= Inclination projection of an axisymmetric stellar velocity tensor
{title2=$\sigma_{\rm los}^2=\sigma_0^2(1-\delta\cos^2i)$}
For an axisymmetric random <velocity dispersion> tensor with $\Pi_{11}=\Pi_{22}=M\sigma_0^2$ and $\Pi_{33}=M\sigma_0^2(1-\delta)$, contracting with the line-of-sight unit vector $(\sin i,0,\cos i)$ gives the displayed mass-weighted <velocity dispersion>. A major-axis rotation amplitude projects as $V_r=V\sin i$. By contrast, an azimuthally averaged line-of-sight streaming root-mean-square is $V\sin i/\sqrt2$ when $V^2$ is the full in-plane streaming mean square. Aperture weights and the definition of the measured rotation statistic must match the intrinsic statistic.
Back to article page
