Polar Motion - Basic Principles

Basic Principles

In the absence of external torques, the vector of the angular momentum M of a rotating system remains constant and is directed toward a fixed point in space. In the case of the Earth, it is almost identical with its axis of rotation. The vector of the figure axis F of the system wobbles around M. This motion is called Euler's free nutation. For a rigid Earth which is an oblate spheroid to a good approximation, the figure axis F is its geometric axis defined by the geographic north and south pole. It is identical with the axis of its polar moment of inertia. The Euler period of free nutation is

(1) τ_E = 1/ν_E = A/(C - A) = 307 sidereal days = 0.84 sidereal years

ν_E = 1.19 is the normalized Euler frequency (in units of reciprocal years), C = 8.04 × 1037 kg m2 is the polar moment of inertia of the Earth, A is its mean equatorial moment of inertia, and C - A = 2.61 × 1035 kg m2.

The observed angle between M and F is a few hundred milliarcseconds (mas) which gives rise to a surface displacement of several meters (100 mas corresponds to 3.09 m) between the figure axis of the Earth and its angular momentum. Using the geometric axis as the primary axis of a new body-fixed coordinate system, one arrives at the Euler equation of a gyroscope describing the apparent motion of the rotation axis about the geometric axis of the Earth. This is the so-called polar motion.

Observations show that the figure axis exhibits an annual wobble forced by surface mass displacement via atmospheric and/or ocean dynamics, while the free nutation is much larger than the Euler period and of the order of 435 to 445 sidereal days. This observed free nutation is called Chandler wobble. There exist, in addition, polar motions with smaller periods of the order of decades. Finally, a secular polar drift of about 0.10 m per year in the direction of 80° west has been observed which is due to mass redistribution within the Earth's interior by continental drift, and/or slow motions within mantle and core which gives rise to changes of the moment of inertia.

The annual variation was discovered by Karl Friedrich Küstner in 1885 by exact measurements of the variation of the latitude of stars, while S.C. Chandler found the free nutation in 1891. Both periods superpose, giving rise to a beat frequency with a period of about 5 to 8 years (see Figure 1).

This polar motion should not be confused with the changing direction of the Earth's spin axis relative to the stars with different periods, caused mostly by the torques on the Geoid due to the gravitational attraction of the Moon and Sun. They are also called nutations, except for the slowest, which is the precession of the equinoxes.