How do we measure the velocity of stars in a galaxy?

[Mohammed_I]

I have read somewhere that the parallax method can be used to measure the distance to stars up to 400 light years away. I did a quick calculation and estimated that it means that our telescopes can resolve an angular movement of 8.15x10^-3 arc seconds.

Taking the distance between the sun and the galactic center to be 24,136 light years, and assuming the sun orbits it at a tangential velocity of 220 km/s. That is a total angular movement of 0.31 arc seconds over a 50 year period, easily measurable I assume.

However, if you try to apply the same method to measure the velocity of a star in say the Triangulum Galaxy, it doesn't seem possible. Let's say the distance between us and the galaxy is 2.725x10⁶ light years, and we are trying to measure the velocity of a star 35 arc minutes away from the center (that is 27,744 light years away). And let's also say that this star is orbiting at a tangential velocity of 200 km/s. That is a total displacement of approximately 0.03 light years over a 50 year period, or if we convert it to angular movement from our perspective, 2.52x10^-3 arc seconds over a 50 year period.

So do we have a telescope with this resolution that has been around for 50 years? And even if that is the case, what do we do with galaxies that are further away?

 

[willem2]

We use the doppler effect to measure the radial component of the velocities of stars (away from or towards us). This is much more accurate than detecting the tangential velocity, which does have to be done by seeing if the star moved compared to the background

 

[Mohammed_I]

We use the doppler effect to measure the radial component of the velocities of stars (away from or towards us). This is much more accurate than detecting the tangential velocity, which does have to be done by seeing if the star moved compared to the background

Thank you for your reply. I have thought a little about what you just mentioned and I concluded a few things, please correct me if I am wrong.

Edit: Ops, I hit post reply instead of preview. I'll continue my reply in a new post since I don't want to delete my reply.

 

[Mohammed_I]

Ok so let's assume that $v_{av}$ is the average velocity of the entire galaxy in the radial direction as calculated from the Doppler effect. And let's say that the disc of the galaxy is contained in a plane inclined on the tangential plane (to the line of sight) with an angle $\theta$ which does not equal zero.

Let's say that for any line $a$ that passes through the center of the galaxy $v_a(\rho) = v_d(\rho) \cos{\rho} - v_{av}$, where $\rho$ is the angular distance between the center of the galaxy and any point along that line, $v_d(\rho)$ is the velocity of a star at point $\rho$ as measured from the Doppler effect, and $v_a(\rho)$ is the velocity of that same star relative to the center of the galaxy as projected on to the radial direction to the center of the galaxy.

There should exist a line $a1$ such that $v_{a1}(\rho) = 0$ and another line $a2$ perpendicular to it where the variation in $v_{a2}(\rho)$ is maximum as $\rho$ changes.

The rotational velocity of the galaxy relative to the center should be calculated as $$v_r(\rho) = \frac{v_{a2}(\rho)}{\sin{\theta}}$$

But then the problem will be, how do we measure $\theta$ ?

And there is also another problem with the error estimation, for example, let's say $\theta = 85^{\circ} \pm 1^{\circ}$ the error would be $-0.13\%$ and $+0.16\%$. But if $\theta = 5^{\circ} \pm 1^{\circ}$ the error would be $-16.6\%$ and $+24.9\%$.

 

[Mohammed_I]

Also, please let me know if the calculation in my first post is correct, as I am starting to think that the difference between the proper distance and the light-travel distance should play a part into it.