Virial sentence

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The virial theorem ( Latin vis 'force') is a relationship between the temporal arithmetic mean values ​​of the kinetic energy  and the potential energy of a closed physical system . The virial theorem was established by Rudolf Clausius in 1870 in the essay On a mechanical theorem applicable to heat .  

The virial is Clausius expression

Here referred to

  • the force acting on the -th particle
  • the position vector of the -th particle
  • the cross line a mean value explained in more detail below, e.g. B. a time or crowd means .

The virial theorem was originally formulated by Clausius as the theorem of classical mechanics (as equality of virial and mean kinetic energy). It enables general estimates of the proportions of potential and kinetic energy, even in complex systems, e.g. B. in multi-body problems in astrophysics . There is also a quantum mechanical virial theorem, a virial theorem of statistical mechanics , from which u. a. the ideal gas law and corrections for real gases were derived, as well as a relativistic virial theorem.

The virial theorem is only valid under certain conditions, for example in the case of the virial theorem of mechanics, that with averaging over time, the locations and velocities of the particles are limited, or that there is thermal equilibrium .

Virial theorem of mechanics

Particle in a conservative force field

A simple case is represented by non-interacting particles in an external force field that is conservative , i.e. derived from a potential (the associated charge is denoted by, it is precisely the mass in the case of gravity ):  

This is the gradient of the field or the potential.

The virial theorem applies, as will be shown below, if the motion remains in the finite, i.e. place and momentum are limited for all times, and reads

in which

  • is the kinetic energy of the particle
  • the dash denotes the mean value for times over time .

If one also assumes a homogeneous potential of degree  in the position variable , i. H. it applies to (values ​​for k can be found below in Consequences and Examples ), then the above equation is simplified with Euler's equation for homogeneous functions :


where is the total potential energy of the particles. The virial theorem is therefore a relationship between mean kinetic and mean potential energy.

Interacting particles

For the derivation of the gas laws and their application in astrophysics, the case of a closed system of interacting particles is of particular interest. As above, assuming a finite motion, the virial theorem results:

It is the resultant of the forces acting -th particles forces of the other are applied particles of the system. Since a closed system is considered, there are no external forces this time. Because of the following, the choice of the origin for the position vectors in the virial is arbitrary. At first glance, the expression in the virial looks complicated, but it can be derived from the assumption that the forces acting in pairs between the particles can each be derived from homogeneous potentials of degree , as above on the form


Conclusions and Examples

With the total energy it follows from the virial theorem:

For the known case ( gravitation , Coulomb force ) z. B .:

In particular, it follows that the total energy for the application of the virial theorem must be negative in the case (since is positive).

For harmonic oscillations ( ) the following applies:


Here the description in the textbook by Landau and Lifschitz is followed, where the virial theorem is discussed in connection with the scaling behavior of mechanical quantities ( mechanical similarity ) . It is only used that the kinetic energy is quadratic in the velocities , and the impulses are formally introduced via . Then by Euler's theorem about homogeneous functions

from what

follows, where the sum of the scalar products from the momenta and the locations of all particles is:

Now the asymptotic limit value of the time average is formed:

In particular, the following applies to the mean value of the time derivative of :

If one is dealing with a system in which the speeds and locations of the particles are limited (e.g. in the case of periodic orbits), then it follows

and with further the virial sentence

if one assumes that the potential is  a homogeneous function of the spatial coordinates of the degree  . From this point of view, the theorem expresses the equality of mean values ​​of kinetic and potential energy with pre-factors resulting from the scaling behavior: 2 for kinetic energy, since the velocities or impulses come in quadratically, for potential, since the position variables come in with a power  .

A similar derivation can already be found in Clausius and in the textbook on classical mechanics by Herbert Goldstein . Goldstein also points out that the virial theorem with the potential term also applies if, in addition to the potential forces, there are frictional forces that are proportional to the speed, since these do not contribute to the virial theorem. However, this only applies if a steady state occurs, i.e. if energy is supplied so that the movement does not come to a complete standstill, because then all time averages would disappear.

Special cases of averaging

As with Clausius, the horizontal line usually denotes the mean value for times over time . In certain special cases, however, this can also be simplified.

Closed tracks

If the trajectories are closed , the time average can be replaced by averaging over a period . The virial theorem follows here directly from the periodicity of the movement.

In two special cases of homogeneous potentials, namely for the potential of the harmonic oscillator ( ) and for the Coulomb potential ( ), one always obtains closed orbits for finite (i.e. not infinite) movements in the one- or two-body problem.

Many-body system

If a many-particle system is in thermal equilibrium , the system can be regarded as ergodic , i.e. That is , the time mean is the same as the cluster mean for all observation quantities . Since this applies in particular to the kinetic and potential energy and the cluster mean of the energies is formed from the sum of the individual energies divided by the number  of objects, the cluster mean can be expressed by the total energies. We therefore get for equilibrium systems

without averaging over time, because the values ​​are constant over time (see also below the treatment of the virial theorem in the context of statistical mechanics).


For the gravitational particle system in astrophysics (e.g. as a model of galaxy and star clusters ) the above The basic requirement in the derivation of the virial theorem, namely that the system remains spatially limited, is not given for long periods of time . All these piles dissolve at some point, as particles repeatedly collect enough energy through mutual interaction (interference) with the others to escape.

However, the periods of time in which this happens are very long: In astrophysics, the relaxation time of a star cluster or a galaxy defines the time in which an equilibrium distribution is established. For the Milky Way it is  years (at an age of  years) and for typical globular clusters it is  years. According to Maxwell's speed distribution , 0.74 percent of the stars reach the escape speed within the period and escape.

Numerical calculations showed that the proportion is even slightly higher and that the virial theorem in the clusters is well fulfilled due to the equilibrium that is established (with a start-up time of two to three relaxation times). After the expires , 90 percent of the stars have migrated.

Application example: determining the mass of astronomical clusters

The virial theorem is used, for example, in astrophysics and celestial mechanics . There one uses the Newtonian gravitational potential , which is homogeneous of degree −1. Then:

The virial theorem makes it possible to find quite good estimates for the total masses of dynamically bound systems such as star clusters , galaxies or galaxy clusters . The total mass of such a heap can then be fully expressed by observations such as radial velocities , angular distances and apparent brightnesses of the individual objects. The only requirement for the application of the virial theorem is to know the distance of the cluster. We want to sketch the procedure here based on the mass determination of such a cluster:

The total kinetic energy of a star or galaxy cluster is through

given. But neither the individual masses  nor the amounts of speed  are observable quantities. To introduce this, the contributions of the individual objects must be expressed by the total mass and suitable mean values. For example, one can assume that the individual masses are  proportional to the individual luminosity and form a luminosity-weighted mean ( indicated by the index ):  

If one assumes that the system is spherically symmetrical and is in equilibrium (one then also says that it is virialized ), then the speeds are evenly distributed over the spatial directions and it applies

where or are the spreads (deviations from the mean) of the velocities, i.e. the spatial or radial velocities relative to the center of gravity of the heap. For example, the galaxies of the Coma cluster have a Gaussian distribution of velocities with a spread of 1000 km / s. This gives:

On the other hand applies to the potential total energy under the condition of spherical symmetry


  • the gravitational constant  ,
  • the total radius of  the system and
  • a factor  which is of the order of magnitude 1 and which depends on the radial distribution function , i.e. the geometry of the pile. For an (albeit unrealistic) uniform distribution within the radius  , for example . In general, the factor is to be determined from the observed angular distances between the individual systems and the cluster center.

Applying the virial theorem for gravity, we get the total mass of the cluster as:

The mass resulting from the observation is called the virial mass. Since it is of the order of magnitude 1, you can also see that the mean speed  corresponds approximately to the escape speed (with an exact match for ).

Although this method of determining the mass is fraught with uncertainties, when measuring strongly differing escape speeds from galaxy clusters and interpreting the redshift, Fritz Zwicky noted as early as 1933 that a large part of the mass could be very dense in the form of dark matter : the sum of Masses of the cluster's visible galaxies were an order of magnitude lower. Because to explain the redshift a 400 times greater mass density is required than the density derived from the masses of luminous matter. “If this were to be the case, the surprising result would be that dark matter is present in much greater density than luminous matter.” In elliptical galaxies, too, the virial mass was found to be 10 to 100 times greater than the luminous mass is. In contrast to spiral galaxies, where the mass can be determined from the rotation curve, the virial method is often the only method of mass determination for elliptical galaxies.

Another astrophysical application is the estimation of jeans mass and the theorem is also used in studies of the stability of gas sphere models for stars. For an ideal gas held together by gravity as a star model, the virial theorem shows that the star cannot cool down in the end phase (when all fusion processes have come to a standstill). If the amount of gravitational binding energy increases  due to the contraction of the star, half of the increase goes into the kinetic energy of the star matter, which is considered to be ideal gas, and thus increases the temperature, the rest is radiated. If the pressure inside becomes too high, the description as a classic ideal gas collapses, however, as a degenerate Fermigas forms (white dwarf).

Tensor form

In the context of continuum mechanics , the tensorial virial theorem from the shock- free Boltzmann equation is proven and used in astrophysics.

If gravitation is again assumed as the interaction , the sentence has the form


  • the inertia tensor 
  • the tensor  of kinetic energy,
  • the stress tensor  and
  • the tensor of potential energy.

In the static case, the time derivative on the left side of the equation is omitted, and since the stress tensor is without a trace, the trace of the equation again gives the scalar virial theorem.

The occurrence of the second time derivative of the inertia tensor can be motivated by the following reformulation of in the scalar case:

with the scalar moment of inertia

Variants in astrophysics

The following form of the virial theorem was used for applications in astrophysics

first derived from Henri Poincaré and Arthur Eddington .

For stationary systems the left side disappears, and in the application considered the potential gravitational energy of the particles of a gas cloud or of the stars in galaxies was:

In celestial mechanics , this form of the virial theorem was already known to Joseph-Louis Lagrange (1772, in a treatise on the three-body problem ) and was generalized by Carl Gustav Jacobi (lectures on dynamics).

A division of the kinetic energy into

  • a share  of hydrodynamic rivers and
  • a proportion of  the accidental heat movement

as well as an additional consideration for the potential energy

returns the virial theorem in the following scalar form:

A tensor form of this virial theorem for astrophysical applications in the presence of magnetic fields was given by Eugene Parker in 1954 and by Subramanyan Chandrasekhar and Enrico Fermi in 1953 . Chandrasekhar also developed specialized virial theorems for his discussion of the equilibrium figures of rotating fluids.

In plasma physics , as an application of the virial theorem, it can be shown that there are no stationary, finite plasma configurations (plasmoids) enclosed by their own magnetic fields. Instead, for the inclusion of the plasma z. B. external walls or external magnetic fields required.

The virial theorem of quantum mechanics

For quantum mechanics , the virial theorem retains its validity, as was shown by Fock .

The Hamiltonian of the system of point particles is

Form the commutator of with , formed from the position operator and the momentum operator of the -th particle:   

If one forms by summation over the particles , it follows

with the kinetic energy  .

According to Heisenberg's equations of motion , the left side is the same . The expected value disappears in a stationary state , so with  

the quantum version of the virial theorem follows, with the pointed brackets standing for quantum mechanical expectation values ​​of the respective operators for a stationary state.

The virial theorem of statistical mechanics

Like the uniform distribution theorem , a version of the virial theorem also belongs to the general statements of classical statistical mechanics.

As a mean formation with the help of the canonical ensemble one obtains (see the equal distribution theorem):

with .

The equation below gives:


thus a contribution per degree of freedom for the mean kinetic energy (uniform distribution theorem).

The lower and upper equations together give the virial theorem of statistical mechanics :

which also applies to quantum statistics .

According to Clausius, it is customary to divide the contribution of the potential into

  • the inner virial , d. H. the contribution of the potential of the internal forces (interaction of the particles with each other) and
  • the outer virial , d. H. the contribution of the wall potential or the forces on the wall.

The outer virial provides:


  • the pressure and
  • the volume .

Thereby the surface (wall) was integrated and the Gaussian integral theorem was applied.

This gives the virial form of the thermal equation of state :


so for particles with the uniform distribution theorem:

This is the ideal gas equation with the virial of the internal forces as an additional term. The virial can be developed according to powers of the particle density  (see: Virial development ) for the development of equations of state for real gases .

The derivation of the gas equation was the main goal of Clausius' original work, using the virial theorem of mechanics as a basis.

The virial theorem of the theory of relativity

There is also a relativistic virial theorem. For particles in interaction with electromagnetic fields it can be found in the textbook on theoretical physics by Landau and Lifschitz, but it can also be formulated for other interactions .

Since the trace of the energy-momentum tensor of the electromagnetic field disappears, one can - using the four-dimensional energy conservation law for systems with limited motion (impulses, coordinates, etc. vary between finite bounds, the electromagnetic fields vanish at infinity) - similar to the classical virial theorem show by averaging over time:


  • the total energy of the system
    • the energy-momentum tensor of  the entire system of particles and fields
    • the four-dimensional index 
    • the track  using Einstein's summation convention .

For small velocities the classical form of the virial theorem for the Coulomb potential results :

whereby the rest energies of the particles are subtracted from the total energy.

Relativistic versions of the virial theorem were already applied by Chandrasekhar to white dwarfs . He also examined versions in general relativity in the context of the post-Newton approximation.


Gives a simple derivation of the scalar virial theorem.
Here one finds the tensorial generalization and applications.
  • Wilhelm Brenig : Statistical theory of heat. 3rd edition, Springer 1992, pp. 144 f. (Virial theorem in statistical mechanics).
  • George W. Collins : The Virial Theorem in Stellar Astrophysics. Pachart Press, 1978, online.
  • R. Becker : Theory of Heat. 1961, p. 85 (on the outer virial).
  • Albrecht Unsöld : The New Cosmos. Springer, 2nd ed., 1974, p. 283, derivation and significance for calculating the structure of stars. (Not in the 1966 BI paperback.)

Web links

References and comments

  1. ^ R. Clausius : About a mechanical theorem applicable to heat. Annalen der Physik, Volume 217, 1870, pp. 124-130.
  2. a b H. Goldstein : Classical Mechanics. Akademische Verlagsgesellschaft, 1978, p. 76 f.
  3. The definitions of the virial vary somewhat, e.g. For example, both Wolfgang Pauli in his lectures on thermodynamics (ETH Zurich 1958) and the book by Honerkamp cited below omit the prefactor −1/2 in the definition of the virial and Pauli also omits the mean formation.
  4. a b J. Honerkamp, H. Römer : Classical Theoretical Physics . Springer, 2012, ISBN 978-3-642-23262-6 ( Chapter 2.12: The virial sentence in the Google book search).
  5. J. Wess : Theoretical Mechanics . Springer-Verlag, 2008, ISBN 978-3-540-74869-4 ( Chapter 13: Homogeneous potentials in the Google book search).
  6. a b c H. Voigt: Outline of astronomy. BI Verlag, 1980, p. 367 ff., P. 487.
  7. Sebastian von Hoerner: Journal for Astrophysics. Volume 50, 1960, 184. Then about five times higher.
  8. Roger Tayler: Galaxies. Construction and development. Vieweg, 1986, p. 120.
  9. A. Unsöld, B. Baschek: The new cosmos. Springer, 1988, p. 346.
  10. ^ F. Zwicky, The red shift of extragalactic nebulae. Helvetica Physica Acta, Volume 6, 1933, p. 125. Online
  11. a b S. Chandrasekhar : An introduction to the study of stellar structure. Chicago 1939, p. 51 ff.
  12. Wolfgang Hillebrandt, Ewald Müller: Introduction to Theoretical Astrophysics. Script of the Technical University of Munich, 2008, Chapter 2 (PDF).
  13. ^ H. Poincaré : Leçons sur les hypothèses cosmogoniques. Paris 1911.
  14. ^ A. Eddington : Monthly Notices Roy. Astron. Soc. 76, 1916, 528.
  15. ^ S. Chandrasekhar: Hydrodynamic and hydromagnetic stability. Oxford University Press, 1961, p. 596.
  16. Henrik Beuther: Star formation. Script, 2009 (PDF).
  17. E. Parker : Tensor Virial Equations. Physical Review 96, 1954, 1686-1689.
  18. ^ S. Chandrasekhar, E. Fermi : Problems of Gravitational Stability in the Presence of a Magnetic Field. Astrophysical Journal, 118, 1953, 116.
  19. ^ S. Chandrasekhar: Ellipsoidal figures of equilibrium. Yale University Press, 2009.
  20. George Schmidt: Physics of High Temperature Plasmas. Academic Press, 1979, p. 72.
  21. WA Fock : Comment on the virial theorem . In: Journal of Physics . 63, No. 11, 1930, pp. 855-858. doi : 10.1007 / BF01339281 .
  22. Landau, Lifschitz, Classical Field Theory. Volume 2, Akademie Verlag, 1977, p. 99 f., § 34.
  23. ^ J. Gaite: The relativistic virial theorem and scale invariance. Physics Uspekhi, Volume 56, 2013, p. 919.
  24. ^ S. Chandrasekhar: The Post-Newtonian Equations of Hydrodynamics in General Relativity. Astrophysical Journal, Volume 142, 1965, pp. 1488-1512, bibcode : 1965ApJ ... 142.1488C .
  25. George W. Collins: The Virial Theorem in Stellar Astrophysics. Pachart Press, 1978, chapter 2.