Earth’s rotation: observations and theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 317

Tabl e 4 . 4 Parameters fitted to successive retrograde free core
nutation (RFCN) resonances at 400-day steps along the time axis.

a 2 mas 2 /cpd f 0 10 − 3 cpd Time in days Q Period (days)

1600 168.91 3.4934 − 2.00415 − 498.965

2000 217.10 8.3631 − 2.26549 − 441.406

2400 204.58 8.3887 − 2.27840 − 438.904

2800 180.64 8.2660 − 2.26412 − 441.673

3200 132.81 8.3479 − 2.27136 − 440.265

3600 53.808 6.4096 − 2.38518 − 419.256

4000 34.536 5.7286 − 2.38140 − 419.921

4400 24.125 5.1931 − 2.42864 − 411.753

4800 19.160 5.2115 − 2.41370 − 414.302

5200 15.594 6.1112 − 2.31616 − 431.749

− −

5600 14.098 5.5330 2.30438 433.956

− −

6000 8.5950 3.6275 2.27682 439.209

− −

6400 7.8865 3.5702 2.25907 442.660

6800 6.8884 2.9255 − 2.14030 − 467.224

A 0 e ± π t / QT A = , (4.116)

where A 0 is the amplitude at time t 0and T is the period (positive for the PFCN,

negative for the RFCN). The positive sign in the exponent refers to the RFCN, the
negative sign in the exponent refers to the PFCN. In free decay, the logarithms of
the amplitudes of the free core nutations then follow the linear relation

=

log A = ct + d , (4.117)

with the slope given by

π log e
QT =−

log e
τ ,

c =± (4.118)

where τ is the e-folding time, and the intercept is d = log A 0 . The half-life is then

t 1/2 = τln 2. (4.119)

A plot of the logarithm of the amplitude of the PFCN in microarcseconds (μas)
against time is shown in Figure 4.18. The straight line, fitted by least squares,
returns the values c 10 − 4 and d

=− ± × = ±

0.0299684.
A similar plot of the logarithm of the amplitude of the RFCN in microarcseconds
(μas) against time is shown in Figure 4.19. The straight line, fitted by least squares,
returns the values c

1.14877 0.0741224 2.3881

10 − 4 and d =− 1.35076 ± 0.0896257 × = 2.6179 ± 0.0403216.