Earth’s rotation: observations and theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 308

r

A

p / 2- a

g

e 0

B

e

y

p - e 0 -

Figure 4.13 Spherical triangle showing the relation of the increases in longitude
and obliquity resulting from a displacement of the celestial pole.

marking the celestial equator. The great circle of intersection of the plane of Earth's
orbit with the celestial sphere is the celestial ecliptic. The celestial equator and the
celestial ecliptic intersect at γ, the vernal equinox or the first point of Aries .
If the rotation axis moves through the angle r at ecliptic longitude α measured
along the ecliptic from the vernal equinox γ, an increase Δ ψ in the ecliptic long-

itude results, given by the side γ B of the spherical triangle A γ B illustrated in Fig-
ure 4.13. At the same time, the obliquity of the ecliptic increases from the reference
value 0 to 0 −Δ .

The equatorial plane rotates about the axis OA orthogonal to the plane POP ,giv-
ing the side A γof the spherical triangle asπ/2

+Δ , giving the angle AB γ of the spherical triangle as π − 0

− α. By the law of sines for spherical

triangles,

sin (π/2 − α) sin Δ ψ

sin r .

−Δ ) =

(4.104)

sin (π − 0

The angles r , Δ are all small quantities whose squares and products can

be neglected. Thus, the right-hand side of (4.104) becomes the simple ratio

Δ ψ and

Δ ψ/ r .
The numerator of the left-hand side of (4.104) is equal to cosα, while the denom-
inator can be successively expanded as

sin (π − 0 −Δ ) = sin (π − 0 ) cos Δ − cos (π − 0 ) sin Δ

= sinπcos 0 − cosπsin 0 −Δ [cosπcos 0 + sinπsin 0 ]

= sin 0 +Δ cos 0 , (4.105)

to first order in small quantities. To the same order, (4.104) then yields

r cosα =Δ ψsin 0 . (4.106)

In a right-handed Cartesian co-ordinate system with the z -axis through the celes-
tial reference pole and the x -axis through the vernal equinox with origin at the