This article explains how to calculate the mass of a globular cluster. The first equation is \begin{equation} v^{2}_{1D} = \sigma^{2}_{r} \end{equation}
Can someone please elaborate why this is the case? I get that you can measure the radial velocities of many stars and calculate the average their velocity-squared. So, I assume $v_{1D}^{2}$ means
\begin{equation} v_{1D}^{2} = \frac{1}{N}\sum_{i=1}^{N} v_{r,i}^{2} \end{equation} where $v_{r,i}$ is the magnitude of the radial velocity for star $i$, $N$ is the number of stars (let's say $N = 10^{6}$).
Also, if we define $\hat{x}, \hat{y}$ directions to be perpendicular to the radial directions ($\hat{r}$), then \begin{equation} v_{1D}^{2} = \frac{1}{N}\sum_{i=1}^{N} v_{r,i}^{2} = \frac{1}{N}\sum_{i=1}^{N} v_{x,i}^{2} = \frac{1}{N}\sum_{i=1}^{N} v_{y,i}^{2}. \end{equation}
Question 1. Then what is the physical meaning of this $\sigma_{r}$ and why is this equal to $v_{1D}^{2}$?
The second equation also puzzles me. \begin{equation} v^{2}_{3D} = 3v^{2}_{1D} \end{equation}
Question 2. What would be the definition of $v^{2}_{3D}$ and why does the equation hold?