Earth’s rotation: observations and theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 306

From the real and imaginary parts of the free motion equation (4.60), m 1 =− σ 0 m 2

and m 2 = σ 0 m 1 , giving

ω =−Ω σ 0 m 2 e 1 +Ω σ 0 m 1 e 2

=Ω σ 0 e 3 × ( m 1 e 1 + m 2 e 2 )

= σ 0 e 3 × ( Ω m 1 e 1 +Ω m 2 e 2 +Ω e 3 )

= σ 0 e 3 × ω . (4.98)

Note that ω is the same, whether measured in the space frame or the rotating frame,
since ω × ω = 0. On equating expressions (4.96) and (4.98) for ω ,wefindthat

L × ω

˙ ξ

| = σ 0 e 3

× ω . (4.99)

| L

Since

L × ω =| L || ω | sinγ and e 3 × ω =| ω | sinβ, (4.100)

we obtain that

σ 0 +Ω

β , ξ = σ 0 sin β

sinγ = σ 0 cos β

tan β
tanγ =

cos β
cosγ

˙

sec 2 (4.101)

cosγ

using expression (4.94).The angular rate σ 0 at which ω moves about e 3 , and the
angular rate ˙ ξ at which ω moves about L , are in inverse proportion to the sines of

the semi-apex angles of the cones. Thus, the body cone (polhode) rolls once per
day without slipping on the space cone (herpolhode), as illustrated in Figure 4.11.
This geometrical representation originated with Poinsot in 1834 and is called the
Poinsot construction . The resulting polar motion is the free Chandler wobble.
The nearly diurnal motion of the rotation axis in space is called sway . For the
free polar motion or Chandler wobble, the angle β is only parts in 10 6 and the
angle γ is even smaller. The tangents are then very close to the angles themselves
and their ratio is closely

γ
β =

σ 0
Ω+ σ 0 .

(4.102)

Similarly, the sines of the angles are very close to the angles themselves and

ξ = σ 0 β

˙

γ =Ω+ σ 0 .

(4.103)

The motion of ω about L is faster than the diurnal rotation by the factor 1 + σ 0 / Ω ,

which at 1.00229 is just in excess of unity.