In
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 objec ...
, 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 acce ...
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 potenti ...
of the system. Mathematically, the
theorem states
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 ...
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 languages, 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 ...
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 princip ...
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; 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. 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 tensor ...
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 potenti ...
that is proportional to some power of the
interparticle distance , the virial theorem takes the simple form
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 princip ...
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 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. A ...
,
Henri Poincaré,
Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar (; ) (19 October 1910 – 21 August 1995) was an Indian-American theoretical physicist who spent his professional life in the United States. He shared the 1983 Nobel Prize for Physics with William A. Fowler for " ...
,
Enrico Fermi,
Paul Ledoux,
Richard Bader and
Eugene Parker.
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 ...
.
Richard Bader 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, compar ...
for the stability of
white dwarf 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
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 formula results in:
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 moment of inertia about the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
is defined by the equation
where and represent the mass and position of the th particle. is the position vector magnitude. The scalar is defined by the equation
where is the
momentum vector of the th particle. Assuming that the masses are constant, is one-half the time derivative of this moment of inertia
In turn, the time derivative of can be written
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 acce ...
of the system according to the velocity of each particle
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
where is the force applied by particle on particle . Hence, the virial can be written
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:
where we have assumed that
Newton's third law of motion 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
which is equal and opposite to , the force applied by particle on particle , as may be confirmed by explicit calculation. Hence,
Thus, we have
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
where the coefficient and the exponent are constants. In such cases, the virial is given by the equation
where is the total potential energy of the system
Thus, we have
For gravitating systems the exponent equals −1, giving Lagrange's identity
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
from which we obtain the exact equation
The virial theorem states that if , then
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 :
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:
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
This general result is useful for complex gravitating systems such as
solar system
The Solar System Capitalization 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 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- ...
. 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 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 the ...
.
Evaluate the
commutator of the
Hamiltonian
with the position operator and the momentum operator
of particle ,
Summing over all particles, one finds for
the commutator amounts to
where
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'',
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 non ...
or
Klein–Gordon equation, is
Pokhozhaev's identity, also known as
Derrick's theorem.
Let
be continuous and real-valued, with
.
Denote
.
Let
be a solution to the equation
in the sense of
distributions.
Then
satisfies the relation
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
and . We have,
The last expression can be simplified to
.
Thus, under the conditions described in earlier sections (including
Newton's third law of motion, , despite relativity), the time average for particles with a power law potential is
In particular, the ratio of kinetic energy to potential energy is no longer fixed, but necessarily falls into an interval:
where the more relativistic systems exhibit the larger ratios.
Generalizations
Lord Rayleigh published a generalization of the virial theorem in 1903.
Henri Poincaré 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 tensor ...
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:
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
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 acce ...
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
and is the
electromagnetic stress tensor,
A
plasmoid 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
where is the speed of the
ion acoustic wave (or the
Alfvén wave, 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:
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.
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:
where
is the speed of light,
is the acceleration field constant,
is the mass density of particles,
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). .
where the energy
considered as the kinetic field energy associated with four-current
, and
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 of a system to its
kinetic
Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to:
* Kinetic theory, describing a gas as particles in random motion
* Kinetic energy, the energy of an object that it possesses due to its motion
Art and ent ...
or
thermal energy. Some common virial relations are
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 t ...
, 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 constan ...
, 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 celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
, the mass and size of a galaxy (or general overdensity) is often defined in terms of the "
virial mass" and "
virial radius
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 ...
" 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 of its gas and stars, assuming
circular Keplerian orbits. Using the virial theorem, the
velocity dispersion 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
Here
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.
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 or a
galaxy cluster, 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
where is the
Hubble parameter 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 which case the virial radius is approximated as
The virial mass is then defined relative to this radius as
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 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 stress Virial stress is a measure of mechanical stress on an atomic scale for homogeneous systems.
The expression of the (local) virial stress can be derived as the functional derivative of the free energy of a molecular system with respect to the deforma ...
*
Virial mass
*
Chandrasekhar tensor
*
Chandrasekhar virial equations In astrophysics, the Chandrasekhar virial equations are a hierarchy of moment equations of the Euler equations, developed by the Indian American astrophysicist Subrahmanyan Chandrasekhar, and the physicist Enrico Fermi and Norman R. Lebovitz.
Mat ...
*
Derrick's theorem
*
Equipartition theorem
*
Ehrenfest's theorem
*
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 Theoremat MathPages
*
', Georgia State University
Physics theorems
Dynamics (mechanics)
Solid mechanics
Concepts in physics
Equations of astronomy