Editing Hydrostatic equilibrium (section)

=== 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>