Tensor virial theorem

For an isolated collisionless system, the tensor virial theorem is $\frac{1}{2}{\ddot{I}}_{jk}=2T_{jk}+{\Pi }_{jk}+W_{jk}$, separating ordered kinetic, random kinetic, and gravitational potential-energy tensors.

Use the position second-moment tensor $I_{ij}=\int \rho x_i{x}_j{d}^{3}x$, not the engineering moment-of-inertia convention. If a density pattern rotates steadily about its third principal axis, $I(t)=R(\Omega t)I(0)R(\Omega t{)}^T$. Differentiating twice at a principal-axis instant gives the displayed tensor. The tensor virial theorem then determines the pattern speed from the difference of its first two diagonal components. This density-pattern speed is distinct from stellar mean streaming; in the axisymmetric limit $I_{11}=I_{22}$, this difference cannot determine it.

For a finite stationary axisymmetric self-gravitating system with similar oblate density surfaces, let $q=b/a$, $s=\sqrt{1-q^2}$, $A_1=q\arccos (q)/s^3-q^2/s^2$, and $A_3=2/s^2-2q\arccos (q)/s^3$. Its stellar potential-energy tensor has $Q=A_1/(q^2{A}_3)$; the common radial integral cancels. The spherical limit has $A_1=A_3=2/3$ and $Q=1$. With mass-weighted in-plane velocity dispersion ${\sigma }_x$ and constant azimuthal streaming $v$, the tensor virial theorem gives $v^2/{\sigma }_x^2=2[(1-\beta )Q-1]$, where ${\sigma }_z^2=(1-\beta ){\sigma }_x^2$. Neither $Q=1/q$ nor the omission of the Cartesian streaming factor is exact.

For a pressure-supported scale-free tracer $\rho \propto (R^2+z^2/q^2{)}^{-\gamma /2}$ in a spherical Newtonian gravitational potential, the tensor virial theorem constrains integrated directional velocity dispersions. Modest oblate flattening and a decreasing radial mass density require greater in-plane random support per direction. The displayed expansion assumes finite global virial integrals; the corresponding ratio of force moments also has a cutoff-independent angular interpretation.

For $h=({\sin }^2\theta +q^{-2}{\cos }^2\theta {)}^{-\gamma /2}$, separation in spherical coordinates makes both stellar potential-energy tensor integrals contain $\int r^{3-\gamma }{\Phi }^{\prime }(r)dr$. Their ratio is the displayed positive finite angular expression. If the radial integrals diverge, identical spherical cutoffs give this same ratio of force moments, but there is no finite global pressure-energy ratio without further hypotheses. A finite-aperture tensor virial theorem retains its surface tensor.

The planar trace of the stellar pressure-energy tensor is the sum of its two Cartesian in-plane components. For an axisymmetric nonstreaming population it equals $\int \rho (⟨v_R^2⟩+⟨v_{\varphi }^2⟩)d^{3}x$, not merely the cylindrical radial second moment. A spherical isotropic population has ${\Pi }_{\perp }=2{\Pi }_{zz}$.

This force-moment tensor enters the tensor virial theorem. For an isolated self-gravitating population it is symmetric by pairwise interchange of the Newtonian interaction; for a tracer in an external field use the force-moment definition directly, with the appropriate symmetric combination in a time-dependent inertia identity.

This tensor integrates the local pressure tensor, where ${\sigma }_{ij}=⟨(v_i-u_i)(v_j-u_j)⟩$ is the velocity covariance. Half its trace is the random kinetic energy. It enters the tensor virial theorem with coefficient one.

For an axisymmetric random velocity dispersion tensor with ${\Pi }_{11}={\Pi }_{22}=M{\sigma }_0^2$ and ${\Pi }_{33}=M{\sigma }_0^2(1-\delta )$, contracting with the line-of-sight unit vector $(\sin i,0,\cos i)$ gives the displayed mass-weighted velocity dispersion. A major-axis rotation amplitude projects as $V_r=V\sin i$. By contrast, an azimuthally averaged line-of-sight streaming root-mean-square is $V\sin i/\sqrt{2}$ when $V^2$ is the full in-plane streaming mean square. Aperture weights and the definition of the measured rotation statistic must match the intrinsic statistic.

This tensor records the ordered kinetic energy of the streaming velocity $u_i=⟨v_i⟩$. It excludes random motion; its vanishing is compatible with a nonzero stellar pressure-energy tensor.

For an isolated self-gravitating fluid obeying ideal magnetohydrodynamics, let $I_{ij}=\int \rho x_i{x}_{j}dV$ and $K_{ij}=\frac{1}{2}\int \rho u_i{u}_{j}dV$. Use the force-stress convention $\rho D_t{u}_i={\partial }_j{T}_{ij}$, with pressure, magnetic and negative Newtonian gravitational stress tensor contributions. Two time derivatives of $I$ give $4K_{ij}$ plus $\int (x_i{\partial }_k{T}_{jk}+x_j{\partial }_k{T}_{ik})dV$. Integration by parts and symmetry yield the theorem when surface terms vanish. This fluid version complements the collisionless formulation of the tensor virial theorem.

At an initially resting instant with negligible pressure, sum the horizontal components of the magnetized-fluid tensor virial theorem. The horizontal magnetic trace is $-B_z^2/{\mu }_0$ and the gravitational force-stress trace is $({\partial }_z\Phi {)}^2/(4\pi G)$. Their negatives give the displayed radial second-moment balance. Vacuum field stresses contribute too; being initially at rest does not imply zero acceleration.