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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.

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 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.

The simplified moment prescription $q={\sigma }_z^2/({\sigma }_x^2+v_p^2)$, with $e=1-q$ and $\beta =1-{\sigma }_z^2/{\sigma }_x^2$, gives the displayed relation by solving for $v_p^2$. The prescription is an approximation, not a consequence of the tensor virial theorem for arbitrary oblate density profiles. For constant azimuthal streaming $v_{\mathrm{az}}$, the one-component streaming second moment is $v_p^2=v_{\mathrm{az}}^2/2$.

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.

Use the stellar position second-moment tensor $I_{ij}=\int \rho x_i{x}_j{d}^{3}x$. This is the $I$ entering the tensor virial theorem, rather than the engineering moment-of-inertia tensor $\mathrm{tr}(I)1-I$. For a density pattern rotating steadily about its third principal axis, choose $t=0$ when its in-plane axes are principal axes. Then $I(0)=\mathrm{diag}(I_1,I_2,I_3)$ andIn particular, $I_{11}=I_1{\cos }^2\Omega t+I_2{\sin }^2\Omega t$, $I_{22}=I_1{\sin }^2\Omega t+I_2{\cos }^2\Omega t$, and $I_{12}=(I_1-I_2)\sin \Omega t\cos \Omega t$, while $I_{33}=I_3$ and $I_{13}=I_{23}=0$. Two differentiations at $t=0$ prove the inertia-tensor acceleration of a steadily rotating stellar pattern:Both the off-diagonal acceleration and the in-plane trace acceleration vanish at this instant.

Write $\mathbf{u}=\overline{\mathbf{v}}$ for the local mean streaming. The stellar kinetic-energy tensor, stellar pressure-energy tensor, and stellar potential-energy tensor have the conventionsFor an isolated self-gravitating system, $\frac{1}{2}{\ddot{I}}_{ij}=2T_{ij}+{\Pi }_{ij}+W_{ij}$. Subtracting its second diagonal component from the first givessoThis quotient requires $I_{11}\ne I_{22}$. In an axisymmetric density distribution it becomes $0/0$: rotating the pattern about its symmetry axis does not change the density, and this equation cannot determine a pattern speed. Stellar streaming need not be rigid rotation with angular frequency $\Omega$.

Next add the first two diagonal virial equations and use the third. Their acceleration terms vanish, soFor $T_{33}=0$, ${\Pi }_{33}=-W_{33}$. With the planar virial trace ${\Pi }_{\perp }={\Pi }_{11}+{\Pi }_{22}$ and $W_{\perp }=W_{11}+W_{22}$, the prescribed mass-weighted quantities satisfy ${\Pi }_{\perp }=2M{\sigma }_0^2$, $2(T_{11}+T_{22})=M{V}_0^2$, and $1-\delta =2{\Pi }_{33}/{\Pi }_{\perp }$. Therefore $W_{33}=-(1-\delta )M{\sigma }_0^2$ and the planar equation becomesEliminating $M{\sigma }_0^2$ givesThe ${\nu }_0$ in the numerator of the PDF's displayed relation is a notation slip: the velocity subsequently defined is $V_0$. The relation follows from the traces, not from replacing a galaxy by a solid rotating body.

For the axisymmetric oblate spheroid, rotational symmetry gives ${\Pi }_{11}={\Pi }_{22}=M{\sigma }_0^2$, ${\Pi }_{33}=M{\sigma }_0^2(1-\delta )$ and zero integrated off-diagonal random stresses. A line-of-sight unit vector can be chosen as $\mathbf{n}=(\sin i,0,\cos i)$, with $i=0$ face-on. Contracting the random stress tensor with this vector gives the inclination projection of an axisymmetric stellar velocity tensor:This is a mass-weighted random velocity dispersion after subtracting streaming, with consistent aperture weights. Along the projected major axis, azimuthal streaming lies along the direction whose line-of-sight projection is $\sin i$; a rotation amplitude $V_0$ consequently projects asThere is a necessary observational convention here. The globally defined $V_0^2=2(T_{11}+T_{22})/M$ is a streaming mean square. The displayed amplitude relation holds if the rotation curve is constant, or if the measured rotation-amplitude statistic is weighted consistently to represent this intrinsic quantity. Axisymmetry alone does not equate an arbitrary observed peak rotation to the global $V_0$. If $V_r$ instead means the full-aperture line-of-sight streaming root-mean-square, azimuthal averaging yieldsand the corresponding root-mean-square is $V_0\sin i/\sqrt{2}$. Keeping amplitude and root-mean-square distinct prevents a spurious factor of two in the kinetic energy.

Finally, let the intrinsic equatorial and polar semiaxes be $a$ and $c$, and put $q_t=c/a=1-{ϵ}_t$. An intrinsic boundary has $(x^2+y^2)/a^2+z^2/c^2=1$. Its sky minor-axis coordinate is $Y=y\cos i-z\sin i$. Eliminating the line-of-sight coordinate, or maximizing $Y$ on this ellipse, gives the projected minor semiaxis $\sqrt{a^2{\cos }^{2}i+c^2{\sin }^{2}i}$; the major semiaxis remains $a$. Thus the projected axis ratio of an oblate spheroid isSubtracting from unity provesThe face-on limit is round with zero projected rotation amplitude; the edge-on limit reveals the intrinsic flattening and in-plane random velocity dispersion. These limits check both the geometry and the convention for inclination.

The stated kinematics do not, by themselves, fix a galaxy's intrinsic axis ratio. The requested curve is the usual simplified oblate-rotator flattening proxy; its additional geometric and streaming conventions should be explicit. Put $q=b/a=1-e$ and adopt the approximate moment prescriptionHere $v_p^2$ is the ordered support assigned to one in-plane component. Using ${\sigma }_z^2=(1-\beta ){\sigma }_x^2$ and solving givesIn this prescription, isotropic random motions give the oblate isotropic rotator curve; positive $\beta$ allows flattening with less rotation. A real solution requires $0\le e<1$, $\beta <1$, and $e\ge \beta$. Equality $e=\beta$ gives no rotation. This is a useful diagnostic approximation, not an exact identity for arbitrary density distributions.

To see what the tensor virial theorem actually requires, take a finite stationary axisymmetric system with mass-weighted constant velocity dispersions and azimuthal streaming speed $v_{\mathrm{az}}$. Since $u_x=-v_{\mathrm{az}}\sin \varphi$, the azimuthal average has $⟨u_x^2⟩=v_{\mathrm{az}}^2/2$. The $xx$ and $zz$ components giveand therefore, with the oblate tensor-virial shape factor $Q=W_{xx}/W_{zz}$,For similar oblate self-gravitating density surfaces, write $s=\sqrt{1-q^2}$. The ellipsoidal force coefficients, obtained by integrating the Newtonian force over those surfaces, areThe common radial integral in the stellar potential-energy tensor cancels, but these shape coefficients remain. At $q=1/2$, $Q\simeq 1.79$, rather than $1/q=2$. With $\beta =0$, the intrinsic azimuthal ratio is about $1.26$, whereas the printed simplified ratio is $1$. Recovering the latter uses both $v_p=v_{\mathrm{az}}/\sqrt{2}$ and $Q\simeq 1/q$. Thus, if the printed $v$ literally means intrinsic azimuthal speed, its claimed exact relation omits necessary model assumptions and a streaming normalization. The next sketch displays the requested approximate curve with that qualification.

A nonrotating elliptical galaxy need not have an isotropic velocity ellipsoid: unequal random-motion stresses in different directions can support an oblate or triaxial shape without substantial mean rotation. The tensor virial theorem, $2T_{ij}+W_{ij}=0$ for a stationary isolated system with appropriate boundary terms, balances this directional kinetic stress against the directional gravitational stress. Anisotropic random motions can supply the flattening, whereas an isotropic nonrotating equilibrium has much stronger restrictions on its shape.

Let $u_i=⟨v_i⟩$, ${\sigma }_{ij}=⟨(v_i-u_i)(v_j-u_j)⟩$, and $P_{ij}=\rho {\sigma }_{ij}$. The Jeans equation obtained by taking the first velocity moment of the Collisionless Boltzmann equation isFor an equilibrium population multiply by $x_i$ and integrate. Integration by parts givesAssume the integrals exist and the boundary flux vanishes, as for an isolated finite system with adequate decay. With the stellar kinetic-energy tensor, stellar pressure-energy tensor and stellar potential-energy tensor defined bywe obtain the equilibrium tensor virial theoremThe stellar kinetic-energy tensor describes ordered motion, whereas the stellar pressure-energy tensor describes random motion. In particular, $T_{ij}=0$ does not mean the stars have no kinetic energy. If a surface flux is retained, the right side of the displayed equilibrium identity is that surface tensor; it cannot be discarded for an arbitrary finite aperture. For an isolated self-gravitating population, $W_{ij}$ is symmetric by pairwise interchange in the Newtonian interaction integral, and the time-dependent identity is $\frac{1}{2}{\ddot{I}}_{ij}=2T_{ij}+{\Pi }_{ij}+W_{ij}$ with $I_{ij}=\int \rho x_i{x}_j{d}^{3}x$.

For the spherical Newtonian gravitational potential, ${\partial }_j\Phi ={\Phi }^{\prime }(r)x_j/r$. In the pressure-supported case the tensor virial theorem gives ${\Pi }_{ii}=\int \rho {\Phi }^{\prime }(r)x_i^2/r{d}^{3}x$. The notation ${\Pi }_{RR}$ used in this question is the planar virial trace ${\Pi }_{xx}+{\Pi }_{yy}$, not just the integral of the local cylindrical radial velocity dispersion. In cylindrical coordinates this trace includes both $v_R^2$ and $v_{\varphi }^2$. Summing the two Cartesian components therefore givesThe distinction explains the factor of two in the spherical limit.

Use ordinary spherical coordinates, $R=r\sin \theta$, $z=r\cos \theta$. The mass density separates as ${\rho }_0{r}^{-\gamma }h(\theta )$, whereBoth potential-energy integrals have the same radial factor $2\pi {\rho }_0{\int }_0^{\infty }{r}^{3-\gamma }{\Phi }^{\prime }(r)dr$. When this factor is finite and positive it cancels, leaving the scale-free tracer virial ratioThe angular factor in the printed intermediate formula has its sine and cosine interchanged. Keeping its sine/cosine weights while using that printed factor would instead produce the opposite sign in the final small-flattening correction. Merely measuring $\theta$ from the equatorial plane cannot fix this: the measure and both moment weights would have to change as well.

The angular ratio is finite and positive for every fixed $q>0$ and finite real $\gamma$, because $h$ is positive and bounded above and below. However, the claim that the unrestricted global virial integrals are always well-defined needs a hypothesis. In a point-mass Newtonian gravitational potential, the common radial factor is proportional to ${\int }_0^{\infty }{r}^{1-\gamma }dr$: convergence at zero requires $\gamma <2$, while convergence at infinity requires $\gamma >2$. There is no such $\gamma$. If the global energies diverge, the boxed angular ratio is still the ratio of the two potential-energy integrals with identical spherical inner and outer cutoffs, independent of the cutoffs. It is not a quotient of two finite global stellar pressure-energy tensors; applying a finite-aperture tensor virial theorem additionally requires its surface terms. These convergence and boundary qualifications are separate from the printed angular typo.

Set $ϵ=q^{-2}-1$. For modest flattening, the Taylor expansion is $h=1-(\gamma /2)ϵ{\cos }^2\theta +O({ϵ}^2)$. The required elementary angular integrals areThese follow directly by $t=\cos \theta$, or from the supplied Gamma function identity. Hence the numerator is $(4/3)[1-\gamma ϵ/10+O({ϵ}^2)]$ and the denominator is $(2/3)[1-3\gamma ϵ/10+O({ϵ}^2)]$. The flattening–anisotropy virial relation becomesFor a decreasing tracer mass density ($\gamma >0$), oblate flattening ($q<1$) requires more random kinetic support per in-plane direction than vertically. In a spherical Newtonian gravitational field, directional orbital anisotropy supplies the flattening rather than an anisotropic force law. A steeper radial mass density amplifies the required velocity dispersion anisotropy. A prolate tracer reverses the leading inequality. The relation constrains integrated velocity dispersions, not a unique local galactic distribution function.

The hypothesis is $|{\Sigma }_M|<\Sigma$, with the physical surface density nonnegative. At height $z>0$, the vertical derivative of an isolated thin-disk gravitational potential has a strictly positive kernel:The same formula with ${\Sigma }_M$ represents ${\partial }_z{\Phi }_M$. The triangle inequality and the strict surface density bound therefore give the positive-kernel comparison of thin-disk fieldsUse ${\partial }_z{\Phi }_M=-\sqrt{4\pi G/{\mu }_0}{B}_z$. The integrand in part (c) is then $({\partial }_z{\Phi }_M{)}^2-({\partial }_z\Phi {)}^2<0$ in the upper vacuum region. Reflection symmetry gives the same result below; the infinitesimally thin disk has zero three-dimensional volume. Thus, with the finite-integral assumptions of the tensor virial theorem,Since the disk starts at rest, ${\dot{I}}_{\perp }=0$ initially and its radial second moment begins to decrease. This is magnetic subcriticality of a razor-thin disk: magnetic support cannot prevent initial contraction in the global virial sense. The conclusion concerns the mass-weighted radial size, and does not by itself prove that every fluid element accelerates inward or that contraction continues indefinitely.

Use a mass-normalized galactic distribution function $F$, so that $\rho =\int F{d}^{3}v$ and $\rho ⟨v_i{v}_j⟩=\int v_i{v}_{j}F{d}^{3}v$. Multiplying the stationary Collisionless Boltzmann equation by $v_j$ and integrating over velocity space givesAssume the galactic distribution function decays sufficiently rapidly for the velocity boundary term to vanish. Integration by parts then gives $\int v_j{\partial }_{v_i}F{d}^{3}v=-{\delta }_{ij}\rho$, and therefore the Jeans equations areThe second moment includes both ordered motion and velocity dispersion; no assumption of zero mean velocity was needed.

To obtain the tensor virial theorem, multiply the $j$th Jeans equation by $x_i$ and integrate over position. For an isolated, finite system with a vanishing spatial surface term,Since the left side is $-2K_{ij}$, $2K_{ij}+W_{ij}=0$. This convention puts all stellar second moments into $K_{ij}$; it does not separate ordered and random kinetic energies.

For a self-gravitating system, the trace $W$ is the Newtonian gravitational potential energy. Indeed, symmetrizing the pair integral givesThus the trace of the tensor virial theorem is $2K+W=0$, and the total energy obeysThese identities require self-gravity without an additional external potential or an omitted confining boundary pressure.

Let $⟨v_I^2⟩$ now denote a mass-weighted average over the entire initial system. Then $K_I=M_I⟨v_I^2⟩/2$. The gravitational radius is defined by $R_I=-G{M}_I^2/W_I$, so the virial theorem immediately yieldsThe gravitational radius measures total binding energy, rather than a particular geometric edge or half-mass radius.

For the accreted systems define the mass-weighted internal mean-square speed by $M_A⟨v_A^2⟩={\sum }_s{M}_s⟨v_s^2⟩$. If each satellite initially satisfies the virial theorem, its internal energy contributes to $E_A=-M_A⟨v_A^2⟩/2$. In parabolic dry-merger energy accounting, the orbital energy at large separation is zero. Assume a dry galaxy merger, no loss of mass or energy through escaping stars, no external work, and a final relaxed system satisfying the virial theorem. Then conservation of energy gives $E_F=E_I+E_A$ and $M_F=M_I+M_A$. Energy transferred by Chandrasekhar dynamical friction remains part of the total energy under these assumptions. Consequently,Applying the final virial theorem and dividing by the initial relation givesFor a mass doubling, $\eta =1$. A merger of two identical equilibrated galaxies has $ϵ=1$, hence $R_F/R_I=2$. For accretion through many minor galaxy mergers whose satellites have much smaller internal mean-square speeds, $ϵ\ll 1$, hence $R_F/R_I\simeq 4$, tending to $4$ in the cold-satellite limit. Equivalently, nearly fixed total energy makes the gravitational radius scale as $M^2$ during cold minor-merger size growth. Small satellite mass alone does not logically imply $ϵ\ll 1$: the factor four also requires this weak-binding assumption. Likewise, equal masses require comparable internal binding to give the factor two. These are the physical limits behind the two stated merger comparisons.

The Schechter function has logarithmic slope $\alpha \simeq -1$ for $M\ll M_{\ast }$ and an exponential cutoff for $M\gg M_{\ast }$. Thus a log-log plot is approximately a straight line of slope $-1$ at low mass and bends sharply downward near $M_{\ast }$.

For a collisionless self-gravitating system, define the inertia tensorTwo time derivatives, the collisionless Boltzmann equation, and integration by parts give the tensor virial theoremwhereFor an isolated steady state, or a long-time average of a bounded system, ${\ddot{I}}_{ij}=0$ and surface terms vanish, soTaking the trace gives $2K+W=0$. Therefore

Introduce the gravitational radius by $W=-G{M}^2/R$. Virial equilibrium givesAssume dissipationless accretion, negligible escaping mass, no external work, and final re-virialization. The accreted systems bringWith $\eta =M_A/M_I$ and $ϵ=⟨v_A^2⟩/⟨v_I^2⟩$,Since $M_F=M_I(1+\eta )$ and $E_F=-M_F⟨v_F^2⟩/2$,Using $E=-G{M}^2/(2R)$ also gives

Many minor galaxy mergers have dynamically cold accreted material, $ϵ\ll 1$. If they double the mass, $\eta =1$, then $R_F/R_I\simeq 4$, andThe steep low-mass side of the Schechter function supplies many small satellites. Repeated dry galaxy mergers can therefore add stars mainly at large radius, increasing size much faster than mass and lowering mean density. This explains how compact, dense redshift-two galaxies can evolve into larger present-day elliptical galaxies without requiring comparable in-situ star formation.

DefineUsing mass conservation to differentiate a material moment givesMultiply the Jeans equation for component $j$ by $x_k$, integrate over space, discard the surface term of an isolated system, and symmetrize in $j,k$. The result isDecompose $⟨v_j{v}_k⟩={\bar{v}}_j{\bar{v}}_k+{\sigma }_{jk}^2$ and defineThis proves the tensor virial theorem

For a steady oblate system rotating about $z$, $T_{zz}=0$, while symmetry gives $T_{xx}=T_{yy}$. The $zz$ and $xx$ equations areUsing ${\Pi }_{zz}=(1-\delta ){\Pi }_{xx}$, $2T_{xx}=M{v}_0^2$, and ${\Pi }_{xx}=M{\sigma }_0^2$ givesWith $W_{xx}/W_{zz}\simeq (1-ϵ{)}^{-0.9}$, the edge-on curves areAt fixed ellipticity, increasing anisotropy lowers $v_0/{\sigma }_0$; the curve starts only once $(1-ϵ{)}^{-0.9}\ge (1-\delta {)}^{-1}$. The $\delta =0$ line is the oblate isotropic rotator reference.

At inclination $i<90^{\circ }$, line-of-sight rotation is reduced approximately by $\sin i$, and the projected ellipticity also decreases as the spheroid is viewed closer to its symmetry axis. A galaxy therefore moves down and left from its edge-on location, with the exact track set by intrinsic thickness and anisotropy.

Low-mass elliptical galaxies mostly occupy the fast-rotating, flattened, nearly isotropic region close to the oblate-rotator line. High-mass systems mostly occupy the slow-rotating region below it and require anisotropy or triaxiality. Gas-rich dissipative evolution retains angular momentum, forms a compact rotating stellar component, and produces low-mass fast rotators. Repeated dry major mergers randomize orbits, lower specific angular momentum, scour central cores through black-hole binaries, and build massive slow rotators.

Four characteristic contrasts are:

These are population trends rather than sharp boundaries in the fast and slow rotator galaxy classification.

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.

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.