Geometric correction for heliocentric velocity dispersion ($A(r,R_{\odot })$)

For tracers sampled uniformly on a Galactocentric shell of radius $r$, with the Sun at radius $R_{\odot }$, the correction in the heliocentric projection of a spherical velocity dispersion isThe limiting value at $r=R_{\odot }$ is $1/2$; it tends to $2/3$ as $r/R_{\odot }\to 0$ and to $2R_{\odot }^2/(3r^2)$ at large radius. Uniform heliocentric solid angle at fixed $r>R_{\odot }$ instead gives $A=2R_{\odot }^2/(3r^2)$ exactly. Specify the selection measure when applying the correction.