Hydrostatic equilibrium

In fluid 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.^[1] For instance, the pressure-gradient force prevents gravity from collapsing Earth's atmosphere into a thin, dense shell, whereas gravity prevents the pressure gradient force from diffusing the atmosphere into space.

Hydrostatic equilibrium is the current distinguishing criterion between dwarf planets and 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. There are 31 observationally confirmed such objects (apart from the Sun), sometimes called planemos,^[2] in the Solar System, seven more^[3] that are virtually certain, and a hundred or so more that are likely.^[3]

Mathematical consideration

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 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 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,

F_{{top}}=-P_{{top}}\cdot A.

Similarly, the force on the volume element from the pressure of the fluid below pushing upwards is

F_{{bottom}}=P_{{bottom}}\cdot A.

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:

F_{{weight}}=-\rho \cdot g\cdot V.

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.

F_{{weight}}=-\rho \cdot g\cdot A\cdot h

By balancing these forces, the total force on the fluid is

\sum F=F_{{bottom}}+F_{{top}}+F_{{weight}}=P_{{bottom}}\cdot A-P_{{top}}\cdot A-\rho \cdot g\cdot A\cdot h.

This sum equals zero if the fluid's velocity is constant. Dividing by A,

0=P_{{bottom}}-P_{{top}}-\rho \cdot g\cdot h.

Or,

P_{{top}}-P_{{bottom}}=-\rho \cdot g\cdot h.

P_top − P_bottom 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 infinitesimally small, the equation can be written in differential form.

dP=-\rho \cdot g\cdot dh.

Density changes with pressure, and gravity changes with height, so the equation would be:

dP=-\rho (P)\cdot g(h)\cdot dh.

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

u=v={\frac {\partial p}{\partial x}}={\frac {\partial p}{\partial y}}=0.

Then the only non-trivial equation is the $z$ -equation, which now reads

{\frac {\partial p}{\partial z}}+\rho g=0.

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

T^{\mu \nu }=(\rho c^{-2}+P)u^{\mu }u^{\nu }+Pg^{\mu \nu }

into the Einstein field equations

R_{{\mu \nu }}={\frac {8\pi G}{c^{4}}}(T_{{\mu \nu }}-{\frac {1}{2}}g_{{\mu \nu }}T)

and using the conservation condition

\nabla _{\mu }T^{\mu \nu }=0

one can derive the Tolman–Oppenheimer–Volkoff equation for the structure of a static, spherically symmetric relativistic star in isotropic coordinates:

{\frac {dP}{dr}}=-{\frac {GM(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)}{rc^{2}}}\right)^{{-1}}

In practice, Ρ and ρ are related by an equation of state of the form f(Ρ,ρ)=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:

M(r)=4\pi \int _{0}^{r}dr'r'^{2}\rho (r').

Per standard procedure in taking the nonrelativistic limit, we let c→∞, so that the factor

\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)}{rc^{2}}}\right)^{{-1}}\rightarrow 1

Therefore, in the nonrelativistic limit the Tolman–Oppenheimer–Volkoff equation reduces to Newton's hydrostatic equilibrium:

\frac{dP}{dr}=-\frac{GM(r)\rho(r)}{r^2}=-g(r)\,\rho(r)\longrightarrow dP = - \rho(h)\,g(h)\, dh

(we have made the trivial notation change h=r and have used f(Ρ,ρ)=0 to express ρ in terms of P).^[4] A similar equation can be computed for rotating, axially symmetric stars, which in its gauge independent form reads:

{\frac {\partial _{i}P}{P+\rho }}-\partial _{i}\ln u^{t}+u_{t}u^{\phi }\partial _{i}{\frac {u_{\phi }}{u_{t}}}=0

Unlike the TOV equilibrium equation, these are two equations (for instance, if as usual when treating stars, one chooses spherical coordinates as basis coordinates $(t,r,\theta,\phi)$ , the index i runs for the coordinates r and $\theta$ ).

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 of their 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% larger at the equator than at the poles. A star with an angular velocity above the critical angular velocity becomes a Jacobi (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.^[5]

If the star has a massive nearby companion object then tidal forces 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.

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 ${\mathcal {L}}_{X}=\Lambda (T_{B})\rho _{B}^{2}$ where $T_{B}$ and $\rho _{B}$ are the temperature and density of the baryonic matter, and $\Lambda (T)$ is some function of temperature and fundamental constants. The baryonic density satisfies the above equation $dP=-\rho gdr$ :

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.

The integral is a measure of the total mass of the cluster, with $r$ being the proper distance to the center of the cluster. Using the ideal gas law $p_{B}=kT_{B}\rho _{B}/m_{B}$ ( $k$ is Boltzmann's constant and $m_{B}$ is a characteristic mass of the baryonic gas particles) and rearranging, we arrive at

{\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.

Multiplying by $r^{2}/\rho _{B}(r)$ and differentiating with respect to $r$ yields

{\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).

If we make the assumption that cold dark matter particles have an isotropic velocity distribution, then the same derivation applies to these particles, and their density $\rho _{D}=\rho _{M}-\rho _{B}$ satisfies the non-linear differential equation

{\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).

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 $\sigma _{D}^{2}$ of the dark matter, which is given by

\sigma _{D}^{2}={\frac {kT_{D}}{m_{D}}}.

The central density ratio $\rho _{B}(0)/\rho _{M}(0)$ is dependent on the redshift $z$ of the cluster and is given by

\rho _{B}(0)/\rho _{M}(0)\propto (1+z)^{2}\left({\frac {\theta }{s}}\right)^{{3/2}}

where $\theta$ is the angular width of the cluster and $s$ the proper distance to the cluster. Values for the ratio range from .11 to .14 for various surveys.^[6]

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, 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 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 Earth. However, in the cases of moons in synchronous orbit, nearly unidirectional tidal forces create a scalene ellipsoid. Also, the dwarf planet Haumea is scalene due to its rapid rotation.

It had been thought that icy objects with a diameter larger than roughly 400 km are usually in hydrostatic equilibrium, whereas those smaller than that are not. Icy objects need less mass for hydrostatic equilibrium than rocky objects. The smallest object that is known to have an equilibrium shape is the icy moon Mimas at 397 km, whereas the largest object known to have an obviously non-equilibrium shape is the rocky asteroid Pallas at 532 km (582 × 556 × 500 ± 18 km). However, Mimas is not actually in hydrostatic equilibrium for its current rotation. The smallest body confirmed to be in hydrostatic equilibrium is the icy moon Rhea, at 1,528 km, whereas the largest body known to not be in hydrostatic equilibrium is the icy moon Iapetus, at 1,470 km.

Because the terrestrial planets and dwarf planets (and likewise the larger satellites, like the Moon and 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 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 Vesta, which may have solidified while in hydrostatic equilibrium but was subsequently significantly deformed by large impacts (now 572.6 × 557.2 × 446.4 km).^[7]

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 Earth and maintaining pressure differences with altitude.

Notes

↑ White (2008). p 63, 66.
↑ Alan Stern considers these all to be "planets", but that conception was rejected by the International Astronomical Union.
1 2 "Astronomer Mike Brown". Gps.caltech.edu. 2013-11-01. Retrieved 2014-06-15.
↑ Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. pp. 451–454. ISBN 9780691145587.
↑ "Gallery : The shape of Planet Earth". Josleys.com. Retrieved 2014-06-15.
↑ Weinberg, Steven (2008). Cosmology. New York: Oxford University Press. pp. 70–71. ISBN 978-0-19-852682-7.
↑ Savage, Don; Jones, Tammy; Villard, Ray (1995-04-19). "Asteroid or Mini-Planet? Hubble Maps the Ancient Surface of Vesta (Key Stages in the Evolution of the Asteroid Vesta)". Hubble Site News Release STScI-1995-20. Retrieved 2006-10-17.

References

White, Frank M. (2008). "Pressure Distribution in a Fluid". Fluid Mechanics. New York: McGraw-Hill. pp. 63–107. ISBN 978-0-07-128645-9.

External links

Strobel, Nick. (May, 2001). Nick Strobel's Astronomy Notes.

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