Precession
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. An important example is the steady change in the orientation of the axis of rotation of the Earth, known as the precession of the equinoxes.
Torque-free or Torque neglected
Torque-free precession implies that no external moment (torque) is applied to the body. In torque-free precession, the
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 follows:[1]
When an object is not perfectly rigid, inelastic dissipation will tend to damp torque-free precession,[3] and the rotation axis will align itself with one of the inertia axes of the body.
For a generic solid object without any axis of symmetry, the evolution of the object's orientation, represented (for example) by a rotation matrix R that transforms internal to external coordinates, may be numerically simulated. Given the object's fixed internal
Torque-induced
Torque-induced precession (gyroscopic precession) is the phenomenon in which the
The device depicted on the right is gimbal mounted. From inside to outside there are three axes of rotation: the hub of the wheel, the gimbal axis, and the vertical pivot.
To distinguish between the two horizontal axes, rotation around the wheel hub will be called spinning, and rotation around the gimbal axis will be called pitching. Rotation around the vertical pivot axis is called rotation.
First, imagine that the entire device is rotating around the (vertical) pivot axis. Then, spinning of the wheel (around the wheelhub) is added. Imagine the gimbal axis to be locked, so that the wheel cannot pitch. The gimbal axis has sensors, that measure whether there is a torque around the gimbal axis.
In the picture, a section of the wheel has been named dm1. At the depicted moment in time, section dm1 is at the
The same reasoning applies for the bottom half of the wheel, but there the arrows point in the opposite direction to that of the top arrows. Combined over the entire wheel, there is a torque around the gimbal axis when some spinning is added to rotation around a vertical axis.
It is important to note that the torque around the gimbal axis arises without any delay; the response is instantaneous.
In the discussion above, the setup was kept unchanging by preventing pitching around the gimbal axis. In the case of a spinning toy top, when the spinning top starts tilting, gravity exerts a torque. However, instead of rolling over, the spinning top just pitches a little. This pitching motion reorients the spinning top with respect to the torque that is being exerted. The result is that the torque exerted by gravity – via the pitching motion – elicits gyroscopic precession (which in turn yields a counter torque against the gravity torque) rather than causing the spinning top to fall to its side.
Precession or gyroscopic considerations have an effect on bicycle performance at high speed. Precession is also the mechanism behind gyrocompasses.
Classical (Newtonian)
Precession is the change of angular velocity and angular momentum produced by a torque. The general equation that relates the torque to the rate of change of angular momentum is:
Due to the way the torque vectors are defined, it is a vector that is perpendicular to the plane of the forces that create it. Thus it may be seen that the angular momentum vector will change perpendicular to those forces. Depending on how the forces are created, they will often rotate with the angular momentum vector, and then circular precession is created.
Under these circumstances the angular velocity of precession is given by: [5]
where Is is the moment of inertia, ωs is the angular velocity of spin about the spin axis, m is the mass, g is the acceleration due to gravity, θ is the angle between the spin axis and the axis of precession and r is the distance between the center of mass and the pivot. The torque vector originates at the center of mass. Using ω = 2π/T, we find that the period of precession is given by:[6]
Where Is is the moment of inertia, Ts is the period of spin about the spin axis, and τ is the torque. In general, the problem is more complicated than this, however.
Relativistic (Einsteinian)
The special and general theories of relativity give three types of corrections to the Newtonian precession, of a gyroscope near a large mass such as Earth, described above. They are:
- Thomas precession, a special-relativistic correction accounting for an object (such as a gyroscope) being accelerated along a curved path.
- de Sitter precession, a general-relativistic correction accounting for the Schwarzschild metric of curved space near a large non-rotating mass.
- Lense–Thirring precession, a general-relativistic correction accounting for the frame dragging by the Kerr metric of curved space near a large rotating mass.
Astronomy
In astronomy, precession refers to any of several gravity-induced, slow and continuous changes in an astronomical body's rotational axis or orbital path. Precession of the equinoxes, perihelion precession, changes in the tilt of Earth's axis to its orbit, and the eccentricity of its orbit over tens of thousands of years are all important parts of the astronomical theory of ice ages. (See Milankovitch cycles.)
Axial precession (precession of the equinoxes)
Axial precession is the movement of the rotational axis of an astronomical body, whereby the axis slowly traces out a cone. In the case of Earth, this type of precession is also known as the precession of the equinoxes, lunisolar precession, or precession of the equator. Earth goes through one such complete precessional cycle in a period of approximately 26,000 years or 1° every 72 years, during which the positions of stars will slowly change in both
The
Apsidal precession
The orbits of planets around the Sun do not really follow an identical ellipse each time, but actually trace out a flower-petal shape because the major axis of each planet's elliptical orbit also precesses within its orbital plane, partly in response to perturbations in the form of the changing gravitational forces exerted by other planets. This is called perihelion precession or apsidal precession.
In the adjunct image, Earth's apsidal precession is illustrated. As the Earth travels around the Sun, its elliptical orbit rotates gradually over time. The eccentricity of its ellipse and the precession rate of its orbit are exaggerated for visualization. Most orbits in the Solar System have a much smaller eccentricity and precess at a much slower rate, making them nearly circular and nearly stationary.
Discrepancies between the observed perihelion precession rate of the planet
Nodal precession
Orbital nodes also precess over time.
See also
- Larmor precession
- Nutation
- Polar motion
- Precession (mechanical)
- Precession as a form of parallel transport
References
- ISBN 9781600860270
- ^ Boal, David (2001). "Lecture 26 – Torque-free rotation – body-fixed axes" (PDF). Retrieved 2008-09-17.
- .
- ISBN 978-1-4020-8988-6.
- ^ Moebs, William; Ling, Samuel J.; Sanny, Jeff (Sep 19, 2016). 11.4 Precession of a Gyroscope - University Physics Volume 1 | OpenStax. Houston, Texas. Retrieved 23 October 2020.
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: CS1 maint: location missing publisher (link) - ^ Moebs, William; Ling, Samuel J.; Sanny, Jeff (Sep 19, 2016). 11.4 Precession of a Gyroscope - University Physics Volume 1 | OpenStax. Houston, Texas. Retrieved 23 October 2020.
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: CS1 maint: location missing publisher (link) - ISBN 978-0-7503-0886-1.
- ^ Swerdlow, Noel (1991). On the cosmical mysteries of Mithras. Classical Philology, 86, (1991), 48–63. p. 59.
- ISBN 0-521-05801-5, p. 220.
- ISBN 978-0-521-53551-9.
- ^ Max Born (1924), Einstein's Theory of Relativity (The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and Earth.)
- ^ "An even larger value for a precession has been found, for a black hole in orbit around a much more massive black hole, amounting to 39 degrees each orbit". Archived from the original on 2018-08-07. Retrieved 2023-11-15.
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External links
- Media related to Precession at Wikimedia Commons
- Explanation and derivation of formula for precession of a top
- Precession and the Milankovich theory From Stargazers to Starships