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I'm currently trying to understand the following plot from Astrophysics for physicists by A.R. Choudhuri. enter image description here

I think I understand the basic concept of measuring the line-of-sight velocities with the redshift and the intensities for specific frequencies to generate the plot (mostly due to the help of another SE post (see here)).

What I still don't understand is how exactly it is possible to have positive and negative $v_R$ on the same side (left or right from the $l=0$ line). No matter how we chose the galactic coordinate system exactly (to determine the sign of $l$), in either case the mapped gas will move away from or move towards me, but not both at the same time, right? So how can we measure negative and positive radial velocities at the same time.

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I suspect what you are forgetting is that the Sun is also in motion around the Galactic centre, not just the gas that is being observed.

The basic equation for determining the line of sight velocity $V$ of a parcel of gas that is moving tangentially (i.e. assuming circular orbits) to the Galactic centre with speed $V_g$ and which is at a distance $r_g$ from the Galactic centre is: $$ V = \frac{V_g}{r_g} r_0 \sin l - V_0 \sin l,$$ (derivation can be found here, but I note that in your previous question you present the same equation from Choudhuri's book, so presumably it is derived there too) where $r_0$ is the distance from the Sun to the Galactic centre, $V_0$ is the (assumed tangential) velocity of the Sun with respect to the Galactic centre and $l$ is the Galactic longitude ($l=0$ is towards the Galactic centre).

Clearly $V$ can be either positive or negative. For $0<l<90^{\circ}$, $V>0$ if $V_g > V_0 r_g/r_0$ and $V<0$ if $V_g < V_0 r_g/r_0$ (and vice versa for $270<l<360$ - note that usually $l$ is defined to run from 0 to 360).

So, whether $V$ is positive or negative depends on $V_g(r)$. To a zeroth order approximation, the tangential velocity as a function of radius is roughly constant at Galactocentric radii within a few kpc of the Sun's radius - i.e. $V_g \simeq V_0$. In that case we have a positive velocity, for $0<l<90^{\circ}$, if $r_g<r_0$ and negative if $r_g>r_0$ (and vice-versa for $270<l<360^{\circ}$).

When we look in a direction that is within about 45 degrees of the Galactic centre, it is possible to see gas that is inside the solar circle (which is on "our side" of the Galaxy and has $r_g < r_0$), but also to see gas which is beyond the solar circle on the other side of the Galaxy with $r_g>r_0$. Thus it is possible to see gas with both positive and negative line of sight velocities.

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