Virial theorem

For a confined Newtonian gravitational system, $\dot{G}=2T+V$ for $G=\sum p_i\cdot r_i$, hence $2⟨T⟩=-⟨V⟩$.

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 $k_{B}T\simeq \mu m_p{v}_c^2/2$.

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.