Stellar asymmetric drift ($v_a=v_c-{\overline{v}}_{\varphi }$)

Stellar asymmetric drift is the lag of a pressure-supported stellar population's mean rotation behind the local circular speed. Neglecting the tilted velocity ellipsoid, radial Jeans equation balance givesA declining density and larger random velocities generally increase the lag; gravitational balance is supplied partly by random-motion stresses.

Reflection symmetry makes $\overline{v_R{v}_z}$ odd in height, so its value at the midplane vanishes while its derivative need not vanish. The radial Jeans equation includes this derivative in the stellar asymmetric drift. For a velocity ellipsoid locally aligned with spherical coordinates, $\overline{v_R{v}_z}\simeq ({\sigma }_R^2-{\sigma }_z^2)z/R$, and its midplane contribution is $-({\sigma }_R^2-{\sigma }_z^2)$. Dropping a mixed moment at the symmetry plane does not justify dropping its derivative.

For a stationary axisymmetric tracer with isotropic velocity dispersion, vanishing tilted stress, density $\nu \propto e^{-R/R_d}$, and ${\sigma }^2\propto e^{-R/R_{\sigma }}$, the radial Jeans equation gives the displayed expression. More generally its right side is $-{\sigma }^{2}d\ln (\nu {\sigma }^2)/d\ln R$. It requires a nonnegative streaming-speed square; the density and dispersion laws must describe the same tracer and the chosen height or consistent vertical average.