3467:
2959:
2356:
4224:
3462:{\displaystyle {\begin{aligned}\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}&=\sum _{k=1}^{N}\sum _{j=1}^{N}\mathbf {F} _{jk}\cdot \mathbf {r} _{k}=\sum _{k=2}^{N}\sum _{j=1}^{k-1}\left(\mathbf {F} _{jk}\cdot \mathbf {r} _{k}+\mathbf {F} _{kj}\cdot \mathbf {r} _{j}\right)\\&=\sum _{k=2}^{N}\sum _{j=1}^{k-1}\left(\mathbf {F} _{jk}\cdot \mathbf {r} _{k}-\mathbf {F} _{jk}\cdot \mathbf {r} _{j}\right)=\sum _{k=2}^{N}\sum _{j=1}^{k-1}\mathbf {F} _{jk}\cdot \left(\mathbf {r} _{k}-\mathbf {r} _{j}\right)\end{aligned}}}
4919:
1956:
3765:
1947:
4530:
8539:
997:
7750:
7114:
8148:
435: 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
2351:{\displaystyle {\begin{aligned}{\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}\\&=2T+\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}\end{aligned}}}
14684:
4219:{\displaystyle {\begin{aligned}\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}&=\sum _{k=2}^{N}\sum _{j=1}^{k-1}\mathbf {F} _{jk}\cdot \left(\mathbf {r} _{k}-\mathbf {r} _{j}\right)\\&=-\sum _{k=2}^{N}\sum _{j=1}^{k-1}{\frac {dV_{jk}}{dr_{jk}}}{\frac {|\mathbf {r} _{k}-\mathbf {r} _{j}|^{2}}{r_{jk}}}\\&=-\sum _{k=2}^{N}\sum _{j=1}^{k-1}{\frac {dV_{jk}}{dr_{jk}}}r_{jk}.\end{aligned}}}
1653:
6055:
8285:
623:
4914:{\displaystyle {\begin{aligned}-{\frac {1}{2}}\,\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}&={\frac {1}{2}}\,\sum _{k=1}^{N}\sum _{j<k}{\frac {dV_{jk}}{dr_{jk}}}r_{jk}\\&={\frac {1}{2}}\,\sum _{k=1}^{N}\sum _{j<k}n\alpha r_{jk}^{n-1}r_{jk}\\&={\frac {1}{2}}\,\sum _{k=1}^{N}\sum _{j<k}nV_{jk}={\frac {n}{2}}\,V_{\text{TOT}}\end{aligned}}}
9099:
7953:
9825:
11408:
5843:
5477:
1311:
5624:
11666:
6219:
10049:
7503:
12076:
9568:
2929:
1942:{\displaystyle {\begin{aligned}{\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\,.\end{aligned}}}
12560:
8928:
2644:
3699:
8534:{\displaystyle {\frac {n}{2}}\left\langle V_{\mathrm {TOT} }\right\rangle _{\tau }=\left\langle \sum _{k=1}^{N}\left({\frac {1+{\sqrt {1-\beta _{k}^{2}}}}{2}}\right)T_{k}\right\rangle _{\tau }=\left\langle \sum _{k=1}^{N}\left({\frac {\gamma _{k}+1}{2\gamma _{k}}}\right)T_{k}\right\rangle _{\tau }\,.}
4439:
992:{\displaystyle \langle T\rangle ={\frac {1}{t_{2}-t_{1}}}\int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}{\frac {1}{2}}m_{k}\left|\mathbf {v} _{k}(t)\right|^{2}dt={\frac {1}{t_{2}-t_{1}}}\int _{t_{1}}^{t_{2}}\left({\frac {1}{2}}m|\mathbf {v} _{1}(t)|^{2}+{\frac {1}{2}}m|\mathbf {v} _{2}(t)|^{2}\right)dt=mv^{2}.}
13760:
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
12084:
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
11257:
152:
5774:
5280:
9322:
9626:
1113:
8143:{\displaystyle {\begin{aligned}{\frac {1}{2}}\mathbf {p} \cdot \mathbf {v} &={\frac {1}{2}}{\boldsymbol {\beta }}\gamma mc\cdot {\boldsymbol {\beta }}c\\&={\frac {1}{2}}\gamma \beta ^{2}mc^{2}\\&=\left({\frac {\gamma \beta ^{2}}{2(\gamma -1)}}\right)T\,.\end{aligned}}}
5482:
8252:
11427:
6075:
12316:
9919:
11857:
7735:
7329:
13127:
11879:
12996:
2762:
6050:{\displaystyle \lim _{\tau \to \infty }\left|\left\langle {\frac {dG^{\mathrm {bound} }}{dt}}\right\rangle _{\tau }\right|=\lim _{\tau \to \infty }\left|{\frac {G(\tau )-G(0)}{\tau }}\right|\leq \lim _{\tau \to \infty }{\frac {G_{\max }-G_{\min }}{\tau }}=0.}
12393:
5140:
8644:
9453:
2462:
5810:. 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
13747:
3531:
10146:
4231:
1498:
12214:
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:
7099:
6304:
6730:
10232:
11248:
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
11182:
13657:
13485:
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
50:
5255:
12387:
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:
5667:
479:
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
13481:
10831:
12833:
12376:, 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
9094:{\displaystyle m\underbrace {\frac {d^{2}x}{dt^{2}}} _{\text{acceleration}}=\underbrace {-kx} _{\text{spring}}\underbrace {-\gamma {\frac {dx}{dt}}} _{\text{friction}}\underbrace {+F\cos(\omega t)} _{\text{external driving}}}
10560:
10650:
6454:
8153:
13501:, 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
402:
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
12220:
6886:
11724:
7583:
1622:
10753:
9820:{\displaystyle {\begin{cases}X&={\sqrt {\frac {F^{2}}{\gamma ^{2}\omega ^{2}+m^{2}(\omega _{0}^{2}-\omega ^{2})^{2}}}}\\\tan \varphi &=-{\frac {\gamma \omega }{m(\omega _{0}^{2}-\omega ^{2})}}\end{cases}}}
7922:
5012:
13021:
10388:
9199:
320:
11253:
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:
8795:
11033:
8740:
7958:
2738:
14121:
11097:
12865:
9400:
11403:{\displaystyle 2\lim _{\tau \to +\infty }\langle T\rangle _{\tau }=\lim _{\tau \to +\infty }\langle U\rangle _{\tau }\qquad {\text{if and only if}}\quad \lim _{\tau \to +\infty }{\tau }^{-2}I(\tau )=0\,.}
13562:
9618:
12185:
10465:
9835:
Consider a container filled with an ideal gas consisting of point masses. The force applied to the point masses is the negative of the forces applied to the wall of the container, which is of the form
5472:{\displaystyle \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 }},}
217:
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
5019:
9885:
8544:
1306:{\displaystyle -{\frac {1}{2}}\sum _{k=1}^{N}{\bigl \langle }\mathbf {F} _{k}\cdot \mathbf {r} _{k}{\bigr \rangle }=-{\frac {1}{2}}(-Fr-Fr)=Fr={\frac {mv^{2}}{r}}\cdot r=mv^{2}=\langle T\rangle ,}
5619:{\displaystyle \left\langle {\frac {dG}{dt}}\right\rangle _{\tau }=2\left\langle T\right\rangle _{\tau }+\sum _{k=1}^{N}\left\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\right\rangle _{\tau }.}
13386:
4535:
3770:
2964:
1961:
1658:
13235:), 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.
6524:
13662:
6946:
11661:{\displaystyle {\frac {1}{2}}{\frac {d^{2}I}{dt^{2}}}+\int _{V}x_{k}{\frac {\partial G_{k}}{\partial t}}\,d^{3}r=2(T+U)+W^{\mathrm {E} }+W^{\mathrm {M} }-\int x_{k}(p_{ik}+T_{ik})\,dS_{i},}
6214:{\displaystyle \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 }.}
11234:
10044:{\displaystyle \langle T\rangle =-{\frac {1}{2}}{\Bigg \langle }\sum _{i}\mathbf {F} _{i}\cdot \mathbf {r} _{i}{\Bigg \rangle }={\frac {P}{2}}\int \mathbf {\hat {n}} \cdot \mathbf {r} dA}
1373:
13756:
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
9151:
9445:
7009:
4517:
7498:{\displaystyle u\in L_{\mathrm {loc} }^{\infty }(\mathbb {R} ^{n}),\qquad \nabla u\in L^{2}(\mathbb {R} ^{n}),\qquad G(u(\cdot ))\in L^{1}(\mathbb {R} ^{n}),\qquad n\in \mathbb {N} ,}
7324:
6234:
12071:{\displaystyle T_{ik}=\left({\frac {\varepsilon _{0}E^{2}}{2}}+{\frac {B^{2}}{2\mu _{0}}}\right)\delta _{ik}-\left(\varepsilon _{0}E_{i}E_{k}+{\frac {B_{i}B_{k}}{\mu _{0}}}\right).}
9563:{\displaystyle \underbrace {\langle {\dot {x}}\;\gamma {\dot {x}}\rangle } _{\text{power dissipated}}=\underbrace {\langle {\dot {x}}\;F\cos \omega t\rangle } _{\text{power input}}}
6785:
6533:
10971:
9914:
7554:
10933:
13587:
12722:
10679:
2924:{\displaystyle -{\frac {1}{2}}\,\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}=-{\frac {1}{2}}\,\sum _{k=1}^{N}\sum _{j=1}^{N}\mathbf {F} _{jk}\cdot \mathbf {r} _{k}\,.}
13765:. 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
5154:
8881:
8692:
13419:
10761:
10058:
12860:
12555:{\displaystyle v_{\mathrm {rms} }=c{\sqrt {1-{\frac {4\pi \eta \rho _{0}r^{2}}{c^{2}\gamma _{c}^{2}\sin ^{2}\left({\frac {r}{c}}{\sqrt {4\pi \eta \rho _{0}}}\right)}}}},}
12638:
12206:, 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.
9191:
6327:
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.
10864:
10281:, and discovered a discrepancy of mass of about 450, which he explained as due to "dark matter". He refined the analysis in 1937, finding a discrepancy of about 500.
8921:
12608:
10267:
7255:
6363:
2639:{\displaystyle 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}}.}
10890:
7220:
1313:
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
12661:
12585:
12085:
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
11105:
10151:
13408:
10327:
10307:
9171:
8901:
8843:
8823:
7578:
6323:. 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.
3694:{\displaystyle \mathbf {F} _{jk}=-\nabla _{\mathbf {r} _{k}}V_{jk}=-{\frac {dV_{jk}}{dr_{jk}}}\left({\frac {\mathbf {r} _{k}-\mathbf {r} _{j}}{r_{jk}}}\right),}
427:
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.
6790:
4434:{\displaystyle {\frac {dG}{dt}}=2T+\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}=2T-\sum _{k=2}^{N}\sum _{j=1}^{k-1}{\frac {dV_{jk}}{dr_{jk}}}r_{jk}.}
1556:
7865:
4933:
1349:
Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step.
12727:
271:
2675:
10470:
10569:
8654:
The virial theorem has a particularly simple form for periodic motion. It can be used to perform perturbative calculation for nonlinear oscillators.
13493:
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
13414:
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.
147:{\displaystyle \left\langle T\right\rangle =-{\frac {1}{2}}\,\sum _{k=1}^{N}{\bigl \langle }\mathbf {F} _{k}\cdot \mathbf {r} _{k}{\bigr \rangle }}
13507:
12144:
5769:{\displaystyle 2\left\langle T\right\rangle _{\tau }=-\sum _{k=1}^{N}\left\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\right\rangle _{\tau }.}
9317:{\displaystyle m\langle {\dot {x}}{\dot {x}}\rangle =k\langle xx\rangle +\gamma \langle x{\dot {x}}\rangle -F\langle \cos(\omega t)x\rangle }
13231:" respectively. Because galaxies and overdensities in continuous fluids can be highly extended (even to infinity in some models, such as an
10684:
6346:
Although originally derived for classical mechanics, the virial theorem also holds for quantum mechanics, as first shown by Fock using the
13343:
10332:
8749:
7174:
In the field of quantum mechanics, there exists another form of the virial theorem, applicable to localized solutions to the stationary
10976:
9450:
To solve the two unknowns, we need another equation. In steady state, the power lost per cycle is equal to the power gained per cycle:
8697:
8247:{\displaystyle \left({\frac {1+{\sqrt {1-\beta ^{2}}}}{2}}\right)T\qquad {\text{or}}\qquad \left({\frac {\gamma +1}{2\gamma }}\right)T}
6468:
510:, acted upon by mutually attractive forces. Suppose the particles are at diametrically opposite points of a circular orbit with radius
14630:
Fedosin, S. G. (2016). "The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept".
11042:
9327:
13580:. A common choice for the factor is 200, which corresponds roughly to the typical over-density in spherical top-hat collapse (see
12311:{\displaystyle \left\langle W_{k}\right\rangle \approx -0.6\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle ,}
11852:{\displaystyle p_{ik}=\Sigma n^{\sigma }m^{\sigma }\langle v_{i}v_{k}\rangle ^{\sigma }-V_{i}V_{k}\Sigma m^{\sigma }n^{\sigma },}
9573:
10393:
7730:{\displaystyle \left({\frac {n-2}{2}}\right)\int _{\mathbb {R} ^{n}}|\nabla u(x)|^{2}\,dx=n\int _{\mathbb {R} ^{n}}G(u(x))\,dx.}
4465:
386:
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
14877:
14848:
14819:
14792:
14564:
13884:
13793:
13122:{\displaystyle {\frac {3}{5}}{\frac {GM}{R}}={\frac {3}{2}}{\frac {k_{\mathrm {B} }T}{m_{\mathrm {p} }}}={\frac {1}{2}}v^{2}}
14580:
Kolár, M.; O'Shea, S. F. (July 1996). "A high-temperature approximation for the path-integral quantum Monte Carlo method".
9838:
8541:
In particular, the ratio of kinetic energy to potential energy is no longer fixed, but necessarily falls into an interval:
6737:
14887:
i̇Pekoğlu, Y.; Turgut, S. (2016). "An elementary derivation of the quantum virial theorem from
Hellmann–Feynman theorem".
12360:, then we can see that in the virial theorem the kinetic energy is related to the potential energy not by the coefficient
7508:
12991:{\displaystyle ~W_{f}={\frac {1}{4\mu _{0}}}\int F_{\alpha \beta }F^{\alpha \beta }{\sqrt {-g}}\,dx^{1}\,dx^{2}\,dx^{3}}
12669:
229:. However, the virial theorem does not depend on the notion of temperature and holds even for systems that are not in
13957:
7797:
7161:
7779:
7175:
7143:
6891:
13905:"Virial Field Relationship for Molecular Charge Distributions and the Spatial Partitioning of Molecular Properties"
11187:
14974:
12666:
Unlike the virial theorem for particles, for the electromagnetic field the virial theorem is written as follows:
8255:
7771:
7135:
6358:
440:
9106:
13798:
9405:
7775:
7139:
5135:{\displaystyle {\frac {dG}{dt}}=2T+\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}=2T-nV_{\text{TOT}}\,.}
11244:
Lord
Rayleigh published a generalization of the virial theorem in 1900 which was partially reprinted in 1903.
8639:{\displaystyle {\frac {2\langle T_{\mathrm {TOT} }\rangle }{n\langle V_{\mathrm {TOT} }\rangle }}\in \left\,,}
14959:
7263:
14969:
13502:
262:
10938:
9890:
13742:{\displaystyle M_{\text{vir}}\approx M_{200}={\frac {4}{3}}\pi r_{200}^{3}\cdot 200\rho _{\text{crit}}.}
12135:
is the larger of the plasma pressure or the magnetic pressure. Equating these two terms and solving for
10895:
14484:
14440:
13232:
10658:
7557:
3470:
448:
10141:{\textstyle \int \mathbf {\hat {n}} \cdot \mathbf {r} dA=\int \nabla \cdot \mathbf {r} dV=3\int dV=3V}
471:
was the first to use the virial theorem to deduce the existence of unseen matter, which is now called
14954:
10973:. He also measured the radial velocities of the galaxies by doppler shifts in galactic spectra to be
8848:
7179:
1493:{\displaystyle I=\sum _{k=1}^{N}m_{k}\left|\mathbf {r} _{k}\right|^{2}=\sum _{k=1}^{N}m_{k}r_{k}^{2}}
9635:
8664:
14964:
13818:
7760:
7183:
7124:
5262:
428:
7094:{\displaystyle 2\langle T\rangle =\sum _{n}\left\langle X_{n}{\frac {dV}{dX_{n}}}\right\rangle ~.}
14936:
14150:
12838:
7764:
7128:
14075:
13173:. Note that these relations are only approximate, and often the leading numerical factors (e.g.
12613:
6299:{\displaystyle \langle T\rangle _{\tau }=-{\frac {1}{2}}\langle V_{\text{TOT}}\rangle _{\tau }.}
13577:
13146:
424:
14809:
14782:
14330:
9176:
6231:
equals −1 and the average kinetic energy equals half of the average negative potential energy
2956:), we split the sum in terms below and above this diagonal and we add them together in pairs:
14487:; Fermi E (1953). "Problems of Gravitational Stability in the Presence of a Magnetic Field".
13808:
11871:
11036:
10843:
8906:
6725:{\displaystyle =X_{n}+P_{n}=i\hbar X_{n}{\frac {dV}{dX_{n}}}-i\hbar {\frac {P_{n}^{2}}{m}}~.}
5258:
226:
218:
12998:
sets the potential field energy found through the components of the electromagnetic tensor.
12590:
10237:
10227:{\textstyle \langle T\rangle =N\langle {\frac {1}{2}}mv^{2}\rangle =N\cdot {\frac {3}{2}}kT}
7225:
14896:
14865:
14706:
14649:
14589:
14496:
14456:
14414:
14377:
14252:
14193:
14087:
14037:
13990:
13916:
13872:
13803:
13223:, the mass and size of a galaxy (or general overdensity) is often defined in terms of the "
13007:
11177:{\displaystyle {\frac {5R\langle v_{r}^{2}\rangle }{G}}\approx 3.6\times 10^{14}M_{\odot }}
10869:
7187:
1367:
375:
represents the total potential energy of the system, i.e., the sum of the potential energy
13652:{\displaystyle r_{\text{vir}}\approx r_{200}=r,\qquad \rho =200\cdot \rho _{\text{crit}}.}
11422:
The virial theorem can be extended to include electric and magnetic fields. The result is
7196:
6063:
is only approximately zero, the virial theorem holds to the same degree of approximation.
8:
13247:
12381:
12350:
of the particles at the center of the system. Under normal conditions we can assume that
6971:
5250:{\displaystyle {\frac {dG}{dt}}={\frac {1}{2}}{\frac {d^{2}I}{dt^{2}}}=2T+V_{\text{TOT}}}
3526:. Since the force is the negative gradient of the potential energy, we have in this case
481:
436:
230:
14908:
14900:
14869:
14780:
14710:
14653:
14593:
14500:
14460:
14418:
14381:
14256:
14230:
14197:
14091:
14041:
13994:
13920:
13876:
13253:
can be used in a similar way. Taking the kinetic energy (per particle) of the system as
12643:
12567:
6334:
holds for the system under consideration, the averaging need not be taken over time; an
14912:
14730:
14696:
14665:
14639:
14276:
14242:
14053:
14025:
14006:
13778:
13487:
13393:
13153:
10312:
10292:
10052:
9156:
8886:
8828:
8808:
7563:
32:
14930:
14601:
13904:
14916:
14873:
14844:
14815:
14788:
14734:
14722:
14560:
14268:
14211:
14162:
14103:
14057:
14010:
13963:
13953:
13945:
13880:
13813:
13476:{\displaystyle {\frac {GM_{\text{vir}}}{R_{\text{vir}}}}\approx \sigma _{\max }^{2}.}
13239:
12199:
11675:
10826:{\displaystyle Nm={\frac {5\langle v^{2}\rangle }{3G\langle {\frac {1}{r}}\rangle }}}
8658:
6347:
1639:
1360:
1093:
407:
of the system is equal to its virial, or that the average kinetic energy is equal to
14669:
14540:
14523:
14346:
14280:
12828:{\textstyle ~E_{kf}=\int A_{\alpha }j^{\alpha }{\sqrt {-g}}\,dx^{1}\,dx^{2}\,dx^{3}}
11245:
444:
14904:
14761:
14714:
14657:
14597:
14535:
14504:
14464:
14422:
14385:
14311:
14260:
14201:
14095:
14045:
13998:
13924:
13847:
6335:
6309:
1357:
241:
40:
14749:
13006:
The virial theorem is frequently applied in astrophysics, especially relating the
14264:
13569:
11683:
10555:{\textstyle \langle U\rangle =-Gm^{2}\sum _{i<j}\langle {1}/{r_{i,j}}\rangle }
9103:
When the oscillator has reached a steady state, it performs a stable oscillation
6320:
399:
211:
11710:
are the electric and magnetic energy content of the volume considered. Finally,
10645:{\textstyle \langle U\rangle =-{\frac {1}{2}}N^{2}Gm^{2}\langle {1}/{r}\rangle }
14781:
BAIDYANATH BASU; TANUKA CHATTOPADHYAY; SUDHINDRA NATH BISWAS (1 January 2010).
14304:
The London, Edinburgh, and Dublin
Philosophical Magazine and Journal of Science
13498:
13243:
13015:
13011:
11691:
7858:
6449:{\displaystyle H=V{\bigl (}\{X_{i}\}{\bigr )}+\sum _{n}{\frac {P_{n}^{2}}{2m}}}
6324:
2417:
423:
the average potential energy. The virial theorem can be obtained directly from
195:
36:
28:
14859:
14718:
14685:"The integral theorem of generalized virial in the relativistic uniform model"
14661:
14315:
13866:
13851:
12203:
14948:
14726:
14272:
14215:
14166:
14107:
13900:
13783:
13762:
13757:
13228:
11719:
is the fluid-pressure tensor expressed in the local moving coordinate system
10681:
as the gravitational potential of a uniform ball of constant density, giving
10562:. Assuming the motion of the stars are all the same over a long enough time (
476:
464:
460:
233:. The virial theorem has been generalized in various ways, most notably to a
14426:
13238:
In galaxy dynamics, the mass of a galaxy is often inferred by measuring the
5778:
There are many reasons why the average of the time derivative might vanish,
27:
provides a general equation that relates the average over time of the total
14766:
13967:
10278:
9916:
is the unit normal vector pointing outwards. Then the virial theorem states
468:
452:
13788:
13581:
13224:
13214:
10277:
In 1933, Fritz Zwicky applied the virial theorem to estimate the mass of
8743:
6881:{\displaystyle {\frac {i}{\hbar }}=2T-\sum _{n}X_{n}{\frac {dV}{dX_{n}}}}
6224:
2648:
485:
472:
456:
222:
3496:
It often happens that the forces can be derived from a potential energy
210:
word for "force" or "energy", and was given its technical definition by
14049:
14002:
10563:
7000:
of this time derivative vanishes in a stationary state, leading to the
6354:
6331:
1617:{\displaystyle G=\sum _{k=1}^{N}\mathbf {p} _{k}\cdot \mathbf {r} _{k}}
600:. The average kinetic energy of the system in an interval of time from
13928:
13410:
is the radius at which the velocity dispersion is being measured, and
10748:{\textstyle \langle U\rangle =-{\frac {3}{5}}{\frac {GN^{2}m^{2}}{R}}}
8254:. Thus, under the conditions described in earlier sections (including
6308:
This general result is useful for complex gravitating systems such as
13220:
7917:{\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
7810:
For a single particle in special relativity, it is not the case that
5007:{\displaystyle V_{\text{TOT}}=\sum _{k=1}^{N}\sum _{j<k}V_{jk}\,.}
20:
14295:
14099:
13838:
Clausius, RJE (1870). "On a
Mechanical Theorem Applicable to Heat".
12835:
considered as the kinetic field energy associated with four-current
10383:{\textstyle \langle T\rangle ={\frac {1}{2}}Nm\langle v^{2}\rangle }
7749:
7113:
1042:. The attractive forces act in opposite directions as positions, so
240:
If the force between any two particles of the system results from a
14701:
14644:
14508:
14469:
14444:
14390:
14365:
14247:
14206:
14181:
12081:
6948:
is the kinetic energy. The left-hand side of this equation is just
1636:
404:
315:{\displaystyle 2\langle T\rangle =n\langle V_{\text{TOT}}\rangle .}
198:
represent the average over time of the enclosed quantity. The word
14445:"The Potentials and the Superpotentials of Homogeneous Ellipsoids"
13208:
8790:{\displaystyle \langle T\rangle =-{\frac {1}{2}}\langle U\rangle }
4527:
are constants. In such cases, the virial is given by the equation
999:
Taking center of mass as the origin, the particles have positions
362:
represents the potential energy between two particles of distance
14026:"Nonlinear scalar field equations, I existence of a ground state"
11028:{\displaystyle \langle v_{r}^{2}\rangle =(1000{\text{km/s}})^{2}}
8735:{\displaystyle \langle T\rangle ={\frac {n}{2}}\langle U\rangle }
2733:{\displaystyle \mathbf {F} _{k}=\sum _{j=1}^{N}\mathbf {F} _{jk}}
44:
11092:{\displaystyle \langle v^{2}\rangle =3\langle v_{r}^{2}\rangle }
13494:
11250:
11102:
By the virial theorem, the total mass of the cluster should be
9395:{\displaystyle F\cos(\varphi )=m(\omega _{0}^{2}-\omega ^{2})X}
8646:
where the more relativistic systems exhibit the larger ratios.
6313:
234:
8805:
Analysis based on. For a one-dimensional oscillator with mass
11236:, meaning the total mass is 450 times that of observed mass.
10390:. The total gravitational potential energy of the cluster is
207:
176:
13557:{\displaystyle \rho _{\text{crit}}={\frac {3H^{2}}{8\pi G}}}
13659:
The virial mass is then defined relative to this radius as
10892:
galaxies in the cluster, each having observed stellar mass
9813:
9613:{\displaystyle \sin \varphi =-{\frac {\gamma X\omega }{F}}}
488:
14335:. Cambridge: Cambridge University Press. pp. 491–493.
12180:{\displaystyle \tau \,\sim {\frac {R}{c_{\mathrm {s} }}},}
10758:
So by the virial theorem, the total mass of the cluster is
10460:{\displaystyle U=-\sum _{i<j}{\frac {Gm^{2}}{r_{i,j}}}}
1650:
is one-half the time derivative of this moment of inertia
13761:
increases even as energy is lost, effectively a negative
10289:
He approximated the Coma cluster as a spherical "gas" of
14439:
5834:, and the average goes to zero in the limit of infinite
5273:
The average of this derivative over a duration of time,
13584:), in which case the virial radius is approximated as
12730:
11417:
10687:
10572:
10473:
10335:
10154:
10061:
7266:
6894:
3760:, as may be confirmed by explicit calculation. Hence,
2668:
is the sum of all the forces from the other particles
2649:
Connection with the potential energy between particles
221:; this average total kinetic energy is related to the
14774:
14182:"On the Masses of Nebulae and of Clusters of Nebulae"
13665:
13590:
13510:
13422:
13396:
13346:
13024:
12868:
12841:
12672:
12646:
12616:
12593:
12570:
12396:
12380:
is not equal to zero and should be considered as the
12223:
12147:
11882:
11727:
11430:
11260:
11190:
11108:
11045:
10979:
10941:
10898:
10872:
10846:
10764:
10661:
10396:
10315:
10295:
10240:
9922:
9893:
9880:{\displaystyle d\mathbf {F} =-\mathbf {\hat {n}} PdA}
9841:
9629:
9576:
9456:
9408:
9330:
9202:
9179:
9159:
9109:
8931:
8909:
8889:
8851:
8831:
8811:
8752:
8742:. In particular, for gravitational or electrostatic (
8700:
8667:
8547:
8288:
8156:
7956:
7868:
7586:
7566:
7511:
7332:
7228:
7199:
7012:
6793:
6740:
6536:
6471:
6366:
6237:
6078:
5846:
5670:
5485:
5283:
5157:
5022:
4936:
4533:
4468:
4234:
3768:
3534:
2962:
2765:
2678:
2465:
1959:
1656:
1559:
1376:
1116:
626:
274:
202:
for the right-hand side of the equation derives from
53:
31:
of a stable system of discrete particles, bound by a
14483:
8800:
6319:
A simple application of the virial theorem concerns
1646:
th particle. Assuming that the masses are constant,
39:
done is independent of path) with that of the total
13381:{\displaystyle {\frac {GM}{R}}\approx \sigma ^{2}.}
12125:. The terms on the right hand side add up to about
4443:
13741:
13651:
13556:
13475:
13402:
13380:
13121:
12990:
12854:
12827:
12716:
12655:
12632:
12602:
12579:
12554:
12310:
12179:
12103:, and the left hand side of the virial theorem is
12070:
11851:
11660:
11402:
11228:
11176:
11091:
11027:
10965:
10935:(suggested by Hubble), and the cluster has radius
10927:
10884:
10858:
10825:
10747:
10673:
10644:
10554:
10459:
10382:
10321:
10301:
10261:
10226:
10140:
10043:
9908:
9879:
9819:
9623:Now we have two equations that yield the solution
9612:
9562:
9439:
9394:
9316:
9185:
9165:
9145:
9093:
8915:
8895:
8875:
8837:
8817:
8789:
8734:
8686:
8638:
8533:
8246:
8142:
7916:
7729:
7572:
7548:
7497:
7318:
7249:
7214:
7093:
6940:
6880:
6779:
6724:
6518:
6448:
6298:
6213:
6049:
5768:
5618:
5471:
5249:
5134:
5006:
4913:
4511:
4433:
4218:
3693:
3461:
2923:
2732:
2638:
2350:
1941:
1616:
1492:
1305:
991:
314:
146:
11698:is the random "thermal" energy of the particles,
10071:
10021:
9994:
9950:
9900:
9862:
6519:{\displaystyle P_{n}=-i\hbar {\frac {d}{dX_{n}}}}
4452:between two particles is proportional to a power
14946:
14557:Mathematical Introduction to Celestial Mechanics
14405:Parker, E.N. (1954). "Tensor Virial Equations".
13460:
11345:
11303:
11265:
6030:
6017:
5994:
5931:
5848:
14886:
14807:
14579:
14229:Bertone, Gianfranco; Hooper, Dan (2018-10-15).
14023:
13209:Galaxies and cosmology (virial mass and radius)
6941:{\textstyle T=\sum _{n}{\frac {P_{n}^{2}}{2m}}}
4448:In a common special case, the potential energy
14619:(Second ed.). Academic Press. p. 72.
14582:Journal of Physics A: Mathematical and General
12209:
6059:Even if the average of the time derivative of
14814:. Cambridge University Press. pp. 242–.
14328:
14293:
14228:
13981:Fock, V. (1930). "Bemerkung zum Virialsatz".
13297:, and the potential energy (per particle) as
11684:momentum density of the electromagnetic field
10148:. And since the average total kinetic energy
6401:
6378:
1549:is the position vector magnitude. The scalar
1187:
1153:
324:Thus, twice the average total kinetic energy
139:
105:
14938:Gravitational Contraction and Star Formation
14559:. Englewood Cliffs, NJ: Prentice–Hall, Inc.
14298:On a theorem analogous to the virial theorem
13899:
12337:exceeds the kinetic energy of the particles
12302:
12272:
11791:
11767:
11328:
11321:
11290:
11283:
11229:{\displaystyle Nm=8\times 10^{11}M_{\odot }}
11136:
11118:
11086:
11068:
11059:
11046:
10998:
10980:
10817:
10804:
10793:
10780:
10694:
10688:
10668:
10662:
10639:
10621:
10579:
10573:
10549:
10518:
10480:
10474:
10377:
10364:
10342:
10336:
10196:
10170:
10161:
10155:
9929:
9923:
9545:
9514:
9492:
9461:
9447:is the natural frequency of the oscillator.
9311:
9287:
9278:
9260:
9251:
9242:
9233:
9206:
8784:
8778:
8759:
8753:
8729:
8723:
8707:
8701:
8604:
8583:
8575:
8554:
8278:, despite relativity), the time average for
7022:
7016:
6396:
6383:
6338:can also be taken, with equivalent results.
6284:
6270:
6245:
6238:
6199:
6185:
6163:
6132:
6086:
6079:
4930:is the total potential energy of the system
1297:
1291:
633:
627:
494:
306:
293:
284:
278:
7778:. Unsourced material may be challenged and
7142:. Unsourced material may be challenged and
1344:
268:, the virial theorem takes the simple form
14861:The Virial Theorem in Stellar Astrophysics
14750:"The Integral Theorem of the Field Energy"
14366:"On the Radial Pulsation of Gaseous Stars"
14161:. Translated by Heinz Andernach: 110–127.
14073:
13893:
13868:The Virial Theorem in Stellar Astrophysics
9529:
9476:
9146:{\displaystyle x=X\cos(\omega t+\varphi )}
8661:. If the central potential is of the form
6734:Summing over all particles, one finds for
183:th particle, which is located at position
14838:
14765:
14700:
14643:
14539:
14468:
14389:
14246:
14205:
13944:
12974:
12960:
12946:
12811:
12797:
12783:
12151:
11641:
11525:
11396:
9440:{\displaystyle \omega _{0}={\sqrt {k/m}}}
8657:It can also be used to study motion in a
8632:
8527:
8150:The last expression can be simplified to
8132:
7798:Learn how and when to remove this message
7717:
7684:
7664:
7620:
7488:
7464:
7408:
7367:
7309:
7162:Learn how and when to remove this message
5420:
5367:
5128:
5000:
4896:
4829:
4728:
4617:
4551:
4512:{\displaystyle V_{jk}=\alpha r_{jk}^{n},}
2917:
2844:
2779:
1931:
1822:
592:. The respective magnitudes are fixed at
338:times the average total potential energy
81:
14801:
14787:. PHI Learning Pvt. Ltd. pp. 365–.
14353:. Paris: Hermann. pp. 90–91 et seq.
14351:Leçons sur les hypothèses cosmogoniques
14345:
13837:
12341:by a factor equal to the Lorentz factor
12097:, then the moment of inertia is roughly
8282:particles with a power law potential is
7739:
7103:
5479:from which we obtain the exact equation
3507:that is a function only of the distance
2933:Since no particle acts on itself (i.e.,
14857:
14747:
14682:
14629:
14614:
14554:
14521:
14151:"The Redshift of Extragalactic Nebulae"
13865:Collins, G. W. (1978). "Introduction".
13864:
13490:sense, or when used self-consistently.
10284:
8019:
8002:
7319:{\textstyle G(s)=\int _{0}^{s}g(t)\,dt}
6341:
2412:is the net force on that particle, and
1522:represent the mass and position of the
14947:
14689:Continuum Mechanics and Thermodynamics
14632:Continuum Mechanics and Thermodynamics
14404:
14363:
14179:
14148:
14024:Berestycki, H.; Lions, P. -L. (1983).
6066:For power-law forces with an exponent
9196:Applying the virial theorem, we have
6780:{\displaystyle Q=\sum _{n}X_{n}P_{n}}
5144:For gravitating systems the exponent
14524:"A sharp form of the virial theorem"
14144:
14142:
14069:
14067:
13980:
13940:
13938:
13018:. Some common virial relations are
12610:is the acceleration field constant,
7776:adding citations to reliable sources
7743:
7222:be continuous and real-valued, with
7140:adding citations to reliable sources
7107:
14617:Physics of High Temperature Plasmas
11418:Inclusion of electromagnetic fields
10966:{\displaystyle R=10^{6}{\text{ly}}}
9909:{\displaystyle \mathbf {\hat {n}} }
8694:, the virial theorem simplifies to
7549:{\displaystyle -\nabla ^{2}u=g(u),}
5816:, is bounded between two extremes,
2759:. Hence, the virial can be written
257:that is proportional to some power
160:is the total kinetic energy of the
43:of the system. Mathematically, the
13:
14832:
14754:Gazi University Journal of Science
14074:Sivardiere, Jean (December 1986).
13088:
13072:
13001:
12640:is the mass density of particles,
12409:
12406:
12403:
12166:
11823:
11744:
11584:
11569:
11516:
11501:
11358:
11316:
11278:
11239:
10928:{\displaystyle m=10^{9}M_{\odot }}
10097:
8598:
8595:
8592:
8569:
8566:
8563:
8316:
8313:
8310:
7637:
7516:
7384:
7357:
7351:
7348:
7345:
6004:
5941:
5896:
5893:
5890:
5887:
5884:
5858:
3557:
431:generalization of the identity to
14:
14986:
14924:
14808:William K. Rose (16 April 1998).
14683:Fedosin, Sergey G. (2018-09-24).
14139:
14064:
13935:
12717:{\displaystyle ~E_{kf}+2W_{f}=0,}
10674:{\displaystyle \langle U\rangle }
8801:Driven damped harmonic oscillator
6974:of motion. The expectation value
6799:
6691:
6645:
6491:
5268:
2751:is the force applied by particle
14843:(2nd ed.). Addison–Wesley.
13952:(2nd ed.). Addison-Wesley.
13246:. Using the virial theorem, the
12292:
12277:
11184:. However, the observed mass is
10104:
10081:
10068:
10031:
10018:
9982:
9967:
9897:
9859:
9846:
9830:
7980:
7972:
7748:
7112:
6152:
6137:
5742:
5727:
5592:
5577:
5093:
5078:
4590:
4575:
4444:Special case of power-law forces
4305:
4290:
4057:
4042:
3916:
3901:
3878:
3811:
3796:
3752:, the force applied by particle
3658:
3643:
3563:
3537:
3440:
3425:
3402:
3334:
3316:
3301:
3283:
3203:
3185:
3170:
3152:
3084:
3066:
3005:
2990:
2907:
2889:
2818:
2803:
2717:
2681:
2612:
2580:
2334:
2319:
2267:
2252:
2205:
2173:
2114:
2088:
2035:
2014:
1949:In turn, the time derivative of
1915:
1900:
1857:
1831:
1771:
1756:
1604:
1589:
1421:
1175:
1160:
928:
874:
749:
127:
112:
16:Theorem of statistical mechanics
14784:AN INTRODUCTION TO ASTROPHYSICS
14741:
14676:
14623:
14608:
14573:
14548:
14541:10.1090/S0002-9904-1964-11175-7
14515:
14477:
14433:
14398:
14357:
14339:
14322:
14287:
14222:
14173:
13909:The Journal of Chemical Physics
13623:
13242:of its gas and stars, assuming
11343:
11337:
8876:{\displaystyle F\cos(\omega t)}
8209:
8203:
7480:
7424:
7383:
3703:which is equal and opposite to
3493:(equal and opposite reaction).
2420:of the system according to the
14114:
14017:
13974:
13858:
13831:
13799:Chandrasekhar virial equations
13008:gravitational potential energy
11638:
11606:
11557:
11545:
11414:term otherwise must be added.
11387:
11381:
11352:
11310:
11272:
11016:
11004:
10272:
9804:
9773:
9725:
9693:
9386:
9355:
9346:
9340:
9305:
9296:
9140:
9125:
9076:
9067:
8870:
8861:
8687:{\displaystyle U\propto r^{n}}
8119:
8107:
7714:
7711:
7705:
7699:
7654:
7649:
7643:
7633:
7540:
7534:
7505:be a solution to the equation
7474:
7459:
7443:
7440:
7434:
7428:
7418:
7403:
7377:
7362:
7306:
7300:
7276:
7270:
7238:
7232:
7209:
7203:
7176:nonlinear Schrödinger equation
6816:
6804:
6626:
6607:
6601:
6582:
6566:
6537:
6070:, the general equation holds:
6001:
5977:
5971:
5962:
5956:
5938:
5855:
5457:
5451:
5442:
5436:
5415:
5409:
5401:
5395:
4069:
4036:
1229:
1208:
949:
944:
938:
922:
895:
890:
884:
868:
765:
759:
1:
14909:10.1088/0143-0807/37/4/045405
14811:Advanced Stellar Astrophysics
14294:Lord Rayleigh (August 1900).
13824:
11872:electromagnetic stress tensor
8923:, the equation of motion is
1341:results in net cancellation.
553:, which are normal to forces
14332:Scientific Papers: 1892–1901
14265:10.1103/RevModPhys.90.045002
12093:is confined within a radius
10309:stars of roughly equal mass
8256:Newton's third law of motion
3518:between the point particles
3471:Newton's third law of motion
7:
14889:European Journal of Physics
14602:10.1088/0305-4470/29/13/018
14180:Zwicky, F. (October 1937).
14080:American Journal of Physics
13772:
13769:equals −1 no longer holds.
12855:{\displaystyle j^{\alpha }}
12210:Relativistic uniform system
8649:
7839:. Instead, it is true that
6456:with the position operator
3469:where we have assumed that
1553:is defined by the equation
1370:is defined by the equation
10:
14991:
14941:, Georgia State University
14076:"Using the virial theorem"
13212:
13205:) are neglected entirely.
12633:{\displaystyle ~\rho _{0}}
8903:, and damping coefficient
6787:the commutator amounts to
6465:and the momentum operator
2459:velocity of each particle
506:particles with equal mass
449:Subrahmanyan Chandrasekhar
393:
14719:10.1007/s00161-018-0715-x
14662:10.1007/s00161-016-0536-8
14370:The Astrophysical Journal
14316:10.1080/14786440009463903
14235:Reviews of Modern Physics
14186:The Astrophysical Journal
14030:Arch. Rational Mech. Anal
13903:; Beddall, P. M. (1972).
13852:10.1080/14786447008640370
13244:circular Keplerian orbits
10866:estimated that there are
495:Illustrative special case
14615:Schmidt, George (1979).
14231:"History of dark matter"
13983:Zeitschrift für Physik A
13751:
9186:{\displaystyle \varphi }
1345:Statement and derivation
14858:Collins, G. W. (1978).
14555:Pollard, Harry (1966).
14427:10.1103/PhysRev.96.1686
12663:is the current radius.
12587:is the speed of light,
10859:{\displaystyle _{1933}}
10835:
8916:{\displaystyle \gamma }
7580:satisfies the relation
14975:Equations of astronomy
14839:Goldstein, H. (1980).
14767:10.5281/zenodo.3252783
14748:Fedosin, S.G. (2019).
14443:; Lebovitz NR (1962).
14329:Lord Rayleigh (1903).
14155:Helvetica Physica Acta
14149:Zwicky, Fritz (1933).
14122:"2.11: Virial Theorem"
13840:Philosophical Magazine
13743:
13653:
13578:gravitational constant
13558:
13477:
13404:
13382:
13123:
12992:
12856:
12829:
12718:
12657:
12634:
12604:
12603:{\displaystyle ~\eta }
12581:
12556:
12312:
12271:
12181:
12072:
11853:
11662:
11404:
11230:
11178:
11093:
11029:
10967:
10929:
10886:
10860:
10827:
10749:
10675:
10646:
10556:
10461:
10384:
10323:
10303:
10263:
10262:{\displaystyle PV=NkT}
10228:
10142:
10045:
9910:
9881:
9821:
9614:
9570:, which simplifies to
9564:
9441:
9396:
9324:, which simplifies to
9318:
9187:
9167:
9147:
9095:
8917:
8897:
8877:
8839:
8819:
8791:
8736:
8688:
8640:
8535:
8460:
8361:
8248:
8144:
7918:
7731:
7574:
7550:
7499:
7320:
7251:
7250:{\displaystyle g(0)=0}
7216:
7095:
7003:quantum virial theorem
6942:
6882:
6781:
6726:
6520:
6450:
6300:
6215:
6131:
6051:
5770:
5718:
5620:
5568:
5473:
5251:
5136:
5075:
5008:
4970:
4915:
4850:
4749:
4638:
4572:
4519:where the coefficient
4513:
4435:
4374:
4347:
4287:
4220:
4155:
4128:
3991:
3964:
3875:
3848:
3793:
3695:
3463:
3399:
3372:
3275:
3248:
3144:
3117:
3063:
3042:
2987:
2925:
2886:
2865:
2800:
2734:
2714:
2640:
2561:
2502:
2352:
2316:
2249:
2154:
2079:
2011:
1943:
1897:
1811:
1743:
1618:
1586:
1494:
1464:
1403:
1307:
1150:
993:
720:
316:
263:interparticle distance
148:
102:
14528:Bull. Amer. Math. Soc
13819:Pokhozhaev's identity
13809:Equipartition theorem
13744:
13654:
13559:
13478:
13405:
13383:
13124:
12993:
12857:
12830:
12719:
12658:
12635:
12605:
12582:
12557:
12313:
12251:
12182:
12073:
11854:
11663:
11405:
11231:
11179:
11094:
11030:
10968:
10930:
10887:
10885:{\displaystyle N=800}
10861:
10828:
10750:
10676:
10647:
10557:
10462:
10385:
10324:
10304:
10264:
10229:
10143:
10046:
9911:
9882:
9822:
9615:
9565:
9442:
9397:
9319:
9188:
9173:is the amplitude and
9168:
9148:
9096:
8918:
8898:
8878:
8840:
8820:
8792:
8737:
8689:
8641:
8536:
8440:
8341:
8249:
8145:
7919:
7740:In special relativity
7732:
7575:
7551:
7500:
7321:
7252:
7217:
7184:Pokhozhaev's identity
7180:Klein–Gordon equation
7104:Pokhozhaev's identity
7096:
6943:
6883:
6782:
6727:
6521:
6451:
6301:
6216:
6111:
6052:
5771:
5698:
5621:
5548:
5474:
5259:Joseph-Louis Lagrange
5257:which was derived by
5252:
5137:
5055:
5009:
4950:
4916:
4830:
4729:
4618:
4552:
4514:
4436:
4348:
4327:
4267:
4221:
4129:
4108:
3965:
3944:
3849:
3828:
3773:
3696:
3464:
3373:
3352:
3249:
3228:
3118:
3097:
3043:
3022:
2967:
2926:
2866:
2845:
2780:
2735:
2694:
2641:
2541:
2482:
2353:
2296:
2229:
2134:
2059:
1991:
1944:
1877:
1791:
1723:
1619:
1566:
1495:
1444:
1383:
1356:point particles, the
1308:
1130:
1038:with fixed magnitude
994:
700:
514:. The velocities are
484:for the stability of
317:
227:equipartition theorem
225:of the system by the
219:statistical mechanics
149:
82:
14960:Dynamics (mechanics)
14522:Pollard, H. (1964).
13794:Chandrasekhar tensor
13663:
13588:
13508:
13420:
13394:
13344:
13145:. The constants are
13022:
12866:
12839:
12728:
12670:
12644:
12614:
12591:
12568:
12394:
12221:
12198:is the speed of the
12145:
11880:
11725:
11428:
11258:
11188:
11106:
11043:
10977:
10939:
10896:
10870:
10844:
10762:
10685:
10659:
10570:
10471:
10394:
10333:
10313:
10293:
10285:Theoretical analysis
10238:
10152:
10059:
9920:
9891:
9839:
9627:
9574:
9454:
9406:
9328:
9200:
9193:is the phase angle.
9177:
9157:
9107:
8929:
8907:
8887:
8849:
8829:
8809:
8750:
8698:
8665:
8545:
8286:
8154:
7954:
7866:
7772:improve this section
7584:
7564:
7509:
7330:
7264:
7226:
7215:{\displaystyle g(s)}
7197:
7136:improve this section
7010:
6892:
6791:
6738:
6534:
6469:
6364:
6342:In quantum mechanics
6235:
6076:
5844:
5668:
5483:
5281:
5155:
5020:
4934:
4531:
4466:
4232:
3766:
3532:
2960:
2763:
2676:
2463:
1957:
1654:
1557:
1374:
1352:For a collection of
1114:
624:
272:
51:
14970:Concepts in physics
14901:2016EJPh...37d5405I
14870:1978vtsa.book.....C
14841:Classical Mechanics
14711:2019CMT....31..627F
14654:2017CMT....29..361F
14594:1996JPhA...29.3471K
14501:1953ApJ...118..116C
14461:1962ApJ...136.1037C
14419:1954PhRv...96.1686P
14382:1945ApJ...102..143L
14364:Ledoux, P. (1945).
14257:2018RvMP...90d5002B
14198:1937ApJ....86..217Z
14092:1986AmJPh..54.1100S
14042:1983ArRMA..82..313B
13995:1930ZPhy...63..855F
13950:Classical mechanics
13921:1972JChPh..56.3320B
13877:1978vtsa.book.....C
13719:
13469:
13248:velocity dispersion
13010:of a system to its
12487:
12382:material derivative
11135:
11085:
11039:of kinetic energy,
10997:
9790:
9710:
9372:
8397:
7361:
7296:
6972:Heisenberg equation
6970:, according to the
6927:
6710:
6435:
5419:
5346:
5150:Lagrange's identity
4795:
4505:
2527:
2369:is the mass of the
1489:
848:
699:
482:Chandrasekhar limit
437:James Clerk Maxwell
425:Lagrange's identity
231:thermal equilibrium
14931:The Virial Theorem
14126:Physics LibreTexts
14050:10.1007/BF00250555
14003:10.1007/BF01339281
13946:Goldstein, Herbert
13779:Virial coefficient
13739:
13705:
13649:
13554:
13488:order of magnitude
13473:
13455:
13400:
13378:
13164:, and proton mass
13154:Boltzmann constant
13141:, and temperature
13119:
12988:
12852:
12825:
12714:
12656:{\displaystyle ~r}
12653:
12630:
12600:
12580:{\displaystyle ~c}
12577:
12552:
12473:
12308:
12177:
12089:. If a total mass
12068:
11849:
11658:
11400:
11362:
11320:
11282:
11226:
11174:
11121:
11089:
11071:
11025:
10983:
10963:
10925:
10882:
10856:
10823:
10745:
10671:
10642:
10552:
10517:
10457:
10421:
10380:
10319:
10299:
10259:
10224:
10138:
10053:divergence theorem
10041:
9964:
9906:
9877:
9817:
9812:
9776:
9696:
9610:
9560:
9559:
9552:
9506:
9499:
9437:
9392:
9358:
9314:
9183:
9163:
9143:
9091:
9090:
9083:
9050:
9043:
9008:
9001:
8980:
8973:
8913:
8893:
8883:, spring constant
8873:
8835:
8815:
8787:
8732:
8684:
8636:
8531:
8383:
8244:
8140:
8138:
7914:
7727:
7570:
7546:
7495:
7339:
7316:
7282:
7247:
7212:
7091:
7037:
6938:
6913:
6910:
6878:
6840:
6777:
6756:
6722:
6696:
6516:
6446:
6421:
6418:
6332:ergodic hypothesis
6296:
6211:
6047:
6008:
5945:
5862:
5766:
5616:
5469:
5387:
5332:
5247:
5148:equals −1, giving
5132:
5004:
4986:
4911:
4909:
4866:
4772:
4765:
4654:
4509:
4488:
4456:of their distance
4431:
4216:
4214:
3691:
3459:
3457:
2921:
2730:
2636:
2513:
2348:
2346:
1939:
1937:
1614:
1490:
1475:
1303:
989:
820:
671:
312:
144:
33:conservative force
14879:978-0-912918-13-6
14864:. Pachart Press.
14850:978-0-201-02918-5
14821:978-0-521-58833-1
14794:978-81-203-4071-8
14588:(13): 3471–3494.
14566:978-0-13-561068-8
14086:(12): 1100–1103.
13929:10.1063/1.1677699
13886:978-0-912918-13-6
13871:. Pachart Press.
13814:Ehrenfest theorem
13804:Derrick's theorem
13733:
13700:
13673:
13643:
13598:
13552:
13518:
13450:
13447:
13436:
13403:{\displaystyle R}
13360:
13240:rotation velocity
13233:isothermal sphere
13147:Newton's constant
13107:
13094:
13061:
13048:
13033:
12944:
12905:
12871:
12781:
12733:
12724:where the energy
12675:
12649:
12619:
12596:
12573:
12547:
12545:
12537:
12514:
12200:ion acoustic wave
12172:
12058:
11963:
11931:
11676:moment of inertia
11523:
11473:
11439:
11344:
11341:
11302:
11264:
11143:
11013:
10961:
10821:
10815:
10743:
10711:
10655:Zwicky estimated
10596:
10502:
10455:
10406:
10356:
10322:{\displaystyle m}
10302:{\displaystyle N}
10216:
10181:
10074:
10024:
10010:
9955:
9946:
9903:
9865:
9808:
9736:
9735:
9608:
9557:
9526:
9512:
9510:
9504:
9489:
9473:
9459:
9457:
9435:
9275:
9230:
9218:
9166:{\displaystyle X}
9088:
9053:
9051:
9048:
9037:
9011:
9009:
9006:
8986:
8984:
8978:
8969:
8937:
8935:
8896:{\displaystyle k}
8838:{\displaystyle x}
8818:{\displaystyle m}
8776:
8721:
8659:central potential
8608:
8500:
8404:
8398:
8297:
8235:
8207:
8194:
8188:
8123:
8044:
7999:
7969:
7912:
7911:
7909:
7808:
7807:
7800:
7607:
7573:{\displaystyle u}
7188:Derrick's theorem
7172:
7171:
7164:
7087:
7078:
7028:
6936:
6901:
6876:
6831:
6802:
6747:
6718:
6714:
6683:
6514:
6444:
6409:
6348:Ehrenfest theorem
6280:
6268:
6195:
6183:
6109:
6039:
5993:
5984:
5930:
5911:
5847:
5509:
5464:
5385:
5365:
5330:
5307:
5244:
5223:
5189:
5176:
5125:
5041:
4971:
4944:
4904:
4894:
4851:
4827:
4750:
4726:
4693:
4639:
4615:
4549:
4523:and the exponent
4413:
4253:
4194:
4093:
4030:
3682:
3632:
2842:
2777:
2631:
2599:
2539:
2480:
2224:
2192:
2107:
2054:
1982:
1850:
1721:
1706:
1689:
1669:
1361:moment of inertia
1264:
1206:
1128:
1094:centripetal force
916:
862:
818:
729:
669:
303:
79:
14982:
14955:Physics theorems
14920:
14883:
14854:
14826:
14825:
14805:
14799:
14798:
14778:
14772:
14771:
14769:
14745:
14739:
14738:
14704:
14680:
14674:
14673:
14647:
14627:
14621:
14620:
14612:
14606:
14605:
14577:
14571:
14570:
14552:
14546:
14545:
14543:
14519:
14513:
14512:
14485:Chandrasekhar, S
14481:
14475:
14474:
14472:
14441:Chandrasekhar, S
14437:
14431:
14430:
14413:(6): 1686–1689.
14402:
14396:
14395:
14393:
14361:
14355:
14354:
14343:
14337:
14336:
14326:
14320:
14319:
14310:(303): 210–213.
14291:
14285:
14284:
14250:
14226:
14220:
14219:
14209:
14177:
14171:
14170:
14146:
14137:
14136:
14134:
14133:
14118:
14112:
14111:
14071:
14062:
14061:
14021:
14015:
14014:
13978:
13972:
13971:
13942:
13933:
13932:
13915:(7): 3320–3329.
13897:
13891:
13890:
13862:
13856:
13855:
13846:(265): 122–127.
13835:
13768:
13748:
13746:
13745:
13740:
13735:
13734:
13731:
13718:
13713:
13701:
13693:
13688:
13687:
13675:
13674:
13671:
13658:
13656:
13655:
13650:
13645:
13644:
13641:
13613:
13612:
13600:
13599:
13596:
13575:
13570:Hubble parameter
13567:
13563:
13561:
13560:
13555:
13553:
13551:
13540:
13539:
13538:
13525:
13520:
13519:
13516:
13503:critical density
13482:
13480:
13479:
13474:
13468:
13463:
13451:
13449:
13448:
13445:
13439:
13438:
13437:
13434:
13424:
13413:
13409:
13407:
13406:
13401:
13387:
13385:
13384:
13379:
13374:
13373:
13361:
13356:
13348:
13337:
13336:
13334:
13333:
13328:
13325:
13317:
13315:
13314:
13311:
13308:
13296:
13292:
13290:
13289:
13286:
13283:
13273:
13271:
13270:
13267:
13264:
13252:
13204:
13202:
13201:
13198:
13195:
13188:
13186:
13185:
13182:
13179:
13172:
13163:
13151:
13144:
13140:
13136:
13132:
13128:
13126:
13125:
13120:
13118:
13117:
13108:
13100:
13095:
13093:
13092:
13091:
13081:
13077:
13076:
13075:
13064:
13062:
13054:
13049:
13044:
13036:
13034:
13026:
12997:
12995:
12994:
12989:
12987:
12986:
12973:
12972:
12959:
12958:
12945:
12937:
12935:
12934:
12922:
12921:
12906:
12904:
12903:
12902:
12886:
12881:
12880:
12869:
12861:
12859:
12858:
12853:
12851:
12850:
12834:
12832:
12831:
12826:
12824:
12823:
12810:
12809:
12796:
12795:
12782:
12774:
12772:
12771:
12762:
12761:
12746:
12745:
12731:
12723:
12721:
12720:
12715:
12704:
12703:
12688:
12687:
12673:
12662:
12660:
12659:
12654:
12647:
12639:
12637:
12636:
12631:
12629:
12628:
12617:
12609:
12607:
12606:
12601:
12594:
12586:
12584:
12583:
12578:
12571:
12561:
12559:
12558:
12553:
12548:
12546:
12544:
12543:
12539:
12538:
12536:
12535:
12517:
12515:
12507:
12497:
12496:
12486:
12481:
12472:
12471:
12461:
12460:
12459:
12450:
12449:
12430:
12422:
12414:
12413:
12412:
12379:
12375:
12373:
12372:
12369:
12366:
12359:
12349:
12340:
12336:
12320:where the value
12317:
12315:
12314:
12309:
12301:
12300:
12295:
12286:
12285:
12280:
12270:
12265:
12241:
12237:
12236:
12197:
12186:
12184:
12183:
12178:
12173:
12171:
12170:
12169:
12156:
12138:
12134:
12130:
12124:
12123:
12121:
12120:
12115:
12112:
12102:
12096:
12092:
12088:
12077:
12075:
12074:
12069:
12064:
12060:
12059:
12057:
12056:
12047:
12046:
12045:
12036:
12035:
12025:
12020:
12019:
12010:
12009:
12000:
11999:
11982:
11981:
11969:
11965:
11964:
11962:
11961:
11960:
11947:
11946:
11937:
11932:
11927:
11926:
11925:
11916:
11915:
11905:
11895:
11894:
11869:
11858:
11856:
11855:
11850:
11845:
11844:
11835:
11834:
11822:
11821:
11812:
11811:
11799:
11798:
11789:
11788:
11779:
11778:
11766:
11765:
11756:
11755:
11740:
11739:
11718:
11709:
11703:
11697:
11694:of the "fluid",
11689:
11681:
11673:
11667:
11665:
11664:
11659:
11654:
11653:
11637:
11636:
11621:
11620:
11605:
11604:
11589:
11588:
11587:
11574:
11573:
11572:
11535:
11534:
11524:
11522:
11514:
11513:
11512:
11499:
11497:
11496:
11487:
11486:
11474:
11472:
11471:
11470:
11457:
11453:
11452:
11442:
11440:
11432:
11409:
11407:
11406:
11401:
11377:
11376:
11368:
11361:
11342:
11339:
11336:
11335:
11319:
11298:
11297:
11281:
11235:
11233:
11232:
11227:
11225:
11224:
11215:
11214:
11183:
11181:
11180:
11175:
11173:
11172:
11163:
11162:
11144:
11139:
11134:
11129:
11110:
11098:
11096:
11095:
11090:
11084:
11079:
11058:
11057:
11034:
11032:
11031:
11026:
11024:
11023:
11014:
11011:
10996:
10991:
10972:
10970:
10969:
10964:
10962:
10959:
10957:
10956:
10934:
10932:
10931:
10926:
10924:
10923:
10914:
10913:
10891:
10889:
10888:
10883:
10865:
10863:
10862:
10857:
10855:
10854:
10832:
10830:
10829:
10824:
10822:
10820:
10816:
10808:
10796:
10792:
10791:
10775:
10754:
10752:
10751:
10746:
10744:
10739:
10738:
10737:
10728:
10727:
10714:
10712:
10704:
10680:
10678:
10677:
10672:
10651:
10649:
10648:
10643:
10638:
10633:
10628:
10620:
10619:
10607:
10606:
10597:
10589:
10561:
10559:
10558:
10553:
10548:
10547:
10546:
10530:
10525:
10516:
10501:
10500:
10466:
10464:
10463:
10458:
10456:
10454:
10453:
10438:
10437:
10436:
10423:
10420:
10389:
10387:
10386:
10381:
10376:
10375:
10357:
10349:
10328:
10326:
10325:
10320:
10308:
10306:
10305:
10300:
10268:
10266:
10265:
10260:
10233:
10231:
10230:
10225:
10217:
10209:
10195:
10194:
10182:
10174:
10147:
10145:
10144:
10139:
10107:
10084:
10076:
10075:
10067:
10050:
10048:
10047:
10042:
10034:
10026:
10025:
10017:
10011:
10003:
9998:
9997:
9991:
9990:
9985:
9976:
9975:
9970:
9963:
9954:
9953:
9947:
9939:
9915:
9913:
9912:
9907:
9905:
9904:
9896:
9886:
9884:
9883:
9878:
9867:
9866:
9858:
9849:
9826:
9824:
9823:
9818:
9816:
9815:
9809:
9807:
9803:
9802:
9789:
9784:
9768:
9760:
9737:
9734:
9733:
9732:
9723:
9722:
9709:
9704:
9692:
9691:
9679:
9678:
9669:
9668:
9658:
9657:
9648:
9647:
9619:
9617:
9616:
9611:
9609:
9604:
9593:
9569:
9567:
9566:
9561:
9558:
9555:
9553:
9548:
9528:
9527:
9519:
9505:
9503:power dissipated
9502:
9500:
9495:
9491:
9490:
9482:
9475:
9474:
9466:
9446:
9444:
9443:
9438:
9436:
9431:
9423:
9418:
9417:
9401:
9399:
9398:
9393:
9385:
9384:
9371:
9366:
9323:
9321:
9320:
9315:
9277:
9276:
9268:
9232:
9231:
9223:
9220:
9219:
9211:
9192:
9190:
9189:
9184:
9172:
9170:
9169:
9164:
9152:
9150:
9149:
9144:
9100:
9098:
9097:
9092:
9089:
9087:external driving
9086:
9084:
9079:
9049:
9046:
9044:
9039:
9038:
9036:
9028:
9020:
9007:
9004:
9002:
8997:
8979:
8976:
8974:
8968:
8967:
8966:
8953:
8949:
8948:
8938:
8922:
8920:
8919:
8914:
8902:
8900:
8899:
8894:
8882:
8880:
8879:
8874:
8845:, driving force
8844:
8842:
8841:
8836:
8824:
8822:
8821:
8816:
8796:
8794:
8793:
8788:
8777:
8769:
8741:
8739:
8738:
8733:
8722:
8714:
8693:
8691:
8690:
8685:
8683:
8682:
8645:
8643:
8642:
8637:
8631:
8627:
8609:
8607:
8603:
8602:
8601:
8578:
8574:
8573:
8572:
8549:
8540:
8538:
8537:
8532:
8526:
8525:
8520:
8516:
8515:
8514:
8505:
8501:
8499:
8498:
8497:
8484:
8477:
8476:
8466:
8459:
8454:
8430:
8429:
8424:
8420:
8419:
8418:
8409:
8405:
8400:
8399:
8396:
8391:
8376:
8367:
8360:
8355:
8331:
8330:
8325:
8321:
8320:
8319:
8298:
8290:
8281:
8277:
8253:
8251:
8250:
8245:
8240:
8236:
8234:
8226:
8215:
8208:
8205:
8199:
8195:
8190:
8189:
8187:
8186:
8171:
8162:
8149:
8147:
8146:
8141:
8139:
8128:
8124:
8122:
8102:
8101:
8100:
8087:
8075:
8071:
8070:
8058:
8057:
8045:
8037:
8029:
8022:
8005:
8000:
7992:
7983:
7975:
7970:
7962:
7949:
7948:
7946:
7945:
7940:
7937:
7923:
7921:
7920:
7915:
7913:
7910:
7908:
7907:
7898:
7897:
7888:
7880:
7876:
7856:
7852:
7838:
7830:
7828:
7827:
7824:
7821:
7803:
7796:
7792:
7789:
7783:
7752:
7744:
7736:
7734:
7733:
7728:
7695:
7694:
7693:
7692:
7687:
7663:
7662:
7657:
7636:
7631:
7630:
7629:
7628:
7623:
7612:
7608:
7603:
7592:
7579:
7577:
7576:
7571:
7556:in the sense of
7555:
7553:
7552:
7547:
7524:
7523:
7504:
7502:
7501:
7496:
7491:
7473:
7472:
7467:
7458:
7457:
7417:
7416:
7411:
7402:
7401:
7376:
7375:
7370:
7360:
7355:
7354:
7325:
7323:
7322:
7317:
7295:
7290:
7256:
7254:
7253:
7248:
7221:
7219:
7218:
7213:
7186:, also known as
7167:
7160:
7156:
7153:
7147:
7116:
7108:
7100:
7098:
7097:
7092:
7085:
7084:
7080:
7079:
7077:
7076:
7075:
7062:
7054:
7052:
7051:
7036:
6999:
6998:
6996:
6994:
6993:
6988:
6985:
6969:
6968:
6966:
6965:
6960:
6957:
6947:
6945:
6944:
6939:
6937:
6935:
6926:
6921:
6912:
6909:
6887:
6885:
6884:
6879:
6877:
6875:
6874:
6873:
6860:
6852:
6850:
6849:
6839:
6803:
6795:
6786:
6784:
6783:
6778:
6776:
6775:
6766:
6765:
6755:
6731:
6729:
6728:
6723:
6716:
6715:
6709:
6704:
6695:
6684:
6682:
6681:
6680:
6667:
6659:
6657:
6656:
6638:
6637:
6625:
6624:
6600:
6599:
6581:
6580:
6565:
6564:
6555:
6554:
6529:
6525:
6523:
6522:
6517:
6515:
6513:
6512:
6511:
6495:
6481:
6480:
6464:
6455:
6453:
6452:
6447:
6445:
6443:
6434:
6429:
6420:
6417:
6405:
6404:
6395:
6394:
6382:
6381:
6336:ensemble average
6305:
6303:
6302:
6297:
6292:
6291:
6282:
6281:
6278:
6269:
6261:
6253:
6252:
6230:
6220:
6218:
6217:
6212:
6207:
6206:
6197:
6196:
6193:
6184:
6176:
6171:
6170:
6161:
6160:
6155:
6146:
6145:
6140:
6130:
6125:
6110:
6102:
6094:
6093:
6069:
6062:
6056:
6054:
6053:
6048:
6040:
6035:
6034:
6033:
6021:
6020:
6010:
6007:
5989:
5985:
5980:
5951:
5944:
5926:
5922:
5921:
5916:
5912:
5910:
5902:
5901:
5900:
5899:
5873:
5861:
5837:
5833:
5824:
5815:
5809:
5802:
5800:
5798:
5797:
5792:
5789:
5775:
5773:
5772:
5767:
5762:
5761:
5756:
5752:
5751:
5750:
5745:
5736:
5735:
5730:
5717:
5712:
5691:
5690:
5685:
5663:
5656:
5654:
5652:
5651:
5646:
5643:
5625:
5623:
5622:
5617:
5612:
5611:
5606:
5602:
5601:
5600:
5595:
5586:
5585:
5580:
5567:
5562:
5544:
5543:
5538:
5520:
5519:
5514:
5510:
5508:
5500:
5492:
5478:
5476:
5475:
5470:
5465:
5460:
5431:
5418:
5404:
5386:
5378:
5366:
5364:
5356:
5348:
5345:
5340:
5331:
5323:
5318:
5317:
5312:
5308:
5306:
5298:
5290:
5277:, is defined as
5276:
5261:and extended by
5256:
5254:
5253:
5248:
5246:
5245:
5242:
5224:
5222:
5221:
5220:
5207:
5203:
5202:
5192:
5190:
5182:
5177:
5175:
5167:
5159:
5147:
5141:
5139:
5138:
5133:
5127:
5126:
5123:
5102:
5101:
5096:
5087:
5086:
5081:
5074:
5069:
5042:
5040:
5032:
5024:
5013:
5011:
5010:
5005:
4999:
4998:
4985:
4969:
4964:
4946:
4945:
4942:
4929:
4920:
4918:
4917:
4912:
4910:
4906:
4905:
4902:
4895:
4887:
4882:
4881:
4865:
4849:
4844:
4828:
4820:
4812:
4808:
4807:
4794:
4783:
4764:
4748:
4743:
4727:
4719:
4711:
4707:
4706:
4694:
4692:
4691:
4690:
4674:
4673:
4672:
4656:
4653:
4637:
4632:
4616:
4608:
4599:
4598:
4593:
4584:
4583:
4578:
4571:
4566:
4550:
4542:
4526:
4522:
4518:
4516:
4515:
4510:
4504:
4499:
4481:
4480:
4462:
4455:
4451:
4440:
4438:
4437:
4432:
4427:
4426:
4414:
4412:
4411:
4410:
4394:
4393:
4392:
4376:
4373:
4362:
4346:
4341:
4314:
4313:
4308:
4299:
4298:
4293:
4286:
4281:
4254:
4252:
4244:
4236:
4225:
4223:
4222:
4217:
4215:
4208:
4207:
4195:
4193:
4192:
4191:
4175:
4174:
4173:
4157:
4154:
4143:
4127:
4122:
4098:
4094:
4092:
4091:
4079:
4078:
4077:
4072:
4066:
4065:
4060:
4051:
4050:
4045:
4039:
4033:
4031:
4029:
4028:
4027:
4011:
4010:
4009:
3993:
3990:
3979:
3963:
3958:
3934:
3930:
3926:
3925:
3924:
3919:
3910:
3909:
3904:
3890:
3889:
3881:
3874:
3863:
3847:
3842:
3820:
3819:
3814:
3805:
3804:
3799:
3792:
3787:
3759:
3755:
3751:
3700:
3698:
3697:
3692:
3687:
3683:
3681:
3680:
3668:
3667:
3666:
3661:
3652:
3651:
3646:
3639:
3633:
3631:
3630:
3629:
3613:
3612:
3611:
3595:
3587:
3586:
3574:
3573:
3572:
3571:
3566:
3549:
3548:
3540:
3525:
3521:
3517:
3506:
3492:
3468:
3466:
3465:
3460:
3458:
3454:
3450:
3449:
3448:
3443:
3434:
3433:
3428:
3414:
3413:
3405:
3398:
3387:
3371:
3366:
3348:
3344:
3343:
3342:
3337:
3328:
3327:
3319:
3310:
3309:
3304:
3295:
3294:
3286:
3274:
3263:
3247:
3242:
3221:
3217:
3213:
3212:
3211:
3206:
3197:
3196:
3188:
3179:
3178:
3173:
3164:
3163:
3155:
3143:
3132:
3116:
3111:
3093:
3092:
3087:
3078:
3077:
3069:
3062:
3057:
3041:
3036:
3014:
3013:
3008:
2999:
2998:
2993:
2986:
2981:
2955:
2944:
2930:
2928:
2927:
2922:
2916:
2915:
2910:
2901:
2900:
2892:
2885:
2880:
2864:
2859:
2843:
2835:
2827:
2826:
2821:
2812:
2811:
2806:
2799:
2794:
2778:
2770:
2758:
2754:
2750:
2739:
2737:
2736:
2731:
2729:
2728:
2720:
2713:
2708:
2690:
2689:
2684:
2671:
2667:
2663:
2653:The total force
2645:
2643:
2642:
2637:
2632:
2630:
2622:
2621:
2620:
2615:
2605:
2600:
2598:
2590:
2589:
2588:
2583:
2573:
2571:
2570:
2560:
2555:
2540:
2532:
2526:
2521:
2512:
2511:
2501:
2496:
2481:
2473:
2458:
2457:
2455:
2454:
2449:
2446:
2415:
2411:
2410:
2408:
2407:
2402:
2399:
2372:
2368:
2357:
2355:
2354:
2349:
2347:
2343:
2342:
2337:
2328:
2327:
2322:
2315:
2310:
2280:
2276:
2275:
2270:
2261:
2260:
2255:
2248:
2243:
2225:
2223:
2215:
2214:
2213:
2208:
2198:
2193:
2191:
2183:
2182:
2181:
2176:
2166:
2164:
2163:
2153:
2148:
2127:
2123:
2122:
2117:
2108:
2106:
2098:
2097:
2096:
2091:
2081:
2078:
2073:
2055:
2053:
2045:
2044:
2043:
2038:
2028:
2023:
2022:
2017:
2010:
2005:
1983:
1981:
1973:
1965:
1952:
1948:
1946:
1945:
1940:
1938:
1924:
1923:
1918:
1909:
1908:
1903:
1896:
1891:
1870:
1866:
1865:
1860:
1851:
1849:
1841:
1840:
1839:
1834:
1824:
1821:
1820:
1810:
1805:
1784:
1780:
1779:
1774:
1765:
1764:
1759:
1753:
1752:
1742:
1737:
1722:
1720:
1709:
1707:
1699:
1690:
1688:
1680:
1672:
1670:
1662:
1649:
1645:
1634:
1623:
1621:
1620:
1615:
1613:
1612:
1607:
1598:
1597:
1592:
1585:
1580:
1552:
1548:
1546:
1525:
1521:
1510:
1499:
1497:
1496:
1491:
1488:
1483:
1474:
1473:
1463:
1458:
1440:
1439:
1434:
1430:
1429:
1424:
1413:
1412:
1402:
1397:
1365:
1355:
1340:
1326:
1312:
1310:
1309:
1304:
1287:
1286:
1265:
1260:
1259:
1258:
1245:
1207:
1199:
1191:
1190:
1184:
1183:
1178:
1169:
1168:
1163:
1157:
1156:
1149:
1144:
1129:
1121:
1109:
1091:
1041:
1037:
1012:
998:
996:
995:
990:
985:
984:
963:
959:
958:
957:
952:
937:
936:
931:
925:
917:
909:
904:
903:
898:
883:
882:
877:
871:
863:
855:
847:
846:
845:
835:
834:
833:
819:
817:
816:
815:
803:
802:
789:
778:
777:
772:
768:
758:
757:
752:
740:
739:
730:
722:
719:
714:
698:
697:
696:
686:
685:
684:
670:
668:
667:
666:
654:
653:
640:
617:
608:
599:
595:
591:
566:
552:
527:
513:
509:
505:
422:
420:
419:
416:
413:
390:equals −1.
389:
385:
374:
365:
361:
350:
349:
337:
333:
332:
321:
319:
318:
313:
305:
304:
301:
267:
260:
256:
242:potential energy
193:
182:
174:
163:
159:
153:
151:
150:
145:
143:
142:
136:
135:
130:
121:
120:
115:
109:
108:
101:
96:
80:
72:
64:
41:potential energy
14990:
14989:
14985:
14984:
14983:
14981:
14980:
14979:
14965:Solid mechanics
14945:
14944:
14927:
14880:
14851:
14835:
14833:Further reading
14830:
14829:
14822:
14806:
14802:
14795:
14779:
14775:
14746:
14742:
14681:
14677:
14628:
14624:
14613:
14609:
14578:
14574:
14567:
14553:
14549:
14520:
14516:
14482:
14478:
14438:
14434:
14407:Physical Review
14403:
14399:
14362:
14358:
14347:Poincaré, Henri
14344:
14340:
14327:
14323:
14292:
14288:
14227:
14223:
14178:
14174:
14147:
14140:
14131:
14129:
14120:
14119:
14115:
14100:10.1119/1.14723
14072:
14065:
14022:
14018:
13989:(11): 855–858.
13979:
13975:
13960:
13943:
13936:
13901:Bader, R. F. W.
13898:
13894:
13887:
13863:
13859:
13836:
13832:
13827:
13775:
13766:
13754:
13730:
13726:
13714:
13709:
13692:
13683:
13679:
13670:
13666:
13664:
13661:
13660:
13640:
13636:
13608:
13604:
13595:
13591:
13589:
13586:
13585:
13573:
13565:
13541:
13534:
13530:
13526:
13524:
13515:
13511:
13509:
13506:
13505:
13464:
13459:
13444:
13440:
13433:
13429:
13425:
13423:
13421:
13418:
13417:
13411:
13395:
13392:
13391:
13369:
13365:
13349:
13347:
13345:
13342:
13341:
13329:
13326:
13321:
13320:
13318:
13312:
13309:
13306:
13305:
13303:
13298:
13287:
13284:
13281:
13280:
13278:
13268:
13265:
13262:
13261:
13259:
13254:
13250:
13217:
13211:
13199:
13196:
13193:
13192:
13190:
13183:
13180:
13177:
13176:
13174:
13171:
13165:
13162:
13156:
13149:
13142:
13138:
13134:
13130:
13113:
13109:
13099:
13087:
13086:
13082:
13071:
13070:
13066:
13065:
13063:
13053:
13037:
13035:
13025:
13023:
13020:
13019:
13004:
13002:In astrophysics
12982:
12978:
12968:
12964:
12954:
12950:
12936:
12927:
12923:
12914:
12910:
12898:
12894:
12890:
12885:
12876:
12872:
12867:
12864:
12863:
12846:
12842:
12840:
12837:
12836:
12819:
12815:
12805:
12801:
12791:
12787:
12773:
12767:
12763:
12757:
12753:
12738:
12734:
12729:
12726:
12725:
12699:
12695:
12680:
12676:
12671:
12668:
12667:
12645:
12642:
12641:
12624:
12620:
12615:
12612:
12611:
12592:
12589:
12588:
12569:
12566:
12565:
12531:
12527:
12516:
12506:
12505:
12501:
12492:
12488:
12482:
12477:
12467:
12463:
12462:
12455:
12451:
12445:
12441:
12431:
12429:
12421:
12402:
12401:
12397:
12395:
12392:
12391:
12377:
12370:
12367:
12364:
12363:
12361:
12356:
12351:
12347:
12342:
12338:
12333:
12326:
12321:
12296:
12291:
12290:
12281:
12276:
12275:
12266:
12255:
12232:
12228:
12224:
12222:
12219:
12218:
12212:
12196:
12190:
12165:
12164:
12160:
12155:
12146:
12143:
12142:
12136:
12132:
12126:
12116:
12113:
12108:
12107:
12105:
12104:
12098:
12094:
12090:
12086:
12052:
12048:
12041:
12037:
12031:
12027:
12026:
12024:
12015:
12011:
12005:
12001:
11995:
11991:
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11974:
11970:
11956:
11952:
11948:
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11938:
11936:
11921:
11917:
11911:
11907:
11906:
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11903:
11899:
11887:
11883:
11881:
11878:
11877:
11867:
11862:
11840:
11836:
11830:
11826:
11817:
11813:
11807:
11803:
11794:
11790:
11784:
11780:
11774:
11770:
11761:
11757:
11751:
11747:
11732:
11728:
11726:
11723:
11722:
11716:
11711:
11705:
11699:
11695:
11687:
11679:
11671:
11649:
11645:
11629:
11625:
11613:
11609:
11600:
11596:
11583:
11582:
11578:
11568:
11567:
11563:
11530:
11526:
11515:
11508:
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11488:
11482:
11478:
11466:
11462:
11458:
11448:
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11441:
11431:
11429:
11426:
11425:
11420:
11369:
11364:
11363:
11348:
11338:
11331:
11327:
11306:
11293:
11289:
11268:
11259:
11256:
11255:
11242:
11240:Generalizations
11220:
11216:
11210:
11206:
11189:
11186:
11185:
11168:
11164:
11158:
11154:
11130:
11125:
11111:
11109:
11107:
11104:
11103:
11080:
11075:
11053:
11049:
11044:
11041:
11040:
11019:
11015:
11010:
10992:
10987:
10978:
10975:
10974:
10958:
10952:
10948:
10940:
10937:
10936:
10919:
10915:
10909:
10905:
10897:
10894:
10893:
10871:
10868:
10867:
10850:
10847:
10845:
10842:
10841:
10838:
10807:
10797:
10787:
10783:
10776:
10774:
10763:
10760:
10759:
10733:
10729:
10723:
10719:
10715:
10713:
10703:
10686:
10683:
10682:
10660:
10657:
10656:
10634:
10629:
10624:
10615:
10611:
10602:
10598:
10588:
10571:
10568:
10567:
10536:
10532:
10531:
10526:
10521:
10506:
10496:
10492:
10472:
10469:
10468:
10443:
10439:
10432:
10428:
10424:
10422:
10410:
10395:
10392:
10391:
10371:
10367:
10348:
10334:
10331:
10330:
10314:
10311:
10310:
10294:
10291:
10290:
10287:
10275:
10239:
10236:
10235:
10208:
10190:
10186:
10173:
10153:
10150:
10149:
10103:
10080:
10066:
10065:
10060:
10057:
10056:
10030:
10016:
10015:
10002:
9993:
9992:
9986:
9981:
9980:
9971:
9966:
9965:
9959:
9949:
9948:
9938:
9921:
9918:
9917:
9895:
9894:
9892:
9889:
9888:
9857:
9856:
9845:
9840:
9837:
9836:
9833:
9811:
9810:
9798:
9794:
9785:
9780:
9769:
9761:
9759:
9751:
9739:
9738:
9728:
9724:
9718:
9714:
9705:
9700:
9687:
9683:
9674:
9670:
9664:
9660:
9659:
9653:
9649:
9646:
9641:
9631:
9630:
9628:
9625:
9624:
9594:
9592:
9575:
9572:
9571:
9554:
9518:
9517:
9513:
9511:
9501:
9481:
9480:
9465:
9464:
9460:
9458:
9455:
9452:
9451:
9427:
9422:
9413:
9409:
9407:
9404:
9403:
9380:
9376:
9367:
9362:
9329:
9326:
9325:
9267:
9266:
9222:
9221:
9210:
9209:
9201:
9198:
9197:
9178:
9175:
9174:
9158:
9155:
9154:
9108:
9105:
9104:
9085:
9054:
9052:
9045:
9029:
9021:
9019:
9012:
9010:
9003:
8987:
8985:
8975:
8962:
8958:
8954:
8944:
8940:
8939:
8936:
8930:
8927:
8926:
8908:
8905:
8904:
8888:
8885:
8884:
8850:
8847:
8846:
8830:
8827:
8826:
8810:
8807:
8806:
8803:
8768:
8751:
8748:
8747:
8713:
8699:
8696:
8695:
8678:
8674:
8666:
8663:
8662:
8652:
8617:
8613:
8591:
8590:
8586:
8579:
8562:
8561:
8557:
8550:
8548:
8546:
8543:
8542:
8521:
8510:
8506:
8493:
8489:
8485:
8472:
8468:
8467:
8465:
8461:
8455:
8444:
8439:
8435:
8434:
8425:
8414:
8410:
8392:
8387:
8375:
8368:
8366:
8362:
8356:
8345:
8340:
8336:
8335:
8326:
8309:
8308:
8304:
8300:
8299:
8289:
8287:
8284:
8283:
8279:
8276:
8267:
8259:
8227:
8216:
8214:
8210:
8204:
8182:
8178:
8170:
8163:
8161:
8157:
8155:
8152:
8151:
8137:
8136:
8103:
8096:
8092:
8088:
8086:
8082:
8073:
8072:
8066:
8062:
8053:
8049:
8036:
8027:
8026:
8018:
8001:
7991:
7984:
7979:
7971:
7961:
7957:
7955:
7952:
7951:
7941:
7938:
7933:
7932:
7930:
7925:
7903:
7899:
7893:
7889:
7887:
7875:
7867:
7864:
7863:
7854:
7840:
7825:
7822:
7819:
7818:
7816:
7811:
7804:
7793:
7787:
7784:
7769:
7753:
7742:
7688:
7683:
7682:
7681:
7677:
7658:
7653:
7652:
7632:
7624:
7619:
7618:
7617:
7613:
7593:
7591:
7587:
7585:
7582:
7581:
7565:
7562:
7561:
7519:
7515:
7510:
7507:
7506:
7487:
7468:
7463:
7462:
7453:
7449:
7412:
7407:
7406:
7397:
7393:
7371:
7366:
7365:
7356:
7344:
7343:
7331:
7328:
7327:
7291:
7286:
7265:
7262:
7261:
7227:
7224:
7223:
7198:
7195:
7194:
7168:
7157:
7151:
7148:
7133:
7117:
7106:
7071:
7067:
7063:
7055:
7053:
7047:
7043:
7042:
7038:
7032:
7011:
7008:
7007:
6989:
6986:
6981:
6980:
6978:
6976:
6975:
6961:
6958:
6953:
6952:
6950:
6949:
6928:
6922:
6917:
6911:
6905:
6893:
6890:
6889:
6869:
6865:
6861:
6853:
6851:
6845:
6841:
6835:
6794:
6792:
6789:
6788:
6771:
6767:
6761:
6757:
6751:
6739:
6736:
6735:
6705:
6700:
6694:
6676:
6672:
6668:
6660:
6658:
6652:
6648:
6633:
6629:
6620:
6616:
6595:
6591:
6576:
6572:
6560:
6556:
6550:
6546:
6535:
6532:
6531:
6527:
6507:
6503:
6499:
6494:
6476:
6472:
6470:
6467:
6466:
6462:
6457:
6436:
6430:
6425:
6419:
6413:
6400:
6399:
6390:
6386:
6377:
6376:
6365:
6362:
6361:
6344:
6321:galaxy clusters
6287:
6283:
6277:
6273:
6260:
6248:
6244:
6236:
6233:
6232:
6228:
6202:
6198:
6192:
6188:
6175:
6166:
6162:
6156:
6151:
6150:
6141:
6136:
6135:
6126:
6115:
6101:
6089:
6085:
6077:
6074:
6073:
6067:
6060:
6029:
6025:
6016:
6012:
6011:
6009:
5997:
5952:
5950:
5946:
5934:
5917:
5903:
5883:
5882:
5878:
5874:
5872:
5868:
5867:
5863:
5851:
5845:
5842:
5841:
5835:
5832:
5826:
5823:
5817:
5811:
5807:
5793:
5790:
5785:
5784:
5782:
5780:
5779:
5757:
5746:
5741:
5740:
5731:
5726:
5725:
5724:
5720:
5719:
5713:
5702:
5686:
5675:
5674:
5669:
5666:
5665:
5661:
5647:
5644:
5639:
5638:
5636:
5634:
5633:
5632:states that if
5607:
5596:
5591:
5590:
5581:
5576:
5575:
5574:
5570:
5569:
5563:
5552:
5539:
5528:
5527:
5515:
5501:
5493:
5491:
5487:
5486:
5484:
5481:
5480:
5432:
5430:
5405:
5391:
5377:
5357:
5349:
5347:
5341:
5336:
5322:
5313:
5299:
5291:
5289:
5285:
5284:
5282:
5279:
5278:
5274:
5271:
5241:
5237:
5216:
5212:
5208:
5198:
5194:
5193:
5191:
5181:
5168:
5160:
5158:
5156:
5153:
5152:
5145:
5122:
5118:
5097:
5092:
5091:
5082:
5077:
5076:
5070:
5059:
5033:
5025:
5023:
5021:
5018:
5017:
4991:
4987:
4975:
4965:
4954:
4941:
4937:
4935:
4932:
4931:
4928:
4922:
4908:
4907:
4901:
4897:
4886:
4874:
4870:
4855:
4845:
4834:
4819:
4810:
4809:
4800:
4796:
4784:
4776:
4754:
4744:
4733:
4718:
4709:
4708:
4699:
4695:
4683:
4679:
4675:
4665:
4661:
4657:
4655:
4643:
4633:
4622:
4607:
4600:
4594:
4589:
4588:
4579:
4574:
4573:
4567:
4556:
4541:
4534:
4532:
4529:
4528:
4524:
4520:
4500:
4492:
4473:
4469:
4467:
4464:
4463:
4461:
4457:
4453:
4449:
4446:
4419:
4415:
4403:
4399:
4395:
4385:
4381:
4377:
4375:
4363:
4352:
4342:
4331:
4309:
4304:
4303:
4294:
4289:
4288:
4282:
4271:
4245:
4237:
4235:
4233:
4230:
4229:
4213:
4212:
4200:
4196:
4184:
4180:
4176:
4166:
4162:
4158:
4156:
4144:
4133:
4123:
4112:
4096:
4095:
4084:
4080:
4073:
4068:
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4020:
4016:
4012:
4002:
3998:
3994:
3992:
3980:
3969:
3959:
3948:
3932:
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3920:
3915:
3914:
3905:
3900:
3899:
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3864:
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3832:
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3809:
3800:
3795:
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3769:
3767:
3764:
3763:
3757:
3753:
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3742:
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3731:
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3673:
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3647:
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3600:
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3508:
3505:
3497:
3491:
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3456:
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3429:
3424:
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3422:
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3406:
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3388:
3377:
3367:
3356:
3338:
3333:
3332:
3320:
3315:
3314:
3305:
3300:
3299:
3287:
3282:
3281:
3280:
3276:
3264:
3253:
3243:
3232:
3219:
3218:
3207:
3202:
3201:
3189:
3184:
3183:
3174:
3169:
3168:
3156:
3151:
3150:
3149:
3145:
3133:
3122:
3112:
3101:
3088:
3083:
3082:
3070:
3065:
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3058:
3047:
3037:
3026:
3015:
3009:
3004:
3003:
2994:
2989:
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2982:
2971:
2963:
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2893:
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2807:
2802:
2801:
2795:
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2591:
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2497:
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2398:
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2374:
2370:
2367:
2359:
2345:
2344:
2338:
2333:
2332:
2323:
2318:
2317:
2311:
2300:
2278:
2277:
2271:
2266:
2265:
2256:
2251:
2250:
2244:
2233:
2216:
2209:
2204:
2203:
2199:
2197:
2184:
2177:
2172:
2171:
2167:
2165:
2159:
2155:
2149:
2138:
2125:
2124:
2118:
2113:
2112:
2099:
2092:
2087:
2086:
2082:
2080:
2074:
2063:
2046:
2039:
2034:
2033:
2029:
2027:
2018:
2013:
2012:
2006:
1995:
1984:
1974:
1966:
1964:
1960:
1958:
1955:
1954:
1953:can be written
1950:
1936:
1935:
1919:
1914:
1913:
1904:
1899:
1898:
1892:
1881:
1868:
1867:
1861:
1856:
1855:
1842:
1835:
1830:
1829:
1825:
1823:
1816:
1812:
1806:
1795:
1782:
1781:
1775:
1770:
1769:
1760:
1755:
1754:
1748:
1744:
1738:
1727:
1713:
1708:
1698:
1691:
1681:
1673:
1671:
1661:
1657:
1655:
1652:
1651:
1647:
1643:
1633:
1625:
1608:
1603:
1602:
1593:
1588:
1587:
1581:
1570:
1558:
1555:
1554:
1550:
1545:
1537:
1535:
1527:
1523:
1520:
1512:
1509:
1501:
1484:
1479:
1469:
1465:
1459:
1448:
1435:
1425:
1420:
1419:
1415:
1414:
1408:
1404:
1398:
1387:
1375:
1372:
1371:
1363:
1353:
1347:
1334:
1328:
1320:
1314:
1282:
1278:
1254:
1250:
1246:
1244:
1198:
1186:
1185:
1179:
1174:
1173:
1164:
1159:
1158:
1152:
1151:
1145:
1134:
1120:
1115:
1112:
1111:
1097:
1092:. Applying the
1082:
1071:
1060:
1049:
1043:
1039:
1031:
1020:
1014:
1006:
1000:
980:
976:
953:
948:
947:
932:
927:
926:
921:
908:
899:
894:
893:
878:
873:
872:
867:
854:
853:
849:
841:
837:
836:
829:
825:
824:
811:
807:
798:
794:
793:
788:
773:
753:
748:
747:
746:
742:
741:
735:
731:
721:
715:
704:
692:
688:
687:
680:
676:
675:
662:
658:
649:
645:
644:
639:
625:
622:
621:
616:
610:
607:
601:
597:
593:
585:
574:
568:
560:
554:
546:
535:
529:
521:
515:
511:
507:
500:
497:
417:
414:
411:
410:
408:
400:Rudolf Clausius
396:
387:
376:
373:
367:
363:
352:
347:
340:
339:
335:
326:
325:
300:
296:
273:
270:
269:
265:
258:
244:
212:Rudolf Clausius
192:
184:
180:
175:represents the
173:
165:
161:
155:
138:
137:
131:
126:
125:
116:
111:
110:
104:
103:
97:
86:
71:
54:
52:
49:
48:
19:In statistical
17:
12:
11:
5:
14988:
14978:
14977:
14972:
14967:
14962:
14957:
14943:
14942:
14934:
14926:
14925:External links
14923:
14922:
14921:
14884:
14878:
14855:
14849:
14834:
14831:
14828:
14827:
14820:
14800:
14793:
14773:
14760:(2): 686–703.
14740:
14695:(3): 627–638.
14675:
14638:(2): 361–371.
14622:
14607:
14572:
14565:
14547:
14534:(5): 703–705.
14514:
14509:10.1086/145732
14476:
14470:10.1086/147456
14432:
14397:
14391:10.1086/144747
14356:
14338:
14321:
14286:
14221:
14207:10.1086/143864
14172:
14138:
14113:
14063:
14036:(4): 313–345.
14016:
13973:
13958:
13934:
13892:
13885:
13857:
13829:
13828:
13826:
13823:
13822:
13821:
13816:
13811:
13806:
13801:
13796:
13791:
13786:
13781:
13774:
13771:
13753:
13750:
13738:
13729:
13725:
13722:
13717:
13712:
13708:
13704:
13699:
13696:
13691:
13686:
13682:
13678:
13669:
13648:
13639:
13635:
13632:
13629:
13626:
13622:
13619:
13616:
13611:
13607:
13603:
13594:
13550:
13547:
13544:
13537:
13533:
13529:
13523:
13514:
13499:galaxy cluster
13472:
13467:
13462:
13458:
13454:
13443:
13432:
13428:
13399:
13377:
13372:
13368:
13364:
13359:
13355:
13352:
13213:Main article:
13210:
13207:
13169:
13160:
13116:
13112:
13106:
13103:
13098:
13090:
13085:
13080:
13074:
13069:
13060:
13057:
13052:
13047:
13043:
13040:
13032:
13029:
13016:thermal energy
13003:
13000:
12985:
12981:
12977:
12971:
12967:
12963:
12957:
12953:
12949:
12943:
12940:
12933:
12930:
12926:
12920:
12917:
12913:
12909:
12901:
12897:
12893:
12889:
12884:
12879:
12875:
12849:
12845:
12822:
12818:
12814:
12808:
12804:
12800:
12794:
12790:
12786:
12780:
12777:
12770:
12766:
12760:
12756:
12752:
12749:
12744:
12741:
12737:
12713:
12710:
12707:
12702:
12698:
12694:
12691:
12686:
12683:
12679:
12652:
12627:
12623:
12599:
12576:
12551:
12542:
12534:
12530:
12526:
12523:
12520:
12513:
12510:
12504:
12500:
12495:
12491:
12485:
12480:
12476:
12470:
12466:
12458:
12454:
12448:
12444:
12440:
12437:
12434:
12428:
12425:
12420:
12417:
12411:
12408:
12405:
12400:
12354:
12345:
12331:
12324:
12307:
12304:
12299:
12294:
12289:
12284:
12279:
12274:
12269:
12264:
12261:
12258:
12254:
12250:
12247:
12244:
12240:
12235:
12231:
12227:
12211:
12208:
12194:
12176:
12168:
12163:
12159:
12154:
12150:
12067:
12063:
12055:
12051:
12044:
12040:
12034:
12030:
12023:
12018:
12014:
12008:
12004:
11998:
11994:
11989:
11985:
11980:
11977:
11973:
11968:
11959:
11955:
11951:
11945:
11941:
11935:
11930:
11924:
11920:
11914:
11910:
11902:
11898:
11893:
11890:
11886:
11865:
11848:
11843:
11839:
11833:
11829:
11825:
11820:
11816:
11810:
11806:
11802:
11797:
11793:
11787:
11783:
11777:
11773:
11769:
11764:
11760:
11754:
11750:
11746:
11743:
11738:
11735:
11731:
11714:
11692:kinetic energy
11657:
11652:
11648:
11644:
11640:
11635:
11632:
11628:
11624:
11619:
11616:
11612:
11608:
11603:
11599:
11595:
11592:
11586:
11581:
11577:
11571:
11566:
11562:
11559:
11556:
11553:
11550:
11547:
11544:
11541:
11538:
11533:
11529:
11521:
11518:
11511:
11507:
11503:
11495:
11491:
11485:
11481:
11477:
11469:
11465:
11461:
11456:
11451:
11447:
11438:
11435:
11419:
11416:
11399:
11395:
11392:
11389:
11386:
11383:
11380:
11375:
11372:
11367:
11360:
11357:
11354:
11351:
11347:
11340:if and only if
11334:
11330:
11326:
11323:
11318:
11315:
11312:
11309:
11305:
11301:
11296:
11292:
11288:
11285:
11280:
11277:
11274:
11271:
11267:
11263:
11246:Henri Poincaré
11241:
11238:
11223:
11219:
11213:
11209:
11205:
11202:
11199:
11196:
11193:
11171:
11167:
11161:
11157:
11153:
11150:
11147:
11142:
11138:
11133:
11128:
11124:
11120:
11117:
11114:
11088:
11083:
11078:
11074:
11070:
11067:
11064:
11061:
11056:
11052:
11048:
11022:
11018:
11009:
11006:
11003:
11000:
10995:
10990:
10986:
10982:
10955:
10951:
10947:
10944:
10922:
10918:
10912:
10908:
10904:
10901:
10881:
10878:
10875:
10853:
10849:
10837:
10834:
10819:
10814:
10811:
10806:
10803:
10800:
10795:
10790:
10786:
10782:
10779:
10773:
10770:
10767:
10742:
10736:
10732:
10726:
10722:
10718:
10710:
10707:
10702:
10699:
10696:
10693:
10690:
10670:
10667:
10664:
10641:
10637:
10632:
10627:
10623:
10618:
10614:
10610:
10605:
10601:
10595:
10592:
10587:
10584:
10581:
10578:
10575:
10551:
10545:
10542:
10539:
10535:
10529:
10524:
10520:
10515:
10512:
10509:
10505:
10499:
10495:
10491:
10488:
10485:
10482:
10479:
10476:
10452:
10449:
10446:
10442:
10435:
10431:
10427:
10419:
10416:
10413:
10409:
10405:
10402:
10399:
10379:
10374:
10370:
10366:
10363:
10360:
10355:
10352:
10347:
10344:
10341:
10338:
10329:, which gives
10318:
10298:
10286:
10283:
10274:
10271:
10258:
10255:
10252:
10249:
10246:
10243:
10223:
10220:
10215:
10212:
10207:
10204:
10201:
10198:
10193:
10189:
10185:
10180:
10177:
10172:
10169:
10166:
10163:
10160:
10157:
10137:
10134:
10131:
10128:
10125:
10122:
10119:
10116:
10113:
10110:
10106:
10102:
10099:
10096:
10093:
10090:
10087:
10083:
10079:
10073:
10070:
10064:
10040:
10037:
10033:
10029:
10023:
10020:
10014:
10009:
10006:
10001:
9996:
9989:
9984:
9979:
9974:
9969:
9962:
9958:
9952:
9945:
9942:
9937:
9934:
9931:
9928:
9925:
9902:
9899:
9876:
9873:
9870:
9864:
9861:
9855:
9852:
9848:
9844:
9832:
9829:
9814:
9806:
9801:
9797:
9793:
9788:
9783:
9779:
9775:
9772:
9767:
9764:
9758:
9755:
9752:
9750:
9747:
9744:
9741:
9740:
9731:
9727:
9721:
9717:
9713:
9708:
9703:
9699:
9695:
9690:
9686:
9682:
9677:
9673:
9667:
9663:
9656:
9652:
9645:
9642:
9640:
9637:
9636:
9634:
9607:
9603:
9600:
9597:
9591:
9588:
9585:
9582:
9579:
9551:
9547:
9544:
9541:
9538:
9535:
9532:
9525:
9522:
9516:
9509:
9498:
9494:
9488:
9485:
9479:
9472:
9469:
9463:
9434:
9430:
9426:
9421:
9416:
9412:
9391:
9388:
9383:
9379:
9375:
9370:
9365:
9361:
9357:
9354:
9351:
9348:
9345:
9342:
9339:
9336:
9333:
9313:
9310:
9307:
9304:
9301:
9298:
9295:
9292:
9289:
9286:
9283:
9280:
9274:
9271:
9265:
9262:
9259:
9256:
9253:
9250:
9247:
9244:
9241:
9238:
9235:
9229:
9226:
9217:
9214:
9208:
9205:
9182:
9162:
9142:
9139:
9136:
9133:
9130:
9127:
9124:
9121:
9118:
9115:
9112:
9082:
9078:
9075:
9072:
9069:
9066:
9063:
9060:
9057:
9042:
9035:
9032:
9027:
9024:
9018:
9015:
9000:
8996:
8993:
8990:
8983:
8972:
8965:
8961:
8957:
8952:
8947:
8943:
8934:
8912:
8892:
8872:
8869:
8866:
8863:
8860:
8857:
8854:
8834:
8814:
8802:
8799:
8786:
8783:
8780:
8775:
8772:
8767:
8764:
8761:
8758:
8755:
8746:) attraction,
8731:
8728:
8725:
8720:
8717:
8712:
8709:
8706:
8703:
8681:
8677:
8673:
8670:
8651:
8648:
8635:
8630:
8626:
8623:
8620:
8616:
8612:
8606:
8600:
8597:
8594:
8589:
8585:
8582:
8577:
8571:
8568:
8565:
8560:
8556:
8553:
8530:
8524:
8519:
8513:
8509:
8504:
8496:
8492:
8488:
8483:
8480:
8475:
8471:
8464:
8458:
8453:
8450:
8447:
8443:
8438:
8433:
8428:
8423:
8417:
8413:
8408:
8403:
8395:
8390:
8386:
8382:
8379:
8374:
8371:
8365:
8359:
8354:
8351:
8348:
8344:
8339:
8334:
8329:
8324:
8318:
8315:
8312:
8307:
8303:
8296:
8293:
8272:
8263:
8243:
8239:
8233:
8230:
8225:
8222:
8219:
8213:
8202:
8198:
8193:
8185:
8181:
8177:
8174:
8169:
8166:
8160:
8135:
8131:
8127:
8121:
8118:
8115:
8112:
8109:
8106:
8099:
8095:
8091:
8085:
8081:
8078:
8076:
8074:
8069:
8065:
8061:
8056:
8052:
8048:
8043:
8040:
8035:
8032:
8030:
8028:
8025:
8021:
8017:
8014:
8011:
8008:
8004:
7998:
7995:
7990:
7987:
7985:
7982:
7978:
7974:
7968:
7965:
7960:
7959:
7906:
7902:
7896:
7892:
7886:
7883:
7879:
7874:
7871:
7859:Lorentz factor
7806:
7805:
7756:
7754:
7747:
7741:
7738:
7726:
7723:
7720:
7716:
7713:
7710:
7707:
7704:
7701:
7698:
7691:
7686:
7680:
7676:
7673:
7670:
7667:
7661:
7656:
7651:
7648:
7645:
7642:
7639:
7635:
7627:
7622:
7616:
7611:
7606:
7602:
7599:
7596:
7590:
7569:
7545:
7542:
7539:
7536:
7533:
7530:
7527:
7522:
7518:
7514:
7494:
7490:
7486:
7483:
7479:
7476:
7471:
7466:
7461:
7456:
7452:
7448:
7445:
7442:
7439:
7436:
7433:
7430:
7427:
7423:
7420:
7415:
7410:
7405:
7400:
7396:
7392:
7389:
7386:
7382:
7379:
7374:
7369:
7364:
7359:
7353:
7350:
7347:
7342:
7338:
7335:
7315:
7312:
7308:
7305:
7302:
7299:
7294:
7289:
7285:
7281:
7278:
7275:
7272:
7269:
7246:
7243:
7240:
7237:
7234:
7231:
7211:
7208:
7205:
7202:
7170:
7169:
7120:
7118:
7111:
7105:
7102:
7090:
7083:
7074:
7070:
7066:
7061:
7058:
7050:
7046:
7041:
7035:
7031:
7027:
7024:
7021:
7018:
7015:
6934:
6931:
6925:
6920:
6916:
6908:
6904:
6900:
6897:
6872:
6868:
6864:
6859:
6856:
6848:
6844:
6838:
6834:
6830:
6827:
6824:
6821:
6818:
6815:
6812:
6809:
6806:
6801:
6798:
6774:
6770:
6764:
6760:
6754:
6750:
6746:
6743:
6721:
6713:
6708:
6703:
6699:
6693:
6690:
6687:
6679:
6675:
6671:
6666:
6663:
6655:
6651:
6647:
6644:
6641:
6636:
6632:
6628:
6623:
6619:
6615:
6612:
6609:
6606:
6603:
6598:
6594:
6590:
6587:
6584:
6579:
6575:
6571:
6568:
6563:
6559:
6553:
6549:
6545:
6542:
6539:
6510:
6506:
6502:
6498:
6493:
6490:
6487:
6484:
6479:
6475:
6460:
6442:
6439:
6433:
6428:
6424:
6416:
6412:
6408:
6403:
6398:
6393:
6389:
6385:
6380:
6375:
6372:
6369:
6343:
6340:
6325:Doppler effect
6295:
6290:
6286:
6276:
6272:
6267:
6264:
6259:
6256:
6251:
6247:
6243:
6240:
6210:
6205:
6201:
6191:
6187:
6182:
6179:
6174:
6169:
6165:
6159:
6154:
6149:
6144:
6139:
6134:
6129:
6124:
6121:
6118:
6114:
6108:
6105:
6100:
6097:
6092:
6088:
6084:
6081:
6046:
6043:
6038:
6032:
6028:
6024:
6019:
6015:
6006:
6003:
6000:
5996:
5992:
5988:
5983:
5979:
5976:
5973:
5970:
5967:
5964:
5961:
5958:
5955:
5949:
5943:
5940:
5937:
5933:
5929:
5925:
5920:
5915:
5909:
5906:
5898:
5895:
5892:
5889:
5886:
5881:
5877:
5871:
5866:
5860:
5857:
5854:
5850:
5830:
5821:
5803:
5765:
5760:
5755:
5749:
5744:
5739:
5734:
5729:
5723:
5716:
5711:
5708:
5705:
5701:
5697:
5694:
5689:
5684:
5681:
5678:
5673:
5657:
5630:virial theorem
5615:
5610:
5605:
5599:
5594:
5589:
5584:
5579:
5573:
5566:
5561:
5558:
5555:
5551:
5547:
5542:
5537:
5534:
5531:
5526:
5523:
5518:
5513:
5507:
5504:
5499:
5496:
5490:
5468:
5463:
5459:
5456:
5453:
5450:
5447:
5444:
5441:
5438:
5435:
5429:
5426:
5423:
5417:
5414:
5411:
5408:
5403:
5400:
5397:
5394:
5390:
5384:
5381:
5376:
5373:
5370:
5363:
5360:
5355:
5352:
5344:
5339:
5335:
5329:
5326:
5321:
5316:
5311:
5305:
5302:
5297:
5294:
5288:
5270:
5269:Time averaging
5267:
5240:
5236:
5233:
5230:
5227:
5219:
5215:
5211:
5206:
5201:
5197:
5188:
5185:
5180:
5174:
5171:
5166:
5163:
5131:
5121:
5117:
5114:
5111:
5108:
5105:
5100:
5095:
5090:
5085:
5080:
5073:
5068:
5065:
5062:
5058:
5054:
5051:
5048:
5045:
5039:
5036:
5031:
5028:
5016:Thus, we have
5003:
4997:
4994:
4990:
4984:
4981:
4978:
4974:
4968:
4963:
4960:
4957:
4953:
4949:
4940:
4926:
4900:
4893:
4890:
4885:
4880:
4877:
4873:
4869:
4864:
4861:
4858:
4854:
4848:
4843:
4840:
4837:
4833:
4826:
4823:
4818:
4815:
4813:
4811:
4806:
4803:
4799:
4793:
4790:
4787:
4782:
4779:
4775:
4771:
4768:
4763:
4760:
4757:
4753:
4747:
4742:
4739:
4736:
4732:
4725:
4722:
4717:
4714:
4712:
4710:
4705:
4702:
4698:
4689:
4686:
4682:
4678:
4671:
4668:
4664:
4660:
4652:
4649:
4646:
4642:
4636:
4631:
4628:
4625:
4621:
4614:
4611:
4606:
4603:
4601:
4597:
4592:
4587:
4582:
4577:
4570:
4565:
4562:
4559:
4555:
4548:
4545:
4540:
4537:
4536:
4508:
4503:
4498:
4495:
4491:
4487:
4484:
4479:
4476:
4472:
4459:
4445:
4442:
4430:
4425:
4422:
4418:
4409:
4406:
4402:
4398:
4391:
4388:
4384:
4380:
4372:
4369:
4366:
4361:
4358:
4355:
4351:
4345:
4340:
4337:
4334:
4330:
4326:
4323:
4320:
4317:
4312:
4307:
4302:
4297:
4292:
4285:
4280:
4277:
4274:
4270:
4266:
4263:
4260:
4257:
4251:
4248:
4243:
4240:
4228:Thus, we have
4211:
4206:
4203:
4199:
4190:
4187:
4183:
4179:
4172:
4169:
4165:
4161:
4153:
4150:
4147:
4142:
4139:
4136:
4132:
4126:
4121:
4118:
4115:
4111:
4107:
4104:
4101:
4099:
4097:
4090:
4087:
4083:
4076:
4071:
4064:
4059:
4054:
4049:
4044:
4038:
4026:
4023:
4019:
4015:
4008:
4005:
4001:
3997:
3989:
3986:
3983:
3978:
3975:
3972:
3968:
3962:
3957:
3954:
3951:
3947:
3943:
3940:
3937:
3935:
3933:
3929:
3923:
3918:
3913:
3908:
3903:
3897:
3893:
3888:
3885:
3880:
3873:
3870:
3867:
3862:
3859:
3856:
3852:
3846:
3841:
3838:
3835:
3831:
3827:
3824:
3822:
3818:
3813:
3808:
3803:
3798:
3791:
3786:
3783:
3780:
3776:
3772:
3771:
3746:
3737:
3733:
3727:
3718:
3714:
3708:
3690:
3686:
3679:
3676:
3672:
3665:
3660:
3655:
3650:
3645:
3637:
3628:
3625:
3621:
3617:
3610:
3607:
3603:
3599:
3593:
3590:
3585:
3582:
3578:
3570:
3565:
3559:
3555:
3552:
3547:
3544:
3539:
3512:
3501:
3487:
3478:
3453:
3447:
3442:
3437:
3432:
3427:
3421:
3417:
3412:
3409:
3404:
3397:
3394:
3391:
3386:
3383:
3380:
3376:
3370:
3365:
3362:
3359:
3355:
3351:
3347:
3341:
3336:
3331:
3326:
3323:
3318:
3313:
3308:
3303:
3298:
3293:
3290:
3285:
3279:
3273:
3270:
3267:
3262:
3259:
3256:
3252:
3246:
3241:
3238:
3235:
3231:
3227:
3224:
3222:
3220:
3216:
3210:
3205:
3200:
3195:
3192:
3187:
3182:
3177:
3172:
3167:
3162:
3159:
3154:
3148:
3142:
3139:
3136:
3131:
3128:
3125:
3121:
3115:
3110:
3107:
3104:
3100:
3096:
3091:
3086:
3081:
3076:
3073:
3068:
3061:
3056:
3053:
3050:
3046:
3040:
3035:
3032:
3029:
3025:
3021:
3018:
3016:
3012:
3007:
3002:
2997:
2992:
2985:
2980:
2977:
2974:
2970:
2966:
2965:
2938:
2920:
2914:
2909:
2904:
2899:
2896:
2891:
2884:
2879:
2876:
2873:
2869:
2863:
2858:
2855:
2852:
2848:
2841:
2838:
2833:
2830:
2825:
2820:
2815:
2810:
2805:
2798:
2793:
2790:
2787:
2783:
2776:
2773:
2768:
2745:
2727:
2724:
2719:
2712:
2707:
2704:
2701:
2697:
2693:
2688:
2683:
2672:in the system
2658:
2650:
2647:
2635:
2629:
2626:
2619:
2614:
2609:
2603:
2597:
2594:
2587:
2582:
2577:
2569:
2565:
2559:
2554:
2551:
2548:
2544:
2538:
2535:
2530:
2525:
2520:
2516:
2510:
2506:
2500:
2495:
2492:
2489:
2485:
2479:
2476:
2471:
2468:
2441:
2425:
2418:kinetic energy
2394:
2378:
2363:
2341:
2336:
2331:
2326:
2321:
2314:
2309:
2306:
2303:
2299:
2295:
2292:
2289:
2286:
2283:
2281:
2279:
2274:
2269:
2264:
2259:
2254:
2247:
2242:
2239:
2236:
2232:
2228:
2222:
2219:
2212:
2207:
2202:
2196:
2190:
2187:
2180:
2175:
2170:
2162:
2158:
2152:
2147:
2144:
2141:
2137:
2133:
2130:
2128:
2126:
2121:
2116:
2111:
2105:
2102:
2095:
2090:
2085:
2077:
2072:
2069:
2066:
2062:
2058:
2052:
2049:
2042:
2037:
2032:
2026:
2021:
2016:
2009:
2004:
2001:
1998:
1994:
1990:
1987:
1985:
1980:
1977:
1972:
1969:
1963:
1962:
1934:
1930:
1927:
1922:
1917:
1912:
1907:
1902:
1895:
1890:
1887:
1884:
1880:
1876:
1873:
1871:
1869:
1864:
1859:
1854:
1848:
1845:
1838:
1833:
1828:
1819:
1815:
1809:
1804:
1801:
1798:
1794:
1790:
1787:
1785:
1783:
1778:
1773:
1768:
1763:
1758:
1751:
1747:
1741:
1736:
1733:
1730:
1726:
1719:
1716:
1712:
1705:
1702:
1697:
1694:
1692:
1687:
1684:
1679:
1676:
1668:
1665:
1660:
1659:
1629:
1611:
1606:
1601:
1596:
1591:
1584:
1579:
1576:
1573:
1569:
1565:
1562:
1541:
1531:
1516:
1505:
1487:
1482:
1478:
1472:
1468:
1462:
1457:
1454:
1451:
1447:
1443:
1438:
1433:
1428:
1423:
1418:
1411:
1407:
1401:
1396:
1393:
1390:
1386:
1382:
1379:
1346:
1343:
1332:
1318:
1302:
1299:
1296:
1293:
1290:
1285:
1281:
1277:
1274:
1271:
1268:
1263:
1257:
1253:
1249:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1205:
1202:
1197:
1194:
1189:
1182:
1177:
1172:
1167:
1162:
1155:
1148:
1143:
1140:
1137:
1133:
1127:
1124:
1119:
1080:
1069:
1058:
1047:
1029:
1018:
1004:
988:
983:
979:
975:
972:
969:
966:
962:
956:
951:
946:
943:
940:
935:
930:
924:
920:
915:
912:
907:
902:
897:
892:
889:
886:
881:
876:
870:
866:
861:
858:
852:
844:
840:
832:
828:
823:
814:
810:
806:
801:
797:
792:
787:
784:
781:
776:
771:
767:
764:
761:
756:
751:
745:
738:
734:
728:
725:
718:
713:
710:
707:
703:
695:
691:
683:
679:
674:
665:
661:
657:
652:
648:
643:
638:
635:
632:
629:
614:
605:
583:
572:
558:
544:
533:
519:
496:
493:
445:Henri Poincaré
395:
392:
371:
345:
311:
308:
299:
295:
292:
289:
286:
283:
280:
277:
196:angle brackets
188:
169:
141:
134:
129:
124:
119:
114:
107:
100:
95:
92:
89:
85:
78:
75:
70:
67:
63:
60:
57:
29:kinetic energy
25:virial theorem
15:
9:
6:
4:
3:
2:
14987:
14976:
14973:
14971:
14968:
14966:
14963:
14961:
14958:
14956:
14953:
14952:
14950:
14940:
14939:
14935:
14932:
14929:
14928:
14918:
14914:
14910:
14906:
14902:
14898:
14895:(4): 045405.
14894:
14890:
14885:
14881:
14875:
14871:
14867:
14863:
14862:
14856:
14852:
14846:
14842:
14837:
14836:
14823:
14817:
14813:
14812:
14804:
14796:
14790:
14786:
14785:
14777:
14768:
14763:
14759:
14755:
14751:
14744:
14736:
14732:
14728:
14724:
14720:
14716:
14712:
14708:
14703:
14698:
14694:
14690:
14686:
14679:
14671:
14667:
14663:
14659:
14655:
14651:
14646:
14641:
14637:
14633:
14626:
14618:
14611:
14603:
14599:
14595:
14591:
14587:
14583:
14576:
14568:
14562:
14558:
14551:
14542:
14537:
14533:
14529:
14525:
14518:
14510:
14506:
14502:
14498:
14494:
14490:
14486:
14480:
14471:
14466:
14462:
14458:
14455:: 1037–1047.
14454:
14450:
14446:
14442:
14436:
14428:
14424:
14420:
14416:
14412:
14408:
14401:
14392:
14387:
14383:
14379:
14375:
14371:
14367:
14360:
14352:
14348:
14342:
14334:
14333:
14325:
14317:
14313:
14309:
14305:
14301:
14299:
14290:
14282:
14278:
14274:
14270:
14266:
14262:
14258:
14254:
14249:
14244:
14241:(4): 045002.
14240:
14236:
14232:
14225:
14217:
14213:
14208:
14203:
14199:
14195:
14191:
14187:
14183:
14176:
14168:
14164:
14160:
14156:
14152:
14145:
14143:
14127:
14123:
14117:
14109:
14105:
14101:
14097:
14093:
14089:
14085:
14081:
14077:
14070:
14068:
14059:
14055:
14051:
14047:
14043:
14039:
14035:
14031:
14027:
14020:
14012:
14008:
14004:
14000:
13996:
13992:
13988:
13984:
13977:
13969:
13965:
13961:
13959:0-201-02918-9
13955:
13951:
13947:
13941:
13939:
13930:
13926:
13922:
13918:
13914:
13910:
13906:
13902:
13896:
13888:
13882:
13878:
13874:
13870:
13869:
13861:
13853:
13849:
13845:
13841:
13834:
13830:
13820:
13817:
13815:
13812:
13810:
13807:
13805:
13802:
13800:
13797:
13795:
13792:
13790:
13787:
13785:
13784:Virial stress
13782:
13780:
13777:
13776:
13770:
13764:
13763:specific heat
13759:
13758:main sequence
13749:
13736:
13727:
13723:
13720:
13715:
13710:
13706:
13702:
13697:
13694:
13689:
13684:
13680:
13676:
13667:
13646:
13637:
13633:
13630:
13627:
13624:
13620:
13617:
13614:
13609:
13605:
13601:
13592:
13583:
13579:
13571:
13548:
13545:
13542:
13535:
13531:
13527:
13521:
13512:
13504:
13500:
13496:
13491:
13489:
13483:
13470:
13465:
13456:
13452:
13441:
13430:
13426:
13415:
13397:
13388:
13375:
13370:
13366:
13362:
13357:
13353:
13350:
13339:
13338:we can write
13332:
13324:
13301:
13295:
13276:
13257:
13249:
13245:
13241:
13236:
13234:
13230:
13229:virial radius
13226:
13222:
13216:
13206:
13168:
13159:
13155:
13148:
13114:
13110:
13104:
13101:
13096:
13083:
13078:
13067:
13058:
13055:
13050:
13045:
13041:
13038:
13030:
13027:
13017:
13013:
13009:
12999:
12983:
12979:
12975:
12969:
12965:
12961:
12955:
12951:
12947:
12941:
12938:
12931:
12928:
12924:
12918:
12915:
12911:
12907:
12899:
12895:
12891:
12887:
12882:
12877:
12873:
12847:
12843:
12820:
12816:
12812:
12806:
12802:
12798:
12792:
12788:
12784:
12778:
12775:
12768:
12764:
12758:
12754:
12750:
12747:
12742:
12739:
12735:
12711:
12708:
12705:
12700:
12696:
12692:
12689:
12684:
12681:
12677:
12664:
12650:
12625:
12621:
12597:
12574:
12562:
12549:
12540:
12532:
12528:
12524:
12521:
12518:
12511:
12508:
12502:
12498:
12493:
12489:
12483:
12478:
12474:
12468:
12464:
12456:
12452:
12446:
12442:
12438:
12435:
12432:
12426:
12423:
12418:
12415:
12398:
12389:
12385:
12383:
12357:
12348:
12335:
12327:
12318:
12305:
12297:
12287:
12282:
12267:
12262:
12259:
12256:
12252:
12248:
12245:
12242:
12238:
12233:
12229:
12225:
12216:
12207:
12205:
12201:
12193:
12187:
12174:
12161:
12157:
12152:
12148:
12140:
12129:
12119:
12111:
12101:
12083:
12078:
12065:
12061:
12053:
12049:
12042:
12038:
12032:
12028:
12021:
12016:
12012:
12006:
12002:
11996:
11992:
11987:
11983:
11978:
11975:
11971:
11966:
11957:
11953:
11949:
11943:
11939:
11933:
11928:
11922:
11918:
11912:
11908:
11900:
11896:
11891:
11888:
11884:
11875:
11873:
11868:
11859:
11846:
11841:
11837:
11831:
11827:
11818:
11814:
11808:
11804:
11800:
11795:
11785:
11781:
11775:
11771:
11762:
11758:
11752:
11748:
11741:
11736:
11733:
11729:
11720:
11717:
11708:
11702:
11693:
11685:
11677:
11668:
11655:
11650:
11646:
11642:
11633:
11630:
11626:
11622:
11617:
11614:
11610:
11601:
11597:
11593:
11590:
11579:
11575:
11564:
11560:
11554:
11551:
11548:
11542:
11539:
11536:
11531:
11527:
11519:
11509:
11505:
11493:
11489:
11483:
11479:
11475:
11467:
11463:
11459:
11454:
11449:
11445:
11436:
11433:
11423:
11415:
11413:
11397:
11393:
11390:
11384:
11378:
11373:
11370:
11365:
11355:
11349:
11332:
11324:
11313:
11307:
11299:
11294:
11286:
11275:
11269:
11261:
11252:
11247:
11237:
11221:
11217:
11211:
11207:
11203:
11200:
11197:
11194:
11191:
11169:
11165:
11159:
11155:
11151:
11148:
11145:
11140:
11131:
11126:
11122:
11115:
11112:
11100:
11081:
11076:
11072:
11065:
11062:
11054:
11050:
11038:
11037:equipartition
11020:
11007:
11001:
10993:
10988:
10984:
10953:
10949:
10945:
10942:
10920:
10916:
10910:
10906:
10902:
10899:
10879:
10876:
10873:
10851:
10848:
10833:
10812:
10809:
10801:
10798:
10788:
10784:
10777:
10771:
10768:
10765:
10756:
10740:
10734:
10730:
10724:
10720:
10716:
10708:
10705:
10700:
10697:
10691:
10665:
10653:
10635:
10630:
10625:
10616:
10612:
10608:
10603:
10599:
10593:
10590:
10585:
10582:
10576:
10565:
10543:
10540:
10537:
10533:
10527:
10522:
10513:
10510:
10507:
10503:
10497:
10493:
10489:
10486:
10483:
10477:
10450:
10447:
10444:
10440:
10433:
10429:
10425:
10417:
10414:
10411:
10407:
10403:
10400:
10397:
10372:
10368:
10361:
10358:
10353:
10350:
10345:
10339:
10316:
10296:
10282:
10280:
10270:
10256:
10253:
10250:
10247:
10244:
10241:
10221:
10218:
10213:
10210:
10205:
10202:
10199:
10191:
10187:
10183:
10178:
10175:
10167:
10164:
10158:
10135:
10132:
10129:
10126:
10123:
10120:
10117:
10114:
10111:
10108:
10100:
10094:
10091:
10088:
10085:
10077:
10062:
10054:
10038:
10035:
10027:
10012:
10007:
10004:
9999:
9987:
9977:
9972:
9960:
9956:
9943:
9940:
9935:
9932:
9926:
9874:
9871:
9868:
9853:
9850:
9842:
9831:Ideal gas law
9828:
9799:
9795:
9791:
9786:
9781:
9777:
9770:
9765:
9762:
9756:
9753:
9748:
9745:
9742:
9729:
9719:
9715:
9711:
9706:
9701:
9697:
9688:
9684:
9680:
9675:
9671:
9665:
9661:
9654:
9650:
9643:
9638:
9632:
9621:
9605:
9601:
9598:
9595:
9589:
9586:
9583:
9580:
9577:
9549:
9542:
9539:
9536:
9533:
9530:
9523:
9520:
9507:
9496:
9486:
9483:
9477:
9470:
9467:
9448:
9432:
9428:
9424:
9419:
9414:
9410:
9389:
9381:
9377:
9373:
9368:
9363:
9359:
9352:
9349:
9343:
9337:
9334:
9331:
9308:
9302:
9299:
9293:
9290:
9284:
9281:
9272:
9269:
9263:
9257:
9254:
9248:
9245:
9239:
9236:
9227:
9224:
9215:
9212:
9203:
9194:
9180:
9160:
9137:
9134:
9131:
9128:
9122:
9119:
9116:
9113:
9110:
9101:
9080:
9073:
9070:
9064:
9061:
9058:
9055:
9040:
9033:
9030:
9025:
9022:
9016:
9013:
8998:
8994:
8991:
8988:
8981:
8970:
8963:
8959:
8955:
8950:
8945:
8941:
8932:
8924:
8910:
8890:
8867:
8864:
8858:
8855:
8852:
8832:
8812:
8798:
8781:
8773:
8770:
8765:
8762:
8756:
8745:
8726:
8718:
8715:
8710:
8704:
8679:
8675:
8671:
8668:
8660:
8655:
8647:
8633:
8628:
8624:
8621:
8618:
8614:
8610:
8587:
8580:
8558:
8551:
8528:
8522:
8517:
8511:
8507:
8502:
8494:
8490:
8486:
8481:
8478:
8473:
8469:
8462:
8456:
8451:
8448:
8445:
8441:
8436:
8431:
8426:
8421:
8415:
8411:
8406:
8401:
8393:
8388:
8384:
8380:
8377:
8372:
8369:
8363:
8357:
8352:
8349:
8346:
8342:
8337:
8332:
8327:
8322:
8305:
8301:
8294:
8291:
8275:
8271:
8266:
8262:
8257:
8241:
8237:
8231:
8228:
8223:
8220:
8217:
8211:
8200:
8196:
8191:
8183:
8179:
8175:
8172:
8167:
8164:
8158:
8133:
8129:
8125:
8116:
8113:
8110:
8104:
8097:
8093:
8089:
8083:
8079:
8077:
8067:
8063:
8059:
8054:
8050:
8046:
8041:
8038:
8033:
8031:
8023:
8015:
8012:
8009:
8006:
7996:
7993:
7988:
7986:
7976:
7966:
7963:
7944:
7936:
7928:
7904:
7900:
7894:
7890:
7884:
7881:
7877:
7872:
7869:
7861:
7860:
7851:
7847:
7843:
7837:
7833:
7814:
7802:
7799:
7791:
7781:
7777:
7773:
7767:
7766:
7762:
7757:This section
7755:
7751:
7746:
7745:
7737:
7724:
7721:
7718:
7708:
7702:
7696:
7689:
7678:
7674:
7671:
7668:
7665:
7659:
7646:
7640:
7625:
7614:
7609:
7604:
7600:
7597:
7594:
7588:
7567:
7559:
7558:distributions
7543:
7537:
7531:
7528:
7525:
7520:
7512:
7492:
7484:
7481:
7477:
7469:
7454:
7450:
7446:
7437:
7431:
7425:
7421:
7413:
7398:
7394:
7390:
7387:
7380:
7372:
7340:
7336:
7333:
7313:
7310:
7303:
7297:
7292:
7287:
7283:
7279:
7273:
7267:
7258:
7244:
7241:
7235:
7229:
7206:
7200:
7191:
7189:
7185:
7181:
7177:
7166:
7163:
7155:
7145:
7141:
7137:
7131:
7130:
7126:
7121:This section
7119:
7115:
7110:
7109:
7101:
7088:
7081:
7072:
7068:
7064:
7059:
7056:
7048:
7044:
7039:
7033:
7029:
7025:
7019:
7013:
7005:
7004:
6992:
6984:
6973:
6964:
6956:
6932:
6929:
6923:
6918:
6914:
6906:
6902:
6898:
6895:
6870:
6866:
6862:
6857:
6854:
6846:
6842:
6836:
6832:
6828:
6825:
6822:
6819:
6813:
6810:
6807:
6796:
6772:
6768:
6762:
6758:
6752:
6748:
6744:
6741:
6732:
6719:
6711:
6706:
6701:
6697:
6688:
6685:
6677:
6673:
6669:
6664:
6661:
6653:
6649:
6642:
6639:
6634:
6630:
6621:
6617:
6613:
6610:
6604:
6596:
6592:
6588:
6585:
6577:
6573:
6569:
6561:
6557:
6551:
6547:
6543:
6540:
6508:
6504:
6500:
6496:
6488:
6485:
6482:
6477:
6473:
6463:
6440:
6437:
6431:
6426:
6422:
6414:
6410:
6406:
6391:
6387:
6373:
6370:
6367:
6360:
6356:
6353:Evaluate the
6351:
6349:
6339:
6337:
6333:
6328:
6326:
6322:
6317:
6315:
6311:
6310:solar systems
6306:
6293:
6288:
6274:
6265:
6262:
6257:
6254:
6249:
6241:
6226:
6225:gravitational
6221:
6208:
6203:
6189:
6180:
6177:
6172:
6167:
6157:
6147:
6142:
6127:
6122:
6119:
6116:
6112:
6106:
6103:
6098:
6095:
6090:
6082:
6071:
6064:
6057:
6044:
6041:
6036:
6026:
6022:
6013:
5998:
5990:
5986:
5981:
5974:
5968:
5965:
5959:
5953:
5947:
5935:
5927:
5923:
5918:
5913:
5907:
5904:
5879:
5875:
5869:
5864:
5852:
5839:
5829:
5820:
5814:
5806:
5796:
5788:
5776:
5763:
5758:
5753:
5747:
5737:
5732:
5721:
5714:
5709:
5706:
5703:
5699:
5695:
5692:
5687:
5682:
5679:
5676:
5671:
5660:
5650:
5642:
5631:
5626:
5613:
5608:
5603:
5597:
5587:
5582:
5571:
5564:
5559:
5556:
5553:
5549:
5545:
5540:
5535:
5532:
5529:
5524:
5521:
5516:
5511:
5505:
5502:
5497:
5494:
5488:
5466:
5461:
5454:
5448:
5445:
5439:
5433:
5427:
5424:
5421:
5412:
5406:
5398:
5392:
5388:
5382:
5379:
5374:
5371:
5368:
5361:
5358:
5353:
5350:
5342:
5337:
5333:
5327:
5324:
5319:
5314:
5309:
5303:
5300:
5295:
5292:
5286:
5266:
5264:
5260:
5238:
5234:
5231:
5228:
5225:
5217:
5213:
5209:
5204:
5199:
5195:
5186:
5183:
5178:
5172:
5169:
5164:
5161:
5151:
5142:
5129:
5119:
5115:
5112:
5109:
5106:
5103:
5098:
5088:
5083:
5071:
5066:
5063:
5060:
5056:
5052:
5049:
5046:
5043:
5037:
5034:
5029:
5026:
5014:
5001:
4995:
4992:
4988:
4982:
4979:
4976:
4972:
4966:
4961:
4958:
4955:
4951:
4947:
4938:
4925:
4898:
4891:
4888:
4883:
4878:
4875:
4871:
4867:
4862:
4859:
4856:
4852:
4846:
4841:
4838:
4835:
4831:
4824:
4821:
4816:
4814:
4804:
4801:
4797:
4791:
4788:
4785:
4780:
4777:
4773:
4769:
4766:
4761:
4758:
4755:
4751:
4745:
4740:
4737:
4734:
4730:
4723:
4720:
4715:
4713:
4703:
4700:
4696:
4687:
4684:
4680:
4676:
4669:
4666:
4662:
4658:
4650:
4647:
4644:
4640:
4634:
4629:
4626:
4623:
4619:
4612:
4609:
4604:
4602:
4595:
4585:
4580:
4568:
4563:
4560:
4557:
4553:
4546:
4543:
4538:
4506:
4501:
4496:
4493:
4489:
4485:
4482:
4477:
4474:
4470:
4441:
4428:
4423:
4420:
4416:
4407:
4404:
4400:
4396:
4389:
4386:
4382:
4378:
4370:
4367:
4364:
4359:
4356:
4353:
4349:
4343:
4338:
4335:
4332:
4328:
4324:
4321:
4318:
4315:
4310:
4300:
4295:
4283:
4278:
4275:
4272:
4268:
4264:
4261:
4258:
4255:
4249:
4246:
4241:
4238:
4226:
4209:
4204:
4201:
4197:
4188:
4185:
4181:
4177:
4170:
4167:
4163:
4159:
4151:
4148:
4145:
4140:
4137:
4134:
4130:
4124:
4119:
4116:
4113:
4109:
4105:
4102:
4100:
4088:
4085:
4081:
4074:
4062:
4052:
4047:
4024:
4021:
4017:
4013:
4006:
4003:
3999:
3995:
3987:
3984:
3981:
3976:
3973:
3970:
3966:
3960:
3955:
3952:
3949:
3945:
3941:
3938:
3936:
3927:
3921:
3911:
3906:
3895:
3891:
3886:
3883:
3871:
3868:
3865:
3860:
3857:
3854:
3850:
3844:
3839:
3836:
3833:
3829:
3825:
3823:
3816:
3806:
3801:
3789:
3784:
3781:
3778:
3774:
3761:
3749:
3745:
3740:
3736:
3730:
3726:
3721:
3717:
3711:
3707:
3701:
3688:
3684:
3677:
3674:
3670:
3663:
3653:
3648:
3635:
3626:
3623:
3619:
3615:
3608:
3605:
3601:
3597:
3591:
3588:
3583:
3580:
3576:
3568:
3553:
3550:
3545:
3542:
3527:
3515:
3511:
3504:
3500:
3494:
3490:
3486:
3481:
3477:
3473:holds, i.e.,
3472:
3451:
3445:
3435:
3430:
3419:
3415:
3410:
3407:
3395:
3392:
3389:
3384:
3381:
3378:
3374:
3368:
3363:
3360:
3357:
3353:
3349:
3345:
3339:
3329:
3324:
3321:
3311:
3306:
3296:
3291:
3288:
3277:
3271:
3268:
3265:
3260:
3257:
3254:
3250:
3244:
3239:
3236:
3233:
3229:
3225:
3223:
3214:
3208:
3198:
3193:
3190:
3180:
3175:
3165:
3160:
3157:
3146:
3140:
3137:
3134:
3129:
3126:
3123:
3119:
3113:
3108:
3105:
3102:
3098:
3094:
3089:
3079:
3074:
3071:
3059:
3054:
3051:
3048:
3044:
3038:
3033:
3030:
3027:
3023:
3019:
3017:
3010:
3000:
2995:
2983:
2978:
2975:
2972:
2968:
2954:
2950:
2941:
2937:
2931:
2918:
2912:
2902:
2897:
2894:
2882:
2877:
2874:
2871:
2867:
2861:
2856:
2853:
2850:
2846:
2839:
2836:
2831:
2828:
2823:
2813:
2808:
2796:
2791:
2788:
2785:
2781:
2774:
2771:
2766:
2748:
2744:
2725:
2722:
2710:
2705:
2702:
2699:
2695:
2691:
2686:
2661:
2657:
2646:
2633:
2627:
2624:
2617:
2607:
2601:
2595:
2592:
2585:
2575:
2567:
2563:
2557:
2552:
2549:
2546:
2542:
2536:
2533:
2528:
2523:
2518:
2514:
2508:
2504:
2498:
2493:
2490:
2487:
2483:
2477:
2474:
2469:
2466:
2453:
2444:
2440:
2437:
2428:
2424:
2419:
2416:is the total
2406:
2397:
2393:
2390:
2381:
2377:
2373:th particle,
2366:
2362:
2339:
2329:
2324:
2312:
2307:
2304:
2301:
2297:
2293:
2290:
2287:
2284:
2282:
2272:
2262:
2257:
2245:
2240:
2237:
2234:
2230:
2226:
2220:
2217:
2210:
2200:
2194:
2188:
2185:
2178:
2168:
2160:
2156:
2150:
2145:
2142:
2139:
2135:
2131:
2129:
2119:
2109:
2103:
2100:
2093:
2083:
2075:
2070:
2067:
2064:
2060:
2056:
2050:
2047:
2040:
2030:
2024:
2019:
2007:
2002:
1999:
1996:
1992:
1988:
1986:
1978:
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1932:
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1594:
1582:
1577:
1574:
1571:
1567:
1563:
1560:
1544:
1540:
1534:
1530:
1526:th particle.
1519:
1515:
1508:
1504:
1485:
1480:
1476:
1470:
1466:
1460:
1455:
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1449:
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1125:
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986:
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941:
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918:
913:
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905:
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879:
864:
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850:
842:
838:
830:
826:
821:
812:
808:
804:
799:
795:
790:
785:
782:
779:
774:
769:
762:
754:
743:
736:
732:
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723:
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708:
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693:
689:
681:
677:
672:
663:
659:
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619:
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589:
582:
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571:
564:
557:
550:
543:
539:
532:
525:
518:
503:
492:
490:
487:
483:
478:
477:Richard Bader
474:
470:
466:
465:Eugene Parker
462:
461:Richard Bader
458:
454:
450:
446:
442:
441:Lord Rayleigh
438:
434:
430:
429:Karl Jacobi's
426:
406:
401:
391:
383:
379:
370:
359:
355:
344:
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132:
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98:
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87:
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73:
68:
65:
61:
58:
55:
46:
42:
38:
34:
30:
26:
22:
14937:
14933:at MathPages
14892:
14888:
14860:
14840:
14810:
14803:
14783:
14776:
14757:
14753:
14743:
14692:
14688:
14678:
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14616:
14610:
14585:
14581:
14575:
14556:
14550:
14531:
14527:
14517:
14492:
14489:Astrophys. J
14488:
14479:
14452:
14449:Astrophys. J
14448:
14435:
14410:
14406:
14400:
14373:
14369:
14359:
14350:
14341:
14331:
14324:
14307:
14303:
14297:
14289:
14238:
14234:
14224:
14189:
14185:
14175:
14158:
14154:
14130:. Retrieved
14128:. 2018-03-22
14125:
14116:
14083:
14079:
14033:
14029:
14019:
13986:
13982:
13976:
13949:
13912:
13908:
13895:
13867:
13860:
13843:
13842:. Series 4.
13839:
13833:
13755:
13492:
13484:
13416:
13389:
13340:
13330:
13322:
13299:
13293:
13274:
13255:
13237:
13218:
13166:
13157:
13005:
12665:
12563:
12390:
12386:
12352:
12343:
12329:
12322:
12319:
12217:
12213:
12191:
12188:
12141:
12127:
12117:
12109:
12099:
12079:
11876:
11863:
11860:
11721:
11712:
11706:
11700:
11669:
11424:
11421:
11411:
11243:
11101:
10839:
10757:
10654:
10288:
10279:Coma Cluster
10276:
9834:
9622:
9449:
9195:
9102:
8977:acceleration
8925:
8804:
8656:
8653:
8273:
8269:
8264:
8260:
7942:
7934:
7926:
7862:
7849:
7845:
7841:
7835:
7831:
7812:
7809:
7794:
7785:
7770:Please help
7758:
7259:
7192:
7173:
7158:
7149:
7134:Please help
7122:
7002:
7001:
6990:
6982:
6962:
6954:
6733:
6526:of particle
6458:
6352:
6345:
6329:
6318:
6307:
6227:attraction,
6222:
6072:
6065:
6058:
5840:
5827:
5818:
5812:
5804:
5794:
5786:
5777:
5658:
5648:
5640:
5629:
5627:
5272:
5149:
5143:
5015:
4923:
4447:
4227:
3762:
3756:on particle
3747:
3743:
3738:
3734:
3728:
3724:
3719:
3715:
3709:
3705:
3702:
3528:
3513:
3509:
3502:
3498:
3495:
3488:
3484:
3479:
3475:
2952:
2948:
2939:
2935:
2932:
2755:on particle
2746:
2742:
2664:on particle
2659:
2655:
2652:
2451:
2442:
2438:
2435:
2426:
2422:
2404:
2395:
2391:
2388:
2379:
2375:
2364:
2360:
1630:
1626:
1542:
1538:
1532:
1528:
1517:
1513:
1506:
1502:
1351:
1348:
1336:
1329:
1322:
1315:
1110:results in:
1106:
1102:
1098:
1088:
1084:
1077:
1073:
1066:
1062:
1055:
1051:
1044:
1033:
1026:
1022:
1015:
1008:
1001:
620:
611:
602:
587:
580:
576:
569:
562:
555:
548:
541:
537:
530:
523:
516:
501:
498:
469:Fritz Zwicky
453:Enrico Fermi
432:
397:
381:
377:
368:
357:
353:
342:
328:
323:
253:
249:
245:
239:
216:
203:
199:
189:
185:
170:
166:
156:
24:
18:
14376:: 143–153.
13789:Virial mass
13582:Virial mass
13225:virial mass
13215:Virial mass
13137:, velocity
13129:for a mass
12204:Alfvén wave
11035:. Assuming
10273:Dark matter
9556:power input
8825:, position
7950:. We have,
6359:Hamiltonian
5263:Carl Jacobi
486:white dwarf
473:dark matter
457:Paul Ledoux
223:temperature
164:particles,
35:(where the
14949:Categories
14702:1912.08683
14645:1801.06453
14248:1605.04909
14132:2023-06-07
13825:References
12139:, we find
10564:ergodicity
10234:, we have
7788:April 2020
7152:April 2020
6355:commutator
1366:about the
351:. Whereas
14917:125030620
14735:125180719
14727:1432-0959
14273:0034-6861
14216:0004-637X
14167:0018-0238
14108:0002-9505
14058:123081616
14011:122502103
13728:ρ
13721:⋅
13703:π
13677:≈
13638:ρ
13634:⋅
13625:ρ
13602:≈
13546:π
13513:ρ
13457:σ
13453:≈
13367:σ
13363:≈
13221:astronomy
13133:, radius
12939:−
12932:β
12929:α
12919:β
12916:α
12908:∫
12896:μ
12848:α
12776:−
12769:α
12759:α
12751:∫
12622:ρ
12598:η
12529:ρ
12525:η
12522:π
12499:
12475:γ
12443:ρ
12439:η
12436:π
12427:−
12303:⟩
12288:⋅
12273:⟨
12253:∑
12246:−
12243:≈
12153:∼
12149:τ
12050:μ
11993:ε
11984:−
11972:δ
11954:μ
11909:ε
11842:σ
11832:σ
11824:Σ
11801:−
11796:σ
11792:⟩
11768:⟨
11763:σ
11753:σ
11745:Σ
11594:∫
11591:−
11517:∂
11502:∂
11480:∫
11385:τ
11371:−
11366:τ
11359:∞
11353:→
11350:τ
11333:τ
11329:⟩
11322:⟨
11317:∞
11311:→
11308:τ
11295:τ
11291:⟩
11284:⟨
11279:∞
11273:→
11270:τ
11222:⊙
11204:×
11170:⊙
11152:×
11146:≈
11137:⟩
11119:⟨
11087:⟩
11069:⟨
11060:⟩
11047:⟨
10999:⟩
10981:⟨
10921:⊙
10818:⟩
10805:⟨
10794:⟩
10781:⟨
10701:−
10695:⟩
10689:⟨
10669:⟩
10663:⟨
10640:⟩
10622:⟨
10586:−
10580:⟩
10574:⟨
10550:⟩
10519:⟨
10504:∑
10487:−
10481:⟩
10475:⟨
10467:, giving
10408:∑
10404:−
10378:⟩
10365:⟨
10343:⟩
10337:⟨
10206:⋅
10197:⟩
10171:⟨
10162:⟩
10156:⟨
10121:∫
10101:⋅
10098:∇
10095:∫
10078:⋅
10072:^
10063:∫
10028:⋅
10022:^
10013:∫
9978:⋅
9957:∑
9936:−
9930:⟩
9924:⟨
9901:^
9863:^
9854:−
9796:ω
9792:−
9778:ω
9766:ω
9763:γ
9757:−
9749:φ
9746:
9716:ω
9712:−
9698:ω
9672:ω
9662:γ
9602:ω
9596:γ
9590:−
9584:φ
9581:
9550:⏟
9546:⟩
9540:ω
9537:
9524:˙
9515:⟨
9497:⏟
9493:⟩
9487:˙
9478:γ
9471:˙
9462:⟨
9411:ω
9378:ω
9374:−
9360:ω
9344:φ
9338:
9312:⟩
9300:ω
9294:
9288:⟨
9282:−
9279:⟩
9273:˙
9261:⟨
9258:γ
9252:⟩
9243:⟨
9234:⟩
9228:˙
9216:˙
9207:⟨
9181:φ
9138:φ
9129:ω
9123:
9081:⏟
9071:ω
9065:
9041:⏟
9017:γ
9014:−
8999:⏟
8989:−
8971:⏟
8911:γ
8865:ω
8859:
8785:⟩
8779:⟨
8766:−
8760:⟩
8754:⟨
8730:⟩
8724:⟨
8708:⟩
8702:⟨
8672:∝
8611:∈
8605:⟩
8584:⟨
8576:⟩
8555:⟨
8523:τ
8491:γ
8470:γ
8442:∑
8427:τ
8385:β
8381:−
8343:∑
8328:τ
8232:γ
8218:γ
8180:β
8176:−
8114:−
8111:γ
8094:β
8090:γ
8051:β
8047:γ
8020:β
8016:⋅
8007:γ
8003:β
7977:⋅
7885:−
7870:γ
7759:does not
7679:∫
7638:∇
7615:∫
7598:−
7517:∇
7513:−
7485:∈
7447:∈
7438:⋅
7391:∈
7385:∇
7358:∞
7337:∈
7284:∫
7123:does not
7030:∑
7023:⟩
7017:⟨
6903:∑
6833:∑
6829:−
6800:ℏ
6749:∑
6692:ℏ
6686:−
6646:ℏ
6492:ℏ
6486:−
6411:∑
6289:τ
6285:⟩
6271:⟨
6258:−
6250:τ
6246:⟩
6239:⟨
6204:τ
6200:⟩
6186:⟨
6168:τ
6164:⟩
6148:⋅
6133:⟨
6113:∑
6099:−
6091:τ
6087:⟩
6080:⟨
6037:τ
6023:−
6005:∞
6002:→
5999:τ
5991:≤
5982:τ
5966:−
5960:τ
5942:∞
5939:→
5936:τ
5919:τ
5859:∞
5856:→
5853:τ
5759:τ
5738:⋅
5700:∑
5696:−
5688:τ
5609:τ
5588:⋅
5550:∑
5541:τ
5517:τ
5462:τ
5446:−
5440:τ
5413:τ
5389:∫
5383:τ
5343:τ
5334:∫
5328:τ
5315:τ
5113:−
5089:⋅
5057:∑
4973:∑
4952:∑
4853:∑
4832:∑
4789:−
4770:α
4752:∑
4731:∑
4641:∑
4620:∑
4586:⋅
4554:∑
4539:−
4486:α
4368:−
4350:∑
4329:∑
4325:−
4301:⋅
4269:∑
4149:−
4131:∑
4110:∑
4106:−
4053:−
3985:−
3967:∑
3946:∑
3942:−
3912:−
3892:⋅
3869:−
3851:∑
3830:∑
3807:⋅
3775:∑
3654:−
3592:−
3558:∇
3554:−
3436:−
3416:⋅
3393:−
3375:∑
3354:∑
3330:⋅
3312:−
3297:⋅
3269:−
3251:∑
3230:∑
3199:⋅
3166:⋅
3138:−
3120:∑
3099:∑
3080:⋅
3045:∑
3024:∑
3001:⋅
2969:∑
2903:⋅
2868:∑
2847:∑
2832:−
2814:⋅
2782:∑
2767:−
2696:∑
2602:⋅
2543:∑
2484:∑
2330:⋅
2298:∑
2263:⋅
2231:∑
2195:⋅
2136:∑
2110:⋅
2061:∑
2025:⋅
1993:∑
1911:⋅
1879:∑
1853:⋅
1793:∑
1767:⋅
1725:∑
1600:⋅
1568:∑
1446:∑
1385:∑
1298:⟩
1292:⟨
1267:⋅
1221:−
1212:−
1196:−
1171:⋅
1132:∑
1118:−
822:∫
805:−
702:∑
673:∫
656:−
634:⟩
628:⟨
499:Consider
398:In 1870,
307:⟩
294:⟨
285:⟩
279:⟨
214:in 1870.
123:⋅
84:∑
69:−
21:mechanics
14670:53692146
14349:(1911).
14281:18596513
13948:(1980).
13773:See also
12239:⟩
12226:⟨
12202:(or the
12131:, where
12082:plasmoid
11412:boundary
9995:⟩
9951:⟨
9887:, where
9402:, where
9153:, where
9047:friction
8650:Examples
8518:⟩
8437:⟨
8422:⟩
8338:⟨
8323:⟩
8302:⟨
7853:, where
7082:⟩
7040:⟨
6997:⟩
6977:⟨
6314:galaxies
5914:⟩
5870:⟨
5801:⟩
5781:⟨
5754:⟩
5722:⟨
5683:⟩
5677:⟨
5655:⟩
5635:⟨
5604:⟩
5572:⟨
5536:⟩
5530:⟨
5512:⟩
5489:⟨
5310:⟩
5287:⟨
1637:momentum
1536:= |
1188:⟩
1154:⟨
1096:formula
405:vis viva
348:⟩
341:⟨
331:⟩
327:⟨
140:⟩
106:⟨
62:⟩
56:⟨
14897:Bibcode
14866:Bibcode
14707:Bibcode
14650:Bibcode
14590:Bibcode
14497:Bibcode
14495:: 116.
14457:Bibcode
14415:Bibcode
14378:Bibcode
14253:Bibcode
14194:Bibcode
14192:: 217.
14088:Bibcode
14038:Bibcode
13991:Bibcode
13968:5675073
13917:Bibcode
13873:Bibcode
13576:is the
13568:is the
13335:
13319:
13316:
13304:
13291:
13279:
13272:
13260:
13227:" and "
13203:
13191:
13187:
13175:
13012:kinetic
12374:
12362:
12122:
12106:
11870:is the
11690:is the
11682:is the
11674:is the
10051:By the
8744:Coulomb
7947:
7931:
7857:is the
7829:
7817:
7780:removed
7765:sources
7560:. Then
7260:Denote
7144:removed
7129:sources
6995:
6979:
6967:
6951:
6357:of the
6330:If the
5799:
5783:
5664:, then
5653:
5637:
2456:
2432:
2409:
2385:
1642:of the
1635:is the
421:
409:
394:History
334:equals
261:of the
179:on the
47:states
45:theorem
14915:
14876:
14847:
14818:
14791:
14733:
14725:
14668:
14563:
14279:
14271:
14214:
14165:
14106:
14056:
14009:
13966:
13956:
13883:
13564:where
13495:galaxy
13152:, the
12870:
12862:, and
12732:
12674:
12648:
12618:
12595:
12572:
12564:where
12189:where
11670:where
11251:tensor
10840:Zwicky
9005:spring
7326:. Let
7086:
6888:where
6717:
4921:where
2740:where
2358:where
1640:vector
1624:where
1547:|
1500:where
1368:origin
1358:scalar
237:form.
235:tensor
206:, the
200:virial
194:, and
154:where
23:, the
14913:S2CID
14731:S2CID
14697:arXiv
14666:S2CID
14640:arXiv
14306:. 5.
14296:"XV.
14277:S2CID
14243:arXiv
14054:S2CID
14007:S2CID
13752:Stars
13497:or a
13390:Here
7848:− 1)
7182:, is
1087:) = −
1025:) = −
579:) = −
540:) = −
489:stars
208:Latin
177:force
14874:ISBN
14845:ISBN
14816:ISBN
14789:ISBN
14723:ISSN
14561:ISBN
14269:ISSN
14212:ISSN
14163:ISSN
14104:ISSN
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5104:=
5099:k
5094:r
5084:k
5079:F
5072:N
5067:1
5064:=
5061:k
5053:+
5050:T
5047:2
5044:=
5038:t
5035:d
5030:G
5027:d
5002:.
4996:k
4993:j
4989:V
4983:k
4977:j
4967:N
4962:1
4959:=
4956:k
4948:=
4939:V
4924:V
4899:V
4892:2
4889:n
4884:=
4879:k
4876:j
4872:V
4868:n
4863:k
4857:j
4847:N
4842:1
4839:=
4836:k
4825:2
4822:1
4817:=
4805:k
4802:j
4798:r
4792:1
4786:n
4781:k
4778:j
4774:r
4767:n
4762:k
4756:j
4746:N
4741:1
4738:=
4735:k
4724:2
4721:1
4716:=
4704:k
4701:j
4697:r
4688:k
4685:j
4681:r
4677:d
4670:k
4667:j
4663:V
4659:d
4651:k
4645:j
4635:N
4630:1
4627:=
4624:k
4613:2
4610:1
4605:=
4596:k
4591:r
4581:k
4576:F
4569:N
4564:1
4561:=
4558:k
4547:2
4544:1
4525:n
4521:α
4507:,
4502:n
4497:k
4494:j
4490:r
4483:=
4478:k
4475:j
4471:V
4458:r
4454:n
4450:V
4429:.
4424:k
4421:j
4417:r
4408:k
4405:j
4401:r
4397:d
4390:k
4387:j
4383:V
4379:d
4371:1
4365:k
4360:1
4357:=
4354:j
4344:N
4339:2
4336:=
4333:k
4322:T
4319:2
4316:=
4311:k
4306:r
4296:k
4291:F
4284:N
4279:1
4276:=
4273:k
4265:+
4262:T
4259:2
4256:=
4250:t
4247:d
4242:G
4239:d
4210:.
4205:k
4202:j
4198:r
4189:k
4186:j
4182:r
4178:d
4171:k
4168:j
4164:V
4160:d
4152:1
4146:k
4141:1
4138:=
4135:j
4125:N
4120:2
4117:=
4114:k
4103:=
4089:k
4086:j
4082:r
4075:2
4070:|
4063:j
4058:r
4048:k
4043:r
4037:|
4025:k
4022:j
4018:r
4014:d
4007:k
4004:j
4000:V
3996:d
3988:1
3982:k
3977:1
3974:=
3971:j
3961:N
3956:2
3953:=
3950:k
3939:=
3928:)
3922:j
3917:r
3907:k
3902:r
3896:(
3887:k
3884:j
3879:F
3872:1
3866:k
3861:1
3858:=
3855:j
3845:N
3840:2
3837:=
3834:k
3826:=
3817:k
3812:r
3802:k
3797:F
3790:N
3785:1
3782:=
3779:k
3758:j
3754:k
3744:V
3739:j
3735:r
3725:V
3720:j
3716:r
3706:F
3689:,
3685:)
3678:k
3675:j
3671:r
3664:j
3659:r
3649:k
3644:r
3636:(
3627:k
3624:j
3620:r
3616:d
3609:k
3606:j
3602:V
3598:d
3589:=
3584:k
3581:j
3577:V
3569:k
3564:r
3551:=
3546:k
3543:j
3538:F
3524:k
3520:j
3510:r
3499:V
3485:F
3476:F
3452:)
3446:j
3441:r
3431:k
3426:r
3420:(
3411:k
3408:j
3403:F
3396:1
3390:k
3385:1
3382:=
3379:j
3369:N
3364:2
3361:=
3358:k
3350:=
3346:)
3340:j
3335:r
3325:k
3322:j
3317:F
3307:k
3302:r
3292:k
3289:j
3284:F
3278:(
3272:1
3266:k
3261:1
3258:=
3255:j
3245:N
3240:2
3237:=
3234:k
3226:=
3215:)
3209:j
3204:r
3194:j
3191:k
3186:F
3181:+
3176:k
3171:r
3161:k
3158:j
3153:F
3147:(
3141:1
3135:k
3130:1
3127:=
3124:j
3114:N
3109:2
3106:=
3103:k
3095:=
3090:k
3085:r
3075:k
3072:j
3067:F
3060:N
3055:1
3052:=
3049:j
3039:N
3034:1
3031:=
3028:k
3020:=
3011:k
3006:r
2996:k
2991:F
2984:N
2979:1
2976:=
2973:k
2953:N
2949:j
2936:F
2919:.
2913:k
2908:r
2898:k
2895:j
2890:F
2883:N
2878:1
2875:=
2872:j
2862:N
2857:1
2854:=
2851:k
2840:2
2837:1
2829:=
2824:k
2819:r
2809:k
2804:F
2797:N
2792:1
2789:=
2786:k
2775:2
2772:1
2757:k
2753:j
2743:F
2726:k
2723:j
2718:F
2711:N
2706:1
2703:=
2700:j
2692:=
2687:k
2682:F
2670:j
2666:k
2660:k
2656:F
2634:.
2628:t
2625:d
2618:k
2613:r
2608:d
2596:t
2593:d
2586:k
2581:r
2576:d
2568:k
2564:m
2558:N
2553:1
2550:=
2547:k
2537:2
2534:1
2529:=
2524:2
2519:k
2515:v
2509:k
2505:m
2499:N
2494:1
2491:=
2488:k
2478:2
2475:1
2470:=
2467:T
2448:/
2443:k
2439:r
2436:d
2427:k
2423:v
2414:T
2401:/
2396:k
2392:p
2389:d
2380:k
2376:F
2371:k
2365:k
2361:m
2340:k
2335:r
2325:k
2320:F
2313:N
2308:1
2305:=
2302:k
2294:+
2291:T
2288:2
2285:=
2273:k
2268:r
2258:k
2253:F
2246:N
2241:1
2238:=
2235:k
2227:+
2221:t
2218:d
2211:k
2206:r
2201:d
2189:t
2186:d
2179:k
2174:r
2169:d
2161:k
2157:m
2151:N
2146:1
2143:=
2140:k
2132:=
2120:k
2115:r
2104:t
2101:d
2094:k
2089:p
2084:d
2076:N
2071:1
2068:=
2065:k
2057:+
2051:t
2048:d
2041:k
2036:r
2031:d
2020:k
2015:p
2008:N
2003:1
2000:=
1997:k
1989:=
1979:t
1976:d
1971:G
1968:d
1951:G
1933:.
1929:G
1926:=
1921:k
1916:r
1906:k
1901:p
1894:N
1889:1
1886:=
1883:k
1875:=
1863:k
1858:r
1847:t
1844:d
1837:k
1832:r
1827:d
1818:k
1814:m
1808:N
1803:1
1800:=
1797:k
1789:=
1777:k
1772:r
1762:k
1757:r
1750:k
1746:m
1740:N
1735:1
1732:=
1729:k
1718:t
1715:d
1711:d
1704:2
1701:1
1696:=
1686:t
1683:d
1678:I
1675:d
1667:2
1664:1
1648:G
1644:k
1631:k
1627:p
1610:k
1605:r
1595:k
1590:p
1583:N
1578:1
1575:=
1572:k
1564:=
1561:G
1551:G
1543:k
1539:r
1533:k
1529:r
1524:k
1518:k
1514:r
1507:k
1503:m
1486:2
1481:k
1477:r
1471:k
1467:m
1461:N
1456:1
1453:=
1450:k
1442:=
1437:2
1432:|
1427:k
1422:r
1417:|
1410:k
1406:m
1400:N
1395:1
1392:=
1389:k
1381:=
1378:I
1364:I
1354:N
1339:)
1337:t
1335:(
1333:2
1330:F
1325:)
1323:t
1321:(
1319:1
1316:F
1301:,
1295:T
1289:=
1284:2
1280:v
1276:m
1273:=
1270:r
1262:r
1256:2
1252:v
1248:m
1242:=
1239:r
1236:F
1233:=
1230:)
1227:r
1224:F
1218:r
1215:F
1209:(
1204:2
1201:1
1193:=
1181:k
1176:r
1166:k
1161:F
1147:N
1142:1
1139:=
1136:k
1126:2
1123:1
1107:r
1105:/
1099:F
1085:t
1083:(
1081:2
1078:r
1074:t
1072:(
1070:2
1067:F
1063:t
1061:(
1059:1
1056:r
1052:t
1050:(
1048:1
1045:F
1040:r
1036:)
1034:t
1032:(
1030:1
1027:r
1023:t
1021:(
1019:2
1016:r
1011:)
1009:t
1007:(
1005:1
1002:r
987:.
982:2
978:v
974:m
971:=
968:t
965:d
961:)
955:2
950:|
945:)
942:t
939:(
934:2
929:v
923:|
919:m
914:2
911:1
906:+
901:2
896:|
891:)
888:t
885:(
880:1
875:v
869:|
865:m
860:2
857:1
851:(
843:2
839:t
831:1
827:t
813:1
809:t
800:2
796:t
791:1
786:=
783:t
780:d
775:2
770:|
766:)
763:t
760:(
755:k
750:v
744:|
737:k
733:m
727:2
724:1
717:N
712:1
709:=
706:k
694:2
690:t
682:1
678:t
664:1
660:t
651:2
647:t
642:1
637:=
631:T
615:2
612:t
606:1
603:t
598:F
594:v
590:)
588:t
586:(
584:1
581:F
577:t
575:(
573:2
570:F
565:)
563:t
561:(
559:1
556:F
551:)
549:t
547:(
545:1
542:v
538:t
536:(
534:2
531:v
526:)
524:t
522:(
520:1
517:v
512:r
508:m
502:N
433:N
418:2
415:/
412:1
388:n
384:)
382:r
380:(
378:V
369:V
364:r
360:)
358:r
356:(
354:V
343:V
336:n
329:T
310:.
298:V
291:n
288:=
282:T
276:2
266:r
259:n
250:r
248:(
246:V
190:k
186:r
181:k
171:k
167:F
162:N
157:T
133:k
128:r
118:k
113:F
99:N
94:1
91:=
88:k
77:2
74:1
66:=
59:T
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