Define
$$
I_{jk}=\int \rho x_jx_k\,d^3x.
$$
Using mass conservation to differentiate a material moment gives
$$
\dot I_{jk}=\int\rho(x_j\langle v_k\rangle+x_k\langle v_j\rangle)\,d^3x.
$$
Multiply 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 is
$$
\frac12\ddot I_{jk}
=\int\rho\langle v_jv_k\rangle\,d^3x
-\frac12\int\rho\left(x_k\Phi_{,j}+x_j\Phi_{,k}\right)d^3x.
$$
Decompose $\langle v_jv_k\rangle=\bar v_j\bar v_k+\sigma_{jk}^2$ and define
$$
T_{jk}=\frac12\int\rho\bar v_j\bar v_k\,d^3x,\qquad
\Pi_{jk}=\int\rho\sigma_{jk}^2\,d^3x,
$$
$$
W_{jk}=-\frac12\int\rho\left(x_k\Phi_{,j}+x_j\Phi_{,k}\right)d^3x.
$$
This proves the <tensor virial theorem>
$$
\boxed{\frac12\ddot I_{jk}=2T_{jk}+\Pi_{jk}+W_{jk}}.
$$

For a steady oblate system rotating about $z$, $T_{zz}=0$, while symmetry gives $T_{xx}=T_{yy}$. The $zz$ and $xx$ equations are
$$
\Pi_{zz}+W_{zz}=0,\qquad 2T_{xx}+\Pi_{xx}+W_{xx}=0.
$$
Using $\Pi_{zz}=(1-\delta)\Pi_{xx}$, $2T_{xx}=Mv_0^2$, and $\Pi_{xx}=M\sigma_0^2$ gives
$$
\boxed{\frac{v_0^2}{\sigma_0^2}=(1-\delta)\frac{W_{xx}}{W_{zz}}-1}.
$$
With $W_{xx}/W_{zz}\simeq(1-\epsilon)^{-0.9}$, the edge-on curves are
$$
\boxed{\frac{v_0}{\sigma_0}
=\left[(1-\delta)(1-\epsilon)^{-0.9}-1\right]^{1/2}},
\qquad
\delta=0,0.1,0.2,0.3.
$$
At fixed ellipticity, increasing anisotropy lowers $v_0/\sigma_0$; the curve starts only once $(1-\epsilon)^{-0.9}\geq(1-\delta)^{-1}$. 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:

* high-mass ellipticals rotate slowly, while low-mass ellipticals are commonly fast rotators;
* high-mass systems are anisotropic and often triaxial or boxy, while low-mass systems are closer to oblate and disky;
* high-mass systems commonly have shallow central cores, while low-mass systems have steep central cusps or extra central light;
* high-mass systems are generally older, redder, more alpha-enhanced, and richer in hot X-ray gas, while lower-mass systems more often show younger populations, cold gas, and residual star formation.

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

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 346 / 1 / Solution

Fast rotators have substantial ordered stellar rotation and are often disky and nearly axisymmetric. Slow rotators have low specific angular momentum and are often massive, anisotropic, triaxial, and core-profiled.


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 Incoming links: Fast and slow rotator galaxy