Earth’s rotation: observations and theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 319

space-fixed invariable axes
reference axes

body cone

rotation axes

space cone

Figure 4.20 Poinsot construction for the PFCN (right) and the RFCN (left).

nearly diurnal retrograde wobbles with angular frequencies close to Ω

is the angular frequency of Earth's rotation. The relation between the nutations and
wobbles is illustrated by the Poinsot constructions in Figure 4.20. For the PFCN,
a small body cone rolls without slipping, once per sidereal day, on the outside of
a large space-fixed cone, as shown on the right of Figure 4.20. The RFCN can be
described as a small body cone rolling, without slipping, once per sidereal day on
the inside of a large space-fixed cone, as shown on the left of Figure 4.20. With
slipping, the body cone in the RFCN would make one revolution backwards for
each traverse of the space cone. For nutation amplitude A N and wobble amplitude
A W , without slipping the body cone rotates forward A N / A W times for each traverse
of the space cone. Then,

−Ω , where

A N

=− Ω

A W −

1

σ N ,

(4.120)

with σ N denoting the signed angular nutation frequency (negative for the RFCN).
For the PFCN, shown on the right of Figure 4.20, with slipping, the body cone
makes one revolution forward for each traverse of the space cone. Without slipping,
the body cone rotates forward A N / A W times for each traverse of the space cone.
Then,

A N
A W +

= Ω

1

σ N ,

(4.121)

with σ N denoting the positive angular nutation frequency. For both modes, the
rotation axis is along the line of contact of the two cones. Its motion in space is the
sum of the motion of the reference axis and its motion with respect to the reference
axis. Thus,

σ N = σ W +Ω , (4.122)