71.41.210.146: Explained mapping for signed integers better.

2014-12-15T01:17:09Z

Explained mapping for signed integers better.

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	~~In [[continuum mechanics]], a [[fluid]]~~ is ~~said to be in '''hydrostatic equilibrium''' or '''hydrostatic balance''' when it~~ is ~~at rest, or when the flow velocity at each point is constant over time~~. ~~This occurs when external forces such as [[gravity]]~~ are ~~balanced~~ by a [[pressure gradient force]].<ref>White (2008). p 63, 66.</ref> For instance, the pressure gradient force prevents gravity from collapsing the [[Earth's atmosphere]] into a thin, dense shell, while gravity prevents the pressure gradient force from diffusing the atmosphere into space.

	~~Hydrostatic equilibrium is the current distinguishing criterion between [[dwarf planet]]s and [[Small Solar System body\|small Solar System bodies]],~~ and ~~has other roles in [[astrophysics]] and [[planetary geology]]. This qualification typically means~~ that the object is symmetrically rounded into a [[spheroid]] or [[ellipsoid]] shape, where any irregular surface features are due to a relatively thin solid [[Crust (geology)\|crust]]. There are [[List of gravitationally rounded objects of the Solar System\|31 observationally confirmed<!--of the DPs, only Ceres, Pluto, and Eris are obs. confirmed. the others rely on mathematical modeling.--> such objects]] (apart from the Sun), sometimes called ''~~[[planemo]]s,''<ref>[[Alan Stern]] considers these all~~ to ~~be "planets", but that conception was rejected by the [[International Astronomical Union]].</ref>~~ in the [[Solar System]], seven more<ref name=brown>http://www.gps.caltech.edu/~mbrown/dps.html</ref><!--3 accepted by the IAU, & 4 more which "must" be according to Mike Brown--> which are virtually certain, and [[List of possible dwarf planets\|a hundred or ~~so more]] which are likely.<ref name=brown/>~~

	~~==Mathematical consideration==~~
	~~[[File:Hydrostatic equilibrium.svg\|thumb\|right\|If the highlighted volume of fluid is not moving, the forces on it upwards must equal the forces downwards.]]~~
	~~===Derivation from force summation===~~
	~~[[Newton's laws of motion]] state that a volume of a fluid which is not in motion or which is in a state of constant velocity must~~ have ~~zero net force on it. This means the sum of the forces~~ in ~~a given direction must be opposed by an equal sum of forces in the opposite direction. This force balance is called a hydrostatic equilibrium.~~

	~~The fluid can be split into a large number of [[cuboid]] volume elements; by considering a single element, the action of the fluid can be derived.~~

	~~There are 3 forces: the force downwards onto the top of the cuboid from the pressure, P, of the fluid above it is, from the definition of [[pressure]],~~
	~~:<math>F_{top} = - P_{top} \cdot A.</math>~~
	~~Similarly, the force on the volume element from the pressure of the fluid below pushing upwards is~~
	~~:<math>F_{bottom} = P_{bottom} \cdot A.</math>~~

	~~Finally, the [[weight]] of the volume element causes a force downwards.~~ ~~If the [[density]] is ρ, the volume is V~~ and ~~g the [[standard gravity]], then:~~
	~~:<math>F_{weight} = -\rho \cdot g \cdot V~~.~~</math>~~
	~~The volume of this cuboid is equal~~ to ~~the area of the top or bottom, times the height — the formula for finding the volume of~~ a ~~cube.~~
	:~~<math>F_{weight} = -\rho \cdot g \cdot A \cdot h</math>~~

	~~By balancing these forces, the total force on the fluid is~~
	:~~<math>\sum F = F_{bottom} + F_{top} + F_{weight} = P_{bottom} \cdot A - P_{top} \cdot A - \rho \cdot g \cdot A \cdot h.<~~/~~math>~~
	~~This sum equals zero if the fluid's velocity is constant. Dividing by A,~~
	~~:<math>0 = P_{bottom} - P_{top} - \rho \cdot g \cdot h.<~~/~~math>~~
	~~Or,~~
	~~:<math>P_{top} - P_{bottom} = - \rho \cdot g \cdot h~~.~~</math>~~
	~~P<sub>top</sub> − P<sub>bottom</sub> is a change in pressure, and h is the height of the volume element – a change in the distance above the ground~~. ~~By saying these changes are [[infinitesimal]]ly small, the equation can be written in [[differential equation\|differential]] form.~~
	~~:<math>dP = - \rho \cdot g \cdot dh.<~~/~~math>~~
	~~Density changes with pressure, and gravity changes with height, so the equation would be:~~
	~~:<math>dP = - \rho(P) \cdot g(h) \cdot dh.<~~/~~math>~~
	~~===Derivation from Navier–Stokes equations===~~
	~~Note finally that this last equation can be derived by solving the three-dimensional [[Navier–Stokes equations]] for the equilibrium situation where~~
	~~:<math>u=v=\frac{\partial~~ p~~}{\partial x}=\frac{\partial p}{\partial y}=0.<~~/~~math>~~
	~~Then the only non-trivial equation is the~~ <~~math~~>z<~~/math~~>-~~equation, which now reads~~
	~~:<math>\frac{\partial p}{\partial z}+\rho g=0.</math>~~
	~~Thus, hydrostatic balance can be regarded as a particularly simple equilibrium solution of the Navier–Stokes equations.~~

	~~==Applications==~~

	~~===Fluids===~~
	The hydrostatic equilibrium pertains to [[hydrostatics]] and the [[principles of equilibrium]] of [[fluid]]s. A hydrostatic balance is a particular balance for weighing substances in water. Hydrostatic balance allows the [[discovery (observation)\|discovery]] of their [[specific gravity\|specific gravities]].

	~~===Astrophysics===~~
	In any given layer of a [[star]], there is a hydrostatic equilibrium between the outward thermal pressure from below and the weight of the material above pressing inward. The [[isotropic]] gravitational field compresses the star into the most compact shape possible. A rotating star in hydrostatic equilibrium is an [[oblate spheroid]] up to a certain (critical) angular velocity. An extreme example of this phenomenon is the star [[Vega]], which has a rotation period of 12.5 hours. Consequently, Vega is about 20% fatter at the equator than at the poles. A star with an angular velocity above the critical angular velocity becomes a Jacobi ([[Ellipsoid\|scalene]]) ellipsoid, and at still faster rotation it is no longer ellipsoidal but [[piriform]] or [[oviform]], with yet other shapes beyond that, though shapes beyond scalene are not stable.<ref>http://~~www~~.~~josleys~~.com/~~show_gallery.php?galid=313<~~/~~ref>~~

	If the star has a massive nearby companion object then [[tidal force]]s come into play as well, distorting the star into a scalene shape when rotation alone would make it a spheroid. An example of this is [[Beta Lyrae]].

	~~Hydrostatic equilibrium is also important for the [[intracluster medium]], where it restricts the amount of fluid that can be present in the core of a [[cluster of galaxies]].~~

	~~===Planetary geology===~~
	The concept of hydrostatic equilibrium has also become important in determining whether an astronomical object is a [[planet]], [[dwarf planet]], or [[small Solar System body]]. According to the [[definition of planet]] adopted by the [[International Astronomical Union]] in 2006, planets and dwarf planets are objects that have sufficient gravity to overcome their own rigidity and assume hydrostatic equilibrium. Such a body will normally have the differentiated interior and geology of a world (a [[planemo]]), though near-hydrostatic bodies such as the proto-planet [[4 Vesta]] may also be differentiated. Sometimes the equilibrium shape is an oblate spheroid, as is the case with the Earth. However, in the cases of moons in synchronous orbit, near unidirectional tidal forces create a scalene ellipsoid, and the dwarf planet {{dp\|Haumea}} appears to be scalene due to its rapid rotation.

	In the Solar System, it appears that icy objects with a diameter larger than ''ca.'' 400 km are usually in hydrostatic equilibrium, while those smaller than that are not. Icy objects, though, can achieve hydrostatic equilibrium at a smaller size than rocky objects. The smallest object known to be in hydrostatic equilibrium is the icy moon [[Mimas (moon)\|Mimas]] at 397 km, while the largest object known not to be is the rocky asteroid [[2 Pallas\|Pallas]] at 532 km (582×556×500±18 km).

	Since the [[terrestrial planet]]s and dwarf planets (and likewise the larger [[natural satellite\|satellites]], like the [[Moon]] and [[Io (moon)\|Io]]) have irregular surfaces, this definition evidently has some flexibility, but a specific means of quantifying an object's shape by this standard has not yet been announced. Local irregularities may be consistent with global equilibrium. For example, the massive base of the tallest<!--as opposed to highest--> mountain on Earth, [[Mauna Kea]], has deformed and depressed the level of the surrounding crust, so that the overall distribution of mass approaches equilibrium. The amount of leeway afforded the definition could affect the classification of the asteroid [[4 Vesta\|Vesta]], which may have solidified while in hydrostatic equilibrium but was subsequently significantly deformed by large impacts (now 572.6×557.2×446.4km).<ref name="Hubble">{{cite web
	~~\|author=Savage, Don~~
	~~\|coauthors=Jones, Tammy; and Villard, Ray~~
	~~\|date=1995-04-19~~
	~~\|title=Asteroid or Mini-Planet? Hubble Maps the Ancient Surface of Vesta (Key Stages in the Evolution of the Asteroid Vesta)~~
	~~\|publisher=Hubble Site News Release STScI-1995-20~~
	~~\|url=http:~~//~~hubblesite.org/newscenter/archive/releases/1995/20/image/c~~
	~~\|accessdate=2006-10-17}}</ref>~~

	~~===Atmospherics===~~
	In the atmosphere, the pressure of the air decreases with increasing altitude. This pressure difference causes an upward force called the [[pressure gradient force]]. The force of gravity balances this out, keeping the atmosphere bound to the earth and maintaining pressure differences with altitude.

	~~==See also==~~
	* [[List of Solar System objects in ~~hydrostatic equilibrium]]~~
	* [[Statics]]
	* [[Two-balloon experiment]]

	~~==Notes==~~
	~~<references />~~

	~~==References==~~
	* {{cite book \|title=Fluid Mechanics \|last=White \|first=Frank M. \|year=2008 \|publisher=McGraw-Hill \|location=New York \|isbn=978-0-07-128645-9 \|pages=63–107 \|chapter=Pressure Distribution in a Fluid }}

	~~==External links==~~
	*[http://www.astronomynotes.com/starsun/s7.htm Strobel, Nick. (May, 2001). Nick Strobel's Astronomy Notes.]

	~~[[Category:Fluid mechanics]]~~
	~~[[Category:Astrophysics]]~~
	~~[[Category:Hydrostatics]]~~
	~~[[Category:Definition of planet]~~]

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