Editing Hydrostatic equilibrium

{{Short description|State of balance between external forces on a fluid and internal pressure gradient}}
{{Broader|Hydrostatics}}
{{More citations needed|date=May 2010}}
[[File:Hydrostatic equilibrium.png|thumb|right|Diagram of a newly formed planet in a state of hydrostatic equilibrium.]]

In [[fluid mechanics]], '''hydrostatic equilibrium''', also called '''hydrostatic balance''' and '''hydrostasy''', is the condition of a [[fluid]] or [[Plasticity (physics)|plastic]] solid at rest, which occurs when external forces, such as [[gravity]], are balanced by a [[pressure-gradient force]].<ref>{{cite book |title=Fluid Mechanics |last=White |first=Frank M. |date=2008 |publisher=McGraw-Hill |location=New York |isbn=978-0-07-128645-9 |chapter=Pressure Distribution in a Fluid | pages = 63, 66}}</ref> In the planetary physics of Earth, the pressure-gradient force prevents gravity from collapsing the [[atmosphere of Earth]] into a thin, dense shell, whereas gravity prevents the pressure-gradient force from diffusing the atmosphere into [[outer space]].<ref>{{Cite book|url=https://books.google.com/books?id=cC0Kye7nHEEC&q=hydrostasy&pg=PA81|title = Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation|isbn = 9781139459969|last1 = Vallis|first1 = Geoffrey K.|date = 6 November 2006| publisher=Cambridge University Press }}</ref><ref>{{Cite book|url=https://books.google.com/books?id=Kr2GDwAAQBAJ&q=hydrostasy&pg=PA45 | title = Ocean Circulation in Three Dimensions|isbn = 9780521768436|last1 = Klinger|first1 = Barry A.|last2 = Haine|first2 = Thomas W. N.|date = 14 March 2019| publisher = Cambridge University Press}}</ref> In general, it is what causes objects in space to be spherical.

Hydrostatic equilibrium is the distinguishing criterion between [[dwarf planet]]s and [[Small Solar System body|small solar system bodies]], and features in [[astrophysics]] and [[planetary geology]]. Said qualification of equilibrium indicates that the shape of the object is symmetrically rounded, mostly due to [[rotation]], into an [[ellipsoid]], where any irregular surface features are consequent to a relatively thin solid [[Crust (geology)|crust]]. In addition to the Sun, there are [[List of gravitationally rounded objects of the Solar System|a dozen or so equilibrium objects confirmed<!--of the DPs, only Ceres is obs. confirmed, Pluto quite likely. the others rely on mathematical modeling. Of the moons, only the Galileans and Titan confirmed, Triton quite likely and Rhea compatible.--> to exist]] in the [[Solar System]].

== Mathematical consideration ==
[[File:Hydrostatic equilibrium.svg|thumb|right|If the highlighted volume of fluid is not accelerating, the forces on it upwards must equal the forces downwards.]]

For a hydrostatic fluid on Earth:
<math display="block">dP = - \rho(P) \, g(h) \, dh</math>

=== Derivation from force summation ===
{{further|Mechanical equilibrium}}
[[Newton's laws of motion]] state that a volume of a fluid that is not in motion or that 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 three 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 display="block">F_\text{top} = - P_\text{top} A</math>
Similarly, the force on the volume element from the pressure of the fluid below pushing upwards is
<math display="block">F_\text{bottom} = P_\text{bottom} 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 display="block">F_\text{weight} = -\rho g V</math>
The volume of this cuboid is equal to the area of the top or bottom, times the height&nbsp;– the formula for finding the volume of a cube.
<math display="block">F_\text{weight} = -\rho g A h</math>

By balancing these forces, the total force on the fluid is
<math display="block">\sum F = F_\text{bottom} + F_\text{top} + F_\text{weight} = P_\text{bottom} A - P_\text{top} A - \rho g A h</math>
This sum equals zero if the fluid's velocity is constant.  Dividing by A,
<math display="block">0 = P_\text{bottom} - P_\text{top} - \rho g h</math>
Or,
<math display="block">P_\text{top} - P_\text{bottom} = - \rho g 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 display="block">dP = - \rho g \, dh</math>
Density changes with pressure, and gravity changes with height, so the equation would be:
<math display="block">dP = - \rho(P) \, g(h) \, 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 display="block">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 display="block">\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.

=== Derivation from general relativity ===
By plugging the [[energy–momentum tensor]] for a [[perfect fluid]]
<math display="block">T^{\mu\nu} = \left(\rho c^{2} + P\right) u^\mu u^\nu + P g^{\mu\nu}</math>
into the [[Einstein field equations]]
<math display="block">R_{\mu\nu} = \frac{8\pi G}{c^4} \left(T_{\mu\nu} - \frac{1}{2} g_{\mu\nu} T\right)</math>
and using the conservation condition
<math display="block">\nabla_\mu T^{\mu\nu} = 0</math>
one can derive the [[Tolman–Oppenheimer–Volkoff equation]] for the structure of a static, spherically symmetric relativistic star in isotropic coordinates:
<math display="block">\frac{dP}{dr} = -\frac{G M(r)\rho(r)}{r^2} \left(1+\frac{P(r)}{\rho(r)c^2}\right) \left(1+\frac{4\pi r^3 P(r)}{M(r) c^2}\right) \left(1 - \frac{2GM(r)}{r c^2}\right)^{-1}</math>
In practice, ''Ρ'' and ''ρ'' are related by an equation of state of the form ''f''(''Ρ'',''ρ'')&nbsp;=&nbsp;0, with ''f'' specific to makeup of the star. ''M''(''r'') is a foliation of spheres weighted by the mass density ''ρ''(''r''), with the largest sphere having radius ''r'':
<math display="block">M(r) = 4\pi \int_0^r dr' \, r'^2 \rho(r').</math>
Per standard procedure in taking the nonrelativistic limit, we let {{nowrap|''c'' → ∞}}, so that the factor
<math display="block">\left(1+\frac{P(r)}{\rho(r)c^2}\right) \left(1+\frac{4\pi r^3P(r)}{M(r)c^2}\right) \left(1-\frac{2GM(r)}{r c^2} \right)^{-1} \rightarrow 1</math>
Therefore, in the nonrelativistic limit the Tolman–Oppenheimer–Volkoff equation reduces to Newton's hydrostatic equilibrium:
<math display="block">\frac{dP}{dr} = -\frac{GM(r)\rho(r)}{r^2} = -g(r)\,\rho(r)\longrightarrow dP = - \rho(h)\,g(h)\, dh</math>
(we have made the trivial notation change ''h''&nbsp;=&nbsp;''r'' and have used ''f''(''Ρ'',''ρ'')&nbsp;=&nbsp;0 to express ''ρ'' in terms of ''P'').<ref>{{cite book|last1=Zee|first1=A.|title=Einstein gravity in a nutshell|date=2013|publisher=Princeton University Press | location=Princeton | isbn=9780691145587|pages=451–454}}</ref> A similar equation can be computed for rotating, axially symmetric stars, which in its gauge independent form reads:
<math display="block">\frac{\partial_i P}{P+\rho} - \partial_i \ln u^t + u_t u^\varphi\partial_i\frac{u_\varphi}{u_t}=0</math> 
Unlike the TOV equilibrium equation, these are two equations (for instance, if as usual when treating stars, one chooses spherical coordinates as basis coordinates <math>(t,r,\theta,\varphi)</math>, the index ''i'' runs for the coordinates ''r'' and <math>\theta</math>).

== Applications ==

=== Fluids ===
The hydrostatic equilibrium pertains to [[hydrostatics]] and the principles of equilibrium of [[fluids]]. 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]]. This equilibrium is strictly applicable when an ideal fluid is in steady horizontal laminar flow, and when any fluid is at rest or in vertical motion at constant speed. It can also be a satisfactory approximation when flow speeds are low enough that acceleration is negligible.

=== Astrophysics and planetary science ===
From the time of [[Isaac Newton]] much work has been done on the subject of the equilibrium attained when a fluid rotates in space. This has application to both stars and objects like planets, which may have been fluid in the past or in which the solid material deforms like a fluid when subjected to very high stresses.

In any given layer of a star, there is a hydrostatic equilibrium between the outward-pushing pressure gradient and the weight of the material above pressing inward. One can also study planets under the assumption of hydrostatic equilibrium. A rotating star or planet in hydrostatic equilibrium is usually an [[oblate spheroid]], an [[ellipsoid]] in which two of the principal axes are equal and longer than the third.

An example of this phenomenon is the star [[Vega]], which has a rotation period of 12.5 hours. Consequently, Vega is about 20% larger at the equator than from pole to pole.

In his 1687 ''[[Philosophiæ Naturalis Principia Mathematica]]'' Newton correctly stated that a rotating fluid of uniform density under the influence of gravity would take the form of a spheroid and that the gravity (including the effect of [[centrifugal force]]) would be weaker at the equator than at the poles by an amount equal (at least [[asymptotically]]) to five fourths the centrifugal force at the equator.<ref>[https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookIII-Prop2 Propositions X-XXIV (Motions of celestial bodies and the sea)], Propositions XIX and XX. [https://la.wikisource.org/wiki/Philosophiae_Naturalis_Principia_Mathematica/Liber_III Original Latin].</ref> In 1742, [[Colin Maclaurin]] published his treatise on fluxions in which he showed that the spheroid was an exact solution. If we designate the equatorial radius by <math>r_e,</math> the polar radius by <math>r_p,</math> and the [[eccentricity (mathematics)|eccentricity]] by <math>\epsilon,</math> with
: <math>\epsilon=\sqrt{1-r_p^2/r_e^2},</math>
he found that the gravity at the poles is<ref>{{cite book |last1=Colin Maclaurin |title=A Treatise on Fluxions |date=1742 |page=125 |url=https://ia800903.us.archive.org/25/items/treatiseonfluxio02macl/treatiseonfluxio02macl.pdf}} Maclaurin does not use modern notation but rather gives his results in geometric terms. The gravity results are in article 646. At one point he makes an erroneous statement equivalent to <math>d(\tan\theta-\theta)/d\tan\theta=\tan^2\theta</math> but his subsequent statements are correct.</ref>
: <math>
\begin{align}
g_p & =4\pi\frac{r_p}{r_e}\frac{\epsilon r_e-r_p\arctan(\epsilon r_e/r_p)}{\epsilon^3}G\rho \\
&=3\frac{\epsilon r_e-r_p\arctan(\epsilon r_e/r_p)}{\epsilon^3 r_e^3}GM \\
\end{align}
</math>
where <math>G</math> is the gravitational constant, <math>\rho</math> is the (uniform) density, and <math>M</math> is the total mass. The ratio of this to <math>g_0,</math> the gravity if the fluid is not rotating, is asymptotic to
: <math>g_p/g_0\sim 1+\frac 1{15}\epsilon^2\sim 1+\frac 2 {15} f </math>
as <math>\epsilon</math> goes to zero, where <math>f</math> is the flattening:
: <math>f=\frac{r_e-r_p}{r_e}.</math>

The gravitational attraction on the equator (not including centrifugal force) is
: <math>
\begin{align}
g_e &=\frac 32\left(\frac 1{r_e r_p}-\frac{\epsilon r_e-r_p\arctan(\epsilon r_e/r_p)}{\epsilon^3 r_e^2 r_p}\right)GM\\
& =\frac 32\frac{r_e\arctan(\epsilon r_e/r_p)-\epsilon r_p}{\epsilon^3 r_e^3} GM \\
\end{align}
</math>

Asymptotically, we have:
: <math>g_e/g_0\sim 1-\frac 1{30}\epsilon^2\sim 1-\frac 1 {15} f </math>

Maclaurin showed (still in the case of uniform density) that the component of gravity toward the axis of rotation depended only on the distance from the axis and was proportional to that distance, and the component in the direction toward the plane of the equator depended only on the distance from that plane and was proportional to that distance. Newton had already pointed out that the gravity felt on the equator (including the lightening due to centrifugal force) has to be <math>\frac{r_p}{r_e}g_p</math> in order to have the same pressure at the bottom of channels from the pole or from the equator to the centre, so the centrifugal force at the equator must be
: <math>g_e-\frac{r_p}{r_e}g_p\sim \frac 25\epsilon^2g_e\sim\frac 45fg_e.</math>

Defining the latitude to be the angle between a tangent to the meridian and axis of rotation, the total gravity felt at latitude <math>\phi</math> (including the effect of centrifugal force) is
: <math>g(\phi)=\frac{g_p(1-f)}{\sqrt{1-(2f-f^2)\sin^2\phi}}.</math>

This spheroid solution is stable up to a certain (critical) [[angular momentum]] (normalized by <math>M\sqrt{G\rho r_e}</math>), but in 1834, [[Carl Jacobi]] showed that it becomes unstable once the eccentricity reaches 0.81267 (or <math>f</math> reaches 0.3302).
Above the critical value, the solution becomes a [[Jacobi ellipsoid|Jacobi, or scalene, ellipsoid]] (one with all three axes different). [[Henri Poincaré]] in 1885 found that at still higher angular momentum it will no longer be ellipsoidal but [[wikt:pyriform|piriform]] or [[oviform]]. The symmetry drops from the 8-fold D{{sub|2h}} [[Point groups in three dimensions|point group]] to the 4-fold C{{sub|2v}}, with its axis perpendicular to the axis of rotation.<ref name=Henri>{{cite journal |last1=Henri Poincaré |title=Les formes d'équilibre d'une masse fluide en rotation |journal=Revue Général des Sciences Pures et Appliquées |date=1892 |url=https://www.yumpu.com/fr/document/view/16685029/les-formes-dequilibre-dune-masse-fluide-en-rotation-universite-}}</ref> Other shapes satisfy the equations beyond that, but are not stable, at least not near the point of [[bifurcation theory|bifurcation]].<ref name=Henri/><ref>{{cite web|url=http://www.josleys.com/show_gallery.php?galid=313 |title=Gallery : The shape of Planet Earth |publisher=Josleys.com |access-date=2014-06-15}}</ref> Poincaré was unsure what would happen at higher angular momentum but concluded that eventually the blob would split into two.

The assumption of uniform density may apply more or less to a molten planet or a rocky planet but does not apply to a star or to a planet like the earth which has a dense metallic core. In 1737, [[Alexis Clairaut]] studied the case of density varying with depth.<ref>{{Cite journal|last1 = Clairaut|first1 = Alexis|last2 = Colson|first2 = John|date = 1737|title = An Inquiry concerning the Figure of Such Planets as Revolve about an Axis, Supposing the Density Continually to Vary, from the Centre towards the Surface|jstor = 103921|journal = Philosophical Transactions}}</ref> [[Clairaut's theorem]] states that the variation of the gravity (including centrifugal force) is proportional to the square of the sine of the latitude, with the proportionality depending linearly on the flattening (<math>f</math>) and the ratio at the equator of centrifugal force to gravitational attraction. (Compare with the exact relation above for the case of uniform density.) Clairaut's theorem is a special case for an oblate spheroid of a connexion found later by [[Pierre-Simon Laplace]] between the shape and the variation of gravity.<ref>See {{cite journal |last1=Sir George Stokes |title=On Attractions, and on Clairaut's Theorem |journal=The Cambridge and Dublin Mathematical Journal |date=1849 |pages=194–219 |url=https://ia801603.us.archive.org/15/items/cambridgeanddub03unkngoog/cambridgeanddub03unkngoog.pdf |author1-link=Sir George Stokes }}</ref>

If the star has a massive nearby companion object, [[tidal forces]] come into play as well, which distort the star into a scalene shape if 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]].

We can also use the principle of hydrostatic equilibrium to estimate the [[velocity dispersion]] of [[dark matter]] in clusters of galaxies. Only [[baryonic]] matter (or, rather, the collisions thereof) emits [[X-ray]] radiation. The absolute X-ray [[luminosity]] per unit volume takes the form <math>\mathcal{L}_X=\Lambda(T_B)\rho_B^2</math> where <math>T_B</math> and <math>\rho_B</math> are the temperature and density of the baryonic matter, and <math>\Lambda(T)</math> is some function of temperature and fundamental constants. The baryonic density satisfies the above equation {{nowrap|<math>dP = -\rho g \, dr</math>:}}
<math display="block">p_B(r+dr)-p_B(r)=-dr\frac{\rho_B(r)G}{r^2}\int_0^r 4\pi r^2\,\rho_M(r)\, dr.</math>
The integral is a measure of the total mass of the cluster, with <math>r</math> being the proper distance to the center of the cluster. Using the [[ideal gas law]] <math>p_B=kT_B\rho_B/m_B</math> (<math>k</math> is the [[Boltzmann constant]] and <math>m_B</math> is a characteristic mass of the baryonic gas particles) and rearranging, we arrive at
<math display="block">\frac{d}{dr}\left(\frac{kT_B(r)\rho_B(r)}{m_B}\right)=-\frac{\rho_B(r)G}{r^2}\int_0^r 4\pi r^2\,\rho_M(r)\, dr.</math>
Multiplying by <math>r^2/\rho_B(r)</math> and differentiating with respect to <math>r</math> yields
<math display="block">\frac{d}{dr}\left[\frac{r^2}{\rho_B(r)}\frac{d}{dr}\left(\frac{kT_B(r)\rho_B(r)}{m_B}\right)\right]=-4\pi Gr^2\rho_M(r).</math>
If we make the assumption that cold dark matter particles have an isotropic velocity distribution, the same derivation applies to these particles, and their density <math>\rho_D=\rho_M-\rho_B</math> satisfies the non-linear differential equation
<math display="block">\frac{d}{dr}\left[\frac{r^2}{\rho_D(r)}\frac{d}{dr}\left(\frac{kT_D(r)\rho_D(r)}{m_D}\right)\right]=-4\pi Gr^2\rho_M(r).</math>
With perfect X-ray and distance data, we could calculate the baryon density at each point in the cluster and thus the dark matter density. We could then calculate the velocity dispersion <math>\sigma^2_D</math> of the dark matter, which is given by 
<math display="block">\sigma^2_D=\frac{kT_D}{m_D}.</math> 
The central density ratio <math>\rho_B(0)/\rho_M(0)</math> is dependent on the [[redshift]] <math>z</math> of the cluster and is given by
<math display="block">\rho_B(0)/\rho_M(0)\propto (1+z)^2\left(\frac{\theta}{s}\right)^{3/2}</math> 
where <math>\theta</math> is the angular width of the cluster and <math>s</math> the proper distance to the cluster. Values for the ratio range from 0.11 to 0.14 for various surveys.<ref>{{cite book|last1=Weinberg|first1=Steven|title=Cosmology|date=2008|publisher=Oxford University Press|location=New York|isbn=978-0-19-852682-7|pages=70–71}}</ref>

=== Planetary geology ===
{{see also|List of gravitationally rounded objects of the Solar System}}
{{further|Clairaut's theorem (gravity)}}

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]] that was adopted by the [[International Astronomical Union]] in 2006, one defining characteristic of planets and dwarf planets is that they are objects that have sufficient gravity to overcome their own rigidity and assume hydrostatic equilibrium. Such a body often has the differentiated interior and geology of a world (a [[planemo]]), but near-hydrostatic or formerly hydrostatic bodies such as the proto-planet [[4 Vesta]] may also be differentiated and some hydrostatic bodies (notably [[Callisto (moon)|Callisto]]) have not thoroughly differentiated since their formation. Often, the equilibrium shape is an [[oblate spheroid]], as is the case with Earth. However, in the cases of moons in synchronous orbit, nearly unidirectional tidal forces create a [[scalene ellipsoid]]. Also, the purported dwarf planet {{dp|Haumea}} is scalene because of its rapid rotation though it may not currently be in equilibrium.

Icy objects were previously believed to need less mass to attain hydrostatic equilibrium than rocky objects. The smallest object that appears to have an equilibrium shape is the icy moon [[Mimas (moon)|Mimas]] at 396&nbsp;km, but the largest icy object known to have an obviously non-equilibrium shape is the icy moon [[Proteus (moon)|Proteus]] at 420&nbsp;km, and the largest rocky bodies in an obviously non-equilibrium shape are the asteroids [[2 Pallas|Pallas]] and [[4 Vesta|Vesta]] at about 520&nbsp;km. However, Mimas is not actually in hydrostatic equilibrium for its current rotation. The smallest body confirmed to be in hydrostatic equilibrium is the dwarf planet [[Ceres (dwarf planet)|Ceres]], which is icy, at 945&nbsp;km, and the largest known body to have a noticeable deviation from hydrostatic equilibrium is [[Iapetus (moon)|Iapetus]] being made of mostly permeable ice and almost no rock.<ref>{{Cite journal |last=Thomas |first=P.C. |date=July 2010 |title=Sizes, shapes, and derived properties of the saturnian satellites after the Cassini nominal mission |url=http://www.ciclops.org/media/sp/2011/6794_16344_0.pdf |journal=Icarus |volume=208 |issue=1 |pages=395–401 |doi=10.1016/j.icarus.2010.01.025 |bibcode=2010Icar..208..395T |archive-date=23 December 2018 |archive-url=https://web.archive.org/web/20181223003125/http://www.ciclops.org/media/sp/2011/6794_16344_0.pdf |url-status=dead }}</ref> At 1,469&nbsp;km Iapetus is neither spherical nor ellipsoid. Instead, it is rather in a strange walnut-like shape due to its unique [[Eequatorial ridge on Iapetus|equatorial ridge]].<ref name="Castillo2007">{{cite journal| last=Castillo-Rogez| first=J. C.|author2=Matson, D. L. |author3=Sotin, C. |author4=Johnson, T. V. |author5=Lunine, Jonathan I. |author6= Thomas, P. C. | title=Iapetus' geophysics: Rotation rate, shape, and equatorial ridge| journal=Icarus| date=2007| volume=190| issue=1| pages=179–202| doi=10.1016/j.icarus.2007.02.018| bibcode=2007Icar..190..179C}}</ref> Some icy bodies may be in equilibrium at least partly due to a subsurface ocean, which is not the definition of equilibrium used by the IAU (gravity overcoming internal rigid-body forces). Even larger bodies deviate from hydrostatic equilibrium, although they are ellipsoidal: examples are Earth's [[Moon]] at 3,474&nbsp;km (mostly rock),<ref>{{cite journal |last1=Garrick-Bethell |first1=I. |last2=Wisdom |first2=J |last3=Zuber |first3=MT |title=Evidence for a Past High-Eccentricity Lunar Orbit |journal=Science |date=4 August 2006 |volume=313 |issue=5787 |pages=652–655 |doi=10.1126/science.1128237 |pmid=16888135 |bibcode=2006Sci...313..652G |s2cid=317360 }}</ref> and the planet [[Mercury (planet)|Mercury]] at 4,880&nbsp;km (mostly metal).<ref name="Mercury">Sean Solomon, Larry Nittler & Brian Anderson, eds. (2018) ''Mercury: The View after MESSENGER''. Cambridge Planetary Science series no. 21, Cambridge University Press, pp. 72–73.</ref>

In 2024, Kiss et al. found that {{dp|Quaoar}} has an ellipsoidal shape incompatible with hydrostatic equilibrium for its current spin. They hypothesised that Quaoar originally had a rapid rotation and was in hydrostatic equilibrium but that its shape became "frozen in" and did not change as it spun down because of tidal forces from its moon [[Weywot]].<ref name="Kiss2024">
{{cite journal
 |display-authors = etal
 |first1 = C. |last1 = Kiss
 |first2 = T. G. |last2 = Müller
 |first3 = G. |last3 = Marton
 |first4 = R. |last4 = Szakáts
 |first5 = A. |last5 = Pál
 |first6 = L. |last6 = Molnár
 |title      = The visible and thermal light curve of the large Kuiper belt object (50000) Quaoar
 |journal    = Astronomy & Astrophysics
 |date       = March 2024
 |volume     = 684
 |issue      = 
 |pages      = A50
 |doi        = 10.1051/0004-6361/202348054
 |arxiv      = 2401.12679
 |bibcode    = 2024A&A...684A..50K
}}</ref> If so, this would resemble the situation of Iapetus, which is too oblate for its current spin.<ref name="Science News">Cowen, R. (2007). Idiosyncratic Iapetus, ''Science News'' vol. 172, pp. 104–106. [http://www.sciencenews.org/articles/20070818/bob8ref.asp references] {{Webarchive|url=https://web.archive.org/web/20071013165655/http://www.sciencenews.org/articles/20070818/bob8ref.asp |date=2007-10-13 }}</ref><ref name="Thomas2010">{{cite journal| doi = 10.1016/j.icarus.2010.01.025| last1 = Thomas| first1 = P. C.| date = July 2010| title = Sizes, shapes, and derived properties of the saturnian satellites after the Cassini nominal mission| journal = Icarus| volume = 208| issue = 1| pages = 395–401| url = http://www.ciclops.org/media/sp/2011/6794_16344_0.pdf| bibcode = 2010Icar..208..395T| access-date = 2015-09-25| archive-date = 2018-12-23| archive-url = https://web.archive.org/web/20181223003125/http://www.ciclops.org/media/sp/2011/6794_16344_0.pdf| url-status = dead}}</ref> Iapetus is generally still considered a [[planetary-mass moon]] nonetheless<ref name=planetarysociety>Emily Lakdawalla et al., [https://www.planetary.org/worlds/what-is-a-planet What Is A Planet?] {{Webarchive|url=https://web.archive.org/web/20220122142140/https://www.planetary.org/worlds/what-is-a-planet |date=2022-01-22}} The Planetary Society, 21 April 2020</ref> though not always.<ref name=ChenKipping>{{cite journal |last1=Chen |first1=Jingjing |last2=Kipping |first2=David |date=2016 |title=Probabilistic Forecasting of the Masses and Radii of Other Worlds |journal=The Astrophysical Journal |volume=834 |issue=1 |page=17 |doi= 10.3847/1538-4357/834/1/17|arxiv=1603.08614 |s2cid=119114880 |doi-access=free}}</ref>

Solid bodies have irregular surfaces, but 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 and so the overall distribution of mass approaches equilibrium.

=== Atmospheric modeling ===
{{further|Atmospheric model}}
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, keeps the atmosphere bound to Earth and maintains pressure differences with altitude.

== See also ==
* [[List of gravitationally rounded objects of the Solar System]]; a list of objects that have a rounded, ellipsoidal shape due to their own gravity (but are not necessarily in hydrostatic equilibrium)
* [[Statics]]
* [[Two-balloon experiment]]

== References ==
{{reflist}}

== External links ==
* [http://www.astronomynotes.com/starsun/s7.htm Strobel, Nick. (May, 2001). Nick Strobel's Astronomy Notes.]
* {{youTube|lkg1p173TAc|Demonstration}} by Richard Pogge, Ohio State University, Department of Astronomy

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[[Category:Concepts in astrophysics]]
[[Category:Concepts in astronomy]]
[[Category:Definition of planet]]
[[Category:Fluid mechanics]]
[[Category:Hydrostatics]]