequency by f 0 /2 Q .
Simply to match this value to that for the Parzen window requires a record length

1.82 2 Q

f 0 =

×

T = 990 years. (4.77)

The maximum entropy spectral analysis method avoids this limitation of record
length, allowing the true value of Q to be determined.