= 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> is
$$
\sigma_{\rm los}^2=\sigma_r^2\cos^2\chi+\sigma_t^2\sin^2\chi
=\sigma_r^2(1-\beta\sin^2\chi),
$$
where $\sigma_t^2$ 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 $A=\langle\sin^2\chi\rangle$.
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