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Tidal force

From Wikipedia, the free encyclopedia
Gravitational effect also known as the differential force and the perturbing force
Figure 1: Tidal interaction between thespiral galaxyNGC 169 and a smaller companion[1]

Thetidal force ortide-generating force is the difference ingravitational attraction between different points in agravitational field, causing bodies to be pulled unevenly and as a result are being stretched towards the attraction. It is thedifferential force of gravity, the net betweengravitational forces, thederivative ofgravitational potential, thegradient of gravitational fields. Therefore tidal forces are aresidual force, a secondary effect of gravity, highlighting its spatial elements, making the closer near-side more attracted than the more distant far-side.

This produces a range oftidal phenomena, such as ocean tides. Earth's tides are mainly produced by the relative close gravitational field of the Moon and to a lesser extent by the stronger, but further away gravitational field of the Sun. The ocean on the side of Earth facing the Moon is being pulled by the gravity of the Moon away fromEarth's crust, while on the other side of Earth there the crust is being pulled away from the ocean, resulting in Earth being stretched, bulging on both sides, and having oppositehigh-tides. Tidal forces viewed from Earth, that is from arotating reference frame, appear ascentripetal andcentrifugal forces, but are not caused by the rotation.[2]

Further tidal phenomena includesolid-earth tides,tidal locking, breaking apart of celestial bodies and formation ofring systems within theRoche limit, and in extreme cases,spaghettification of objects. Tidal forces have also been shown to be fundamentally related togravitational waves.[3]

Incelestial mechanics, the expressiontidal force can refer to a situation in which a body or material (for example, tidal water) is mainly under the gravitational influence of a second body (for example, the Earth), but is also perturbed by the gravitational effects of a third body (for example, the Moon). The perturbing force is sometimes in such cases called a tidal force[4] (for example, theperturbing force on the Moon): it is the difference between the force exerted by the third body on the second and the force exerted by the third body on the first.[5]

Explanation

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Figure 2: Shown in red, the Moon's gravityresidual field at the surface of the Earth is known (along with another and weaker differential effect due to the Sun) as thetide generating force. This is the primary mechanism driving tidal action, explaining two simultaneous tidal bulges. Earth's rotation accounts further for the occurrence of two high tides per day on the same location. In this figure, the Earth is the central black circle while the Moon is far off to the right. It shows both the tidal field (thick red arrows) and the gravity field (thin blue arrows) exerted on Earth's surface and center (label O) by the Moon (label S). Theoutward direction of the arrows on the right and left of the Earth indicates that where the Moon is atzenith or atnadir.

When a body (body 1) is acted on by the gravity of another body (body 2), the field can vary significantly on body 1 between the side of the body facing body 2 and the side facing away from body 2. Figure 2 shows the differential force of gravity on a spherical body (body 1) exerted by another body (body 2).

Thesetidal forces cause strains on both bodies and may distort them or even, in extreme cases, break one or the other apart.[6] TheRoche limit is the distance from a planet at which tidal effects would cause an object to disintegrate because the differential force of gravity from the planet overcomes the attraction of the parts of the object for one another.[7] These strains would not occur if the gravitational field were uniform, because a uniformfield only causes the entire body to accelerate together in the same direction and at the same rate.

Size and distance

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The relationship of an astronomical body's size, to its distance from another body, strongly influences the magnitude of tidal force.[8] The tidal force acting on an astronomical body, such as the Earth, is directly proportional to the diameter of the Earth and inversely proportional to the cube of the distance from another body producing a gravitational attraction, such as the Moon or the Sun. Tidal action on bath tubs, swimming pools, lakes, and other small bodies of water is negligible.[9]

Figure 3: Graph showing how gravitational attraction drops off with increasing distance from a body

Figure 3 is a graph showing how gravitational force declines with distance. In this graph, the attractive force decreases in proportion to the square of the distance (Y = 1/X2), while the slope (Y′ = −2/X3) is inversely proportional to the cube of the distance.

The tidal force corresponds to the difference in Y between two points on the graph, with one point on the near side of the body, and the other point on the far side. The tidal force becomes larger, when the two points are either farther apart, or when they are more to the left on the graph, meaning closer to the attracting body.

For example, even though the Sun has a stronger overall gravitational pull on Earth, the Moon creates a larger tidal bulge because the Moon is closer. This difference is due to the way gravity weakens with distance: the Moon's closer proximity creates a steeper decline in its gravitational pull as you move across Earth (compared to the Sun's very gradual decline from its vast distance). This steeper gradient in the Moon's pull results in a larger difference in force between the near and far sides of Earth, which is what creates the bigger tidal bulge.

Gravitational attraction is inversely proportional to the square of the distance from the source. The attraction will be stronger on the side of a body facing the source, and weaker on the side away from the source. The tidal force is proportional to the difference.[9]

Sun, Earth, and Moon

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The Sun is about 20 million times the Moon's mass, and acts on the Earth over a distance about 400 times larger than that of the Moon. Because of the cubic dependence on distance, this results in the solar tidal force on the Earth being about half that of the lunar tidal force.

Gravitational body causing tidal forceBody subjected to tidal forceTidal acceleration
BodyMass (m{\displaystyle m})BodyRadius (r{\displaystyle r})Distance (d{\displaystyle d})Gm 2rd3{\displaystyle Gm~{\frac {2r}{d^{3}}}}
Sun1.99×1030 kgEarth6.37×106 m1.50×1011 m5.05×10−7 m⋅s−2
Moon7.34×1022 kgEarth6.37×106 m3.84×108 m1.10×10−6 m⋅s−2
Earth5.97×1024 kgMoon1.74×106 m3.84×108 m2.44×10−5 m⋅s−2
G is thegravitational constant =6.674×10−11 m3⋅kg−1⋅s−2‍[10]

Effects

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Figure 4:Saturn's rings are inside the orbits of its principal moons. Tidal forces oppose gravitational coalescence of the material in the rings to form moons.[11]

In the case of an infinitesimally small elastic sphere, the effect of a tidal force is to distort the shape of the body without any change in volume. The sphere becomes anellipsoid with two bulges, pointing towards and away from the other body. Larger objects distort into anovoid, and are slightly compressed, which is what happens to the Earth's oceans under the action of the Moon. All parts of the Earth are subject to the Moon's gravitational forces, causing the water in the oceans to redistribute, forming bulges on the sides near the Moon and far from the Moon.[12]

When a body rotates while subject to tidal forces, internal friction results in the gradual dissipation of its rotational kinetic energy as heat. In the case for the Earth, and Earth's Moon, the loss of rotational kinetic energy results in a gain of about 2 milliseconds per century. If the body is close enough to its primary, this can result in a rotation which istidally locked to the orbital motion, as in the case of the Earth's moon.Tidal heating produces dramatic volcanic effects on Jupiter's moonIo.Stresses caused by tidal forces also cause a regular monthly pattern ofmoonquakes on Earth's Moon.[8]

Tidal forces contribute to ocean currents, which moderate global temperatures by transporting heat energy toward the poles. It has been suggested that variations in tidal forces correlate with cool periods in the global temperature record at 6- to 10-year intervals,[13] and thatharmonic beat variations in tidal forcing may contribute to millennial climate changes. No strong link to millennial climate changes has been found to date.[14]

Figure 5:Comet Shoemaker-Levy 9 in 1994 after breaking up under the influence ofJupiter's tidal forces during a previous pass in 1992.

Tidal effects become particularly pronounced near small bodies of high mass, such asneutron stars orblack holes, where they are responsible for the "spaghettification" of infalling matter. Tidal forces create the oceanictide ofEarth's oceans, where the attracting bodies are theMoon and, to a lesser extent, theSun. Tidal forces are also responsible fortidal locking,tidal acceleration, and tidal heating.Tides may also induce seismicity.

By generating conducting fluids within the interior of the Earth, tidal forces also affect theEarth's magnetic field.[15]

Figure 6: This simulation shows astar getting torn apart by the gravitational tides of asupermassive black hole.

Formulation

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Figure 7: Tidal force is responsible for the merge of galactic pairMRK 1034.[16]
Figure 8: Graphic of tidal forces. The top picture shows the gravity field of a body to the right (not shown); the lower shows their residual gravity once the field at the centre of the sphere is subtracted; this is the tidal force. For visualization purposes, the top arrows may be assumed as equal to 1 N, 2 N, and 3 N (from left to right); the resulting bottom arrows would equal, respectively, −1 N (negative, thus 180-degree rotated), 0 N (invisible), and 1 N. See Figure 2 for a more detailed version

For a given (externally generated) gravitational field, thetidal acceleration at a point with respect to a body is obtained byvector subtraction of the gravitational acceleration at the center of the body (due to the given externally generated field) from the gravitational acceleration (due to the same field) at the given point. Correspondingly, the termtidal force is used to describe the forces due to tidal acceleration. Note that for these purposes the only gravitational field considered is the external one; the gravitational field of the body (as shown in the graphic) is not relevant. (In other words, the comparison is with the conditions at the given point as they would be if there were no externally generated field acting unequally at the given point and at the center of the reference body. The externally generated field is usually that produced by a perturbing third body, often the Sun or the Moon in the frequent example-cases of points on or above the Earth's surface in a geocentric reference frame.)

Tidal acceleration does not require rotation or orbiting bodies; for example, the body may befreefalling in a straight line under the influence of a gravitational field while still being influenced by (changing) tidal acceleration.

ByNewton's law of universal gravitation and laws of motion, a body of massm at distanceR from the center of a sphere of massM feels a forceF→g{\textstyle {\vec {F}}_{g}},

F→g=−r^ G MmR2{\displaystyle {\vec {F}}_{g}=-{\hat {r}}~G~{\frac {Mm}{R^{2}}}}

equivalent to an accelerationa→g{\textstyle {\vec {a}}_{g}},

a→g=−r^ G MR2{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}}

wherer^{\textstyle {\hat {r}}} is aunit vector pointing from the bodyM to the bodym (here, acceleration fromm towardsM has negative sign).

Consider now the acceleration due to the sphere of massM experienced by a particle in the vicinity of the body of massm. WithR as the distance from the center ofM to the center ofm, let ∆r be the (relatively small) distance of the particle from the center of the body of massm. For simplicity, distances are first considered only in the direction pointing towards or away from the sphere of massM. If the body of massm is itself a sphere of radius ∆r, then the new particle considered may be located on its surface, at a distance (R ±∆r) from the centre of the sphere of massM, and∆r may be taken as positive where the particle's distance fromM is greater thanR. Leaving aside whatever gravitational acceleration may be experienced by the particle towardsm on account ofm's own mass, we have the acceleration on the particle due to gravitational force towardsM as:

a→g=−r^ G M(R±Δr)2{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{(R\pm \Delta r)^{2}}}}

Pulling out theR2 term from the denominator gives:

a→g=−r^ G MR2 1(1±ΔrR)2{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}~{\frac {1}{\left(1\pm {\frac {\Delta r}{R}}\right)^{2}}}}

TheMaclaurin series of1/(1±x)2{\textstyle 1/(1\pm x)^{2}} is1∓2x+3x2∓⋯{\textstyle 1\mp 2x+3x^{2}\mp \cdots } which gives a series expansion of:

a→g=−r^ G MR2±r^ G 2MR2 ΔrR+⋯{\displaystyle {\vec {a}}_{g}=-{\hat {r}}~G~{\frac {M}{R^{2}}}\pm {\hat {r}}~G~{\frac {2M}{R^{2}}}~{\frac {\Delta r}{R}}+\cdots }

The first term is the gravitational acceleration due toM at the center of the reference bodym{\textstyle m}, i.e., at the point whereΔr{\textstyle \Delta r} is zero. This term does not affect the observed acceleration of particles on the surface ofm because with respect toM,m (and everything on its surface) is in free fall. When the force on the far particle is subtracted from the force on the near particle, this first term cancels, as do all other even-order terms. The remaining (residual) terms represent the difference mentioned above and are tidal force (acceleration) terms. When ∆r is small compared toR, the terms after the first residual term are very small and can be neglected, giving the approximate tidal accelerationa→t,axial{\textstyle {\vec {a}}_{t,{\text{axial}}}} for the distances ∆r considered, along the axis joining the centers ofm andM:

a→t,axial≈±r^ 2Δr G MR3{\displaystyle {\vec {a}}_{t,{\text{axial}}}\approx \pm {\hat {r}}~2\Delta r~G~{\frac {M}{R^{3}}}}

When calculated in this way for the case where ∆r is a distance along the axis joining the centers ofm andM,a→t{\textstyle {\vec {a}}_{t}} is directed outwards from to the center ofm (where ∆r is zero).

Tidal accelerations can also be calculated away from the axis connecting the bodiesm andM, requiring avector calculation. In the plane perpendicular to that axis, the tidal acceleration is directed inwards (towards the center where ∆r is zero), and its magnitude is12|a→t,axial|{\textstyle {\frac {1}{2}}\left|{\vec {a}}_{t,{\text{axial}}}\right|} in linear approximation as in Figure 2.

The tidal accelerations at the surfaces of planets in the Solar System are generally very small. For example, the lunar tidal acceleration at the Earth's surface along the Moon–Earth axis is about1.1×10−7 g, while the solar tidal acceleration at the Earth's surface along the Sun–Earth axis is about0.52×10−7 g, whereg is thegravitational acceleration at the Earth's surface. Hence the tide-raising force (acceleration) due to the Sun is about 45% of that due to the Moon.[17] The solar tidal acceleration at the Earth's surface was first given by Newton in thePrincipia.[18]

See also

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References

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  1. ^"Hubble Views a Cosmic Interaction".nasa.gov. NASA. February 11, 2022. Retrieved2022-07-09.
  2. ^Matsuda, Takuya; Isaka, Hiromu; Boffin, Henri M. J. (2015),Confusion around the tidal force and the centrifugal force,arXiv:1506.04085, retrieved2025-02-14
  3. ^arXiv, Emerging Technology from the (2019-12-14)."Tidal forces carry the mathematical signature of gravitational waves".MIT Technology Review. Retrieved2023-11-12.
  4. ^"On the tidal force", I. N. Avsiuk, in "Soviet Astronomy Letters", vol. 3 (1977), pp. 96–99.
  5. ^See p. 509 in"Astronomy: a physical perspective", M. L. Kutner (2003).
  6. ^R Penrose (1999).The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics.Oxford University Press. p. 264.ISBN 978-0-19-286198-6.tidal force.
  7. ^Thérèse Encrenaz; J -P Bibring; M Blanc (2003).The Solar System. Springer. p. 16.ISBN 978-3-540-00241-3.
  8. ^ab"The Tidal Force | Neil deGrasse Tyson".www.haydenplanetarium.org. Retrieved2016-10-10.
  9. ^abSawicki, Mikolaj (1999). "Myths about gravity and tides".The Physics Teacher.37 (7):438–441.Bibcode:1999PhTea..37..438S.CiteSeerX 10.1.1.695.8981.doi:10.1119/1.880345.ISSN 0031-921X.
  10. ^"2022 CODATA Value: Newtonian constant of gravitation".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
  11. ^R. S. MacKay; J. D. Meiss (1987).Hamiltonian Dynamical Systems: A Reprint Selection.CRC Press. p. 36.ISBN 978-0-85274-205-1.
  12. ^Rollin A Harris (1920).The Encyclopedia Americana: A Library of Universal Knowledge. Vol. 26. Encyclopedia Americana Corp. pp. 611–617.
  13. ^Keeling, C. D.; Whorf, T. P. (5 August 1997)."Possible forcing of global temperature by the oceanic tides".Proceedings of the National Academy of Sciences.94 (16):8321–8328.Bibcode:1997PNAS...94.8321K.doi:10.1073/pnas.94.16.8321.PMC 33744.PMID 11607740.
  14. ^Munk, Walter; Dzieciuch, Matthew; Jayne, Steven (February 2002)."Millennial Climate Variability: Is There a Tidal Connection?".Journal of Climate.15 (4):370–385.Bibcode:2002JCli...15..370M.doi:10.1175/1520-0442(2002)015<0370:MCVITA>2.0.CO;2.
  15. ^"Hungry for Power in Space".New Scientist.123: 52. 23 September 1989. Retrieved14 March 2016.
  16. ^"Inseparable galactic twins".ESA/Hubble Picture of the Week. Retrieved12 July 2013.
  17. ^The Admiralty (1987).Admiralty manual of navigation. Vol. 1.The Stationery Office. p. 277.ISBN 978-0-11-772880-6.,Chapter 11, p. 277
  18. ^Newton, Isaac (1729).The mathematical principles of natural philosophy. Vol. 2. p. 307.ISBN 978-0-11-772880-6.{{cite book}}:ISBN / Date incompatibility (help),Book 3, Proposition 36, Page 307 Newton put the force to depress the sea at places 90 degrees distant from the Sun at "1 to 38604600" (in terms ofg), and wrote that the force to raise the sea along the Sun-Earth axis is "twice as great" (i.e., 2 to 38604600) which comes to about 0.52 × 10−7g as expressed in the text.

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