The tidal force is an apparent force that stretches a body towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for the diverse phenomena, including tides, tidal locking, breaking apart of celestial bodies and formation of ring systems within Roche limit, and in extreme cases, spaghettification of objects. It arises because the gravitational force 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 force. 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 (for example, the 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.
1 Explanation 2 Effects of tidal forces 3 Mathematical treatment 4 See also 5 References 6 External links
Figure 2: The Moon's gravity differential field at the surface of the
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). 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. The
Figure 3: 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.
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
Figure 4: Graphic of tidal forces. The top picture shows the gravity field of a body to the right, the lower shows their residual once the field at the centre of the sphere is subtracted; this is the tidal force. See Figure 2 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
vectorially subtracting 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
displaystyle vec F _ g
displaystyle vec F _ g =- hat r ~G~ frac Mm R^ 2
equivalent to an acceleration
displaystyle vec a _ g
displaystyle vec a _ g =- hat r ~G~ frac M R^ 2
displaystyle hat r
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:
( R ± Δ r
displaystyle vec a _ g =- hat r ~G~ frac M (Rpm Delta r)^ 2
Pulling out the R2 term from the denominator gives:
( 1 ± Δ r
displaystyle vec a _ g =- hat r ~G~ frac M R^ 2 ~ frac 1 (1pm Delta r/R)^ 2
( 1 ± x
displaystyle 1/(1pm x)^ 2
1 ∓ 2 x + 3
displaystyle 1mp 2x+3x^ 2 mp cdots
which gives a series expansion of:
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 to M at the center of the reference body
, i.e., at the point where
displaystyle 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
displaystyle vec a _ t
(axial) for the distances ∆r considered, along the axis joining the centers of m and M:
displaystyle vec a _ t
2 Δ r G
displaystyle ~approx ~pm ~ hat r ~2Delta 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 of m and M,
displaystyle vec a _ 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 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
displaystyle vec a _ t
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-
Tidal tensor Amphidromic point Disrupted planet Galactic tide Tidal resonance Spacetime curvature
^ "On the tidal force", I. N. Avsiuk, in "Soviet Astronomy Letters",
vol. 3 (1977), pp. 96–99.
^ See p. 509 in "Astronomy: a physical perspective", M. L. Kutner
^ R Penrose (1999). The Emperor's New Mind: Concerning Computers,
Minds, and the Laws of Physics. Oxford University Press. p. 264.
^ Thérèse Encrenaz; J -P Bibring; M Blanc (2003). The Solar System.
Springer. p. 16. ISBN 3-540-00241-3.
^ R. S. MacKay; J. D. Meiss (1987). Hamiltonian Dynamical Systems: A
Reprint Selection. CRC Press. p. 36.
^ Rollin A Harris (1920). The Encyclopedia Americana: A Library of
Universal Knowledge. 26. Encyclopedia Americana Corp.
^ "The Tidal Force Neil deGrasse Tyson". www.haydenplanetarium.org.
^ "Millennial Climate Variability: Is There a Tidal
^ "Hungry for Power in Space". New Scientist. New Science Pub. 123:
52. 23 September 1989. Retrieved 14 March 2016.
^ "Inseparable galactic twins". ESA/Hubble Picture of the Week.
Retrieved 12 July 2013.
^ The Admiralty (1987). Admiralty manual of navigation. 1. The
Stationery Office. p. 277. ISBN 0-11-772880-2. ,
Chapter 11, p. 277
^ Newton, Isaac (1729). The mathematical principles of natural
philosophy. 2. p. 307. ISBN 0-11-772880-2. , Book 3,
Proposition 36, Page 307 Newton put the force to depress the sea at
places 90 degrees distant from the
Gravitational Tides by J. Christopher Mihos of Case Western Reserve
Audio: Cain/Gay – Astronomy Cast Tidal Forces – July 2007.
Gray, Meghan; Merrifield, Michael. "Tidal Forces". Sixty Symbols.
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