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| In [[mechanics]], the '''virial theorem''' provides a general equation that relates the average over time of the total [[kinetic energy]], <math>\left\langle T \right\rangle</math>, of a stable system consisting of ''N'' particles, bound by potential forces, with that of the total potential energy, <math>\left\langle V_\text{TOT} \right\rangle</math>, where angle brackets represent the average over time of the enclosed quantity. Mathematically, the [[theorem]] states
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| | |
| :<math>
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| 2 \left\langle T \right\rangle = -\sum_{k=1}^N \left\langle \mathbf{F}_k \cdot \mathbf{r}_k \right\rangle
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| </math>
| |
| | |
| where '''F'''<sub>''k''</sub> represents the [[force]] on the ''k''th particle, which is located at position '''r'''<sub>''k''</sub>. The word "virial" derives from ''vis'', the [[Latin]] word for "force" or "energy", and was given its technical definition by [[Rudolf Clausius]] in 1870.<ref>{{cite journal | last = Clausius | first = RJE | year = 1870 | title = On a Mechanical Theorem Applicable to Heat | journal = Philosophical Magazine, Ser. 4 | volume = 40 | pages = 122–127}}</ref>
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| | |
| 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]] 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]]. The virial theorem has been generalized in various ways, most notably to a [[tensor]] form.
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| | |
| If the force between any two particles of the system results from a [[potential energy]] ''V''(''r'') = ''αr<sup> n</sup>'' that is proportional to some power ''n'' of the [[mean inter-particle distance|inter-particle distance]] ''r'', the virial theorem takes the simple form
| |
| | |
| :<math>
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| 2 \langle T \rangle = n \langle V_\text{TOT} \rangle.
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| </math>
| |
| | |
| Thus, twice the average total kinetic energy <math>\left\langle T \right\rangle</math> equals ''n'' times the average total potential energy <math>\left\langle V_\text{TOT} \right\rangle</math>. Whereas ''V''(''r'') represents the potential energy between two particles, ''V''<sub>TOT</sub> represents the total potential energy of the system, i.e., the sum of the potential energy ''V''(''r'') 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 ''n'' equals −1.
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| | |
| Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step.
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| | |
| ==History==
| |
| In 1870, [[Rudolf Clausius]] 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 1/2 the average potential energy. The virial theorem can be obtained directly from [[Lagrange's identity|Lagrange's Identity]] 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. [[Carl Gustav Jacob Jacobi|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.<ref>Collins, G. W. (1978). The Virial Theorem in Stellar Astrophysics. Pachart Press. Introduction</ref> The theorem was later utilized, popularized, generalized and further developed by [[James Clerk Maxwell]], [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]], [[Henri Poincaré]], [[Subrahmanyan Chandrasekhar]], [[Enrico Fermi]], [[Paul Ledoux]] and [[Eugene Parker]]. [[Fritz Zwicky]] was the first to use the virial theorem to deduce the existence of unseen matter, which is now called [[dark matter]]. As another example of its many applications, the virial theorem has been used to derive the [[Chandrasekhar limit]] for the stability of [[white dwarf]] [[star]]s.
| |
| | |
| ==Statement and derivation==
| |
| | |
| ===Definitions of the virial and its time derivative===
| |
| For a collection of ''N'' point particles, the [[scalar (physics)|scalar]] [[moment of inertia]] ''I'' about the [[origin (mathematics)|origin]] is defined by the equation
| |
| :<math>
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| I = \sum_{k=1}^{N} m_{k} |\mathbf{r}_{k}|^{2} = \sum_{k=1}^{N} m_{k} r_{k}^{2}
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| </math>
| |
| where ''m''<sub>''k''</sub> and '''r'''<sub>''k''</sub> represent the mass and position of the ''k''th particle. ''r''<sub>''k''</sub>=|'''r'''<sub>''k''</sub>| is the position vector magnitude. The scalar '''virial''' ''G'' is defined by the equation
| |
| :<math>
| |
| G = \sum_{k=1}^N \mathbf{p}_k \cdot \mathbf{r}_k
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| </math>
| |
| where '''p'''<sub>''k''</sub> is the [[momentum]] [[vector (geometry)|vector]] of the ''k''th particle. Assuming that the masses are constant, the '''virial''' ''G'' is one-half the time derivative of this moment of inertia
| |
| :<math>
| |
| \frac{1}{2} \frac{dI}{dt} = \frac{1}{2} \frac{d}{dt} \sum_{k=1}^N m_{k} \, \mathbf{r}_k \cdot \mathbf{r}_k = \sum_{k=1}^N m_{k} \, \frac{d\mathbf{r}_k}{dt} \cdot \mathbf{r}_k = \sum_{k=1}^N \mathbf{p}_k \cdot \mathbf{r}_k = G\,.
| |
| </math>
| |
| In turn, the time derivative of the virial ''G'' can be written
| |
| :<math>
| |
| \begin{align}
| |
| \frac{dG}{dt} & = \sum_{k=1}^N \mathbf{p}_k \cdot \frac{d\mathbf{r}_k}{dt} +
| |
| \sum_{k=1}^N \frac{d\mathbf{p}_k}{dt} \cdot \mathbf{r}_k \\
| |
| & = \sum_{k=1}^N m_k \frac{d\mathbf{r}_{k}}{dt} \cdot \frac{d\mathbf{r}_k}{dt} + \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k \\
| |
| & = 2 T + \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k\,,
| |
| \end{align}
| |
| </math>
| |
| where ''m''<sub>''k''</sub> is the mass of the ''k''-th particle, <math>\mathbf{F}_k = \frac{d\mathbf{p}_k}{dt}</math> is the net force on that particle, and ''T'' is the total [[kinetic energy]] of the system
| |
| :<math>
| |
| T = \frac{1}{2} \sum_{k=1}^N m_k v_k^2 =
| |
| \frac{1}{2} \sum_{k=1}^N m_k \frac{d\mathbf{r}_k}{dt} \cdot \frac{d\mathbf{r}_k}{dt}.
| |
| </math>
| |
| | |
| ===Connection with the potential energy between particles===
| |
| | |
| The total force <math>\mathbf{F}_k</math> on particle ''k'' is the sum of all the forces from the other particles ''j'' in the system
| |
| | |
| :<math>
| |
| \mathbf{F}_k = \sum_{j=1}^N \mathbf{F}_{jk}
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| </math>
| |
| | |
| where <math>\mathbf{F}_{jk}</math> is the force applied by particle ''j'' on particle ''k''. Hence, the force term of the virial time derivative can be written
| |
| | |
| :<math>
| |
| \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k =
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| \sum_{k=1}^N \sum_{j=1}^N \mathbf{F}_{jk} \cdot \mathbf{r}_k.
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| </math>
| |
| | |
| Since no particle acts on itself (i.e., <math>\mathbf{F}_{jk} = 0</math> whenever <math>j=k</math>), we have
| |
| | |
| :<math>
| |
| \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k =
| |
| \sum_{k=1}^N \sum_{j<k} \mathbf{F}_{jk} \cdot \mathbf{r}_k +
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| \sum_{k=1}^N \sum_{j>k} \mathbf{F}_{jk} \cdot \mathbf{r}_k =
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| \sum_{k=1}^N \sum_{j<k} \mathbf{F}_{jk} \cdot \left( \mathbf{r}_k - \mathbf{r}_j \right).
| |
| </math><ref>[[Media:Virial in the form between paritcles.pdf|Proof of this equation]]</ref>
| |
| | |
| where we have assumed that [[Newton's laws of motion|Newton's third law of motion]] holds, i.e., <math>\mathbf{F}_{jk} = -\mathbf{F}_{kj}</math> (equal and opposite reaction).
| |
| | |
| It often happens that the forces can be derived from a potential energy ''V'' that is a function only of the distance ''r''<sub>''jk''</sub> between the point particles ''j'' and ''k''. Since the force is the negative gradient of the potential energy, we have in this case
| |
| | |
| :<math>
| |
| \mathbf{F}_{jk} = -\nabla_{\mathbf{r}_k} V =
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| - \frac{dV}{dr} \left( \frac{\mathbf{r}_k - \mathbf{r}_j}{r_{jk}} \right),
| |
| </math>
| |
| | |
| which is clearly equal and opposite to <math>\mathbf{F}_{kj} = -\nabla_{\mathbf{r}_j} V</math>, the force applied by particle <math>k</math> on particle ''j'', as may be confirmed by explicit calculation. Hence, the force term of the virial time derivative is
| |
| | |
| :<math>
| |
| \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k =
| |
| \sum_{k=1}^N \sum_{j<k} \mathbf{F}_{jk} \cdot \left( \mathbf{r}_k - \mathbf{r}_j \right) =
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| -\sum_{k=1}^N \sum_{j<k} \frac{dV}{dr} \frac{\left( \mathbf{r}_k - \mathbf{r}_j \right)^2}{r_{jk}} =
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| -\sum_{k=1}^N \sum_{j<k} \frac{dV}{dr} r_{jk}.
| |
| </math>
| |
| | |
| Thus, we have
| |
| | |
| :<math>
| |
| \frac{dG}{dt} = 2 T +
| |
| \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k = 2 T -
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| \sum_{k=1}^N \sum_{j<k} \frac{dV}{dr} r_{jk}.
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| </math>
| |
| | |
| ===Special case of power-law forces===
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| In a common special case, the potential energy ''V'' between two particles is proportional to a power ''n'' of their distance ''r''
| |
| | |
| :<math>
| |
| V(r_{jk}) = \alpha r_{jk}^n,
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| </math>
| |
| | |
| where the coefficient α and the exponent ''n'' are constants. In such cases, the force term of the virial time derivative is given by the equation
| |
| | |
| :<math>
| |
| -\sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k =
| |
| \sum_{k=1}^N \sum_{j<k} \frac{dV}{dr} r_{jk} =
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| \sum_{k=1}^N \sum_{j<k} n V(r_{jk}) = n V_\text{TOT}
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| </math>
| |
| | |
| where ''V''<sub>TOT</sub> is the total potential energy of the system
| |
| | |
| :<math>
| |
| V_\text{TOT} = \sum_{k=1}^N \sum_{j<k} V(r_{jk}).
| |
| </math>
| |
| | |
| Thus, we have
| |
| | |
| :<math>
| |
| \frac{dG}{dt} = 2 T +
| |
| \sum_{k=1}^N \mathbf{F}_k \cdot \mathbf{r}_k = 2 T - n V_\text{TOT}.
| |
| </math>
| |
| | |
| For gravitating systems and also for [[electrostatics|electrostatic systems]], the exponent ''n'' equals −1, giving '''Lagrange's identity''' | |
| | |
| :<math>
| |
| \frac{dG}{dt} = \frac{1}{2} \frac{d^2 I}{dt^2} = 2 T + V_\text{TOT}
| |
| </math>
| |
| | |
| which was derived by Lagrange and extended by Jacobi.
| |
| | |
| ===Time averaging===
| |
| | |
| The average of this derivative over a time, ''τ'', is defined as
| |
| | |
| :<math>
| |
| \left\langle \frac{dG}{dt} \right\rangle_\tau = \frac{1}\tau \int_{0}^\tau \frac{dG}{dt}\,dt = \frac{1}{\tau} \int_{G(0)}^{G(\tau)} \, dG = \frac{G(\tau) - G(0)}{\tau},
| |
| </math>
| |
| | |
| from which we obtain the exact equation
| |
| | |
| :<math>
| |
| \left\langle \frac{dG}{dt} \right\rangle_\tau =
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| 2 \left\langle T \right\rangle_\tau + \sum_{k=1}^N \left\langle \mathbf{F}_k \cdot \mathbf{r}_k \right\rangle_\tau.
| |
| </math>
| |
| | |
| The '''virial theorem''' states that, '''if''' <math>\left\langle {dG}/{dt} \right\rangle_\tau = 0</math>, then
| |
| | |
| :<math>
| |
| 2 \left\langle T \right\rangle_\tau = -\sum_{k=1}^N \left\langle \mathbf{F}_k \cdot \mathbf{r}_k \right\rangle_\tau.
| |
| </math>
| |
| | |
| There are many reasons why the average of the time derivative might vanish, i.e., <math>\left\langle {dG}/{dt} \right\rangle_{\tau} = 0</math>. One often-cited reason applies to ''stably bound systems'', i.e., 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 the virial, ''G''<sup>bound</sup>, is bounded between two extremes, ''G''<sub>min</sub> and ''G''<sub>max</sub>, and the average goes to zero in the limit of very long times ''τ''
| |
| | |
| :<math>
| |
| \lim_{\tau \rightarrow \infty} \left| \left\langle \frac{dG^{\mathrm{bound}}}{dt} \right\rangle_\tau \right| =
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| \lim_{\tau \rightarrow \infty} \left| \frac{G(\tau) - G(0)}{\tau} \right| \le
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| \lim_{\tau \rightarrow \infty} \frac{G_\max - G_\min}{\tau} = 0.
| |
| </math>
| |
| | |
| Even if the average of the time derivative of ''G'' is only approximately zero, the virial theorem holds to the same degree of approximation.
| |
| | |
| For power-law forces with an exponent ''n'', the general equation holds
| |
| | |
| :<math>
| |
| \langle T \rangle_\tau = -\frac{1}{2} \sum_{k=1}^N \langle \mathbf{F}_k \cdot \mathbf{r}_k \rangle_\tau = \frac{n}{2} \langle V_\text{TOT} \rangle_\tau.
| |
| </math>
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|
| |
| For [[gravitation|gravitational attraction]], ''n'' equals −1 and the average kinetic energy equals half of the average negative potential energy
| |
| | |
| :<math>
| |
| \langle T \rangle_\tau = -\frac{1}{2} \langle V_\text{TOT} \rangle_\tau.
| |
| </math>
| |
| | |
| This general result is useful for complex gravitating systems such as [[solar system]]s or [[galaxy|galaxies]].
| |
| | |
| A simple application of the virial theorem concerns galaxy clusters. 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 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.
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| The averaging need not be taken over time; an [[ensemble average]] can also be taken, with equivalent results.
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| | |
| Although derived for classical mechanics, the virial theorem also holds for quantum mechanics, which was proved by Fock<ref>{{cite journal | last = Fock | first = V. | year = 1930 | title = Bemerkung zum Virialsatz | journal = Zeitschrift für Physik A | volume = 63 | issue = 11 | pages = 855–858 | doi = 10.1007/BF01339281|bibcode = 1930ZPhy...63..855F }}</ref> (the quantum equivalent of the l.h.s. <math>\left\langle {dG}/{dt} \right\rangle_\tau</math> vanishes for energy eigenstates).
| |
| | |
| ==In special relativity==
| |
| For a single particle in special relativity, it is not the case that <math>T = \frac 12 \mathbf{p} \cdot \mathbf{v}</math>. Instead, it is true that <math>T = (\gamma - 1) mc^2\,</math> and
| |
| :<math>\begin{align}
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| \frac 12 \mathbf{p} \cdot \mathbf{v} &
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| = \frac 12 \vec{\beta} \gamma mc \cdot \vec{\beta} c
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| = \frac 12 \gamma \beta^2 mc^2
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| = \left( \frac{\gamma \beta^2}{2(\gamma-1)}\right) T
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| \,.\end{align}</math>
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| The last expression can be simplified to either <math>\left(\frac{1 + \sqrt{1-\beta^2}}{2}\right) T</math> or <math>\left(\frac{\gamma + 1}{2 \gamma}\right) T</math>.
| |
| | |
| Thus, under the conditions described in earlier sections (including [[Newton's third law of motion]], <math>\mathbf{F}_{jk} = -\mathbf{F}_{kj}</math>, despite relativity), the time average for <math>N</math> particles with a power law potential is
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| :<math>\frac n2 \langle V_\mathrm{TOT} \rangle_\tau
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| = \left\langle \sum_{k=1}^N \left(\frac{1 + \sqrt{1-\beta_k^2}}{2}\right) T_k \right\rangle_\tau
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| = \left\langle \sum_{k=1}^N \left(\frac{\gamma_k + 1}{2 \gamma_k}\right) T_k \right\rangle_\tau
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| \,.</math>
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| In particular, the ratio of kinetic energy to potential energy is no longer fixed, but necessarily falls into an interval:
| |
| :<math>\frac{2 \langle T_\mathrm{TOT} \rangle}{n \langle V_\mathrm{TOT} \rangle} \in \left[1, 2\right]\,,</math>
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| where the more relativistic systems exhibit the larger ratios.
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| | |
| ==Generalizations==
| |
| | |
| Lord Rayleigh published a generalization of the virial theorem in 1903.<ref>{{cite journal | last = [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]] | year = 1903 | title = Unknown}}</ref> [[Henri Poincaré]] applied a form of the virial theorem in 1911 to the problem of determining cosmological stability.<ref>{{cite book | last = Poincaré | first = H | authorlink = Henri Poincaré | title = Lectures on Cosmological Theories | publisher = Hermann | location = Paris}}</ref> A variational form of the virial theorem was developed in 1945 by Ledoux.<ref>{{cite journal
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| | last = Ledoux
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| | first = P.
| |
| | year = 1945
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| | title = On the Radial Pulsation of Gaseous Stars
| |
| | journal = The Astrophysical Journal
| |
| | volume = 102
| |
| | pages = 143–153
| |
| | url = http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1945ApJ...102..143L&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf
| |
| | accessdate = March 24, 2012
| |
| | doi = 10.1086/144747
| |
| | bibcode = 1945ApJ...102..143L
| |
| }}</ref> A [[tensor]] form of the virial theorem was developed by Parker,<ref>{{cite journal
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| | last = Parker
| |
| | first = E.N.
| |
| | year = 1954
| |
| | title = Tensor Virial Equations
| |
| | journal = Physical Review
| |
| | volume = 96
| |
| | issue = 6
| |
| | pages = 1686–1689
| |
| | format = PDF
| |
| | url = http://prola.aps.org/pdf/PR/v96/i6/p1686_1
| |
| | accessdate = March 24, 2012
| |
| | doi = 10.1103/PhysRev.96.1686
| |
| | bibcode = 1954PhRv...96.1686P
| |
| }}</ref> Chandrasekhar<ref>{{cite journal
| |
| | last = Chandrasekhar
| |
| | first = S
| |
| | authorlink = Subrahmanyan Chandrasekhar
| |
| | coauthors = Lebovitz NR
| |
| | year = 1962
| |
| | title = The Potentials and the Superpotentials of Homogeneous Ellipsoids
| |
| | journal = Ap. J.
| |
| | volume = 136
| |
| | pages = 1037–1047
| |
| | format = PDF
| |
| | url = http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1962ApJ...136.1037C&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf
| |
| | accessdate = March 24, 2012
| |
| | doi = 10.1086/147456
| |
| | bibcode = 1962ApJ...136.1037C
| |
| }}</ref> and Fermi.<ref>{{cite journal
| |
| | last = Chandrasekhar
| |
| | first = S
| |
| | authorlink = Subrahmanyan Chandrasekhar
| |
| | coauthors = Fermi E
| |
| | year = 1953
| |
| | title = Problems of Gravitational Stability in the Presence of a Magnetic Field
| |
| | journal = Ap. J.
| |
| | volume = 118
| |
| | pages = 116
| |
| | format = PDF
| |
| | url = http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1953ApJ...118..116C&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf
| |
| | accessdate = March 24, 2012
| |
| | doi = 10.1086/145732
| |
| | bibcode = 1953ApJ...118..116C
| |
| }}</ref> The following generalization of the virial theorem has been established by Pollard in 1964 for the case of the inverse square law:<ref>{{cite journal
| |
| | last = Pollard
| |
| | first= H.
| |
| | year = 1964
| |
| | title = A sharp form of the virial theorem
| |
| | journal = Bull. Amer. Math. Soc.
| |
| | volume = LXX
| |
| | pages = 703–705
| |
| | format = PDF
| |
| | url = http://www.ams.org/journals/bull/1964-70-05/S0002-9904-1964-11175-7/S0002-9904-1964-11175-7.pdf
| |
| | accessdate = March 24, 2012
| |
| | doi = 10.1090/S0002-9904-1964-11175-7
| |
| | issue = 5
| |
| }}</ref><ref>{{cite book
| |
| | last = Pollard
| |
| | first = Harry
| |
| | title = Mathematical Introduction to Celestial Mechanics
| |
| | publisher = Prentice–Hall, Inc.
| |
| | location = Englewood Cliffs, NJ
| |
| | year = 1966
| |
| }}</ref> the statement <math>2\lim\limits_{\tau\rightarrow+\infty}\langle T\rangle_\tau = \lim\limits_{\tau\rightarrow+\infty}\langle U\rangle_\tau</math> is true if and only if <math>\lim\limits_{\tau\rightarrow+\infty}{\tau}^{-2}I(\tau)=0.</math> A ''boundary'' term otherwise must be added, such as in Ref.<ref>{{cite journal
| |
| | last1 = Kolár
| |
| | first1 = M.
| |
| | last2 = O'Shea
| |
| | first2 = S. F.
| |
| |date=July 1996
| |
| | title = A high-temperature approximation for the path-integral quantum Monte Carlo method
| |
| | journal = Journal of Physics A: Mathematical and General
| |
| | volume = 29
| |
| | issue = 13
| |
| | pages = 3471–3494
| |
| | format = PDF
| |
| | bibcode = 1996JPhA...29.3471K
| |
| | doi = 10.1088/0305-4470/29/13/018
| |
| | accessdate = March 24, 2012
| |
| | url = http://iopscience.iop.org/0305-4470/29/13/018/pdf/0305-4470_29_13_018.pdf
| |
| }}</ref>
| |
| | |
| ==Inclusion of electromagnetic fields==
| |
| | |
| The virial theorem can be extended to include electric and magnetic fields. The result is<ref>{{cite book |first=George |last=Schmidt |title=Physics of High Temperature Plasmas |edition=Second |publisher=Academic Press |year=1979 |pages=72}}</ref>
| |
| | |
| :<math>
| |
| \frac{1}{2}\frac{d^2I}{dt^2}
| |
| + \int_Vx_k\frac{\partial G_k}{\partial t} \, d^3r
| |
| = 2(T+U) + W^E + W^M - \int x_k(p_{ik}+T_{ik}) \, dS_i,
| |
| </math>
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| where ''I'' is the [[moment of inertia]], ''G'' is the [[Poynting vector|momentum density of the electromagnetic field]], ''T'' is the [[kinetic energy]] of the "fluid", ''U'' is the random "thermal" energy of the particles, ''W<sup>E</sup>'' and ''W<sup>M</sup>'' are the electric and magnetic energy content of the volume considered. Finally, ''p<sub>ik</sub>'' is the fluid-pressure tensor expressed in the local moving coordinate system
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| :<math>
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| p_{ik}
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| = \Sigma n^\sigma m^\sigma \langle v_iv_k\rangle^\sigma
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| - V_iV_k\Sigma m^\sigma n^\sigma, | |
| </math>
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| and ''T<sub>ik</sub>'' is the electromagnetic stress tensor,
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| :<math>
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| T_{ik}
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| = \left( \frac{\varepsilon_0E^2}{2} + \frac{B^2}{2\mu_0} \right) \delta_{ik}
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| - \left( \varepsilon_0E_iE_k + \frac{B_iB_k}{\mu_0} \right).
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| </math> | |
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| 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 τ. If a total mass ''M'' is confined within a radius ''R'', then the moment of inertia is roughly ''MR''<sup>2</sup>, and the left hand side of the virial theorem is ''MR''<sup>2</sup>/τ<sup>2</sup>. The terms on the right hand side add up to about ''pR''<sup>3</sup>, where ''p'' is the larger of the plasma pressure or the magnetic pressure. Equating these two terms and solving for τ, we find
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| :<math>\tau\,\sim R/c_s,</math> | |
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| where ''c<sub>s</sub>'' 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.
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| ==In astrophysics==
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| The virial theorem is frequently applied in astrophysics, especially relating the [[potential energy#Gravitational potential energy|gravitational potential energy]] of a system to its [[kinetic energy|kinetic]] or [[thermal energy]]. Some common virial relations are,
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| <math>\frac{3}{5} \frac{GM}{R} = \frac{3}{2} \frac{k_B T}{m_p} = \frac{1}{2} v^2 </math>,
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| for a mass <math>M</math>, radius <math>R</math>, velocity <math>v</math>, and temperature <math>T</math>. And the constants are [[Gravitational constant|Newton's constant]] <math>G</math>, the [[Boltzmann constant]] <math>k_B</math>, and proton mass <math>m_p</math>. Note that these relations are only approximate, and often the leading numerical factors (e.g. 3/5 or 1/2) are neglected entirely.
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| === Galaxies and cosmology (virial mass and radius) ===
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| In [[astronomy]], the mass and size of a galaxy (or general overdensity) is often defined in terms of the "virial radius" and "virial mass" respectively. Because galaxies and overdensities in continuous fluids can be highly extended (even to infinity in some models—e.g. an [[singular isothermal sphere|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.
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| In galaxy dynamics, the mass of a galaxy is often inferred by measuring the [[rotation velocity]] of its gas and stars, assuming [[circular orbit|circular Keplerian orbits]]. Using the virial theorem, the [[Velocity dispersion|dispersion velocity]] <math>\sigma</math> can be used in a similar way. Taking the kinetic energy (per particle) of the system as, T = (1/2) v<sup>2</sup> ~ (3/2) <math>\sigma</math><sup>2</sup>, and the potential energy (per particle) as, U ~ (3/5)(GM/R), we can write
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| <math> \frac{GM}{R} \approx \sigma^2 </math>.
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| Here <math>R</math> is the radius at which the velocity dispersion is being measured, and <math>M</math> 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.
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| <math> \frac{GM_\text{vir}}{R_\text{vir}} \approx \sigma_\text{max}^2 </math>.
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| 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]] sense, or when used self-consistently.
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| 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 (cosmology)|critical density]], <math>\rho_\text{crit}=\frac{3H^2}{8\pi G}</math>. Where <math>H</math> is the [[Hubble's law|Hubble parameter]] and <math>G</math> is the [[gravitational constant]]. A common (although mostly arbitrary) choice for the factor is 200, in which case the virial radius is approximated as <math>r_\text{vir} \approx r_{200}= r(\rho = 200 \cdot \rho_\text{crit})</math>. The virial mass is then defined relative to this radius as <math>M_\text{vir} \approx M_{200} = (4/3)\pi r_{200}^3 \cdot 200 \rho_\text{crit} </math>.
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| ==See also==
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| * [[Virial coefficient]]
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| * [[Virial stress]]
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| * [[Virial mass]]
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| * [[Equipartition theorem]]
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| * [[List of plasma (physics) articles]]
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| *{{Cite book |last=Goldstein |first=H. |year=1980 |title=Classical Mechanics |edition=2nd |publisher=Addison–Wesley |isbn=0-201-02918-9 |postscript=<!--None--> }}
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| *{{Cite journal |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/ |postscript=<!--None--> }}
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| ==External links==
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| * [http://www.mathpages.com/home/kmath572/kmath572.htm The Virial Theorem] at MathPages
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| * ''[http://hyperphysics.phy-astr.gsu.edu/hbase/astro/gravc.html#c2 Gravitational Contraction and Star Formation]'', Georgia State University
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| [[Category:Physics theorems]]
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| [[Category:Dynamics]]
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| [[Category:Solid mechanics]]
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| [[Category:Concepts in physics]]
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