# Precession (Torque-free)

## Description

Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called nutation. In physics, there are two types of precession: torque-free and torque-induced.

In astronomy, “precession” refers to any of several slow changes in an astronomical body’s rotational or orbital parameters, and especially to Earth’s precession of the equinoxes. (See section Astronomy below.)

Torque-free precession implies that no external moment (torque) is applied to the body. In torque-free precession, the angular momentum is a constant, but the angular velocity vector changes orientation with time. What makes this possible is a time-varying moment of inertia, or more precisely, a time-varying inertia matrix. The inertia matrix is composed of the moments of inertia of a body calculated with respect to separate coordinate axes (e.g. x, y, z). If an object is asymmetric about its principal axis of rotation, the moment of inertia with respect to each coordinate direction will change with time, while preserving angular momentum. The result is that the component of the angular velocities of the body about each axis will vary inversely with each axis’ moment of inertia.

The torque-free precession rate of an object with an axis of symmetry, such as a disk, spinning about an axis not aligned with that axis of symmetry can be calculated as shown here.

Related formulas

## Variables

 ωp precession rate (°/s) Is moment of inertia about the axis of symmetry (km*m2) ωs spin rate about the axis of symmetry (°/s) Ip moment of inertia about either of the other two equal perpendicular principal axes (km*m2) a angle between the moment of inertia direction and the symmetry axis (deg)