Virial theorem
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mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, the virial theorem provides a general equation that relates the average over time of the total
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
of a stable system of discrete particles, bound by potential forces, with that of the total
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
of the system. Mathematically, the
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
states \left\langle T \right\rangle = -\frac12\,\sum_^N \bigl\langle \mathbf_k \cdot \mathbf_k \bigr\rangle where is the total kinetic energy of the particles, represents the
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
on the th particle, which is located at position , and
angle brackets A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
represent the average over time of the enclosed quantity. The word virial for the right-hand side of the equation derives from ''vis'', the
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
word for "force" or "energy", and was given its technical definition by
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Sadi Carnot's principle ...
in 1870. The significance of the virial theorem is that it allows the average total kinetic energy to be calculated even for very complicated systems that defy an exact solution, such as those considered in
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
; this average total kinetic energy is related to the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
of the system by the
equipartition theorem In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...
. However, the virial theorem does not depend on the notion of temperature and holds even for systems that are not in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
. The virial theorem has been generalized in various ways, most notably to a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
form. If the force between any two particles of the system results from a
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
that is proportional to some power of the interparticle distance , the virial theorem takes the simple form 2 \langle T \rangle = n \langle V_\text \rangle. Thus, twice the average total kinetic energy equals times the average total potential energy . Whereas represents the potential energy between two particles of distance , represents the total potential energy of the system, i.e., the sum of the potential energy over all pairs of particles in the system. A common example of such a system is a star held together by its own gravity, where equals −1.


History

In 1870,
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Sadi Carnot's principle ...
delivered the lecture "On a Mechanical Theorem Applicable to Heat" to the Association for Natural and Medical Sciences of the Lower Rhine, following a 20-year study of thermodynamics. The lecture stated that the mean
vis viva ''Vis viva'' (from the Latin for "living force") is a historical term used for the first recorded description of what we now call kinetic energy in an early formulation of the principle of conservation of energy. Overview Proposed by Gottfried L ...
of the system is equal to its virial, or that the average kinetic energy is equal to the average potential energy. The virial theorem can be obtained directly from
Lagrange's identity In algebra, Lagrange's identity, named after Joseph Louis Lagrange, is: \begin \left( \sum_^n a_k^2\right) \left(\sum_^n b_k^2\right) - \left(\sum_^n a_k b_k\right)^2 & = \sum_^ \sum_^n \left(a_i b_j - a_j b_i\right)^2 \\ & \left(= \frac \sum_^n ...
as applied in classical gravitational dynamics, the original form of which was included in Lagrange's "Essay on the Problem of Three Bodies" published in 1772. Karl Jacobi's generalization of the identity to ''N'' bodies and to the present form of Laplace's identity closely resembles the classical virial theorem. However, the interpretations leading to the development of the equations were very different, since at the time of development, statistical dynamics had not yet unified the separate studies of thermodynamics and classical dynamics. The theorem was later utilized, popularized, generalized and further developed by
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
,
Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Am ...
,
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
, Subrahmanyan Chandrasekhar,
Enrico Fermi Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and ...
,
Paul Ledoux Paul Ledoux (8 August 1914 – 6 October 1988) was a Belgian astrophysicist best known for his work on stellar stability and variability. With Theodore Walraven, he co-authored a seminal work on stellar oscillations. In 1964 Ledoux was awar ...
,
Richard Bader Richard F. W. Bader (October 15, 1931 – January 15, 2012) was a Canadian quantum chemist, noted for his work on the Atoms in molecules theory. This theory attempts to establish a physical basis for many of the working concepts of chemistry ...
and
Eugene Parker Eugene Newman Parker (June 10, 1927 – March 15, 2022) was an American solar and plasma physicist. In the 1950s he proposed the existence of the solar wind and that the magnetic field in the outer Solar System would be in the shape of a Pa ...
.
Fritz Zwicky Fritz Zwicky (; ; February 14, 1898 – February 8, 1974) was a Swiss astronomer. He worked most of his life at the California Institute of Technology in the United States of America, where he made many important contributions in theoretical an ...
was the first to use the virial theorem to deduce the existence of unseen matter, which is now called
dark matter Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not a ...
.
Richard Bader Richard F. W. Bader (October 15, 1931 – January 15, 2012) was a Canadian quantum chemist, noted for his work on the Atoms in molecules theory. This theory attempts to establish a physical basis for many of the working concepts of chemistry ...
showed the charge distribution of a total system can be partitioned into its kinetic and potential energies that obey the virial theorem. As another example of its many applications, the virial theorem has been used to derive the
Chandrasekhar limit The Chandrasekhar limit () is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about (). White dwarfs resist gravitational collapse primarily through electron degeneracy pressure, compa ...
for the stability of
white dwarf A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes fro ...
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
s.


Illustrative special case

Consider particles with equal mass , acted upon by mutually attractive forces. Suppose the particles are at diametrically opposite points of a circular orbit with radius . The velocities are and , which are normal to forces and . The respective magnitudes are fixed at and . The average kinetic energy of the system is \langle T \rangle = \sum_^N \frac12 m_k \left, \mathbf_k \^2 = \frac12 m, \mathbf_1, ^2 + \frac12 m, \mathbf_2, ^2 = mv^2. Taking center of mass as the origin, the particles have positions and with fixed magnitude . The attractive forces act in opposite directions as positions, so . Applying the
centripetal force A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
formula results in: -\frac12 \sum_^N \bigl\langle \mathbf_k \cdot \mathbf_k \bigr\rangle = -\frac12(-Fr - Fr) = Fr = \frac \cdot r = mv^2 = \langle T \rangle, as required. Note: If the origin is displaced then we'd obtain the same result. This is because the dot product of the displacement with equal and opposite forces , results in net cancellation.


Statement and derivation

Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step. For a collection of point particles, the
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
moment of inertia about the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
is defined by the equation I = \sum_^N m_k \left, \mathbf_k \^2 = \sum_^N m_k r_k^2 where and represent the mass and position of the th particle. is the position vector magnitude. The scalar is defined by the equation G = \sum_^N \mathbf_k \cdot \mathbf_k where is the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
of the th particle. Assuming that the masses are constant, is one-half the time derivative of this moment of inertia \begin \frac12 \frac &= \frac12 \frac \sum_^N m_k \mathbf_k \cdot \mathbf_k \\ &= \sum_^N m_k \, \frac \cdot \mathbf_k \\ &= \sum_^N \mathbf_k \cdot \mathbf_k = G\,. \end In turn, the time derivative of can be written \begin \frac & = \sum_^N \mathbf_k \cdot \frac + \sum_^N \frac \cdot \mathbf_k \\ & = \sum_^N m_k \frac \cdot \frac + \sum_^N \mathbf_k \cdot \mathbf_k \\ & = 2 T + \sum_^N \mathbf_k \cdot \mathbf_k \end where is the mass of the th particle, is the net force on that particle, and is the total
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
of the system according to the velocity of each particle T = \frac12 \sum_^N m_k v_k^2 = \frac12 \sum_^N m_k \frac \cdot \frac.


Connection with the potential energy between particles

The total force on particle is the sum of all the forces from the other particles in the system \mathbf_k = \sum_^N \mathbf_ where is the force applied by particle on particle . Hence, the virial can be written -\frac12\,\sum_^N \mathbf_k \cdot \mathbf_k = -\frac12\,\sum_^N \sum_^N \mathbf_ \cdot \mathbf_k \,. Since no particle acts on itself (i.e., for ), we split the sum in terms below and above this diagonal and we add them together in pairs: \begin \sum_^N \mathbf_k \cdot \mathbf_k & = \sum_^N \sum_^N \mathbf_ \cdot \mathbf_k = \sum_^N \sum_^ \left( \mathbf_ \cdot \mathbf_k + \mathbf_ \cdot \mathbf_j \right) \\ & = \sum_^N \sum_^ \left( \mathbf_ \cdot \mathbf_k - \mathbf_ \cdot \mathbf_j \right) = \sum_^N \sum_^ \mathbf_ \cdot \left( \mathbf_k - \mathbf_j \right) \end where we have assumed that
Newton's third law of motion Newton's laws of motion are three basic Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at re ...
holds, i.e., (equal and opposite reaction). It often happens that the forces can be derived from a potential energy that is a function only of the distance between the point particles and . Since the force is the negative gradient of the potential energy, we have in this case \mathbf_ = -\nabla_ V_ = - \frac \left( \frac \right), which is equal and opposite to , the force applied by particle on particle , as may be confirmed by explicit calculation. Hence, \begin \sum_^N \mathbf_k \cdot \mathbf_k &=\sum_^N \sum_^ \mathbf_ \cdot \left( \mathbf_k - \mathbf_j \right) \\ &=-\sum_^N \sum_^ \frac \frac \\ &=-\sum_^N \sum_^ \frac r_. \end Thus, we have \frac = 2 T + \sum_^N \mathbf_k \cdot \mathbf_k = 2 T - \sum_^N \sum_^ \frac r_.


Special case of power-law forces

In a common special case, the potential energy between two particles is proportional to a power of their distance V_ = \alpha r_^n, where the coefficient and the exponent are constants. In such cases, the virial is given by the equation \begin -\frac12\,\sum_^N \mathbf_k \cdot \mathbf_k &=\frac12\,\sum_^N \sum_ \frac r_ \\ &=\frac12\,\sum_^N \sum_ n \alpha r_^ r_ \\ &=\frac12\,\sum_^N \sum_ n V_ = \frac\, V_\text \end where is the total potential energy of the system V_\text = \sum_^N \sum_ V_ \,. Thus, we have \frac = 2 T + \sum_^N \mathbf_k \cdot \mathbf_k = 2 T - n V_\text \,. For gravitating systems the exponent equals −1, giving Lagrange's identity \frac = \frac12 \frac = 2 T + V_\text which was derived by Joseph-Louis Lagrange and extended by Carl Jacobi.


Time averaging

The average of this derivative over a duration of time, , is defined as \left\langle \frac \right\rangle_\tau = \frac\tau \int_0^\tau \frac\,dt = \frac \int_^ \, dG = \frac, from which we obtain the exact equation \left\langle \frac \right\rangle_\tau = 2 \left\langle T \right\rangle_\tau + \sum_^N \left\langle \mathbf_k \cdot \mathbf_k \right\rangle_\tau. The virial theorem states that if , then 2 \left\langle T \right\rangle_\tau = -\sum_^N \left\langle \mathbf_k \cdot \mathbf_k \right\rangle_\tau. There are many reasons why the average of the time derivative might vanish, . One often-cited reason applies to stably-bound systems, that is to say systems that hang together forever and whose parameters are finite. In that case, velocities and coordinates of the particles of the system have upper and lower limits so that , is bounded between two extremes, and , and the average goes to zero in the limit of infinite : \lim_ \left, \left\langle \frac \right\rangle_\tau \ = \lim_ \left, \frac \ \le \lim_ \frac = 0. Even if the average of the time derivative of is only approximately zero, the virial theorem holds to the same degree of approximation. For power-law forces with an exponent , the general equation holds: \langle T \rangle_\tau = -\frac12 \sum_^N \langle \mathbf_k \cdot \mathbf_k \rangle_\tau = \frac \langle V_\text \rangle_\tau. For
gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
al attraction, equals −1 and the average kinetic energy equals half of the average negative potential energy \langle T \rangle_\tau = -\frac12 \langle V_\text \rangle_\tau. This general result is useful for complex gravitating systems such as
solar system The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar S ...
s or galaxies. A simple application of the virial theorem concerns
galaxy clusters A galaxy cluster, or a cluster of galaxies, is a structure that consists of anywhere from hundreds to thousands of galaxies that are bound together by gravity, with typical masses ranging from 1014 to 1015 solar masses. They are the second-l ...
. If a region of space is unusually full of galaxies, it is safe to assume that they have been together for a long time, and the virial theorem can be applied. Doppler effect measurements give lower bounds for their relative velocities, and the virial theorem gives a lower bound for the total mass of the cluster, including any dark matter. If the ergodic hypothesis holds for the system under consideration, the averaging need not be taken over time; an
ensemble average In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents ...
can also be taken, with equivalent results.


In quantum mechanics

Although originally derived for classical mechanics, the virial theorem also holds for quantum mechanics, as first shown by Fock using the
Ehrenfest theorem The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of th ...
. Evaluate the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
of the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
H=V\bigl(\\bigr)+\sum_n \frac with the position operator and the momentum operator P_n=-i\hbar \frac of particle , ,X_nP_nX_n ,P_n ,X_n_n=i\hbar X_n\frac-i\hbar\frac~. Summing over all particles, one finds for Q=\sum_n X_nP_n the commutator amounts to \frac ,Q2 T-\sum_n X_n\frac where T=\sum_n \frac is the kinetic energy. The left-hand side of this equation is just , according to the
Heisenberg equation In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators ( observables and others) incorporate a dependency on time, b ...
of motion. The expectation value of this time derivative vanishes in a stationary state, leading to the ''quantum virial theorem'', 2\langle T\rangle = \sum_n\left\langle X_n \frac\right\rangle ~.


Pokhozhaev's identity

In the field of quantum mechanics, there exists another form of the virial theorem, applicable to localized solutions to the stationary
nonlinear Schrödinger equation In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlin ...
or
Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. ...
, is Pokhozhaev's identity, also known as Derrick's theorem. Let g(s) be continuous and real-valued, with g(0)=0. Denote G(s)=\int_0^s g(t)\,dt. Let u\in L^\infty_(\R^n), \qquad \nabla u\in L^2(\R^n), \qquad G(u(\cdot))\in L^1(\R^n), \qquad n\in\N, be a solution to the equation -\nabla^2 u=g(u), in the sense of distributions. Then u satisfies the relation (n-2)\int_, \nabla u(x), ^2\,dx=n\int_G(u(x))\,dx.


In special relativity

For a single particle in special relativity, it is not the case that . Instead, it is true that , where is the
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
\gamma = \frac and . We have, \begin \frac 12 \mathbf \cdot \mathbf &= \frac 12 \boldsymbol \gamma mc \cdot \boldsymbol c \\ pt&= \frac 12 \gamma \beta^2 mc^2 \\ pt&= \left( \frac\right) T \,. \end The last expression can be simplified to \left(\frac\right) T \qquad \text \qquad \left(\frac\right) T. Thus, under the conditions described in earlier sections (including
Newton's third law of motion Newton's laws of motion are three basic Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at re ...
, , despite relativity), the time average for particles with a power law potential is \frac \left\langle V_\mathrm \right\rangle_\tau = \left\langle \sum_^N \left(\frac\right) T_k \right\rangle_\tau = \left\langle \sum_^N \left(\frac\right) T_k \right\rangle_\tau \,. In particular, the ratio of kinetic energy to potential energy is no longer fixed, but necessarily falls into an interval: \frac \in \left , 2\right,, where the more relativistic systems exhibit the larger ratios.


Generalizations

Lord Rayleigh published a generalization of the virial theorem in 1903.
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
proved and applied a form of the virial theorem in 1911 to the problem of formation of the Solar System from a proto-stellar cloud (then known as cosmogony). A variational form of the virial theorem was developed in 1945 by Ledoux. A
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
form of the virial theorem was developed by Parker, Chandrasekhar and Fermi. The following generalization of the virial theorem has been established by Pollard in 1964 for the case of the inverse square law: 2\lim_\langle T\rangle_\tau = \lim_\langle U\rangle_\tau \qquad \text \quad \lim_^I(\tau)=0\,. A ''boundary'' term otherwise must be added.


Inclusion of electromagnetic fields

The virial theorem can be extended to include electric and magnetic fields. The result is \frac12\frac + \int_Vx_k\frac \, d^3r = 2(T+U) + W^\mathrm + W^\mathrm - \int x_k(p_+T_) \, dS_i, where is the moment of inertia, is the momentum density of the electromagnetic field, is the
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
of the "fluid", is the random "thermal" energy of the particles, and are the electric and magnetic energy content of the volume considered. Finally, is the fluid-pressure tensor expressed in the local moving coordinate system p_ = \Sigma n^\sigma m^\sigma \langle v_iv_k\rangle^\sigma - V_iV_k\Sigma m^\sigma n^\sigma, and is the
electromagnetic stress tensor The Maxwell stress tensor (named after James Clerk Maxwell) is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a ...
, T_ = \left( \frac + \frac \right) \delta_ - \left( \varepsilon_0E_iE_k + \frac \right). A
plasmoid A plasmoid is a coherent structure of plasma and magnetic fields. Plasmoids have been proposed to explain natural phenomena such as ball lightning, magnetic bubbles in the magnetosphere, and objects in cometary tails, in the solar wind, in th ...
is a finite configuration of magnetic fields and plasma. With the virial theorem it is easy to see that any such configuration will expand if not contained by external forces. In a finite configuration without pressure-bearing walls or magnetic coils, the surface integral will vanish. Since all the other terms on the right hand side are positive, the acceleration of the moment of inertia will also be positive. It is also easy to estimate the expansion time . If a total mass is confined within a radius , then the moment of inertia is roughly , and the left hand side of the virial theorem is . The terms on the right hand side add up to about , where is the larger of the plasma pressure or the magnetic pressure. Equating these two terms and solving for , we find \tau\,\sim \frac, where is the speed of the
ion acoustic wave In plasma physics, an ion acoustic wave is one type of longitudinal oscillation of the ions and electrons in a plasma, much like acoustic waves traveling in neutral gas. However, because the waves propagate through positively charged ions, ion aco ...
(or the
Alfvén wave In plasma physics, an Alfvén wave, named after Hannes Alfvén, is a type of plasma wave in which ions oscillate in response to a restoring force provided by an effective tension on the magnetic field lines. Definition An Alfvén wave is ...
, if the magnetic pressure is higher than the plasma pressure). Thus the lifetime of a plasmoid is expected to be on the order of the acoustic (or Alfvén) transit time.


Relativistic uniform system

In case when in the physical system the pressure field, the electromagnetic and gravitational fields are taken into account, as well as the field of particles’ acceleration, the virial theorem is written in the relativistic form as follows: \left\langle W_k \right\rangle \approx - 0.6 \sum_^N\langle\mathbf_k\cdot\mathbf_k\rangle , where the value exceeds the kinetic energy of the particles by a factor equal to the Lorentz factor of the particles at the center of the system. Under normal conditions we can assume that , then we can see that in the virial theorem the kinetic energy is related to the potential energy not by the coefficient , but rather by the coefficient close to 0.6. The difference from the classical case arises due to considering the pressure field and the field of particles’ acceleration inside the system, while the derivative of the scalar is not equal to zero and should be considered as the
material derivative In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material der ...
. An analysis of the integral theorem of generalized virial makes it possible to find, on the basis of field theory, a formula for the root-mean-square speed of typical particles of a system without using the notion of temperature: v_\mathrm = c \sqrt , where ~ c is the speed of light, ~ \eta is the acceleration field constant, ~ \rho_0 is the mass density of particles, ~ r is the current radius. Unlike the virial theorem for particles, for the electromagnetic field the virial theorem is written as follows:Fedosin S.G
The Integral Theorem of the Field Energy.
Gazi University Journal of Science. Vol. 32, No. 2, pp. 686-703 (2019). .
~ E_ + 2 W_f =0 , where the energy ~ E_ = \int A_\alpha j^\alpha \sqrt \,dx^1 \,dx^2 \,dx^3 considered as the kinetic field energy associated with four-current ~ j^\alpha , and ~ W_f = \frac \int F_ F^ \sqrt \,dx^1 \,dx^2 \,dx^3 sets the potential field energy found through the components of the electromagnetic tensor.


In astrophysics

The virial theorem is frequently applied in astrophysics, especially relating the
gravitational potential energy Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conver ...
of a system to its
kinetic Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory of gases, Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to i ...
or
thermal energy The term "thermal energy" is used loosely in various contexts in physics and engineering. It can refer to several different well-defined physical concepts. These include the internal energy or enthalpy of a body of matter and radiation; heat, d ...
. Some common virial relations are \frac35 \frac = \frac32 \frac = \frac12 v^2 for a mass , radius , velocity , and temperature . The constants are
Newton's constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
, the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, and proton mass . Note that these relations are only approximate, and often the leading numerical factors (e.g. or ) are neglected entirely.


Galaxies and cosmology (virial mass and radius)

In
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, the mass and size of a galaxy (or general overdensity) is often defined in terms of the "
virial mass In astrophysics, the virial mass is the mass of a gravitationally bound astrophysical system, assuming the virial theorem applies. In the context of galaxy formation and dark matter halos, the virial mass is defined as the mass enclosed within th ...
" and " virial radius" respectively. Because galaxies and overdensities in continuous fluids can be highly extended (even to infinity in some models, such as an isothermal sphere), it can be hard to define specific, finite measures of their mass and size. The virial theorem, and related concepts, provide an often convenient means by which to quantify these properties. In galaxy dynamics, the mass of a galaxy is often inferred by measuring the
rotation velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an object ...
of its gas and stars, assuming circular Keplerian orbits. Using the virial theorem, the
velocity dispersion In astronomy, the velocity dispersion (''σ'') is the statistical dispersion of velocities about the mean velocity for a group of astronomical objects, such as an open cluster, globular cluster, galaxy, galaxy cluster, or supercluster. By measurin ...
can be used in a similar way. Taking the kinetic energy (per particle) of the system as , and the potential energy (per particle) as we can write \frac \approx \sigma^2. Here R is the radius at which the velocity dispersion is being measured, and is the mass within that radius. The virial mass and radius are generally defined for the radius at which the velocity dispersion is a maximum, i.e. \frac \approx \sigma_\max^2. As numerous approximations have been made, in addition to the approximate nature of these definitions, order-unity proportionality constants are often omitted (as in the above equations). These relations are thus only accurate in an
order of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic di ...
sense, or when used self-consistently. An alternate definition of the virial mass and radius is often used in cosmology where it is used to refer to the radius of a sphere, centered on a
galaxy A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...
or a
galaxy cluster A galaxy cluster, or a cluster of galaxies, is a structure that consists of anywhere from hundreds to thousands of galaxies that are bound together by gravity, with typical masses ranging from 1014 to 1015 solar masses. They are the second-lar ...
, within which virial equilibrium holds. Since this radius is difficult to determine observationally, it is often approximated as the radius within which the average density is greater, by a specified factor, than the critical density \rho_\text=\frac where is the
Hubble parameter Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving a ...
and is the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
. A common choice for the factor is 200, which corresponds roughly to the typical over-density in spherical top-hat collapse (see
Virial mass In astrophysics, the virial mass is the mass of a gravitationally bound astrophysical system, assuming the virial theorem applies. In the context of galaxy formation and dark matter halos, the virial mass is defined as the mass enclosed within th ...
), in which case the virial radius is approximated as r_\text \approx r_= r, \qquad \rho = 200 \cdot \rho_\text. The virial mass is then defined relative to this radius as M_\text \approx M_ = \frac43\pi r_^3 \cdot 200 \rho_\text .


In stars

The virial theorem is applicable to the cores of stars, by establishing a relation between gravitational potential energy and thermal kinetic energy (i.e. temperature). As stars on the
main sequence In astronomy, the main sequence is a continuous and distinctive band of stars that appears on plots of stellar color versus brightness. These color-magnitude plots are known as Hertzsprung–Russell diagrams after their co-developers, Ejnar Her ...
convert hydrogen into helium in their cores, the mean molecular weight of the core increases and it must contract to maintain enough pressure to support its own weight. This contraction decreases its potential energy and, the virial theorem states, increases its thermal energy. The core temperature increases even as energy is lost, effectively a negative
specific heat In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
. This continues beyond the main sequence, unless the core becomes degenerate since that causes the pressure to become independent of temperature and the virial relation with equals −1 no longer holds.


See also

*
Virial coefficient Virial coefficients B_i appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law. They are characteristic of the interaction potential ...
* Virial stress *
Virial mass In astrophysics, the virial mass is the mass of a gravitationally bound astrophysical system, assuming the virial theorem applies. In the context of galaxy formation and dark matter halos, the virial mass is defined as the mass enclosed within th ...
* Chandrasekhar tensor * Chandrasekhar virial equations * Derrick's theorem *
Equipartition theorem In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...
*
Ehrenfest's theorem The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of th ...
* Pokhozhaev's identity


References


Further reading

* * {{Cite book , last=Collins , first=G. W. , year=1978 , title=The Virial Theorem in Stellar Astrophysics , publisher=Pachart Press , url=http://ads.harvard.edu/books/1978vtsa.book/ , bibcode=1978vtsa.book.....C , isbn=978-0-912918-13-6


External links


The Virial Theorem
at MathPages *

', Georgia State University Physics theorems Dynamics (mechanics) Solid mechanics Concepts in physics Equations of astronomy