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{{Short description|A gravitational effect also known as the differential force and the perturbing force}}
[[File:Arp 282.png|right|300px|thumb|upright=1.5|Tidal interaction between the [[barred spiral galaxy]] [[NGC 169]] and a smaller companion<ref>
{{cite web
 | title=Hubble Views a Cosmic Interaction
 | date=February 11, 2022 | publisher=NASA | website=nasa.gov
 | url=https://www.nasa.gov/image-feature/goddard/2022/hubble-views-a-cosmic-interaction
 | access-date=2022-07-09
}}</ref>]]

The '''tidal force''' or '''tide-generating force''' is a [[gravitational]] effect that stretches a body along the line towards and away from the [[center of mass]] of another body due to spatial variations in strength in [[gravitational field]] from the other body. It is responsible for the [[tide]]s and related phenomena, including [[solid-earth tide]]s, [[tidal locking]], breaking apart of celestial bodies and formation of [[ring system]]s within the [[Roche limit]], and in extreme cases, [[spaghettification]] of objects. It arises because the gravitational field exerted on one body by another is not constant across its parts: the nearer side is attracted more strongly than the farther side. The difference is positive in the near side and negative in the far side, which causes a body to get stretched. Thus, the tidal force is also known as the differential force, residual force, or secondary effect of the gravitational field.

In [[celestial mechanics]], the expression ''tidal 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<ref>[http://adsabs.harvard.edu/full/1977SvAL....3...96A "On the tidal force"], I. N. Avsiuk, in "Soviet Astronomy Letters", vol. 3 (1977), pp. 96–99.</ref> (for example, the [[Lunar theory#Newton|perturbing 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.<ref>See p. 509 in [https://archive.org/details/astronomyphysica00kutn/page/509 <!-- quote="tidal force" perturb. --> "Astronomy: a physical perspective"], M. L. Kutner (2003).</ref>

Tidal forces have also been shown to be fundamentally related to [[gravitational wave]]s.<ref name="arXiv 2019 p440">{{cite web | last=arXiv | first=Emerging Technology from the | title=Tidal forces carry the mathematical signature of gravitational waves | website=MIT Technology Review | date=2019-12-14 | url=https://www.technologyreview.com/2019/12/14/131574/tidal-forces-carry-the-mathematical-signature-of-gravitational-waves/ | access-date=2023-11-12}}</ref>

== Explanation ==
[[File:Tidal field and gravity field.svg |thumb |upright=1.3 |Figure 4: The Moon's gravity ''residual'' field at the surface of the Earth is known (along with another and weaker differential effect due to the Sun) as the ''tide 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).  The ''outward'' direction of the arrows on the right and left of the Earth indicates that where the Moon is at [[zenith]] or at [[nadir]].]]

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 4 shows the differential force of gravity on a spherical body (body 1) exerted by another body (body 2). These so-called ''tidal forces'' cause strains on both bodies and may distort them or even, in extreme cases, break one or the other apart.<ref name=Penrose>
{{cite book
 | title=The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics
 | author=R Penrose
 | page=[https://archive.org/details/emperorsnewmindc1999penr/page/264 264]
 | url=https://archive.org/details/emperorsnewmindc1999penr
 | url-access=registration
 | quote=tidal force.
 | isbn=978-0-19-286198-6
 | date=1999
 | publisher=[[Oxford University Press]]
}}</ref> The [[Roche 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.<ref name=Blanc>
{{cite book
 | title=The Solar System
 | page=16
 | url=https://books.google.com/books?id=Je61Y7UbqWgC&q=tidal+force&pg=PA16
 | author=Thérèse Encrenaz|author-link=Thérèse Encrenaz
 | author2=J -P Bibring
 | author3=M Blanc
 | isbn=978-3-540-00241-3
 | date=2003
 | publisher=Springer
}}</ref> These strains would not occur if the gravitational field were uniform, because a uniform [[field (physics)|field]] only causes the entire body to accelerate together in the same direction and at the same rate.

== Size and distance ==
The relationship of an astronomical body's size, to its distance from another body, strongly influences the magnitude of tidal force.<ref name="Tyson"/>  The tidal force acting on an astronomical body, such as the Earth, is directly proportional to the diameter of that astronomical body 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.<ref name="Sawicki1999"/>

[[File:Inverse x squaired.png|thumb|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 ({{nowrap|1=''Y'' = 1/''X''<sup>2</sup>}}), while the slope ({{nowrap|1=''Y''{{′}} = −2/''X''<sup>3</sup>}}) 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, the Moon produces a greater tidal force on the Earth than the Sun, even though the Sun exerts a greater gravitational attraction on the Earth than the Moon, because the gradient is less.

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.<ref name="Sawicki1999">
{{cite journal
 |last1=Sawicki |first1=Mikolaj
 |title=Myths about gravity and tides
 |journal=[[The Physics Teacher]]
 |volume=37 |issue=7
 |year=1999
 |pages=438–441
 |issn=0031-921X |doi=10.1119/1.880345 |citeseerx=10.1.1.695.8981 |bibcode=1999PhTea..37..438S
}}</ref>

=== Sun, Earth, and Moon ===
The Earth is 81 times more massive than the Moon but has roughly 4 times its radius. As a result, at the same distance, the tidal force of the Earth at the surface of the Moon is about 20 times stronger than that of the Moon at the Earth's surface.<ref>
{{cite book
 |title=Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity
 |edition=illustrated
 |first1=Bernard |last1=Schutz
 |publisher=Cambridge University Press
 |year=2003
 |isbn=978-0-521-45506-0
 |page=45
 |url=https://books.google.com/books?id=P_T0xxhDcsIC
}} [https://books.google.com/books?id=P_T0xxhDcsIC&pg=PA45 Extract of page 45]</ref>

{| class="wikitable" 
! colspan="2"|Gravitational body causing tidal force !!colspan="3"| Body subjected to tidal force   !! Tidal acceleration
|-
! Body || Mass (<math>m</math>) || Body || Radius (<math>r</math>) || Distance (<math>d</math>) || <math> Gm ~ \frac{2r}{d^3} </math>
|-
| style="text-align: center" | '''Sun''' || style="text-align: center" | {{val|1.99|e=30|u=kg}} || '''Earth''' || style="text-align: center" | {{val|6.37|e=6|u=m}} || style="text-align: center" | {{val|1.50|e=11|u=m}} || style="text-align: center" | {{val|5.05|e=-7|u=m.s-2}}
|-
| style="text-align: center" | '''Moon''' || style="text-align: center" | {{val|7.34|e=22|u=kg}} || '''Earth''' || style="text-align: center" | {{val|6.37|e=6|u=m}} || style="text-align: center" | {{val|3.84|e=8|u=m}} || style="text-align: center" | {{val|1.10|e=-6|u=m.s-2}}
|-
| style="text-align: center" | '''Earth''' || style="text-align: center" | {{val|5.97|e=24|u=kg}} || '''Moon''' || style="text-align: center" | {{val|1.74|e=6|u=m}} || style="text-align: center" | {{val|3.84|e=8|u=m}} || style="text-align: center" | {{val|2.44|e=-5|u=m.s-2}}
|-
| colspan=6 style="background: #ffffff; font-size: small; font-weight: normal;" | ''G'' is the [[gravitational constant]] = {{physconst|G|round=3}}
|}

== Effects ==
[[File:Saturn PIA06077.jpg|thumb|left|Figure 5: [[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.<ref name=MacKay>{{cite book |author=R. S. MacKay |author2=J. D. Meiss |title=Hamiltonian Dynamical Systems: A Reprint Selection |page=36 |url=https://books.google.com/books?id=uTeqNsyj86QC&q=tidal+force&pg=PA36 |isbn=978-0-85274-205-1 |date=1987 |publisher=[[CRC Press]]}}</ref>]]

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 an [[ellipsoid]] with two bulges, pointing towards and away from the other body. Larger objects distort into an [[ovoid]], and are slightly compressed, which is what happens to the Earth's oceans under the action of the Moon. The Earth and Moon orbit about their common center of mass or [[barycenter]], and their gravitational attraction provides the [[centripetal force]] necessary to maintain this motion. To an observer on the Earth, very close to this barycenter, the situation is one of the Earth as body 1 acted upon by the gravity of the Moon as body 2. 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.<ref name=Americana>
{{cite book
 | title=The Encyclopedia Americana: A Library of Universal Knowledge
 | author=Rollin A Harris
 | pages=611–617
 | url=https://books.google.com/books?id=r8BPAAAAMAAJ&q=tidal+force&pg=PA612
 | publisher=Encyclopedia Americana Corp.
 | date=1920
 | volume=26
}}</ref>

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 is [[Tidal locking|tidally locked]] to the orbital motion, as in the case of the Earth's moon. [[Tidal heating]] produces dramatic volcanic effects on Jupiter's moon [[Io (moon)|Io]]. {{anchor|Stress}}[[Stress (mechanics)|Stresses]] caused by tidal forces also cause a regular monthly pattern of [[moonquake]]s on Earth's Moon.<ref name="Tyson">{{Cite web|url=http://www.haydenplanetarium.org/tyson/read/1995/11/01/the-tidal-force|title=The Tidal Force {{!}} Neil deGrasse Tyson|website=www.haydenplanetarium.org|access-date=2016-10-10}}</ref>

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,<ref>{{cite journal |last1=Keeling |first1=C. D. |last2=Whorf |first2=T. P. |title=Possible forcing of global temperature by the oceanic tides |journal=Proceedings of the National Academy of Sciences |date=5 August 1997 |volume=94 |issue=16 |pages=8321–8328 |doi=10.1073/pnas.94.16.8321 |pmid=11607740 |bibcode=1997PNAS...94.8321K |pmc=33744 |doi-access=free }}</ref> and that [[beat (acoustics)|harmonic beat]] variations in tidal forcing may contribute to millennial climate changes. No strong link to millennial climate changes has been found to date.<ref>{{cite journal |last1=Munk |first1=Walter |last2=Dzieciuch |first2=Matthew |last3=Jayne |first3=Steven |title=Millennial Climate Variability: Is There a Tidal Connection? |journal=Journal of Climate |date=February 2002 |volume=15 |issue=4 |pages=370–385 |doi=10.1175/1520-0442(2002)015<0370:MCVITA>2.0.CO;2 |bibcode=2002JCli...15..370M |doi-access=free }}</ref>
[[File:Shoemaker-levy-tidal-forces.jpg|thumb|upright=1.25|Figure 1: [[Comet Shoemaker-Levy 9]] in 1994 after breaking up under the influence of [[Jupiter]]'s tidal forces during a previous pass in 1992.]]

Tidal effects become particularly pronounced near small bodies of high mass, such as [[neutron star]]s or [[black hole]]s, where they are responsible for the "[[spaghettification]]" of infalling matter. Tidal forces create the oceanic [[tide]] of [[Earth]]'s oceans, where the attracting bodies are the [[Moon]] and, to a lesser extent, the [[Sun]]. Tidal forces are also responsible for [[tidal locking]], [[tidal acceleration]], and tidal heating. [[Tidal triggering of earthquakes|Tides may also induce seismicity]].

By generating conducting fluids within the interior of the Earth, tidal forces also affect the [[Earth's magnetic field]].<ref>{{cite journal | date=23 September 1989 | title=Hungry for Power in Space | url=https://books.google.com/books?id=FPEUAQAAMAAJ | journal=New Scientist | volume=123 | pages=52 | access-date=14 March 2016}}</ref>

[[File:Supermassive black hole rips star apart (simulation).webm|thumb|Figure 2: This simulation shows a [[star]] getting torn apart by the gravitational tides of a [[supermassive black hole]].]]

== Formulation ==
[[File:Inseparable galactic twins.jpg|thumb|Figure 6: Tidal force is responsible for the merge of galactic pair [[MRK 1034]].<ref>{{cite news|title=Inseparable galactic twins|url=http://www.spacetelescope.org/images/potw1325a/|access-date=12 July 2013|newspaper=ESA/Hubble Picture of the Week}}</ref> ]]
[[File:Tidal-forces.svg|thumb|right|Figure 7: 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 4 for a more detailed version]]

For a given (externally generated) gravitational field, the '''tidal acceleration''' at a point with respect to a body is obtained by [[Euclidean vector#Addition and subtraction|vector 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 term ''tidal 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 be [[freefall]]ing in a straight line under the influence of a gravitational field while still being influenced by (changing) tidal acceleration.

By [[Newton's law of universal gravitation]] and laws of motion, a body of mass ''m'' at distance ''R'' from the center of a sphere of mass ''M'' feels a force <math>\vec{F}_g</math>,
: <math>\vec{F}_g = - \hat{r} ~ G ~ \frac{M m}{R^2}</math>
equivalent to an acceleration <math>\vec{a}_g</math>,
: <math>\vec{a}_g = - \hat{r} ~ G ~ \frac{M}{R^2}</math>
where <math>\hat{r}</math> is a [[unit vector]] pointing from the body ''M'' to the body ''m'' (here, acceleration from ''m'' towards ''M'' has negative sign).

Consider now the acceleration due to the sphere of mass ''M'' experienced by a particle in the vicinity of the body of mass ''m''. With ''R'' as the distance from the center of ''M'' to the center of ''m'', let ∆''r'' be the (relatively small) distance of the particle from the center of the body of mass ''m''. For simplicity, distances are first considered only in the direction pointing towards or away from the sphere of mass ''M''. If the body of mass ''m'' 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 mass ''M'', and ''∆r'' may be taken as positive where the particle's distance from ''M'' is greater than ''R''. Leaving aside whatever gravitational acceleration may be experienced by the particle towards ''m'' on account of ''m''{{'}}s own mass, we have the acceleration on the particle due to gravitational force towards ''M'' as:
: <math>\vec{a}_g = - \hat{r} ~ G ~ \frac{M}{(R \pm \Delta r)^2}</math>

Pulling out the ''R''<sup>2</sup> term from the denominator gives:
: <math>\vec{a}_g = -\hat{r} ~ G ~ \frac{M}{R^2} ~ \frac{1}{\left(1 \pm \frac{\Delta r}{R}\right)^2}</math>

The [[Maclaurin series]] of <math>1/(1 \pm x)^2</math> is <math>1 \mp 2x + 3x^2 \mp \cdots</math> which gives a series expansion of:
: <math>\vec{a}_g = - \hat{r} ~ G ~ \frac{M}{R^2} \pm \hat{r} ~ G ~ \frac{2 M }{R^2} ~ \frac{\Delta r}{R} + \cdots </math>

The first term is the gravitational acceleration due to ''M'' at the center of the reference body <math>m</math>, i.e., at the point where <math>\Delta r</math> is zero. This term does not affect the observed acceleration of particles on the surface of ''m'' because with respect to ''M'', ''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 to ''R'', the terms after the first residual term are very small and can be neglected, giving the approximate tidal acceleration <math>\vec{a}_{t,\text{axial}}</math> for the distances ∆''r'' considered, along the axis joining the centers of ''m'' and ''M'':
: <math>\vec{a}_{t,\text{axial}} \approx \pm \hat{r} ~ 2 \Delta r ~ G ~ \frac{M}{R^3} </math>
When calculated in this way for the case where ∆''r'' is a distance along the axis joining the centers of ''m'' and ''M'', <math>\vec{a}_t</math> is directed outwards from to the center of ''m'' (where ∆''r'' is zero).

Tidal accelerations can also be calculated away from the axis connecting the bodies ''m'' and ''M'', requiring a [[Euclidean vector|vector]] calculation. In the plane perpendicular to that axis, the tidal acceleration is directed inwards (towards the center where ∆''r'' is zero), and its magnitude is <math display="inline">\frac{1}{2}\left|\vec{a}_{t,\text{axial}} \right|</math> in linear approximation as in Figure 4.

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 about {{val|1.1|e=-7|u=''g''}}, while the solar tidal acceleration at the Earth's surface along the Sun–Earth axis is about {{val|0.52|e=-7|u=''g''}}, where ''g'' is the [[standard gravity|gravitational 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.<ref>
{{cite book
 | title=Admiralty manual of navigation
 | volume=1
 | author=The Admiralty
 | publisher=[[The Stationery Office]]
 | date=1987
 | isbn=978-0-11-772880-6
 | page=277
 | url=https://books.google.com/books?id=GCgXCxG4VLcC
}}, [https://books.google.com/books?id=GCgXCxG4VLcC&pg=PA277 Chapter 11, p. 277]
</ref> The solar tidal acceleration at the Earth's surface was first given by Newton in the ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]''.<ref>
{{cite book
 | title=The mathematical principles of natural philosophy
 | volume=2
 | first=Isaac | last=Newton
 | date=1729
 | isbn=978-0-11-772880-6
 | page=307
 | url=https://books.google.com/books?id=6EqxPav3vIsC
}}, [https://archive.org/details/bub_gb_6EqxPav3vIsC/page/307 <!-- pg=307 --> 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 of ''g''), 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<sup>−7</sup> ''g'' as expressed in the text.
</ref>

== See also ==
* [[Amphidromic point]]
* [[Disrupted planet]]
* [[Galactic tide]]
* [[Tidal resonance]]
* [[Tidal stripping]]
* [[Tidal tensor]]
* [[Spacetime curvature]]

== References ==
{{Reflist|30em}}

== External links ==
* [https://geomatix.net/tides/analyzer.htm Analysis and Prediction of Tides: GeoTide]
* [http://burro.astr.cwru.edu/Academics/Astr221/Gravity/tides.html Gravitational Tides] by J. Christopher Mihos of [[Case Western Reserve University]]
* [http://www.astronomycast.com/solar-system/episode-47-tidal-forces/ Audio: Cain/Gay – Astronomy Cast] Tidal Forces – July 2007.
* {{cite web|last=Gray|first=Meghan|title=Tidal Forces|url=http://www.sixtysymbols.com/videos/tides.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|author2=Merrifield, Michael }}
* {{cite web| url=http://astro-gr.org/stellar-collisions-tidal-disruption-star-massive-black-hole/| title=Stellar collisions: Tidal disruption of a star by a massive black hole | author=Pau Amaro Seoane | access-date=2018-12-28}}
* [https://web.archive.org/web/20130921060553/http://www.jal.cc.il.us/~mikolajsawicki/tides_new2.pdf Myths about Gravity and Tides] by Mikolaj Sawicki of John A. Logan College and the University of Colorado.
* [https://web.archive.org/web/20160508041318/http://www.lhup.edu/~dsimanek/scenario/tides.htm Tidal Misconceptions] by Donald E. Simanek

{{physical oceanography}}
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[[Category:Tidal forces| ]]
[[Category:Tides]]
[[Category:Gravity]]
[[Category:Force]]
[[Category:Effects of gravitation]]
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