 Solution

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.