moment of inertia and precession

moment of inertia and precession Moment of inertia is a fundamental property in rotational mechanics. It is the ratio of torque to angular acceleration and is analogous to the role of mass in linear mechanics (ratio of force to linear acceleration). For solid bodies it is a measure of the distribution of mass. The greater the concentration of mass towards the centre of a sphere the smaller is its moment of inertia. In the 1930s K. E. Bullen recognized that a value for the moment of inertia of the Earth was available from astronomical observations of the motions of the Earth and Moon. Using this information he was able to develop the first realistic models of the Earth's density structure (see density distribution within the Earth).

The moment of inertia of a point mass m about an axis at distance r from it is mr². For a solid body with a known distribution of density the moment of inertia about any axis can be obtained by integration of this formula. We are normally interested in moment of inertia about an axis through the centre of mass, and we can compare the Earth with two simple standard results. For spheres of mass m and radius r:

Hollow spherical shell	I = (2/3) mr2
Uniform solid sphere	I = (2/5) mr2
Earth	I = 0.330695 mr2

To gain an idea of the mass concentration needed to give the terrestrial value, consider a model Earth with a uniform mantle of density ρ and radius r surrounding a uniform core of radius 0.546r and density fρ. We would then require f = 3.01, or, in other words, a core with a density about three times greater than that of the mantle. Of course, neither the mantle nor the core of the real Earth is uniform, but this radius ratio is the observed one, and we can thus see that there must be a strong concentration of density towards the centre.

To measure the moment of inertia of a body we need to observe the angular acceleration produced by a known torque. In the case of the Earth the only torques of sufficient magnitude to cause observable effects are those from gravitational interactions with the Moon and Sun. If the Earth had perfect spherical symmetry then no such torques would be possible because all gravitational interactions with a sphere are equivalent to interactions with an equal mass at its centre. Thus the lunar and solar torques on the Earth act on the asymmetry of the mass distribution, specifically the equatorial bulge. They act in a direction that tends to pull the bulge into alignment with themselves.

The geometrical problem is represented in Fig. 1. At the equinoxes, on the y-axis, the Sun is directly above the Equator and no torque arises. The torque is greater when the Sun is at A or B on the x′; axis at an angle θ = 23.5° to the equatorial plane. The Earth then experiences a torque tending to pull the equatorial (x) axis towards x′;. The sense of this torque is the same when the Sun is at both of the positions A and B. Thus the solar torque oscillates in strength between the maximum and zero, but never reverses in sign. It is a torque that switches on and off in a semi-annual cycle. There is similarly a torque due to the Moon, which oscillates in strength in a semi-monthly cycle.

When a rotating body experiences a torque that tends to turn its axis of rotation, the response is a shift of the axis in a perpendicular sense. This is precession. The rotational axis does not turn towards the pole of the ecliptic, that is, the normal to the orbital plane (Ox′;y in the figure), but precesses about it. It describes a cone of semi-angle 23.5°, about the ecliptic pole, completing the cycle in 25 730 years.

The lunar and solar precessional torques are proportional to the difference between moments of inertia of the Earth about the rotational axis (C) and an axis in the equatorial plane (A). There is thus a torque proportional to (C–A) which acts on the Earth's rotational an axis at distance, Cω, where ω is the angular speed of rotation. The rate of change in the direction of the rotational axis is the ratio of torque to angular momentum and so gives a value for the precessional constant. The torque on the Earth exerted by any external body is balanced by a torque on that body, causing a precession of its orbit. The effect is well observed for the Moon but is now more precisely seen in the orbits of artificial satellites. The nodes of an orbit, that is the points at which it crosses the equatorial plane, slowly regress round the Equator and the rate gives a measure of the asymmetry of the Earth's mass. We can thus obtain a value for the axial moment of inertia, C, of 0.330695 Ma², M being the mass of the Earth and a its radius.

Frank D. Stacey

Bibliography

Stacey, F. D. (1992). Physics of the Earth (3rd edn). Brookfield Press, Brisbane.