Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 72 / 2

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.

The rotation-anisotropy diagram below uses the oblate-rotator flattening proxy for $\beta =0,0.2,0.4$. Each curve starts at $e=\beta$ and rises as $e$ increases. The plotted domains cover typical spheroid values, roughly $0\le e\lesssim 0.7$ and $0\le v/\sigma \lesssim 1.5$, with the upper axis extended to show the reference curve's rise. The shaded regions are schematic overlapping populations, not measurements or sharp classification boundaries.

Giant elliptical galaxies commonly have low ordered rotation for their observed flattening, lying below the oblate isotropic rotator reference. Anisotropic velocity dispersions and, in many cases, triaxial structure provide support. Lower-luminosity elliptical galaxies are more commonly rotating, disky systems near the isotropic reference. Classical galactic bulges of spiral galaxies often overlap this rotating-spheroid locus; disk-like bulges can be more strongly rotation supported. These are population trends with substantial overlap, not universal rules for individual galaxies.

Observations usually use projected ellipticity and a line-of-sight rotation measure such as peak speed divided by a central velocity dispersion. The intrinsic $e$ and global dispersions used in the derivation are different quantities. Inclination, apertures, and unresolved streaming can change both coordinates. Consequently a point below the reference suggests extra random-motion support, but does not uniquely measure $\beta$.

Use cylindrical coordinates and tracer density $\nu (R,z)$. A steady axisymmetric Jeans equation, with no mean radial or vertical flow, isWrite $\overline{v_R^2}={\sigma }_R^2$, $\overline{v_{\varphi }^2}={\bar{v}}_{\varphi }^2+{\sigma }_{\varphi }^2$, and $V_c^2=R{\partial }_R\Phi$. Rearrangement gives the stellar asymmetric drift equationFor an isotropic velocity ellipsoid, all three dispersions equal $\sigma$ and the cross stress vanishes. Therefore the required general result isThe positive root is for a prograde galactic disk. A declining radial stress gives $A>0$: random motions supply some support, so mean stellar rotation is slower than cold-gas circular rotation. For $A{\sigma }^2\ll V_c^2$, $V_c-{\bar{v}}_{\varphi }\simeq A{\sigma }^2/(2V_c)$. The condition $\sigma <V_c$ alone is insufficient; one also needs $A{\sigma }^2\le V_c^2$.

There is no universal numerical coefficient expressible from $V_c$ and $\sigma$ alone. For constant velocity dispersion and $\nu \propto R^{-p}$, $A=p$. Thus $p=2$ gives the often-used estimate ${\bar{v}}_{\varphi }=\sqrt{V_c^2-2{\sigma }^2}$, while $p=1$ gives a different answer with the same $V_c$ and $\sigma$. For an exponential galactic disk tracer with $\nu \propto e^{-R/R_d}$ and ${\sigma }^2\propto e^{-R/R_{\sigma }}$, the exponential-disk asymmetric drift has $A=R(R_d^{-1}+R_{\sigma }^{-1})$.

Real galactic disks generally have anisotropic velocity dispersions, often ${\sigma }_R>{\sigma }_{\varphi }>{\sigma }_z$. In the small-random-motion epicyclic motion limit for a flat galaxy rotation curve, ${\sigma }_{\varphi }^2/{\sigma }_R^2\simeq 1/2$, so even the non-gradient term matters. The velocity ellipsoid can tilt, and bars or spiral structure can violate steady axisymmetry. Age-dependent galactic disk heating makes the lag population dependent. Gas has finite turbulent and thermal stresses too, though a cold gas disk often approximates $V_c$ much better than a hot stellar population.

For a concrete Milky-Way-like illustration, takeand isotropic tracer velocity dispersionWith local tracer density $\nu \propto e^{-R/R_d}$, the exponential-disk asymmetric drift then predictsThe sketch shows the cold gas following $V_c$, rising in the inner galaxy and becoming approximately flat, while the stellar curve sits below it. Only radii $R\ge 1\mathrm{kpc}$ are used for the stellar approximation; a central galactic bulge needs a separate model.

At $R=8\mathrm{kpc}$, $A\simeq 4.22$ and $\sigma =30\mathrm{km}{\mathrm{s}}^{-1}$, giving gas speed about $220\mathrm{km}{\mathrm{s}}^{-1}$, stellar speed about $211\mathrm{km}{\mathrm{s}}^{-1}$, and a lag about $9\mathrm{km}{\mathrm{s}}^{-1}$. These are representative scales for the requested schematic, not a fit to the Milky Way. Hotter old stellar populations would have a larger lag: doubling this dispersion at the same radius gives a stellar speed about $182\mathrm{km}{\mathrm{s}}^{-1}$. The specified radial laws and isotropy make the example reproducible while retaining the qualifications in the preceding derivation.