# Precession

There are two types of precession, torque-free and torque-induced, the latter being discussed here in more detail.

## Torque-free precession

Only solid objects can be in torque-free precession. For example, when a plate is thrown, the plate may have some rotation around an axis that is not its axis of symmetry. When the object is not perfectly solid, internal vortices will tend to damp torque-free precession.

## Torque-induced precession

Torque-induced precession (gyroscopic precession) is the phenomenon by which the axis of a spinning object (e.g. a part of a gyroscope) "wobbles" when a torque is applied to it. The phenomenon is commonly seen in a spinning toy top, but all rotating objects can undergo precession. As a spinning object precesses, the tilt of its axis goes around in a circle in the opposite direction that the object is spinning. If the speed of the rotation and the magnitude of the torque are constant the axis will describe a cone, its movement at any instant being at right angles to the direction of the torque. In the case of a toy top, if the axis is not perfectly vertical the torque is applied by the force of gravity trying to tip it over. A rolling wheel will tend to remain upright due to precession. When the wheel tilts to one side, the particles at the top are pushed to one side and the particles at the bottom are pushed the other way. However, since the wheel is rotating, these particles eventually switch places and cancel one another out. Precession or gyroscopic considerations may have a minor effect on bicycle performance at high speed. Precession is also the mechanism behind gyrocompasses.

This concept is easier to understand by examining the effects of inertia, which is often stated by the phrase "A body in motion tends to stay in motion." In this case the "motion" of a rotating body is in its rotation. If an external force pushes upon the rotating body, the body will resist the force by pushing back against it, but the reaction is delayed such that it occurs at a point 90 degrees later in its rotation. If you push a spinning top to the right, it will move forward (assuming the top is spinning counter-clockwise).

Gyroscopic precession also plays a large role in the flight controls on helicopters. Since the driving force behind helicopters is the rotor head (which rotates), gyroscopic precession comes into play. If the rotor head is tilted to the right, its counter-clockwise movement forces the aircraft to fly forward. To ensure the pilot's inputs are correct the aircraft has corrective linkages which tilt the rotor head to the right when the pilots push the "cyclic stick" forward, or to the left when the stick is pulled to the back.

## The physics of precession

Precession is due to the fact that the resultant of the angular velocity of rotation and the angular velocity produced by the torque is an angular velocity about a line which makes an angle with the permanent rotation axis, and this angle lies in a plane at right angles to the plane of the couple producing the torque. The permanent axis must turn towards this line, since the body cannot continue to rotate about any line which is not a principal axis of maximum moment of inertia; that is, the permanent axis turns in a direction at right angles to that in which the torque might be expected to turn it. If the rotating body is symmetrical and its motion unconstrained, and if the torque on the spin axis is at right angles to that axis, the axis of precession will be perpendicular to both the spin axis and torque axis. Under these circumstances the period of precession is given by:

$T_p = \frac{4\pi^2I_s}{QT_s}$

In which Is is the moment of inertia, Ts is the period of spin about the spin axis, and Q is the torque. In general the problem is more complicated than this, however.

For a layman’s explanation of Precession: we will have to imagine the wheel of a gyroscope as a group of particles that are being forced to move in circle. Remember the particles want to move is a straight line. In order for the particles to move in a curved line there must be a force. This force is provided by the structure of the wheel holding the particles within the wheel.

Now let’s see what happens to our accelerating particles when a torque is applied to the spinning wheel. Assume the axis of rotation created by the torque is thru the center of the wheel at 90 degrees to the primary rotation of the wheel. Let’s look at a particle that is on this axis of rotation. Since the particle is on the axis of rotation there is no direct motion applied to the particle at the instant of the applied torque. But let’s look at what will need to happen at the next moment in time. The particle is now going to be forced to curve again. This time in the direction of the curve is to accommodate the tilt of the wheel. Now we have a particle that is already moving and it wants to keep moving in a straight line. So the particle will exert a force on the wheel. If you look at a particle on the other side of the wheel you will see that the force of the second particle is in the opposite direction of the first particle. That pair of unmatched forced is what causes the precession torque that is 90 degrees to the applied torque.

## Precession of the equinoxes

The Earth goes through one complete precession cycle in a period of approximately 25,800 years, during which the positions of stars as measured in the equatorial coordinate system will slowly change; the change is actually due to the change of the coordinates. Over this cycle the Earth's north axial pole moves from where it is now, within 1° of Polaris, in a circle around the ecliptic pole, with an angular radius of 23 degrees 27 arcminutes [1], or about 23.5 degrees. The shift is 1 degree in 180 years (the angle is taken from the observer, not from the center of the circle).

The explanation of this is: The axis of the Earth undergoes precession due to a combination of the Earth's nonspherical shape (it is an oblate spheroid, bulging outward at the equator) and the gravitational tidal forces of the Moon and Sun applying torque as they attempt to pull the equatorial bulge into the plane of the ecliptic.

### A changing north star

Polaris is not particularly well-suited for marking the north celestial pole, as its visual magnitude, which is variable, hovers around 2.1, fairly far down the list of brightest stars in the sky. On the other hand, in 3000 BC the faint star Thuban in the constellation Draco was the pole star; at magnitude 3.67 it is five times fainter than Polaris; today it is all but invisible in light-polluted urban skies. The brightest star known to have been North Star or to be predictable as taking that role in the future is the brilliant Vega in the constellation Lyra, which will be the pole star around the year AD 14,000. When viewed looking down onto the Earth from the north, the direction of precession is clockwise. When standing on Earth looking outward, the axis appears to move counter-clockwise across the sky. This sense of precession, against the sense of Earth's own axial rotation, is opposite to the precession of a top on a table. The reason is that the torques imposed on the Earth by the Sun and Moon act in the sense of trying to align its axis normal to the ecliptic, i.e. to stand up more vertically in regards to the ecliptic plane, while the torque on a top spinning on a hard surface acts in the sense of trying to make the top fall over, rather than to stand up straighter.

Polaris is not exactly at the pole; any long-exposure unguided photo will show it having a short trail. It is close enough for most practical purposes, though. The south celestial pole precesses too, always remaining exactly opposite the north pole. The south pole is in a particularly bland portion of the sky, and the nominal south pole star is Sigma Octantis, which, while fairly close to the pole, is even weaker than Thuban -- magnitude 5.5, which is barely visible even under a properly dark sky. The precession of the Earth is not entirely regular due to the fact that the Sun and Moon are not in the same plane and move relative to each other, causing the torque they apply to Earth to vary. This varying torque produces a slight irregular motion in the poles called nutation.

Precession of the Earth's axis is a very slow effect, but at the level of accuracy at which astronomers work, it does need to be taken into account. Note that precession has no effect on the inclination ("tilt") of the plane of the Earth's equator (and thus its axis of rotation) on its orbital plane. It is 23.5 degrees and precession does not change that. The inclination of the equator on the ecliptic does change due to gravitational torque, but its period is different (main period about 41000 years).

The following figure illustrates the effects of axial precession on the seasons, relative to perihelion and aphelion. The precession of the equinoxes can cause periodic climate change (see Milankovitch cycles), because the hemisphere that experiences summer at perihelion and winter at aphelion (as the southern hemisphere does presently) is in principle prone to more severe seasons than the opposite hemisphere.

Hipparchus first estimated Earth's precession around 130 BC, adding his own observations to those of Babylonian and Chaldean astronomers in the preceding centuries. In particular they measured the distance of the stars like Spica to the Moon and Sun at the time of lunar eclipses, and because he could compute the distance of the Moon and Sun from the equinox at these moments, he noticed that Spica and other stars appeared to have moved over the centuries.

Precession causes the cycle of seasons (tropical year) to be about 20.4 minutes less than the period for the earth to return to the same position with respect to the stars as one year previously (sidereal year). This results in a slow change (one day per 58 calendar years) in the position of the sun with respect to the stars at an equinox. It is significant for calendars and their leap year rules.

## Precession of planetary orbits

File:Perihelion precession.jpg
Precession of the perihelion (very exaggerated)

The revolution of a planet in its orbit around the Sun is also a form of rotary motion. (In this case, the combined system of Earth and Sun is rotating.) So the axis of a planet's orbital plane will also precess over time.

The major axis of each planet's elliptical orbit also precesses within its orbital plane, in response to perturbations in the form of the changing gravitational forces exerted by other planets. This is called "perihelion precession". Discrepancies between the observed perihelion precession rate of the planet Mercury and that predicted by classical mechanics were prominent among the forms of experimental evidence leading to the acceptance of Einstein's Theory of Relativity, which predicted the anomalies accurately.

It is generally understood that the gravitational pulls of the sun and the moon cause the precession of the equinoxes on Earth which operate on cycles of 23,000 and 19,000 years. The precession of the orbit of the Earth is an important part of the astronomical theory of ice ages.

Precession is also an important consideration in the dynamics of atoms and molecules.

## Notes

1. ^  Cook, David R. (1999), "U.S. Department of Energy, Environmental Earth Science Archive, Ask A Scientist" [3]

## References

1. "Moon and Spica", StarDate July 14, 2005, University of Texas McDonald Observatory, [4]da:Præcession
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