The tidal force is a gravitational effect that stretches a body along the line towards the

^{2} term from the denominator gives:
: $\backslash vec\_g\; =\; -\backslash hat\; ~\; G\; ~\; \backslash frac\; ~\; \backslash frac$
The

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^{−7} ''g'' as expressed in the text.

Gravitational Tides

by J. Christopher Mihos of

Audio: Cain/Gay – Astronomy Cast

Tidal Forces – July 2007. * *

Myths about Gravity and Tides

by Mikolaj Sawicki of John A. Logan College and the University of Colorado.

by Donald E. Simanek {{Authority control Tides Gravity Force Effects of gravitation Concepts in astronomy

center of mass
In physics, the center of mass of a distribution of mass
Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...

of another body due to a gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

(difference in strength) in gravitational field
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

from the other body; it is responsible for diverse phenomena, including tide
Tides are the rise and fall of sea level
Mean sea level (MSL) (often shortened to sea level) is an average
In colloquial, ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of ...

s, tidal locking
and Charon are tidally locked to each other. Charon is massive enough that the barycenter of Pluto's system lies outside of Pluto; thus Pluto and Charon are sometimes considered to be a binary system.
Tidal locking (also called gravitational loc ...

, breaking apart of celestial bodies and formation of ring system
A ring system is a disc or ring, orbiting an astronomical object
In astronomy, an astronomical object or celestial object is a naturally occurring physical object, physical entity, association, or structure that exists in the observable u ...

s within the Roche limit
In celestial mechanics, the Roche limit, also called Roche radius, is the distance from a celestial body within which a second celestial body, held together only by its own force of gravity, will disintegrate because the first body's tidal forces ...

, and in extreme cases, spaghettification
In astrophysics
Astrophysics is a science that employs the methods and principles of physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the nat ...

of objects. It arises because the gravitational field exerted on one body by another is not constant across its parts: the nearest side is attracted more strongly than the farthest side. It is this difference that causes a body to get stretched. Thus, the tidal force is also known as the differential force, as well as a secondary effect of the gravitational field.
In celestial mechanics
Celestial mechanics is the branch of astronomy
Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and cel ...

, 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 (for example, the ): 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.
Explanation

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. TheRoche limit
In celestial mechanics, the Roche limit, also called Roche radius, is the distance from a celestial body within which a second celestial body, held together only by its own force of gravity, will disintegrate because the first body's tidal forces ...

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.
These strains would not occur if the gravitational field were uniform, because a uniform field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

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. 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. 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, while the slope relative to value decreases in direct proportion to the distance. This is why the gradient or tidal force at any point 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.Sun, Earth, and Moon

As expected, the table below shows that the distance from the Moon to the Earth, is the same as the distance from the Earth to the 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.Effects

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
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a Surface (mathemat ...

with two bulges, pointing towards and away from the other body. Larger objects distort into an ovoid
An oval (from Latin ''ovum'', "egg") is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas ( projective geometry, technical drawing, etc.) it is given a more precise definition, wh ...

, and are slightly compressed, which is what happens to the Earth's oceans under the action of the Moon. The Earth and Moon rotate about their common center of mass or barycenter
In astronomy
Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses ...

, and their gravitational attraction provides the centripetal force
A centripetal force (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power ...

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.
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 tidally locked
and Charon are tidally locked to each other. Charon is massive enough that the barycenter of Pluto's system lies outside of Pluto; thus Pluto and Charon are sometimes considered to be a binary system.
Tidal locking (also called gravitational loc ...

to the orbital motion, as in the case of the Earth's moon. Tidal heating
Tidal heating (also known as tidal working or tidal flexing) occurs through the tidal friction processes: orbital and rotational energy is dissipated as heat in either (or both) the surface ocean or interior of a planet or satellite. When an object ...

produces dramatic volcanic effects on Jupiter's moon Io. Stresses caused by tidal forces also cause a regular monthly pattern of moonquake
A quake is the result when the surface of a planet
A planet is an astronomical body orbiting a star or Stellar evolution#Stellar remnants, stellar remnant that is massive enough to be Hydrostatic equilibrium, rounded by its own gravity, is n ...

s on Earth's Moon.
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, and that harmonic beat variations in tidal forcing may contribute to millennial climate changes. No strong link to millennial climate changes has been found to date.
Tidal effects become particularly pronounced near small bodies of high mass, such as neutron star
A neutron star is the collapsed core
Core or cores may refer to:
Science and technology
* Core (anatomy)
In common parlance, the core of the body is broadly considered to be the torso. Functional movements are highly dependent on this par ...

s or s, where they are responsible for the "spaghettification
In astrophysics
Astrophysics is a science that employs the methods and principles of physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the nat ...

" of infalling matter. Tidal forces create the oceanic tide
Tides are the rise and fall of sea level
Mean sea level (MSL) (often shortened to sea level) is an average
In colloquial, ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of ...

of Earth
Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

's oceans, where the attracting bodies are the Moon
The Moon is Earth's only natural satellite. At about one-quarter the diameter of Earth (comparable to the width of Australia (continent), Australia), it is the largest natural satellite in the Solar System relative to the size of its plane ...

and, to a lesser extent, the Sun
The Sun is the star
A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many othe ...

. Tidal forces are also responsible for tidal locking
and Charon are tidally locked to each other. Charon is massive enough that the barycenter of Pluto's system lies outside of Pluto; thus Pluto and Charon are sometimes considered to be a binary system.
Tidal locking (also called gravitational loc ...

, tidal acceleration and the Moon
The Moon is Earth's only natural satellite. At about one-quarter the diameter of Earth (comparable to the width of Australia (continent), Australia), it is the largest natural satellite in the Solar System relative to the size ...

, and tidal heating. Tides may also induce seismicity.
By generating conducting fluids within the interior of the Earth, tidal forces also affect the Earth's magnetic field
Earth's magnetic field, also known as the geomagnetic field, is the magnetic field
A magnetic field is a vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. Fo ...

.
Formulation

For a given (externally generated) gravitational field, the tidal acceleration at a point with respect to a body is obtained by 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 befreefall
In Newtonian physics, free fall is any motion of a body
Body may refer to:
In science
* Physical body, an object in physics that represents a large amount, has mass or takes up space
* Body (biology), the physical material of an organism
* ...

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
Newton's law of universal gravitation is usually stated as that every particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can ...

and laws of motion, a body of mass ''m'' at distance ''R'' from the center of a sphere of mass ''M'' feels a force $\backslash vec\_g$,
: $\backslash vec\_g\; =\; -\; \backslash hat\; ~\; G\; ~\; \backslash frac$
equivalent to an acceleration $\backslash vec\_g$,
: $\backslash vec\_g\; =\; -\; \backslash hat\; ~\; G\; ~\; \backslash frac$
where $\backslash hat$ is a unit vector
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

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:
: $\backslash vec\_g\; =\; -\; \backslash hat\; ~\; G\; ~\; \backslash frac$
Pulling out the ''R''Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include:
* Colin Maclaurin (1698–1746), Scottish mathematician
* Normand MacLaurin (1835–1914), Australian politician and university administrator
* Henry Normand MacLaurin (1 ...

of $1/(1\; \backslash pm\; x)^2$ is $1\; \backslash mp\; 2x\; +\; 3x^2\; \backslash mp\; \backslash cdots$ which gives a series expansion of:
: $\backslash vec\_g\; =\; -\; \backslash hat\; ~\; G\; ~\; \backslash frac\; \backslash pm\; \backslash hat\; ~\; G\; ~\; \backslash frac\; ~\; \backslash frac\; +\; \backslash cdots$
The first term is the gravitational acceleration due to ''M'' at the center of the reference body $m$, i.e., at the point where $\backslash Delta\; r$ 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 $\backslash vec\_$ for the distances ∆''r'' considered, along the axis joining the centers of ''m'' and ''M'':
: $\backslash vec\_\; \backslash approx\; \backslash pm\; \backslash hat\; ~\; 2\; \backslash Delta\; r\; ~\; G\; ~\; \backslash frac$
When calculated in this way for the case where ∆''r'' is a distance along the axis joining the centers of ''m'' and ''M'', $\backslash vec\_t$ 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 vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

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 $\backslash frac\backslash left,\; \backslash vec\_\; \backslash $ 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 , while the solar tidal acceleration at the Earth's surface along the Sun–Earth axis is about , where ''g'' is the gravitational acceleration
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

at the Earth's surface. Hence the tide-raising force (acceleration) due to the Sun is about 45% of that due to the Moon. The solar tidal acceleration at the Earth's surface was first given by Newton in the '' Principia''.
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

See also

*Tidal tensor
In Newton's theory of gravitation and in various relativistic classical theories of gravitation, such as general relativity, the tidal tensor represents
#''tidal accelerations'' of a cloud of (electrically neutral, nonspinning) test particles,
#''t ...

* Amphidromic point
An amphidromic point, also called a tidal node, is a geographical location which has zero tidal for one of the . The (the , or height difference between high tide and low tide) for that harmonic constituent increases with distance from this po ...

* Disrupted planet
In astronomy, a disrupted planet is a planet or exoplanet or, perhaps on a somewhat smaller scale, a planetesimal, moon, exomoon or asteroid that has been disrupted or destroyed by a nearby or passing astronomical body or object such as a star. Ne ...

* Galactic tide
A galactic tide is a experienced by objects subject to the of a such as the . Particular areas of interest concerning galactic tides include , the disruption of or , and the Milky Way's tidal effect on the of the .
Effects on external galax ...

* Tidal resonance
* Spacetime curvature
General relativity, also known as the general theory of relativity, is the geometric
Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic
Arithmetic (from the Ancient Greek, Gr ...

References

External links

Gravitational Tides

by J. Christopher Mihos of

Case Western Reserve University
Case Western Reserve University (CWRU) is a private
Private or privates may refer to:
Music
* "In Private
"In Private" was the third single in a row to be a charting success for United Kingdom, British singer Dusty Springfield, after an abse ...

Audio: Cain/Gay – Astronomy Cast

Tidal Forces – July 2007. * *

Myths about Gravity and Tides

by Mikolaj Sawicki of John A. Logan College and the University of Colorado.

by Donald E. Simanek {{Authority control Tides Gravity Force Effects of gravitation Concepts in astronomy