= Solution
For a concrete Milky-Way-like illustration, take
$$
V_c(R)=220\tanh(R/1.5\,\mathrm{kpc})\,\mathrm{km\,s^{-1}},\quad R_d=2.6\,\mathrm{kpc},\quad R_\sigma=7\,\mathrm{kpc},
$$
and isotropic tracer <velocity dispersion>
$$
\sigma(R)=30\exp\left[-\frac{R-8\,\mathrm{kpc}}{2R_\sigma}\right]\,\mathrm{km\,s^{-1}}.
$$
With local tracer density $\nu\propto e^{-R/R_d}$, the <exponential-disk asymmetric drift> then predicts
$$
\bar v_\phi(R)=\left[V_c^2(R)-\sigma^2(R)R\left(\frac1{R_d}+\frac1{R_\sigma}\right)\right]^{1/2}.
$$
The 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 $R\ge1\,\mathrm{kpc}$ are used for the stellar approximation; a central <galactic bulge> needs a separate model.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-72-disc-rotation.png]
{title=Milky-Way-like cold-gas circular rotation and a stellar rotation curve with explicitly specified asymmetric drift}
At $R=8\,\mathrm{kpc}$, $A\simeq4.22$ and $\sigma=30\,\mathrm{km\,s^{-1}}$, giving \b[gas speed about $220\,\mathrm{km\,s^{-1}}$, stellar speed about $211\,\mathrm{km\,s^{-1}}$, and a lag about $9\,\mathrm{km\,s^{-1}}$]. 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 $182\,\mathrm{km\,s^{-1}}$. The specified radial laws and isotropy make the example reproducible while retaining the qualifications in the preceding derivation.
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