Precession

The precession is the name given to the gradual change of orientation of the Axis of rotation of an object or, in a more general way, a Vecteur under the action of the environment, for example, in the case of the axis of rotation of an object, when a couple is applied to him. This phenomenon is easily observable with a spinning top but all the objects in rotation can undergo the precession. At the time of the precession, the angle which forms the axis of rotation or the vector with a given direction remains fixed. The vector or the axis of rotation describes thus has course of time a cone whose axis is the fixed direction. This cone is traversed with a angular Velocity constant which is determined by the facts of the case. The direction in which the precession occurs depends on the problem considered. In the case of a spinning top, the precession is done in the direction opposed to that of rotation.

Fundamental formula

The mathematical formula which describes the precession of a quantity $\{\backslash \; mathbf\; \{NR\}\}$ writes

$\backslash \; frac\; =\; \{\backslash \; mathbf\; \{\backslash \; Omega\}\}\; \backslash \; wedge\; \{\backslash \; mathbf\; \{NR\}\}$,

where $\{\backslash \; mathbf\; \backslash \; Omega\}$ is a constant vector quantity (or possibly slowly variable). Its direction determines the axis of the cone of precession and its standard is homogeneous with a angular Velocity. In such a case, the precession is carried out at the angular velocity

$\backslash \; Omega\_p\; =\; |\{\backslash \; mathbf\; \backslash \; Omega\}|$,

and in the trigonometrical direction in the plan directed by $\{\backslash \; mathbf\; \backslash \; Omega\}$.

Various types of precession

A great physical number of situations give place to a phenomenon of precession:

• In Mécanique, an object turning a such Toupie or a Gyroscope will know a movement of precession if it is not balanced, i.e. if the resultant of the moments of the forces which are exerted on him is not null.
• the plan of oscillation of a Pendulum of Foucault precess during time, thus expressing rotation Ground stre. The physical effect in the beginning of this is the Force of Coriolis.
• In Astronomy, a body rotating can be seen like a Gyroscope and can be caused to précesser. It is for example the case of the Earth, of which the axis of the poles precess because of the gravitational interactions with the Sun. This phenomenon was discovered by the Greek Astronome Hipparque, shortly after the year -150.
• Always in astronomy, a body in orbit will have, in addition to its own rotation, a orbital kinetic Moment resulting from its circular motion or elliptic around the central body. The direction of the orbital kinetic moment represents the normal in the plan of the orbit of the star. This plan can possibly précesser under the influence of other celestial bodies.
• a body in elliptic orbit can see its orbit disturbed in its plan by other stars. One of the disturbances tends to vary the axis determined by the half main roads of the orbit (the Vecteur of Laplace-Runge-Lenz). Thus, the direction determined by the point of the orbit nearest to the central body varies it during time. One speaks about Précession of the perihelion, or more generally, apart from the Solar system, of Précession of the pericenter (or pericenter advances). The advance of the pericenter can be produced by the interactions with other bodies, but can also the being by a variation with the sphericity of the central body. A third possible cause in advance of the pericenter is predicted by the General relativity, of which one of the effects most easily observable is an advance of the pericenter being added to the other causes enumerated above. The advance of the perihelion of the Planet Mercure was the first checking of the theory of the General relativity discovered by Albert Einstein. The system presenting the greatest relativistic advance of the pericenter is the Pulsar doubles PSR J0737-3039 (more than 16 degree S per annum).
• In Physical atomic, a particle having a magnetic Moment will see this one précesser when the particle is plunged in a Magnetic field. One speaks then about Précession of Larmor, whose frequency can be measured.
• Toujours in atomic physics, the clean kinetic moment (the Spin) of a particle also goes précesser if the particle is accelerated. This result, consequence of the restricted Relativity, for the first time was correctly explained by Llewellyn Thomas in the current of the Années 1920 and bears the name of Précession of Thomas.
• a particle plunged in a gravitational Champ also will see its kinetic moment proper precession because of existence of this one. One speaks about Effet of Sitter, predicted for the first time in 1916 by Willem de Sitter.
• the combination of the precession of Thomas and the effect Sitter bears the name of geodetic Précession.
• general relativity also predicts that a turning body causes an domino effect of the Espace-temps in the direction of its rotation. This effect, often called of its English name of frame-dragging is the Effet Lense-Thirring, discovered by Joseph Lense and Hans Thirring in 1918 causes an additional precession of the orbital kinetic moment of a body if the plan of the orbit is not perpendicular to the axis of rotation of the central body, as well as an additional precession of the pericenter and clean kinetic moment of the bodies subjected to the influence of the central body. In this last case, one speaks sometimes about Précession of Schiff. The Lense-Thirring effect can be detectable in theory indirectly by the study of the discs of accretion of compact objects. Its precise measurement in the gravitational field Ground stre is the object of the satellite mission Gravity Probe B of NASA launched in 2004 and whose results should be announced the April 14th 2007. The Lense-Thirring effect belongs to the manifestations of the Gravitomagnétisme, a formal and imperfect analogy between certain aspects of general relativity and the electromagnetism.

The mechanical precession

When an object undergoes a couple, its axis of rotation changes during time. This phenomenon results from the Théorème of the kinetic moment, consequence of the Basic principle of dynamics formulated in second half of the 17th century by Isaac Newton. When this couple is exerted by a force of constant direction (for example the Pesanteur Ground stre) on an object whose kinetic Moment is sufficiently important and whose axis of rotation passes by the point of application of the force, then the object goes précesser, i.e. its kinetic moment will keep a constant intensity, but will see its direction précesser around the direction of the force.

In practice, a spinning top launched with a sufficiently high speed will enter these assumptions. Its axis of rotation thus will keep a constant angle with the vertical (direction of the force of gravity), but will turn at constant speed (if one neglects the Frottement S).

The physics of the precession

Under these conditions, the period of precession is the following one:

$T\_p\; =\; \backslash \; frac\; \{4\; \backslash \; pi^2\; I\_s\}\; \{C\; T\_s\}$,

In which I is the moment of inertia, T the period of rotation around the axis of rotation, and C is the couple. This expression can rewrite in term of the angular velocities corresponding. By noting ω angular velocity of the body ($\backslash \; omega\_s\; =\; 2\; \backslash \; pi/T\_s$) and Ω that of the precession ($\backslash \; Omega\_p\; =\; 2\; \backslash \; pi/T\_p$), one has

$\backslash \; Omega\_p\; \backslash \; omega\_s\; =\; \backslash \; frac\; \{C\}\; \{I\_s\}$.

See too

• Gyroscope

• Spinning top
• Movement of Lagrange of the spinning top
• Pendulum of Foucault
• Precession of the equinoxes
• Nutation
• Precession of Larmor
• Precession of Thomas
• Effect Sitter
• geodetic Precession
• Effect Lense-Thirring
• Precession of Schiff

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