Earth’s rotation: observations and theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 277

motion does not a ff ect the latitude φ. The law of sines for the spherical triangle

OPP then gives

sin m ⊥
sin x =

sin (π/2 − φ) = cosφ. (4.4)

Since the angles m ⊥ and x are very small, this expression closely approximates to

m ⊥
cosφ .

x = (4.5)

The law of sines for the spherical triangle OEE gives

sin t

sin x =

Δ

sinφ, (4.6)

where Δ t is the increase in the sidereal time angle. Again, the angles Δ t and x are

very small, and thus to close approximation,

m 1 sinλ −

m 2 cosλ tanφ. Δ t = m ⊥ tanφ = (4.7)

The raw, instantaneous Universal Time, UT 0 i , is given by the angle measured
eastward along the instantaneous equator from the vernal equinox to the Greenwich
meridian, while Universal Time corrected for polar motion is UT 1. Co-ordinated
Universal Time, UTC , is the time kept by atomic clocks. Thus, two observables are

tanφ m 1 sinλ − m 2 cosλ , UT 0 i − UTC = UT 1 − UTC + (4.8)

and

φ i − φ = m 1 cosλ + m 2 sinλ, (4.9)

with φ i representing the instantaneous latitude of the observatory.
Traditionally, the co-ordinates of the pole are given as ( x ,y) with x = m 1 and

y =− m 2 . In addition, the longitude is expressed as the west longitude ψ = 2π − λ.

Then, the increase in observed latitude is

φ i − φ = x cos (2π − ψ) − ysin (2π − ψ) = x cosψ + ysinψ, (4.10)

and

tanφ x sinψ + ycosψ . UT 0 i − UTC = UT 1 − UTC + − (4.11)