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Precession

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Periodic change in the direction of a rotation axis

For other uses, seePrecession (disambiguation).

Precession of agyroscope^{[clarification needed]}

Rotation
  Precession
  Nutation
inobliquity of a planet

Precession is a change in theorientation of the rotational axis of arotating body. In an appropriatereference frame it can be defined as a change in the firstEuler angle, whereas the third Euler angle defines therotation 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 callednutation. Inphysics, 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 theEarth, known as theprecession of the equinoxes.

Torque-free or torque neglected

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Torque-free precession implies that no external moment (torque) is applied to the body. In torque-free precession, theangular momentum is a constant, but theangular velocity vector changes orientation with time. What makes this possible is a time-varyingmoment of inertia, or more precisely, a time-varyinginertia matrix. The inertia matrix is composed of the moments of inertia of a body calculated with respect to separatecoordinate 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 thecomponent 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 follows:^[1] ${\boldsymbol {\omega }}_{\mathrm {p} }={\frac {{\boldsymbol {I}}_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }}{{\boldsymbol {I}}_{\mathrm {p} }\cos({\boldsymbol {\alpha }})}}$ whereω_p is the precession rate,ω_s is the spin rate about the axis of symmetry,I_s is the moment of inertia about the axis of symmetry,I_p is moment of inertia about either of the other two equal perpendicular principal axes, andα is the angle between the moment of inertia direction and the symmetry axis.^[2]

When an object is not perfectlyrigid, 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 matrixR that transforms internal to external coordinates, may be numerically simulated. Given the object's fixed internalmoment of inertia tensorI₀ and fixed external angular momentumL, the instantaneous angular velocity is ${\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)={\boldsymbol {R}}{\boldsymbol {I}}_{0}^{-1}{\boldsymbol {R}}^{T}{\boldsymbol {L}}$ Precession occurs by repeatedly recalculatingω and applying a smallrotation vectorω dt for the short timedt; e.g.: ${\boldsymbol {R}}_{\text{new}}=\exp \left(\left[{\boldsymbol {\omega }}\left({\boldsymbol {R}}_{\text{old}}\right)\right]_{\times }dt\right){\boldsymbol {R}}_{\text{old}}$ for theskew-symmetric matrix[ω]_×. The errors induced by finite time steps tend to increase the rotational kinetic energy: $E\left({\boldsymbol {R}}\right)={\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\cdot {\frac {\boldsymbol {L}}{2}}$ this unphysical tendency can be counteracted by repeatedly applying a small rotation vectorv perpendicular to bothω andL, noting that $E\left(\exp \left(\left[{\boldsymbol {v}}\right]_{\times }\right){\boldsymbol {R}}\right)\approx E\left({\boldsymbol {R}}\right)+\left({\boldsymbol {\omega }}\left({\boldsymbol {R}}\right)\times {\boldsymbol {L}}\right)\cdot {\boldsymbol {v}}$

Torque-induced

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Torque-induced precession (gyroscopic precession) is the phenomenon in which theaxis of a spinning object (e.g., agyroscope) describes acone in space when an externaltorque is applied to it. The phenomenon is commonly seen in aspinning toy top, but all rotating objects can undergo precession. If thespeed of the rotation and themagnitude of the external torque are constant, the spin axis will move atright angles to thedirection that would intuitively result from the external torque. In the case of a toy top, its weight is acting downwards from itscenter of mass and thenormal force (reaction) of the ground is pushing up on it at the point of contact with the support. These two opposite forces produce a torque which causes the top to precess.

The response of a rotating system to an applied torque. When the device swivels, and some roll is added, the wheel tends to pitch.

The device depicted on the right isgimbal 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 calledspinning, and rotation around the gimbal axis will be calledpitching. Rotation around the vertical pivot axis is calledrotation.

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 atorque around the gimbal axis.

In the picture, a section of the wheel has been nameddm₁. At the depicted moment in time, sectiondm₁ is at theperimeter of the rotating motion around the (vertical) pivot axis. Sectiondm₁, therefore, has a lot of angular rotatingvelocity with respect to the rotation around the pivot axis, and asdm₁ is forced closer to the pivot axis of the rotation (by the wheel spinning further), because of theCoriolis effect, with respect to the vertical pivot axis,dm₁ tends to move in the direction of the top-left arrow in the diagram (shown at 45°) in the direction of rotation around the pivot axis.^[4] Sectiondm₂ of the wheel is moving away from the pivot axis, and so a force (again, a Coriolis force) acts in the same direction as in the case ofdm₁. Note that both arrows point in the same direction.

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 onbicycle performance at high speed. Precession is also the mechanism behindgyrocompasses.

Classical (Newtonian)

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Thetorque caused by the normal force –F_g and the weight of the top causes a change in theangular momentumL in the direction of that torque. This causes the top to precess.

Precession is the change ofangular velocity andangular momentum produced by a torque. The general equation that relates the torque to the rate of change of angular momentum is: ${\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}$ where ${\boldsymbol {\tau }}$ and $\mathbf {L}$ are the torque and angular momentum vectors respectively.

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]

{\boldsymbol {\omega }}_{\mathrm {p} }={\frac {\ mgr}{I_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }}}={\frac {\tau }{I_{\mathrm {s} }{\boldsymbol {\omega }}_{\mathrm {s} }\sin(\theta )}}

whereI_s is themoment of inertia,ω_s 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 andr is the distance between the center of mass and the pivot. The torque vector originates at the center of mass. Usingω =⁠2π/T⁠, we find that theperiod of precession is given by:^[6] $T_{\mathrm {p} }={\frac {4\pi ^{2}I_{\mathrm {s} }}{\ mgrT_{\mathrm {s} }}}={\frac {4\pi ^{2}I_{\mathrm {s} }\sin(\theta )}{\ \tau T_{\mathrm {s} }}}$

WhereI_s is themoment of inertia,T_s is the period of spin about the spin axis, andτ is thetorque. In general, the problem is more complicated than this, however.

Relativistic (Einsteinian)

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The special and general theories ofrelativity 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.

TheSchwarzschild geodesics (sometimes Schwarzschild precession) is used in the prediction of theanomalous perihelion precession of the planets, most notably for the accurate prediction of theapsidal precession of Mercury.

Astronomy

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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 thetilt of Earth's axis to its orbit, and theeccentricity of its orbit over tens of thousands of years are all important parts of the astronomical theory ofice ages.(SeeMilankovitch cycles.)

Axial precession (precession of the equinoxes)

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Main article:Axial precession

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 theprecession of the equinoxes,lunisolar precession, orprecession 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 bothequatorial coordinates andecliptic longitude. Over this cycle, Earth's north axial pole moves from where it is now, within 1° ofPolaris, in a circle around theecliptic pole, with an angular radius of about 23.5°.

Theancient Greek astronomer Hipparchus (c. 190–120 BC) is generally accepted to be the earliest known astronomer to recognize and assess the precession of the equinoxes at about 1° per century (which is not far from the actual value for antiquity, 1.38°),^[7] although there is some minor dispute about whether he was.^[8] Inancient China, theJin-dynasty scholar-officialYu Xi (fl. 307–345 AD) made a similar discovery centuries later, noting that the position of the Sun during the winter solstice had drifted roughly one degree over the course of fifty years relative to the position of the stars.^[9] The precession of Earth's axis was later explained byNewtonian physics. Being anoblate spheroid, Earth has a non-spherical shape, bulging outward at the equator. The gravitationaltidal forces of theMoon andSun apply torque to the equator, attempting to pull theequatorial bulge into the plane of theecliptic, but instead causing it to precess. The torque exerted by the planets, particularlyJupiter, also plays a role.^[10]

Small white disks representing the northern stars on a black background, overlaid by a circle showing the position of the north pole over time

Precessional movement of the axis (left), precession of the equinox in relation to the distant stars (middle), and the path of the north celestial pole among the stars due to the precession. Vega is the bright star near the bottom (right).

Apsidal precession

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Main article:Apsidal precession

Theorbits of planets around theSun 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 orapsidal 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 planetMercury and that predicted byclassical mechanics were prominent among the forms of experimental evidence leading to the acceptance ofEinstein'sTheory of Relativity (in particular, hisGeneral Theory of Relativity), which accurately predicted the anomalies.^[11]^[12] Deviating from Newton's law, Einstein's theory of gravitation predicts an extra term of⁠A/r⁴⁠, which accurately gives the observed excess turning rate of 43arcseconds every 100 years.

Nodal precession

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Main article:Nodal precession

Orbital nodes alsoprecess over time.

For the precession of the Moon's orbit, seelunar precession.

References

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^Schaub, Hanspeter (2003),Analytical Mechanics of Space Systems, AIAA, pp. 149–150,ISBN 9781600860270
^Boal, David (2001)."Lecture 26 – Torque-free rotation – body-fixed axes"(PDF). Retrieved2008-09-17.
^Sharma, Ishan; Burns, Joseph A.; Hui, C.-H. (2005)."Nutational damping times in solids of revolution".Monthly Notices of the Royal Astronomical Society.359 (1): 79.Bibcode:2005MNRAS.359...79S.doi:10.1111/j.1365-2966.2005.08864.x.
^Teodorescu, Petre P (2002).Mechanical Systems, Classical Models: Volume II: Mechanics of Discrete and Continuous Systems. Springer Science & Business Media. p. 420.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. Retrieved23 October 2020.{{cite book}}: 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. Retrieved23 October 2020.{{cite book}}: CS1 maint: location missing publisher (link)
^Barbieri, Cesare (2007).Fundamentals of Astronomy. New York: Taylor and Francis Group. p. 71.ISBN 978-0-7503-0886-1.
^Swerdlow, Noel (1991).On the cosmical mysteries of Mithras. Classical Philology, 86, (1991), 48–63. p. 59.
^Sun, Kwok. (2017).Our Place in the Universe: Understanding Fundamental Astronomy from Ancient Discoveries, second edition. Cham, Switzerland: Springer.ISBN 978-3-319-54171-6, p. 120; see also Needham, Joseph; Wang, Ling. (1995) [1959].Science and Civilization in China: Mathematics and the Sciences of the Heavens and the Earth, vol. 3, reprint edition. Cambridge: Cambridge University Press.ISBN 0-521-05801-5, p. 220.
^Bradt, Hale (2007).Astronomy Methods.Cambridge University Press. p. 66.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". 18 March 2008. Archived from the original on 2018-08-07. Retrieved2023-11-15.{{cite web}}: CS1 maint: bot: original URL status unknown (link)