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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.