 Heliocentric projection of a spherical velocity dispersion

For a non-streaming spherical tracer population, the line of sight makes angle $\chi$ with the Galactocentric radial direction. The projected velocity dispersion iswhere ${\sigma }_t^2$ is one tangential variance and $\beta =1-{\sigma }_t^2/{\sigma }_r^2$ is the velocity-anisotropy parameter. At a fixed radius, averaging with the observational angular weights gives ${\sigma }_{\odot }={\sigma }_r\sqrt{1-\beta A}$, where $A=⟨{\sin }^2\chi ⟩$.