Virial theorem ($2⟨K⟩+⟨U⟩=0$)

The virial theorem relates the average value of a function along a trajectory to the forces that bind it. For an isolated system of particles interacting through Newtonian gravitational potential energy, it gives $2⟨K⟩+⟨U⟩=0$: the average kinetic energy is half the magnitude of the average negative potential energy. An eccentric Kepler orbit makes the distinction vivid: kinetic energy and potential energy vary throughout the orbit, although their sum is constant. The relation concerns their time averages, rather than their values at every instant. It underlies the virial mass estimate and the negative heat capacity of a virialized gravitational system.

Use constant positive particle masses $m_i$, positions ${\mathbf{r}}_i$, mechanical momenta ${\mathbf{p}}_i=m_i{\dot{\mathbf{r}}}_i$, and total kinetic energy $K={\sum }_i{\mathbf{p}}_i^2/(2m_i)$. The auxiliary quantity $G(t)$ is a sum of dot products, not the gravitational constant, which we denote here by $G_N$:This finite-$T$ identity is exact wherever Newton's second law holds along a regular trajectory; ${\mathbf{F}}_i$ must include every force. The endpoint term is what a finite observation window leaves behind. For the infinite-time theorem, suppose the averages on the left exist and $(G(T)-G(0))/T\to 0$. A sufficient condition is uniformly bounded positions and velocities for finitely many particles. More generally $G(T)=o(T)$ suffices, even without those individual bounds. Bounded positions alone do not supply a bound on momenta, especially near a singular potential energy; nor does a small endpoint term prove thermodynamic equilibrium. Periodic motion with $G(T)=G(0)$ gives the identity without an endpoint term at that finite $T$. For an isolated system, use an inertial centre of mass frame so translational drift does not contribute bulk kinetic energy.^[ref]

For a differentiable potential energy that is a homogeneous function of degree $q$, choose its additive constant so that the scaling law really holds. With conservative forces, the Euler theorem for homogeneous functions givesThus a harmonic oscillator has $q=2$ and $⟨K⟩=⟨U⟩$. An isolated system with Newtonian gravitational potential energy hasEach interacting pair occurs once, and $U=0$ at infinite separation. Here the total energy $E=K+U$ is conserved. An external field or a mixture of different scaling powers requires its own force term; one cannot assign the whole potential energy a single degree $q$. The virial theorem for a homogeneous pair potential states the corresponding general energy relation.

A dynamical mass estimate inferred from an equilibrium size and velocity scale. For a pressure-supported spheroid, $M\sim C{R}_e{\sigma }_{\mathrm{los}}^2/G$ depends on the structural coefficient, orbital anisotropy and observational aperture.

For a hydrostatic monatomic ideal gas with negligible external pressure, $2U+{\Omega }_f=0$, where gravitational potential energy is negative. The final total energy is therefore ${\Omega }_f/2$, giving the displayed radiated energy when no mass or other energy carrier escapes. The common $-{\Omega }_f/2$ result additionally takes the initial total energy to be negligible. Initial rest does not remove initial self-gravitational binding energy.

virialization is relaxation toward a gravitationally supported bulk state satisfying an appropriate virial balance. For an isolated approximately stationary Newtonian system with negligible boundary terms, $2K+W\simeq 0$. It does not require every collisionless constituent to reach collisional thermal equilibrium, and ongoing infall, tides or pressure at a boundary modify the balance.

Consider the exact two-body problem with masses $m_1,m_2$. In the inertial centre of mass frame, its internal kinetic energy is $K=m|\dot{\mathbf{r}}{|}^2/2$, where $\mathbf{r}$ is their relative position, $m=m_1{m}_2/(m_1+m_2)$ is the reduced mass, and $U=-\kappa /r$ with $\kappa =G_N{m}_1{m}_2$. Take a noncollision elliptic Kepler orbit with semi-major axis $a$ and orbital eccentricity $0<e<1$. Write $\eta$ for eccentric anomaly, to avoid confusing it with total energy $E$. With mean motion $n=\sqrt{\kappa /(m{a}^3)}$ and orbital period $P=2\pi /n$, Kepler's equation and the orbit geometry giveThe time origin is periapsis. Since $dt=(1-e\cos \eta )d\eta /n$, the time average of $1/r$ over one orbital period is exactly $1/a$. Therefore $⟨K{⟩}_P=\kappa /(2a)$ and $⟨U{⟩}_P=-\kappa /a$, independently of orbital eccentricity. Equal steps in eccentric anomaly are not equal time steps; unweighted averaging of a drawing of the ellipse gives the wrong answer.

The original dimensionless example in both diagrams has $m=a=\kappa =1$ and $e=0.6$. At periapsis, $r=0.4$, $K=2$, $U=-2.5$; at apoapsis, $r=1.6$, $K=0.125$, $U=-0.625$. Both have $E=-0.5$, but $2K+U$ is respectively $1.5$ and $-0.375$. Only the orbital period averages are $⟨K{⟩}_P=0.5$ and $⟨U{⟩}_P=-1$. The instantaneous zero of $2K+U$ occurs at $r=a$, not at every point of the orbit.

The endpoint term can also be evaluated exactly: $G=m\mathbf{r}\cdot \dot{\mathbf{r}}=mn{a}^{2}e\sin \eta$. Starting at periapsis, a window ending at eccentric anomaly $\eta$ hasAt $T=P/4$, Kepler's equation gives $\eta \simeq 2.09133$ and a positive residual $\simeq 0.3314\kappa /a$. At $T=P/2$ or $P$, $G$ returns to zero and the residual vanishes exactly. The lower panel shows the running residual and its endpoint expression coinciding; the shaded quarter-orbital period window explains why a short observation need not satisfy the infinite-time theorem.

Motion can weigh a system whose mass is otherwise inaccessible. For a self-gravitating isolated system in virial equilibrium, let $K$ and $U$ denote equilibrium averages and use its gravitational radius $R_g=-G_N{M}^2/U$ and a mass-weighted mean squared speed relative to its centre of mass. Then $K=M⟨v^2⟩/2$ gives $M=R_g⟨v^2⟩/G_N$. If random velocities are isotropic and mean streaming has been removed, $⟨v^2⟩=3{\sigma }_{\mathrm{los}}^2$, where ${\sigma }_{\mathrm{los}}$ is the one-dimensional line-of-sight velocity dispersion. Observations sample many objects at one epoch; replacing a time average by that sample requires a representative, approximately stationary population, beyond the trajectory identity itself.^[ref]

A practical measurement route uses an optical telescope to locate members and a spectrograph to measure Doppler shifts of identified spectral lines. Their shifts relative to the system's mean give radial velocities; angular separations plus a distance estimate constrain its size. Subtracting instrumental broadening and measurement variance, testing membership, and looking for coherent rotation or subgroups help control the inference. The projected luminous size need not equal $R_g$, and luminous tracers need not have the same distribution as all gravitating mass. Unknown velocity-anisotropy parameters, interlopers, mergers, external tidal forces and appreciable boundary stresses can bias a naive estimate. A large velocity dispersion alone does not establish virial equilibrium.

An external pressure changes the balance even if the system remains confined. For a material boundary acted on by spatially uniform inward pressure $P_s$, the surface force contributes $-P_s{\int }_{\partial \mathcal{V}}\mathbf{r}\cdot \mathbf{n}dS=-3P_s\mathcal{V}$, by the divergence theorem. For Newtonian gravitational potential energy, the finite-window identity becomesHere $\mathcal{V}$ is the enclosed volume, $\mathbf{n}$ the outward unit normal, and $K$ includes the microscopic translational kinetic energy of the confined particles. Do not also add the same random motions as a separate internal pressure term. A nonuniform external stress needs its surface integral; a fixed observational boundary crossed by matter also needs flux terms.^[ref]

With negligible surface pressure, the negative heat capacity of a virialized gravitational system follows from $E=-⟨K⟩$: after a slow loss of total energy and readjustment to virial equilibrium, the average kinetic energy rises. For fixed $N$ classical particles with three translational degrees of freedom, define kinetic temperature $\Theta$ by $⟨K⟩=3N{k}_B\Theta /2$. Along this isolated equilibrium sequence, $dE/d\Theta =-3N{k}_B/2$. The system contracts and its binding potential energy becomes more negative, supplying both the outgoing energy and the increased kinetic energy. This is an endpoint comparison, not a claim that every transient contraction obeys the average relation. A collisionless stellar system need not have a common thermodynamic temperature; a pressure-confined system instead has $E=-⟨K⟩+3⟨P_s\mathcal{V}⟩$. A stable canonical ensemble in contact with a large thermal reservoir does not support this negative heat capacity branch.^[ref]

The proof is a single time derivative. Apply the product rule to each dot product and use Newton's second law:Integrating from $0$ to $T$ gives the endpoint identity; the stated sublinear condition removes the endpoint term only in the infinite-time limit. Equivalently, for the scalar inertia $I=\frac{1}{2}{\sum }_i{m}_i|{\mathbf{r}}_i{|}^2$ used in the pairwise moment-of-inertia identity, $\dot{I}=G$ and $\ddot{I}=2K+{\sum }_i{\mathbf{r}}_i\cdot {\mathbf{F}}_i$. Specifying this factor of one half avoids confusion with another common inertia convention.

In On a Mechanical Theorem applicable to Heat by R. Clausius (1870, pp. 123-124), the author defines stationary motion through bounded positions and velocities, includes planetary orbits and elastic vibrations, and names the average negative half of the force-position sum the “virial”. That historical convention differs from the auxiliary $G(t)$ defined above. His proof on pp. 126-127 keeps an endpoint term, then explains why it vanishes over a complete period or becomes small in a long bounded irregular motion. The same paper separates internal and external forces and includes uniform boundary pressure on pp. 125-126: confinement was already part of the formulation, rather than a reason to omit external stresses.^[ref]

For an isolated equilibrium of equal-mass points, the virial theorem gives $⟨v^2⟩=-W/M$. A particle's instantaneous escape velocity satisfies $v_{\mathrm{esc}}^2=-2\Phi$ with zero potential at infinity. Since $W=\frac{1}{2}{\sum }_{i}m{\Phi }_i$, averaging gives $⟨v_{\mathrm{esc}}^2⟩=-4W/M$. These are three-dimensional mean-square speeds, not one-component velocity dispersions.

For a differentiable total pair potential energy homogeneous of degree $q$ in all particle positions, the Euler theorem for homogeneous functions gives ${\sum }_i{\mathbf{r}}_i\cdot {\nabla }_{i}W=qW$. Since ${\mathbf{F}}_i=-{\nabla }_{i}W$, the force virial is $-qW$. The bounded-motion time-averaged virial theorem consequently gives the displayed equation. For $q\ne 0$, total energy is $E=(q+2)⟨K⟩/q$. A positive kinetic-temperature coefficient makes its heat capacity negative exactly when $-2<q<0$. For a pair potential proportional to $r^{p+1}$ this means $-3<p<-1$.

An isolated Newtonian system in virial equilibrium has total energy $E=K+W=-K$. Along an equilibrium sequence at fixed particle number, define the kinetic temperature by $K=3N{k}_B\Theta /2$. Its heat capacity is therefore negative, as displayed: losing total energy increases the kinetic temperature. A collisionless stellar system need not have a thermodynamic temperature; this is a kinetic-temperature statement unless thermal equilibrium is additionally specified. Boundary pressure and external confinement can change the energy relation.

For an isolated final stellar population with isotropic one-dimensional velocity dispersion $\sigma$, kinetic energy is $K=3M{\sigma }^2/2$. The virial theorem gives $W=-2K$ and total energy $E=-K$. Collapse from a diffuse nearly unbound gas reservoir therefore requires removal of approximately $K$ of energy. The radiated fraction can differ if there are outflows, appreciable initial binding energy, or boundary stresses.

A bounded gravitational system in statistical equilibrium satisfies the time-averaged virial theorem. For inverse-distance gravity with negligible boundary stresses, $2K+W=0$. This relation does not require static particles or a homogeneous density profile, and reaching it does not erase the system's dynamical history.

For an isolated Newtonian point-mass system, $\ddot{I}=T+E\ge E>0$. Twice integrating gives a quadratic lower bound on $I$. The maximum-separation inertia bounds then give at-least-linear asymptotic growth of the diameter $R$ for a trajectory existing for arbitrarily large times. The bound does not assert instantaneous monotonicity or escape of every particle.

For positive masses in their centre of mass frame, ${\sum }_{j<k}{m}_j{m}_k|{\mathbf{r}}_j-{\mathbf{r}}_k{|}^2=M{\sum }_k{m}_k{r}_k^2$. Thus the scalar inertia $I=\frac{1}{2}{\sum }_k{m}_k{r}_k^2$ equals $(2M{)}^{-1}{\sum }_{j<k}{m}_j{m}_k{r}_{jk}^2$. The identity compares total spatial extent with pairwise separations.

Let $R$ be the actual largest pairwise separation of positive masses in their centre of mass frame. The pairwise moment-of-inertia identity gives $m_{(1)}{m}_{(2)}{R}^2/(2M)\le I\le R^2{\sum }_{j<k}{m}_j{m}_k/(2M)$. Consequently the scalar inertia is comparable to $R^2$, independently of configuration.

The virial temperature characterizes the thermal energy of gas supported in a gravitational potential. It is of order $\mu m_{p}GM/(k_{B}R)$, with a structural coefficient determined by the density profile. For a uniform self-gravitating gas sphere the virial theorem gives $k_{B}T=GM\mu m_p/(5R)$; a conventional halo circular-speed definition often uses $k_{B}T\simeq \mu m_p{v}_c^2/2$.

Suppose an isothermal gas with ${\rho }_g\propto r^{-2}$ is supported in a spherical halo with $M(r)=Mr/R$. Then hydrostatic equilibrium and $P={\rho }_g{k}_{B}T/\bar{m}$ give $-2{\rho }_g{k}_{B}T/(\bar{m}r)=-{\rho }_{g}GM/(Rr)$, hence $k_{B}T=\bar{m}GM/(2R)$. Here $\bar{m}$ is the dimensionful mean particle mass. The factor two follows from the density slope and isothermal pressure response; it is not the uniform-sphere virial coefficient. The singular profile and finite boundary are idealizations.

Combining a virial-temperature relation with the density of a uniform self-gravitating sphere gives $M\propto T^{3/2}{n}_H^{-1/2}$. Thus contours of constant mass in a density-temperature plane satisfy $n_H\propto T^3/M^2$. Structural coefficients, composition and the distinction between baryonic and total gravitating mass matter when giving numerical labels.

Two finite isolated steady gravitational systems at the same conserved energy obey $2T+W=0$ at each endpoint, so both have $T=-E$ and $W=2E$. Redistribution can increase central concentration without making the global potential energy more negative. Escaping matter or an external reservoir requires a different energy accounting.

For a self-gravitating system of mass $M$ and Newtonian gravitational potential energy $W<0$, its gravitational radius is defined byIt converts the total binding energy into a characteristic size without assuming a particular density profile.

For identical point masses of total mass $M$, let $W=-G{M}^2/R_g$ exclude self-interactions. The virial theorem gives $⟨v_{\mathrm{eq}}^2⟩=GM/R_g$. Since the sum of individual potential energies counts each pair twice, the mass-weighted mean frozen-potential escape velocity squared is $-4W/M$. A dissipation-free collapse from zero total energy instead has kinetic energy $-W$, hence mean squared speed $-2W/M$. These are three-dimensional speeds relative to the centre of mass; a coherent collapse speed is not a local random velocity dispersion.