Tidal force

The tidal force is a force that stretches a body towards and away from 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 diverse phenomena, including tides, tidal locking, breaking apart of celestial bodies and formation of ring systems 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 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, 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[1] (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.[2]

Explanation

Field tidal
Figure 4: The Moon's gravity differential 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 tidal equipotential bulges, and accounting for two high tides per day. In this figure, the Earth is the central blue circle while the Moon is far off to the right. The outward direction of the arrows on the right and left indicates that where the Moon is overhead (or at the nadir) its perturbing force opposes that between the earth and ocean.

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.[3] 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.[4] These strains would not occur if the gravitational field were uniform, because a uniform 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.[5] 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.[6]

Inverse x squaired
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, 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. The Moon produces a greater tidal force on the Earth, than the tidal force of the Earth on the Moon. The distance is the same, but the diameter of the Earth is greater than the diameter of the Moon, resulting in a greater tidal force.

What matters is not the total gravitational attraction on a body, but the difference from one side to the other. The greater the diameter of the body, the more difference there will be from one side to the other.[4]

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.[6]

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 per unit mass of the Earth on the Moon is about 20 times stronger than that of the Moon on the Earth.

Thus the Earth was able to lock the Moon's rotation to its orbit around the Earth but not vice versa.

Gravitational body causing tidal force Body subjected to tidal force Diameter and distance Tidal force per unit mass
Body Mass (m) Body Radius (r) Distance (d)
Sun 1.99E+30 Earth 6.37E+06 1.50E+11 3.81E-27 5.05E-07
Moon 7.34E+22 Earth 6.37E+06 3.84E+08 2.24E-19 1.10E-06
Earth 5.97E+24 Moon 1.74E+06 3.84E+08 6.12E-20 2.44E-05
m is mass in kilograms; r is radius in meters; d is distance in meters

diameter = 2r

G is the gravitational constant = 6.674×10−11 N·kg–2·m2

Effects

Saturn PIA06077
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.[7]

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 rotate 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.[8]

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 to the orbital motion, as in the case of the Earth's moon. Tidal heating produces dramatic volcanic effects on Jupiter's moon Io. Stresses caused by tidal forces also cause a regular monthly pattern of moonquakes on Earth's Moon.[5]

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,[9] 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.[10]

Tidal effects become particularly pronounced near small bodies of high mass, such as neutron stars or black holes, 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. Tides may also induce seismicity.

By generating conducting fluids within the interior of the Earth, tidal forces also affect the Earth's magnetic field.[11]

Shoemaker-levy-tidal-forces
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.
Figure 2: This simulation shows a star getting torn apart by the gravitational tides of a supermassive black hole.

Formulation

Inseparable galactic twins
Figure 6: Tidal force is responsible for the merge of galactic pair MRK 1034.[12]
Tidal-forces
Figure 7: 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 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 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 freefalling 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 ,

equivalent to an acceleration ,

where 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:

Pulling out the R2 term from the denominator gives:

The Maclaurin series of is which gives a series expansion of:

The first term is the gravitational acceleration due to M at the center of the reference body , i.e., at the point where 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 for the distances ∆r considered, along the axis joining the centers of m and M:

When calculated in this way for the case where ∆r is a distance along the axis joining the centers of m and M, 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 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 1.1 × 10−7 g, while the solar tidal acceleration at the Earth's surface along the Sun-Earth axis is about 0.52 × 10−7 g, where g is the 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.[13] The solar tidal acceleration at the Earth's surface was first given by Newton in the Principia.[14]

See also

References

  1. ^ "On the tidal force", I. N. Avsiuk, in "Soviet Astronomy Letters", vol. 3 (1977), pp. 96–99.
  2. ^ See p. 509 in "Astronomy: a physical perspective", M. L. Kutner (2003).
  3. ^ 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.
  4. ^ a b Thérèse Encrenaz; J -P Bibring; M Blanc (2003). The Solar System. Springer. p. 16. ISBN 978-3-540-00241-3.
  5. ^ a b "The Tidal Force | Neil deGrasse Tyson". www.haydenplanetarium.org. Retrieved 2016-10-10.
  6. ^ a b Sawicki, 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.
  7. ^ R. S. MacKay; J. D. Meiss (1987). Hamiltonian Dynamical Systems: A Reprint Selection. CRC Press. p. 36. ISBN 978-0-85274-205-1.
  8. ^ Rollin A Harris (1920). The Encyclopedia Americana: A Library of Universal Knowledge. 26. Encyclopedia Americana Corp. pp. 611–617.
  9. ^ 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.
  10. ^ 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.
  11. ^ "Hungry for Power in Space". New Scientist. 123: 52. 23 September 1989. Retrieved 14 March 2016.
  12. ^ "Inseparable galactic twins". ESA/Hubble Picture of the Week. Retrieved 12 July 2013.
  13. ^ The Admiralty (1987). Admiralty manual of navigation. 1. The Stationery Office. p. 277. ISBN 978-0-11-772880-6., Chapter 11, p. 277
  14. ^ Newton, Isaac (1729). The mathematical principles of natural philosophy. 2. p. 307. ISBN 978-0-11-772880-6., 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.

External links

Galactic tide

A galactic tide is a tidal force experienced by objects subject to the gravitational field of a galaxy such as the Milky Way. Particular areas of interest concerning galactic tides include galactic collisions, the disruption of dwarf or satellite galaxies, and the Milky Way's tidal effect on the Oort cloud of the Solar System.

Geodesic deviation

In general relativity, geodesic deviation describes the tendency of objects to approach or recede from one another while moving under the influence of a spatially varying gravitational field. Put another way, if two objects are set in motion along two initially parallel trajectories, the presence of a tidal gravitational force will cause the trajectories to bend towards or away from each other, producing a relative acceleration between the objects.Mathematically, the tidal force in general relativity is described by the Riemann curvature tensor, and the trajectory of an object solely under the influence of gravity is called a geodesic. The geodesic deviation equation relates the Riemann curvature tensor to the relative acceleration of two neighboring geodesics. In differential geometry, the geodesic deviation equation is more commonly known as the Jacobi equation.

Hersheypark

Hersheypark (known as Hershey Park until 1970) is a family theme park situated in Hershey, Pennsylvania, United States, about 15 miles (24 km) east of Harrisburg, and 95 miles (153 km) west of Philadelphia. Founded in 1906 by Milton S. Hershey, as a leisure park for the employees of the Hershey Chocolate Company, it is wholly and privately owned by Hershey Entertainment & Resorts Company as of 2016. It has won several awards, including the IAAPA Applause Award.The park opened its first roller coaster in 1923, the Wild Cat, an early Philadelphia Toboggan Company coaster. In 1970, it began a redevelopment plan, which led to new rides, an expansion, and its renaming. The 1970s brought SooperDooperLooper, an early complete-circuit looping roller coaster, as well as a 330-foot-tall (100 m) observation tower, the Kissing Tower. Beginning in the mid-1980s, the park rapidly expanded. Between 1991 and 2008, it added eight roller coasters and the Boardwalk at Hersheypark water park. As of 2016, its area covers over 110 acres (45 ha), containing 70 rides and attractions, as well as a zoo called ZooAmerica – North American Wildlife Park. Adjacent is Hershey's Chocolate World, a visitors' center that is open to the public and that contains shops, restaurants, and a chocolate factory-themed tour ride.

Miranda (moon)

Miranda, also designated Uranus V, is the smallest and innermost of Uranus's five round satellites. It was discovered by Gerard Kuiper on 16 February 1948 at McDonald Observatory in Texas, and named after Miranda from William Shakespeare's play The Tempest. Like the other large moons of Uranus, Miranda orbits close to its planet's equatorial plane. Because Uranus orbits the Sun on its side, Miranda's orbit is perpendicular to the ecliptic and shares Uranus' extreme seasonal cycle.

At just 470 km in diameter, Miranda is one of the smallest closely observed objects in the Solar System that might be in hydrostatic equilibrium (spherical under its own gravity). The only close-up images of Miranda are from the Voyager 2 probe, which made observations of Miranda during its Uranus flyby in January 1986. During the flyby, Miranda's southern hemisphere pointed towards the Sun, so only that part was studied.

Miranda probably formed from an accretion disc that surrounded the planet shortly after its formation, and, like other large moons, it is likely differentiated, with an inner core of rock surrounded by a mantle of ice. Miranda has one of the most extreme and varied topographies of any object in the Solar System, including Verona Rupes, a 20-kilometer-high scarp that is the highest cliff in the Solar System, and chevron-shaped tectonic features called coronae. The origin and evolution of this varied geology, the most of any Uranian satellite, are still not fully understood, and multiple hypotheses exist regarding Miranda's evolution.

Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field), that measures the extent to which the metric tensor is not locally isometric to that of Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation.

The curvature tensor is given in terms of the Levi-Civita connection by the following formula:

where [u,v] is the Lie bracket of vector fields. For each pair of tangent vectors u, v, R(u,v) is a linear transformation of the tangent space of the manifold. It is linear in u and v, and so defines a tensor. Occasionally, the curvature tensor is defined with the opposite sign.

If and are coordinate vector fields then and therefore the formula simplifies to

The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space). The linear transformation is also called the curvature transformation or endomorphism.

The curvature formula can also be expressed in terms of the second covariant derivative defined as:

which is linear in u and v. Then:

Thus in the general case of non-coordinate vectors u and v, the curvature tensor measures the noncommutativity of the second covariant derivative.

Roche limit

In celestial mechanics, the Roche limit, also called Roche radius, is the distance within which a celestial body, held together only by its own force of gravity, will disintegrate due to a second celestial body's tidal forces exceeding the first body's gravitational self-attraction. Inside the Roche limit, orbiting material disperses and forms rings whereas outside the limit material tends to coalesce. The term is named after Édouard Roche (pronounced [ʁɔʃ] (French), rawsh (English)), who was the French astronomer who first calculated this theoretical limit in 1848.

Rolf Brahde

Carl Rolf Brahde (15 March 1918 – 25 May 2009) was a Norwegian astronomer.

He was born in Tvedestrand as Carl Rolf Henriksen. He was the son of teacher Bredo Henriksen (1888–1974) and Edna Lucy Brahde Davidsen (1891–1967). He changed his last name to Brahde in 1933. He finished his secondary education in Skien in 1937, and enrolled at the University of Oslo in the same year. His studies were interrupted by World War II. He fought in the Norwegian Campaign in 1940, and during the subsequent German occupation of Norway he was a member of the clandestine resistance group XU. He was eventually discovered, but fled to Sweden in 1944. He was decorated with the Defence Medal 1940 – 1945. He continued his studies in Stockholm, and was an assistant for Bertil Lindblad at Stockholm Observatory from 1944 to 1946. He then returned to Norway and graduated with the cand.real. degree. He was married to Luisa Aall Barricelli, a daughter of Marna Aall and professor Maurizio Barricelli, from 1948 to 1961. Brahde married stenographer Elsie Unn Axelsen in 1963.Already one year after graduating, in 1948, Brahde was hired as observer at the University of Oslo. In this position, he was responsible for the construction of Harestua Solar Observatory north of Oslo. He lectured in celestial mechanics and spherical astronomy, and also researched optics related to astronomical instruments as well as calculating the tidal force and the lunar eclipse. He pioneered the use of computers in astronomy in Norway. He also published popular science books and articles, he was a common guest in radio and television programmes and provided commentary for the Norwegian Broadcasting Corporation during the television coverage of the Apollo 11 moon landing. He was promoted to professor in 1987, but retired when reaching the age limit of 70 in 1988. He was a professor emeritus for the rest of his life. He continued doing the calculations for the official Norwegian almanac until 1991, having started in 1967. During this entire period the almanac was edited by Eberhart Jensen.Brahde lived at Haslum. He died in May 2009.

Roller Soaker

Roller Soaker was a suspended roller coaster at Hersheypark in Hershey, Pennsylvania. Manufactured by Setpoint USA, the interactive ride opened to the public on May 4, 2002. It was located in the Boardwalk at Hersheypark behind Tidal Force and was the second roller coaster of this type to be built, following Flying Super Saturator at Carowinds installed in 2000. On December 18, 2012, Hersheypark announced through its Facebook page that Roller Soaker would be removed to make way for new water attractions in 2013. The coaster's station is currently occupied by Breaker's Edge, a hydromagnetic water coaster added in 2018.

The nine cars were capable of holding four riders total: two riders facing forward, and two riders facing backwards. Cars moved in and out of the ride in groups of three. The course was filled with gentle slopes and turns.

Roller Soaker was unique in that each rider was given 4 US gallons (15 L) of water which totals 16 US gallons (61 L) of water for each car. The water could be dropped from the car toward park guests waiting in line or over Intercoastal Waterway lazy river. There were also several water cannons which could be fired at riders from individuals on the ground, and several places where guests were guaranteed to get wet by traveling under a waterfall. For guests of the park who did not wish to get soaked, a "dry path" was marked, which was achieved by covering the path with a wide margin to allow for splash.

Spaghettification

In astrophysics, spaghettification (sometimes referred to as the noodle effect) is the vertical stretching and horizontal compression of objects into long thin shapes (rather like spaghetti) in a very strong non-homogeneous gravitational field; it is caused by extreme tidal forces. In the most extreme cases, near black holes, the stretching is so powerful that no object can withstand it, no matter how strong its components. Within a small region the horizontal compression balances the vertical stretching so that small objects being spaghettified experience no net change in volume.

Stephen Hawking described the flight of a fictional astronaut who, passing within a black hole's event horizon, is "stretched like spaghetti" by the gravitational gradient (difference in strength) from head to toe. The reason this happens would be that the gravity force exerted by the singularity would be much stronger at one end of the body than the other. If one were to fall into a black hole feet first, the gravity at their feet would be much stronger than at their head, causing the person to be vertically stretched. Along with that, the right side of the body will be pulled to the left, and the left side of the body will be pulled to the right, horizontally compressing the person. However, the term "spaghettification" was established well before this. Spaghettification of a star was imaged for the first time in 2018 by researchers observing a pair of colliding galaxies approximately 150 million light-years from Earth.

Taurids

The Taurids are an annual meteor shower, associated with the comet Encke. The Taurids are actually two separate showers, with a Southern and a Northern component. The Southern Taurids originated from Comet Encke, while the Northern Taurids originated from the asteroid 2004 TG10. They are named after their radiant point in the constellation Taurus, where they are seen to come from in the sky. Because of their occurrence in late October and early November, they are also called Halloween fireballs.

Encke and the Taurids are believed to be remnants of a much larger comet, which has disintegrated over the past 20,000 to 30,000 years, breaking into several pieces and releasing material by normal cometary activity or perhaps occasionally by close encounters with the tidal force of Earth or other planets (Whipple, 1940; Klačka, 1999). In total, this stream of matter is the largest in the inner solar system. Since the meteor stream is rather spread out in space, Earth takes several weeks to pass through it, causing an extended period of meteor activity, compared with the much smaller periods of activity in other showers. The Taurids are also made up of weightier material, pebbles instead of dust grains.

Tidal disruption event

A tidal disruption event (also known as a tidal disruption flare) is an astronomical phenomenon that occurs when a star approaches sufficiently close to a supermassive black hole that it is pulled apart by the black hole's tidal force, experiencing spaghettification. A portion of the star's mass can be captured into an accretion disk around the black hole, resulting in a temporary flare of electromagnetic radiation as matter in the disk is consumed by the black hole.

Tidal heating

Tidal heating (also known as tidal working or tidal flexing) occurs through the tidal friction processes: orbital energy is dissipated as heat in either the surface ocean or interior of a planet or satellite. When an object is in an elliptical orbit, the tidal forces acting on it are stronger near periapsis than near apoapsis. Thus the deformation of the body due to tidal forces (i.e. the tidal bulge) varies over the course of its orbit, generating internal friction which heats its interior. This energy gained by the object comes from its gravitational energy, so over time in a two-body system, the initial elliptical orbit decays into a circular orbit (tidal circularization). Sustained tidal heating occurs when the elliptical orbit is prevented from circularizing due to additional gravitational forces from other bodies that keep tugging the object back into an elliptical orbit. In this more complex system, gravitational energy still is being converted to thermal energy; however, now the orbit's semimajor axis would shrink rather than its eccentricity.

Tidal heating is responsible for the geologic activity of the most volcanically active body in the Solar System: Io, a moon of Jupiter. Io's eccentricity persists as the result of its orbital resonances with the Galilean moons Europa and Ganymede. The same mechanism has provided the energy to melt the lower layers of the ice surrounding the rocky mantle of Jupiter's next closest large moon, Europa. However, the heating of the latter is weaker, because of reduced flexing—Europa has half Io's orbital frequency and a 14% smaller radius; also, while Europa's orbit is about twice as eccentric as Io's, tidal force falls off with the cube of distance and is only a quarter as strong at Europa. Jupiter maintains the moons' orbits via tides they raise on it and thus its rotational energy ultimately powers the system. Saturn's moon Enceladus is similarly thought to have a liquid water ocean beneath its icy crust due to tidal heating related to its resonance with Dione. The water vapor geysers which eject material from Enceladus are thought to be powered by friction generated within its interior.

The tidal heating rate, , in a satellite that is spin-synchronous and has an eccentric orbit is given by:

where , , and are respectively the satellite's mean radius, mean orbital motion, and eccentricity. is the imaginary portion of the second-order Love number which measures the efficiency of body dissipation within the satellite. This imaginary portion is defined by interplay of the body's rheology and self-gravitation. It, therefore, is a function of the body's radius, density, and rheological parameters (the shear modulus, viscosity, and others -- dependent upon the rheological model). The rheological parameters' values, in turn, depend upon the temperature and the concentration of partial melt in the body's interior.

The tidally dissipated power in a nonsynchronised rotator is given by a more complex expression.

Tidal locking

Tidal locking (also called gravitational locking, captured rotation and spin-orbit locking), in the most well-known case, occurs when an orbiting astronomical body always has the same face toward the object it is orbiting. This is known as synchronous rotation: the tidally locked body takes just as long to rotate around its own axis as it does to revolve around its partner. For example, the same side of the Moon always faces the Earth, although there is some variability because the Moon's orbit is not perfectly circular. Usually, only the satellite is tidally locked to the larger body. However, if both the difference in mass between the two bodies and the distance between them are relatively small, each may be tidally locked to the other; this is the case for Pluto and Charon.

The effect arises between two bodies when their gravitational interaction slows a body's rotation until it becomes tidally locked. Over many millions of years, the interaction forces changes to their orbits and rotation rates as a result of energy exchange and heat dissipation. When one of the bodies reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit, it is said to be tidally locked. The object tends to stay in this state when leaving it would require adding energy back into the system. The object's orbit may migrate over time so as to undo the tidal lock, for example, if a giant planet perturbs the object.

Not every case of tidal locking involves synchronous rotation. With Mercury, for example, this tidally locked planet completes three rotations for every two revolutions around the Sun, a 3:2 spin-orbit resonance. In the special case where an orbit is nearly circular and the body's rotation axis is not significantly tilted, such as the Moon, tidal locking results in the same hemisphere of the revolving object constantly facing its partner.

However, in this case the exact same portion of the body does not always face the partner on all orbits. There can be some shifting due to variations in the locked body's orbital velocity and the inclination of its rotation axis.

Tidal scour

Tidal scour is “sea-floor erosion caused by strong tidal currents resulting in the removal of inshore sediments and formation of deep holes and channels”. Examples of this hydrological process can be found globally. Two locations in the United States where tidal scour is the predominant shaping force is the San Francisco Bay and the Elkhorn Slough. Tidal force can also contribute to bridge scour.

Tidal shock

A tidal shock occurs when a star cluster or other distributed astronomical object passes by a large mass

such as an interstellar cloud, resulting in gravitational perturbation on a time scale that is much less than the mean time for a star to complete an orbit within the cluster. The tidal force from this event can increase the dynamic energy of the cluster, in effect heating it up. This causes the cluster to expand and shed some of the outer stars.Tidal shocks occur, for example, when a globular cluster passes through the galactic plane or near the core of the Milky Way. These events are an important factor during the early evolution of a globular cluster. They work to truncate the outer part of clusters, thereby limiting the impact of future tidal shocks. Streams of stars shed from a globular cluster as a result of tidal shock can form what are termed tidal tails. These are extended streams of stars that lead away from the cluster. Such streams can be used to trace the orbital path of the cluster.

Tide

Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon and the Sun, and the rotation of the Earth.

Tide tables can be used for any given locale to find the predicted times and amplitude (or "tidal range"). The predictions are influenced by many factors including the alignment of the Sun and Moon, the phase and amplitude of the tide (pattern of tides in the deep ocean), the amphidromic systems of the oceans, and the shape of the coastline and near-shore bathymetry (see Timing). They are however only predictions, the actual time and height of the tide is affected by wind and atmospheric pressure. Many shorelines experience semi-diurnal tides—two nearly equal high and low tides each day. Other locations have a diurnal tide—one high and low tide each day. A "mixed tide"—two uneven magnitude tides a day—is a third regular category.Tides vary on timescales ranging from hours to years due to a number of factors, which determine the lunitidal interval. To make accurate records, tide gauges at fixed stations measure water level over time. Gauges ignore variations caused by waves with periods shorter than minutes. These data are compared to the reference (or datum) level usually called mean sea level.While tides are usually the largest source of short-term sea-level fluctuations, sea levels are also subject to forces such as wind and barometric pressure changes, resulting in storm surges, especially in shallow seas and near coasts.

Tidal phenomena are not limited to the oceans, but can occur in other systems whenever a gravitational field that varies in time and space is present. For example, the shape of the solid part of the Earth is affected slightly by Earth tide, though this is not as easily seen as the water tidal movements.

Ultra diffuse galaxy

An ultra diffuse galaxy (UDG) is an extremely low luminosity galaxy, the first example of which was discovered in the nearby Virgo Cluster by Allan Sandage and Bruno Binggeli in 1984. UDGs are a subset of dwarf spheroidals and dwarf ellipticals and as such are a redundant name for an already well studied galaxy type. Such a galaxy may have the same size and mass as the Milky Way but a visible star count of only 1%. Their lack of luminosity is due to the lack of star-forming gas in the galaxy. This results in old stellar populations.Some ultra diffuse galaxies found in the Coma Cluster, about 330 million light years from Earth, have diameters of 60 kly (18 kpc) with 1% of the stars of the Milky Way Galaxy. The distribution of ultra diffuse galaxies in the Coma Cluster is the same as luminous galaxies; this suggests that the cluster environment strips the gas from the galaxies, while allowing them to populate the cluster the same as more luminous galaxies. The similar distribution in the higher tidal force zones suggests a larger dark matter fraction to hold the galaxies together under the higher stress.Dragonfly 44, an ultra diffuse galaxy in the Coma Cluster, is one example. Observations of the rotational speed suggest a mass of about one trillion solar masses, about the same as the mass of the Milky Way. This is also consistent with about 90 globular clusters observed around Dragonfly 44. However, the galaxy emits only 1% of the light emitted by the Milky Way. On 25 August 2016, astronomers reported that Dragonfly 44 may be made almost entirely of dark matter. In 2018 the same authors reported the discovery of a dark matter-free UDG (NGC 1052-DF2, which was already identified on photoplates by Igor Karachentsev) based on velocity measurements of ~10 globular cluster system. The authors concluded that this may rule out modified gravity theories like MOND, but other theories such as the External Field Effect are also possibilities.

Wave base

The wave base, in physical oceanography, is the maximum depth at which a water wave's passage causes significant water motion. For water depths deeper than the wave base, bottom sediments and the seafloor are no longer stirred by the wave motion above.

Weyl tensor

In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. It is a tensor that has the same symmetries as the Riemann tensor with the extra condition that it be trace-free: metric contraction on any pair of indices yields zero.

In general relativity, the Weyl curvature is the only part of the curvature that exists in free space—a solution of the vacuum Einstein equation—and it governs the propagation of gravitational waves through regions of space devoid of matter. More generally, the Weyl curvature is the only component of curvature for Ricci-flat manifolds and always governs the characteristics of the field equations of an Einstein manifold.In dimensions 2 and 3 the Weyl curvature tensor vanishes identically. In dimensions ≥ 4, the Weyl curvature is generally nonzero. If the Weyl tensor vanishes in dimension ≥ 4, then the metric is locally conformally flat: there exists a local coordinate system in which the metric tensor is proportional to a constant tensor. This fact was a key component of Nordström's theory of gravitation, which was a precursor of general relativity.

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