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In this paper ("The Apache Point Observatory Galactic Evolution Experiment: First Detection of High Velocity Milky Way Bar Stars") they plot and calculate $v_{GSR}$. $GSR$ stands for Galactic Standard of Rest, but I am not sure which velocity they mean. Is it the Radial Velocity, with respect to the GSR, or something else?

Maybe to clarify: suppose I have a database of stars with Cartesian location and velocities, how do I calculate the $v_{GSR}$ of a single star?

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The $v_{GSR}$ in the linked paper seems to be the radial velocity (RV, in $km.s^{-1}$) with respect to the GSR. This information is mentioned in the Abstract section.

I don't know exactly what kind of coordinates you have and how exactly are your velocities expressed (Cartesian vectors?), but what you probably need to do is to convert the velocity vector of a star into the radial component using the formula

$r=\sqrt{x^2 + y^2}$

You need to make sure where the origin of your coordinates is.

This link clarifies what GSR is. Basically there are several frames of reference that are used on the galactic scales. LSR (Local Standard of Rest) is co-rotating with the gas and dust of the galaxy (so an average star is at rest in LSR), while GSR is not rotating with the galaxy and can be calculated from LSR using a formula

$v_{GSR}=v_{LSR}+220sin(l)cos(b)$

where l and b are the Galactic longitude and latitude.

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But calculating from LSR makes it even more complicated, because I have a set of Cartesian (so in GSR) coordinates. And it's not the absolute speed (as in $\sqrt{v_x^2 +...}$ because they have negative velocities –  Mathias711 Jun 5 at 14:27
    
@Mathias711 I think I will need to edit my answer, because the linked paper is using a different definition than I was assuming at first. –  mpv Jun 5 at 19:58
    
Okay, thanks. So it is indeed the radial velocity. My coordinates or so that $x=y=0$ is located in the center. $v_{GRS}$ will be $\frac{v\cdot r}{|r|}$. Then I understood correctly :) –  Mathias711 Jun 5 at 20:48

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