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79-621: 1) In cosmological models where dark matter structure grows hierarchically from weak initial fluctuations, dark matter halos are almost self-similar; halo regions which are close to dynamical equilibrium are adequately represented for all masses and at all times by a simple analytic formula with only two free parameters, a characteristic density and a characteristic size. 2) These two parameters are related with rather little scatter; larger halos are less dense. The size-density relation depends on cosmological parameters and so can be used to constrain these observationally. 3) The characteristic density of

158-442: A tensor form. If the force between any two particles of the system results from a potential energy V ( r ) = αr that is proportional to some power n of the interparticle distance r , the virial theorem takes the simple form 2 ⟨ T ⟩ = n ⟨ V TOT ⟩ . {\displaystyle 2\langle T\rangle =n\langle V_{\text{TOT}}\rangle .} Thus, twice

237-1752: A common special case, the potential energy V between two particles is proportional to a power n of their distance r ij : V j k = α r j k n , {\displaystyle V_{jk}=\alpha r_{jk}^{n},} where the coefficient α and the exponent n are constants. In such cases, the virial is − 1 2 ∑ k = 1 N F k ⋅ r k = 1 2 ∑ k = 1 N ∑ j < k d V j k d r j k r j k = 1 2 ∑ k = 1 N ∑ j < k n α r j k n − 1 r j k = 1 2 ∑ k = 1 N ∑ j < k n V j k = n 2 V TOT , {\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}}} where V TOT = ∑ k = 1 N ∑ j < k V j k {\displaystyle V_{\text{TOT}}=\sum _{k=1}^{N}\sum _{j<k}V_{jk}}

316-445: 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 d F = − n ^ P d A {\displaystyle d\mathbf {F} =-{\hat {\mathbf {n} }}PdA} , where n ^ {\displaystyle {\hat {\mathbf {n} }}}

395-692: A duration τ is defined as ⟨ d G d t ⟩ τ = 1 τ ∫ 0 τ d G d t d t = 1 τ ∫ G ( 0 ) G ( τ ) d G = G ( τ ) − G ( 0 ) τ , {\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 }},} from which we obtain

474-549: A finite central density, unlike the NFW profile which has a divergent (infinite) central density. Because of the limited resolution of N-body simulations, it is not yet known which model provides the best description of the central densities of simulated dark-matter halos. Simulations assuming different cosmological initial conditions produce halo populations in which the two parameters of the NFW profile follow different mass-concentration relations, depending on cosmological properties such as

553-658: A general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by a conservative force (where the work done is independent of path) with that of the total potential energy of the system. Mathematically, the theorem states that ⟨ T ⟩ = − 1 2 ∑ k = 1 N ⟨ F k ⋅ r k ⟩ , {\displaystyle \langle T\rangle =-{\frac {1}{2}}\,\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle ,} where T

632-612: A halo is linked to the mean density of the universe at its epoch of maximal growth. Thus the size-density relation reflects the fact that larger halos typically assembled at later times. In the NFW profile, the density of dark matter as a function of radius is given by: ρ ( r ) = ρ 0 r R s ( 1   +   r R s ) 2 {\displaystyle \rho (r)={\frac {\rho _{0}}{{\frac {r}{R_{s}}}\left(1~+~{\frac {r}{R_{s}}}\right)^{2}}}} where ρ 0 and

711-412: A one-dimensional oscillator with mass m {\displaystyle m} , position x {\displaystyle x} , driving force F cos ⁡ ( ω t ) {\displaystyle F\cos(\omega t)} , spring constant k {\displaystyle k} , and damping coefficient γ {\displaystyle \gamma } ,

790-1589: A single particle in special relativity, it is not the case that T = ⁠ 1 / 2 ⁠ p · v . Instead, it is true that T = ( γ − 1) mc , where γ is the Lorentz factor γ = 1 1 − v 2 c 2 , {\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},} and β = ⁠ v / c ⁠ . We have 1 2 p ⋅ v = 1 2 β γ m c ⋅ β c = 1 2 γ β 2 m c 2 = ( γ β 2 2 ( γ − 1 ) ) T . {\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}\\[5pt]&=\left({\frac {\gamma \beta ^{2}}{2(\gamma -1)}}\right)T.\end{aligned}}} The last expression can be simplified to ( 1 + 1 − β 2 2 ) T = ( γ + 1 2 γ ) T . {\displaystyle \left({\frac {1+{\sqrt {1-\beta ^{2}}}}{2}}\right)T=\left({\frac {\gamma +1}{2\gamma }}\right)T.} Thus, under

869-730: A solution to the equation − ∇ 2 u = g ( u ) , {\displaystyle -\nabla ^{2}u=g(u),} in the sense of distributions . Then u {\displaystyle u} satisfies the relation ( n − 2 2 ) ∫ R n | ∇ u ( x ) | 2 d x = n ∫ R n G ( u ( x ) ) d x . {\displaystyle \left({\frac {n-2}{2}}\right)\int _{\mathbb {R} ^{n}}|\nabla u(x)|^{2}\,dx=n\int _{\mathbb {R} ^{n}}G{\big (}u(x){\big )}\,dx.} For

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948-509: A system is a star held together by its own gravity, where n = −1 . In 1870, Rudolf Clausius delivered the lecture "On a Mechanical Theorem Applicable to Heat" to the Association for Natural and Medical Sciences of the Lower Rhine, following a 20-year study of thermodynamics. The lecture stated that the mean vis viva of the system is equal to its virial, or that the average kinetic energy

1027-672: Is 1 2 d 2 I d t 2 + ∫ V x k ∂ G k ∂ t d 3 r = 2 ( T + U ) + W E + W M − ∫ x k ( p i k + T i k ) d S i , {\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},} where I

1106-543: Is also related to the characteristic density and length scale of NFW profile: V c i r c max ≈ 1.64 R s G ρ s {\displaystyle V_{\mathrm {circ} }^{\max }\approx 1.64R_{s}{\sqrt {G\rho _{s}}}} Over a broad range of halo mass and redshift, the NFW profile approximates the equilibrium configuration of dark matter halos produced in simulations of collisionless dark matter particles by numerous groups of scientists. Before

1185-462: Is divergent, but it is often useful to take the edge of the halo to be the virial radius , R vir , which is related to the "concentration parameter", c , and scale radius via R v i r = c R s {\displaystyle R_{\mathrm {vir} }=cR_{s}} (Alternatively, one can define a radius at which the average density within this radius is Δ {\displaystyle \Delta } times

1264-2126: Is equal and opposite to F kj = −∇ r j V kj = −∇ r j V jk , the force applied by particle k on particle j , as may be confirmed by explicit calculation. Hence, ∑ k = 1 N F k ⋅ r k = ∑ k = 2 N ∑ j = 1 k − 1 F j k ⋅ ( r k − r j ) = − ∑ k = 2 N ∑ j = 1 k − 1 d V j k d r j k | r k − r j | 2 r j k = − ∑ k = 2 N ∑ j = 1 k − 1 d V j k d r j k r j k . {\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 (\mathbf {r} _{k}-\mathbf {r} _{j})\\&=-\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}}} Thus d G d t = 2 T + ∑ k = 1 N F k ⋅ r k = 2 T − ∑ k = 2 N ∑ j = 1 k − 1 d V j k d r j k r j k . {\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}.} In

1343-417: Is known as the cusp-core or cuspy halo problem . It is currently debated whether this discrepancy is a consequence of the nature of the dark matter, of the influence of dynamical processes during galaxy formation, or of shortcomings in dynamical modelling of the observational data. Virial theorem#Galaxies and cosmology (virial mass and radius) In statistical mechanics , the virial theorem provides

1422-507: Is one half of the average potential energy. The virial theorem can be obtained directly from Lagrange's identity as applied in classical gravitational dynamics, the original form of which was included in Lagrange's "Essay on the Problem of Three Bodies" published in 1772. Carl Jacobi's generalization of the identity to N  bodies and to the present form of Laplace's identity closely resembles

1501-701: Is roughly 10 or 15 for the Milky Way, and may range from 4 to 40 for halos of various sizes. This can then be used to define a dark matter halo in terms of its mean density, solving the above equation for ρ 0 {\displaystyle \rho _{0}} and substituting it into the original equation. This gives ρ ( r ) = ρ halo 3 A NFW x ( c − 1 + x ) 2 {\displaystyle \rho (r)={\frac {\rho _{\text{halo}}}{3A_{\text{NFW}}\,x(c^{-1}+x)^{2}}}} where The integral of

1580-627: Is the kinetic energy. The left-hand side of this equation is just dQ / dt , according to the Heisenberg equation of motion. The expectation value ⟨ dQ / dt ⟩ of this time derivative vanishes in a stationary state, leading to the quantum virial theorem : 2 ⟨ T ⟩ = ∑ n ⟨ X n d V d X n ⟩ . {\displaystyle 2\langle T\rangle =\sum _{n}\left\langle X_{n}{\frac {dV}{dX_{n}}}\right\rangle .} In

1659-844: Is the mass of the k th particle, F k = ⁠ d p k / dt ⁠ is the net force on that particle, and T is the total kinetic energy of the system according to the v k = ⁠ d r k / dt ⁠ velocity of each particle, T = 1 2 ∑ k = 1 N m k v k 2 = 1 2 ∑ k = 1 N m k d r k d t ⋅ d r k d t . {\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}}.} The total force F k on particle k

SECTION 20

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1738-940: Is the natural frequency of the oscillator. 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: ⟨ x ˙ γ x ˙ ⟩ ⏟ power dissipated = ⟨ x ˙ F cos ⁡ ω t ⟩ ⏟ power input , {\displaystyle \underbrace {\langle {\dot {x}}\,\gamma {\dot {x}}\rangle } _{\text{power dissipated}}=\underbrace {\langle {\dot {x}}\,F\cos \omega t\rangle } _{\text{power input}},} which simplifies to sin ⁡ φ = − γ X ω F {\displaystyle \sin \varphi =-{\frac {\gamma X\omega }{F}}} . Now we have two equations that yield

1817-442: Is the position vector and M vir = 4 π 3 r vir 3 200 ρ crit {\displaystyle M_{\text{vir}}={\frac {4\pi }{3}}r_{\text{vir}}^{3}200\rho _{\text{crit}}} . The radius of the maximum circular velocity (confusingly sometimes also referred to as R max {\displaystyle R_{\max }} ) can be found from

1896-993: Is the sum of all the forces from the other particles j in the system: F k = ∑ j = 1 N F j k , {\displaystyle \mathbf {F} _{k}=\sum _{j=1}^{N}\mathbf {F} _{jk},} where F jk is the force applied by particle j on particle k . Hence, the virial can be written as − 1 2 ∑ k = 1 N F k ⋅ r k = − 1 2 ∑ k = 1 N ∑ j = 1 N F j k ⋅ r k . {\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}.} Since no particle acts on itself (i.e., F jj = 0 for 1 ≤ j ≤ N ), we split

1975-492: Is the total kinetic energy of the N particles, F k represents the force on the k th particle, which is located at position r k , and angle brackets represent the average over time of the enclosed quantity. The word virial for the right-hand side of the equation derives from vis , the Latin word for "force" or "energy", and was given its technical definition by Rudolf Clausius in 1870. The significance of

2054-425: Is the total potential energy of the system. Thus d G d t = 2 T + ∑ k = 1 N F k ⋅ r k = 2 T − n V TOT . {\displaystyle {\frac {dG}{dt}}=2T+\sum _{k=1}^{N}\mathbf {F} _{k}\cdot \mathbf {r} _{k}=2T-nV_{\text{TOT}}.} For gravitating systems

2133-595: Is the unit normal vector pointing outwards. Then the virial theorem states that ⟨ T ⟩ = − 1 2 ⟨ ∑ i F i ⋅ r i ⟩ = P 2 ∫ n ^ ⋅ r d A . {\displaystyle \langle T\rangle =-{\frac {1}{2}}{\Big \langle }\sum _{i}\mathbf {F} _{i}\cdot \mathbf {r} _{i}{\Big \rangle }={\frac {P}{2}}\int {\hat {\mathbf {n} }}\cdot \mathbf {r} \,dA.} By

2212-485: Is too noisy to give useful results for individual objects, but accurate measurements can still be made by averaging the profiles of many similar systems. For the main body of the halos, the agreement with the predictions remains good down to halo masses as small as those of the halos surrounding isolated galaxies like our own. The inner regions of halos are beyond the reach of lensing measurements, however, and other techniques give results which disagree with NFW predictions for

2291-817: The divergence theorem , ∫ n ^ ⋅ r d A = ∫ ∇ ⋅ r d V = 3 ∫ d V = 3 V {\textstyle \int {\hat {\mathbf {n} }}\cdot \mathbf {r} \,dA=\int \nabla \cdot \mathbf {r} \,dV=3\int dV=3V} . And since the average total kinetic energy ⟨ T ⟩ = N ⟨ 1 2 m v 2 ⟩ = N ⋅ 3 2 k T {\textstyle \langle T\rangle =N{\big \langle }{\frac {1}{2}}mv^{2}{\big \rangle }=N\cdot {\frac {3}{2}}kT} , we have P V = N k T {\displaystyle PV=NkT} . In 1933, Fritz Zwicky applied

2370-1026: The k th particle. Assuming that the masses are constant, G is one-half the time derivative of this moment of inertia: 1 2 d I d t = 1 2 d d t ∑ k = 1 N m k r k ⋅ r k = ∑ k = 1 N m k d r k d t ⋅ r k = ∑ k = 1 N p k ⋅ r k = G . {\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}}} In turn,

2449-444: The scalar moment of inertia I about the origin is I = ∑ k = 1 N m k | r k | 2 = ∑ k = 1 N m k r k 2 , {\displaystyle I=\sum _{k=1}^{N}m_{k}|\mathbf {r} _{k}|^{2}=\sum _{k=1}^{N}m_{k}r_{k}^{2},} where m k and r k represent

Navarro–Frenk–White profile - Misplaced Pages Continue

2528-598: The squared density is ∫ 0 R max 4 π r 2 ρ ( r ) 2 d r = 4 π 3 R s 3 ρ 0 2 [ 1 − R s 3 ( R s + R max ) 3 ] {\displaystyle \int _{0}^{R_{\max }}4\pi r^{2}\rho (r)^{2}\,dr={\frac {4\pi }{3}}R_{s}^{3}\rho _{0}^{2}\left[1-{\frac {R_{s}^{3}}{(R_{s}+R_{\max })^{3}}}\right]} so that

2607-777: The "scale radius", R s , are parameters which vary from halo to halo. The integrated mass within some radius R max is M = ∫ 0 R max 4 π r 2 ρ ( r ) d r = 4 π ρ 0 R s 3 [ ln ⁡ ( R s + R max R s ) − R max R s + R max ] {\displaystyle M=\int _{0}^{R_{\max }}4\pi r^{2}\rho (r)\,dr=4\pi \rho _{0}R_{s}^{3}\left[\ln \left({\frac {R_{s}+R_{\max }}{R_{s}}}\right)-{\frac {R_{\max }}{R_{s}+R_{\max }}}\right]} The total mass

2686-691: The NFW potential is: a = − ∇ Φ NFW ( r ) = G M vir ln ⁡ ( 1 + c ) − c / ( 1 + c ) r / ( r + R s ) − ln ⁡ ( 1 + r / R s ) r 3 r {\displaystyle \mathbf {a} =-\nabla {\Phi _{\text{NFW}}(\mathbf {r} )}=G{\frac {M_{\text{vir}}}{\ln {(1+c)}-c/(1+c)}}{\frac {r/(r+R_{s})-\ln {(1+r/R_{s})}}{r^{3}}}\mathbf {r} } where r {\displaystyle \mathbf {r} }

2765-492: The average kinetic energy equals half of the average negative potential energy: ⟨ T ⟩ τ = − 1 2 ⟨ V TOT ⟩ τ . {\displaystyle \langle T\rangle _{\tau }=-{\frac {1}{2}}\langle V_{\text{TOT}}\rangle _{\tau }.} This general result is useful for complex gravitating systems such as planetary systems or galaxies . A simple application of

2844-1196: The average of the time derivative might vanish. One often-cited reason applies to stably bound systems, that is, to systems that hang together forever and whose parameters are finite. In this case, velocities and coordinates of the particles of the system have upper and lower limits, so that G is bounded between two extremes, G min and G max , and the average goes to zero in the limit of infinite τ : lim τ → ∞ | ⟨ d G bound d t ⟩ τ | = lim τ → ∞ | G ( τ ) − G ( 0 ) τ | ≤ lim τ → ∞ G max − G min τ = 0. {\displaystyle \lim _{\tau \to \infty }\left|\left\langle {\frac {dG^{\text{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.} Even if

2923-797: The average of the time derivative of G is only approximately zero, the virial theorem holds to the same degree of approximation. For power-law forces with an exponent n , the general equation holds: ⟨ T ⟩ τ = − 1 2 ∑ k = 1 N ⟨ F k ⋅ r k ⟩ τ = n 2 ⟨ V TOT ⟩ τ . {\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 }.} For gravitational attraction, n = −1 , and

3002-411: The average total kinetic energy ⟨ T ⟩ equals n times the average total potential energy ⟨ V TOT ⟩ . Whereas V ( r ) represents the potential energy between two particles of distance r , V TOT represents the total potential energy of the system, i.e., the sum of the potential energy V ( r ) over all pairs of particles in the system. A common example of such

3081-514: The classical virial theorem. However, the interpretations leading to the development of the equations were very different, since at the time of development, statistical dynamics had not yet unified the separate studies of thermodynamics and classical dynamics. The theorem was later utilized, popularized, generalized and further developed by James Clerk Maxwell , Lord Rayleigh , Henri Poincaré , Subrahmanyan Chandrasekhar , Enrico Fermi , Paul Ledoux , Richard Bader and Eugene Parker . Fritz Zwicky

3160-531: The cluster is U = − ∑ i < j G m 2 r i , j {\displaystyle U=-\sum _{i<j}{\frac {Gm^{2}}{r_{i,j}}}} , giving ⟨ U ⟩ = − G m 2 ∑ i < j ⟨ 1 / r i , j ⟩ {\textstyle \langle U\rangle =-Gm^{2}\sum _{i<j}\langle {1}/{r_{i,j}}\rangle } . Assuming

3239-843: The cluster, each having observed stellar mass m = 10 9 M ⊙ {\displaystyle m=10^{9}M_{\odot }} (suggested by Hubble), and the cluster has radius R = 10 6 ly {\displaystyle R=10^{6}{\text{ly}}} . He also measured the radial velocities of the galaxies by doppler shifts in galactic spectra to be ⟨ v r 2 ⟩ = ( 1000 km/s ) 2 {\displaystyle \langle v_{r}^{2}\rangle =(1000{\text{km/s}})^{2}} . Assuming equipartition of kinetic energy, ⟨ v 2 ⟩ = 3 ⟨ v r 2 ⟩ {\displaystyle \langle v^{2}\rangle =3\langle v_{r}^{2}\rangle } . By

Navarro–Frenk–White profile - Misplaced Pages Continue

3318-453: The commutator is i ℏ [ H , Q ] = 2 T − ∑ n X n d V d X n , {\displaystyle {\frac {i}{\hbar }}[H,Q]=2T-\sum _{n}X_{n}{\frac {dV}{dX_{n}}},} where T = ∑ n P n 2 / 2 m n {\textstyle T=\sum _{n}P_{n}^{2}/2m_{n}}

3397-1075: The conditions described in earlier sections (including Newton's third law of motion , F jk = − F kj , despite relativity), the time average for N particles with a power law potential is n 2 ⟨ V TOT ⟩ τ = ⟨ ∑ k = 1 N ( 1 + 1 − β k 2 2 ) T k ⟩ τ = ⟨ ∑ k = 1 N ( γ k + 1 2 γ k ) T k ⟩ τ . {\displaystyle {\frac {n}{2}}\left\langle V_{\text{TOT}}\right\rangle _{\tau }=\left\langle \sum _{k=1}^{N}\left({\tfrac {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 }.} In particular,

3476-633: The critical or mean density of the universe , resulting in a similar relation: R Δ = c Δ R s {\displaystyle R_{\Delta }=c_{\Delta }R_{s}} . The virial radius will lie around R 200 {\displaystyle R_{200}} to R 500 {\displaystyle R_{500}} , though values of Δ = 1000 {\displaystyle \Delta =1000} are used in X-ray astronomy, for example, due to higher concentrations.) The total mass in

3555-487: The dark matter virializes , the distribution of dark matter deviates from an NFW profile, and significant substructure is observed in simulations both during and after the collapse of the halos. Alternative models, in particular the Einasto profile , have been shown to represent the dark matter profiles of simulated halos as well as or better than the NFW profile by including an additional third parameter. The Einasto profile has

3634-463: The dark matter distribution inside the visible galaxies which lie at halo centers. Observations of the inner regions of bright galaxies like the Milky Way and M31 may be compatible with the NFW profile, but this is open to debate. The NFW dark matter profile is not consistent with observations of the inner regions of low surface brightness galaxies, which have less central mass than predicted. This

3713-459: The density of the universe and the nature of the very early process which created all structure. Observational measurements of this relation thus offer a route to constraining these properties. The dark matter density profiles of massive galaxy clusters can be measured directly by gravitational lensing and agree well with the NFW profiles predicted for cosmologies with the parameters inferred from other data. For lower mass halos, gravitational lensing

3792-844: The equation of motion is m d 2 x d t 2 ⏟ acceleration = − k x d d ⏟ spring   −   γ d x d t ⏟ friction   +   F cos ⁡ ( ω t ) d d ⏟ external driving . {\displaystyle m\underbrace {\frac {d^{2}x}{dt^{2}}} _{\text{acceleration}}=\underbrace {-kx{\vphantom {\frac {d}{d}}}} _{\text{spring}}\ \underbrace {-\ \gamma {\frac {dx}{dt}}} _{\text{friction}}\ \underbrace {+\ F\cos(\omega t){\vphantom {\frac {d}{d}}}} _{\text{external driving}}.} When

3871-1007: The exact equation ⟨ d G d t ⟩ τ = 2 ⟨ T ⟩ τ + ∑ k = 1 N ⟨ F k ⋅ r k ⟩ τ . {\displaystyle \left\langle {\frac {dG}{dt}}\right\rangle _{\tau }=2\langle T\rangle _{\tau }+\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle _{\tau }.} The virial theorem states that if ⟨ dG / dt ⟩ τ = 0 , then 2 ⟨ T ⟩ τ = − ∑ k = 1 N ⟨ F k ⋅ r k ⟩ τ . {\displaystyle 2\langle T\rangle _{\tau }=-\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle _{\tau }.} There are many reasons why

3950-449: The exponent n equals −1, giving Lagrange's identity d G d t = 1 2 d 2 I d t 2 = 2 T + V TOT , {\displaystyle {\frac {dG}{dt}}={\frac {1}{2}}{\frac {d^{2}I}{dt^{2}}}=2T+V_{\text{TOT}},} which was derived by Joseph-Louis Lagrange and extended by Carl Jacobi . The average of this derivative over

4029-1307: The field of quantum mechanics, there exists another form of the virial theorem, applicable to localized solutions to the stationary nonlinear Schrödinger equation or Klein–Gordon equation , is Pokhozhaev's identity , also known as Derrick's theorem . Let g ( s ) {\displaystyle g(s)} be continuous and real-valued, with g ( 0 ) = 0 {\displaystyle g(0)=0} . Denote G ( s ) = ∫ 0 s g ( t ) d t {\textstyle G(s)=\int _{0}^{s}g(t)\,dt} . Let u ∈ L loc ∞ ( R n ) , ∇ u ∈ L 2 ( R n ) , G ( u ( ⋅ ) ) ∈ L 1 ( R n ) , n ∈ N {\displaystyle u\in L_{\text{loc}}^{\infty }(\mathbb {R} ^{n}),\quad \nabla u\in L^{2}(\mathbb {R} ^{n}),\quad G(u(\cdot ))\in L^{1}(\mathbb {R} ^{n}),\quad n\in \mathbb {N} } be

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4108-746: The forces can be derived from a potential energy V jk that is a function only of the distance r jk between the point particles j and k . Since the force is the negative gradient of the potential energy, we have in this case F j k = − ∇ r k V j k = − d V j k d r j k ( r k − r j r j k ) , {\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),} which

4187-771: The gravitational potential of a uniform ball of constant density, giving ⟨ U ⟩ = − 3 5 G N 2 m 2 R {\textstyle \langle U\rangle =-{\frac {3}{5}}{\frac {GN^{2}m^{2}}{R}}} . So by the virial theorem, the total mass of the cluster is N m = 5 ⟨ v 2 ⟩ 3 G ⟨ 1 r ⟩ {\displaystyle Nm={\frac {5\langle v^{2}\rangle }{3G\langle {\frac {1}{r}}\rangle }}} Zwicky 1933 {\displaystyle _{1933}} estimated that there are N = 800 {\displaystyle N=800} galaxies in

4266-627: The halo within R v i r {\displaystyle R_{\mathrm {vir} }} is M = ∫ 0 R v i r 4 π r 2 ρ ( r ) d r = 4 π ρ 0 R s 3 [ ln ⁡ ( 1 + c ) − c 1 + c ] . {\displaystyle M=\int _{0}^{R_{\mathrm {vir} }}4\pi r^{2}\rho (r)\,dr=4\pi \rho _{0}R_{s}^{3}\left[\ln(1+c)-{\frac {c}{1+c}}\right].} The specific value of c

4345-881: The larger ratios. The virial theorem has a particularly simple form for periodic motion. It can be used to perform perturbative calculation for nonlinear oscillators. It can also be used to study motion in a central potential . If the central potential is of the form U ∝ r n {\displaystyle U\propto r^{n}} , the virial theorem simplifies to ⟨ T ⟩ = n 2 ⟨ U ⟩ {\displaystyle \langle T\rangle ={\frac {n}{2}}\langle U\rangle } . In particular, for gravitational or electrostatic ( Coulomb ) attraction, ⟨ T ⟩ = − 1 2 ⟨ U ⟩ {\displaystyle \langle T\rangle =-{\frac {1}{2}}\langle U\rangle } . Analysis based on Sivardiere, 1986. For

4424-407: The limits lim r → ∞ Φ = 0 {\displaystyle \lim _{r\to \infty }\Phi =0} and lim r → 0 Φ = − 4 π G ρ 0 R s 2 {\displaystyle \lim _{r\to 0}\Phi =-4\pi G\rho _{0}R_{s}^{2}} . The acceleration due to

4503-399: The mass and position of the k th particle. r k = | r k | is the position vector magnitude. Consider the scalar G = ∑ k = 1 N p k ⋅ r k , {\displaystyle G=\sum _{k=1}^{N}\mathbf {p} _{k}\cdot \mathbf {r} _{k},} where p k is the momentum vector of

4582-712: The maximum of M ( r ) / r {\displaystyle M(r)/r} as R c i r c max = α R s {\displaystyle R_{\mathrm {circ} }^{\max }=\alpha R_{s}} where α ≈ 2.16258 {\displaystyle \alpha \approx 2.16258} is the positive root of ln ⁡ ( 1 + α ) = α ( 1 + 2 α ) ( 1 + α ) 2 . {\displaystyle \ln \left(1+\alpha \right)={\frac {\alpha (1+2\alpha )}{(1+\alpha )^{2}}}.} Maximum circular velocity

4661-570: The mean squared density inside of R max is ⟨ ρ 2 ⟩ R max = R s 3 ρ 0 2 R max 3 [ 1 − R s 3 ( R s + R max ) 3 ] {\displaystyle \langle \rho ^{2}\rangle _{R_{\max }}={\frac {R_{s}^{3}\rho _{0}^{2}}{R_{\max }^{3}}}\left[1-{\frac {R_{s}^{3}}{(R_{s}+R_{\max })^{3}}}\right]} which for

4740-702: The mean squared density inside the scale radius is simply ⟨ ρ 2 ⟩ R s = 7 8 ρ 0 2 {\displaystyle \langle \rho ^{2}\rangle _{R_{s}}={\frac {7}{8}}\rho _{0}^{2}} Solving Poisson's equation gives the gravitational potential Φ ( r ) = − 4 π G ρ 0 R s 3 r ln ⁡ ( 1 + r R s ) {\displaystyle \Phi (r)=-{\frac {4\pi G\rho _{0}R_{s}^{3}}{r}}\ln \left(1+{\frac {r}{R_{s}}}\right)} with

4819-456: The motion of the stars are all the same over a long enough time ( ergodicity ), ⟨ U ⟩ = − 1 2 N 2 G m 2 ⟨ 1 / r ⟩ {\textstyle \langle U\rangle =-{\frac {1}{2}}N^{2}Gm^{2}\langle {1}/{r}\rangle } . Zwicky estimated ⟨ U ⟩ {\displaystyle \langle U\rangle } as

SECTION 60

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4898-403: 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 F 1 ( t ) , F 2 ( t ) results in net cancellation. Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step. For a collection of N point particles,

4977-951: The origin, the particles have positions r 1 ( t ) and r 2 ( t ) = − r 1 ( t ) with fixed magnitude r . The attractive forces act in opposite directions as positions, so F 1 ( t ) ⋅ r 1 ( t ) = F 2 ( t ) ⋅ r 2 ( t ) = − Fr . Applying the centripetal force formula F = mv / r results in − 1 2 ∑ k = 1 N ⟨ F k ⋅ r k ⟩ = − 1 2 ( − F r − F r ) = F r = m v 2 r ⋅ r = m v 2 = ⟨ T ⟩ , {\displaystyle -{\frac {1}{2}}\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle =-{\frac {1}{2}}(-Fr-Fr)=Fr={\frac {mv^{2}}{r}}\cdot r=mv^{2}=\langle T\rangle ,} as required. Note: If

5056-1272: The oscillator has reached a steady state, it performs a stable oscillation x = X cos ⁡ ( ω t + φ ) {\displaystyle x=X\cos(\omega t+\varphi )} , where X {\displaystyle X} is the amplitude, and φ {\displaystyle \varphi } is the phase angle. Applying the virial theorem, we have m ⟨ x ˙ x ˙ ⟩ = k ⟨ x x ⟩ + γ ⟨ x x ˙ ⟩ − F ⟨ cos ⁡ ( ω t ) x ⟩ {\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 } , which simplifies to F cos ⁡ ( φ ) = m ( ω 0 2 − ω 2 ) X {\displaystyle F\cos(\varphi )=m(\omega _{0}^{2}-\omega ^{2})X} , where ω 0 = k / m {\displaystyle \omega _{0}={\sqrt {k/m}}}

5135-1439: The particles are at diametrically opposite points of a circular orbit with radius r . The velocities are v 1 ( t ) and v 2 ( t ) = − v 1 ( t ) , which are normal to forces F 1 ( t ) and F 2 ( t ) = − F 1 ( t ) . The respective magnitudes are fixed at v and F . The average kinetic energy of the system in an interval of time from t 1 to t 2 is ⟨ T ⟩ = 1 t 2 − t 1 ∫ t 1 t 2 ∑ k = 1 N 1 2 m k | v k ( t ) | 2 d t = 1 t 2 − t 1 ∫ t 1 t 2 ( 1 2 m | v 1 ( t ) | 2 + 1 2 m | v 2 ( t ) | 2 ) d t = m v 2 . {\displaystyle \langle T\rangle ={\frac {1}{t_{2}-t_{1}}}\int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}{\frac {1}{2}}m_{k}|\mathbf {v} _{k}(t)|^{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}.} Taking center of mass as

5214-920: The position operator X n and the momentum operator P n = − i ℏ d d X n {\displaystyle P_{n}=-i\hbar {\frac {d}{dX_{n}}}} of particle n , [ H , X n P n ] = X n [ H , P n ] + [ H , X n ] P n = i ℏ X n d V d X n − i ℏ P n 2 m n . {\displaystyle [H,X_{n}P_{n}]=X_{n}[H,P_{n}]+[H,X_{n}]P_{n}=i\hbar X_{n}{\frac {dV}{dX_{n}}}-i\hbar {\frac {P_{n}^{2}}{m_{n}}}.} Summing over all particles, one finds that for Q = ∑ n X n P n {\displaystyle Q=\sum _{n}X_{n}P_{n}}

5293-435: The ratio of kinetic energy to potential energy is no longer fixed, but necessarily falls into an interval: 2 ⟨ T TOT ⟩ n ⟨ V TOT ⟩ ∈ [ 1 , 2 ] , {\displaystyle {\frac {2\langle T_{\text{TOT}}\rangle }{n\langle V_{\text{TOT}}\rangle }}\in [1,2],} where the more relativistic systems exhibit

5372-706: The solution { X = F 2 γ 2 ω 2 + m 2 ( ω 0 2 − ω 2 ) 2 , tan ⁡ φ = − γ ω m ( ω 0 2 − ω 2 ) . {\displaystyle {\begin{cases}X={\sqrt {\dfrac {F^{2}}{\gamma ^{2}\omega ^{2}+m^{2}(\omega _{0}^{2}-\omega ^{2})^{2}}}},\\\tan \varphi =-{\dfrac {\gamma \omega }{m(\omega _{0}^{2}-\omega ^{2})}}.\end{cases}}} Consider

5451-1777: The sum in terms below and above this diagonal and add them together in pairs: ∑ k = 1 N F k ⋅ r k = ∑ k = 1 N ∑ j = 1 N F j k ⋅ r k = ∑ k = 2 N ∑ j = 1 k − 1 ( F j k ⋅ r k + F k j ⋅ r j ) = ∑ k = 2 N ∑ j = 1 k − 1 ( F j k ⋅ r k − F j k ⋅ r j ) = ∑ k = 2 N ∑ j = 1 k − 1 F j k ⋅ ( r k − r j ) , {\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}(\mathbf {F} _{jk}\cdot \mathbf {r} _{k}+\mathbf {F} _{kj}\cdot \mathbf {r} _{j})\\&=\sum _{k=2}^{N}\sum _{j=1}^{k-1}(\mathbf {F} _{jk}\cdot \mathbf {r} _{k}-\mathbf {F} _{jk}\cdot \mathbf {r} _{j})=\sum _{k=2}^{N}\sum _{j=1}^{k-1}\mathbf {F} _{jk}\cdot (\mathbf {r} _{k}-\mathbf {r} _{j}),\end{aligned}}} where we have used Newton's third law of motion , i.e., F jk = − F kj (equal and opposite reaction). It often happens that

5530-729: The system under consideration, the averaging need not be taken over time; an ensemble average can also be taken, with equivalent results. Although originally derived for classical mechanics, the virial theorem also holds for quantum mechanics, as first shown by Fock using the Ehrenfest theorem . Evaluate the commutator of the Hamiltonian H = V ( { X i } ) + ∑ n P n 2 2 m n {\displaystyle H=V{\bigl (}\{X_{i}\}{\bigr )}+\sum _{n}{\frac {P_{n}^{2}}{2m_{n}}}} with

5609-1273: The time derivative of G is d G d t = ∑ k = 1 N p k ⋅ d r k d t + ∑ k = 1 N d p k d t ⋅ r k = ∑ k = 1 N m k d r k d t ⋅ d r k d t + ∑ k = 1 N F k ⋅ r k = 2 T + ∑ k = 1 N F k ⋅ r k , {\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}}} where m k

5688-492: The total mass is 450 times that of observed mass. Lord Rayleigh published a generalization of the virial theorem in 1900, which was partially reprinted in 1903. Henri Poincaré proved and applied a form of the virial theorem in 1911 to the problem of formation of the Solar System from a proto-stellar cloud (then known as cosmogony ). A variational form of the virial theorem was developed in 1945 by Ledoux. A tensor form of

5767-549: The virial radius simplifies to ⟨ ρ 2 ⟩ R v i r = ρ 0 2 c 3 [ 1 − 1 ( 1 + c ) 3 ] ≈ ρ 0 2 c 3 {\displaystyle \langle \rho ^{2}\rangle _{R_{\mathrm {vir} }}={\frac {\rho _{0}^{2}}{c^{3}}}\left[1-{\frac {1}{(1+c)^{3}}}\right]\approx {\frac {\rho _{0}^{2}}{c^{3}}}} and

5846-428: The virial theorem concerns galaxy clusters . If a region of space is unusually full of galaxies, it is safe to assume that they have been together for a long time, and the virial theorem can be applied. Doppler effect measurements give lower bounds for their relative velocities, and the virial theorem gives a lower bound for the total mass of the cluster, including any dark matter. If the ergodic hypothesis holds for

5925-526: The virial theorem is that it allows the average total kinetic energy to be calculated even for very complicated systems that defy an exact solution, such as those considered in statistical mechanics ; this average total kinetic energy is related to the temperature of the system by the equipartition theorem . However, the virial theorem does not depend on the notion of temperature and holds even for systems that are not in thermal equilibrium . The virial theorem has been generalized in various ways, most notably to

6004-709: The virial theorem to estimate the mass of Coma Cluster , 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. He approximated the Coma cluster as a spherical "gas" of N {\displaystyle N} stars of roughly equal mass m {\displaystyle m} , which gives ⟨ T ⟩ = 1 2 N m ⟨ v 2 ⟩ {\textstyle \langle T\rangle ={\frac {1}{2}}Nm\langle v^{2}\rangle } . The total gravitational potential energy of

6083-961: The virial theorem was developed by Parker, Chandrasekhar and Fermi. The following generalization of the virial theorem has been established by Pollard in 1964 for the case of the inverse square law: 2 lim τ → + ∞ ⟨ T ⟩ τ = lim τ → + ∞ ⟨ U ⟩ τ if and only if lim τ → + ∞ τ − 2 I ( τ ) = 0. {\displaystyle 2\lim _{\tau \to +\infty }\langle T\rangle _{\tau }=\lim _{\tau \to +\infty }\langle U\rangle _{\tau }\quad {\text{if and only if}}\quad \lim _{\tau \to +\infty }{\tau }^{-2}I(\tau )=0.} A boundary term otherwise must be added. The virial theorem can be extended to include electric and magnetic fields. The result

6162-520: The virial theorem, the total mass of the cluster should be 5 R ⟨ v r 2 ⟩ G ≈ 3.6 × 10 14 M ⊙ {\displaystyle {\frac {5R\langle v_{r}^{2}\rangle }{G}}\approx 3.6\times 10^{14}M_{\odot }} . However, the observed mass is N m = 8 × 10 11 M ⊙ {\displaystyle Nm=8\times 10^{11}M_{\odot }} , meaning

6241-591: Was the first to use the virial theorem to deduce the existence of unseen matter, which is now called dark matter . Richard Bader showed that the charge distribution of a total system can be partitioned into its kinetic and potential energies that obey the virial theorem. As another example of its many applications, the virial theorem has been used to derive the Chandrasekhar limit for the stability of white dwarf stars . Consider N = 2 particles with equal mass m , acted upon by mutually attractive forces. Suppose

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