# Average velocity of stars in globular cluster

This article explains how to calculate the mass of a globular cluster. The first equation is $$$$v^{2}_{1D} = \sigma^{2}_{r}$$$$

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

$$$$v_{1D}^{2} = \frac{1}{N}\sum_{i=1}^{N} v_{r,i}^{2}$$$$ where $$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 $$$$v_{1D}^{2} = \frac{1}{N}\sum_{i=1}^{N} v_{r,i}^{2} = \frac{1}{N}\sum_{i=1}^{N} v_{x,i}^{2} = \frac{1}{N}\sum_{i=1}^{N} v_{y,i}^{2}.$$$$

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. $$$$v^{2}_{3D} = 3v^{2}_{1D}$$$$

Question 2. What would be the definition of $$v^{2}_{3D}$$ and why does the equation hold?

$$\sigma^2$$ is the variance and the (general) definition of variance (for any set of data) is $$\sigma^2=\frac{\sum x^2}{N} - \bar x^2$$ That is the mean of the squares minus the square of the mean.
But if you have subtracted the systemic velocity from the data, then $$\bar x$$ is zero, so the variance in the radial velocity is the average of the squares of the residual velocities after the mean has been subtracted.
The fact that $$v^{2}_{3D} = 3v^{2}_{1D}$$ is just an application of Pythagoras' theorem. If a star (or group of stars) has residual velocity $$v$$ towards Earth, then assuming it has the same velocity in the two tangential directions, then by Pythagoras, it's 3d velocity relative to the system is $$v_{3D}=\sqrt{v^2+v^2+v^2}.$$