How is the Rotational Velocity of Galaxies Measured?

[ChrisVer]

I was wondering, how can we measure the rotational velocity of a galaxy?
In practice knowing the mass distributions and so on, we could calculate it by classical mechanics (or maybe GR). However people measured the rotational velocity of the galaxies and found that it doesn't correspond to the expected curve (DM or MOND). How did they do it? In a telescope I have the feeling that a galaxy appears as a single dot -1 dimensional object (impossible to see whether it's rotating or not).

 

[Orodruin]

Doppler measurements relating red/blue shift of stars moving near the galactic plane. Naturally, your instrument must have enough resolution to perform such measurements.

 

[Matterwave]

The answer is given by Orodruin, but I want to amend something to it. In fact our telescopes (the ones that astronomers use) are of sufficient resolution such that quite many galaxies will actually be resolved. Have you seen the pictures by HST? Many of those show explicitly that we can see spiral galaxies, or others etc.

I think you are thinking of stars when you think that a galaxy will simply appear as a dot. It is far harder to resolve a star in our own galaxy than it is to resolve a nearby-ish galaxy. We can look at the angular resolutions needed. Let's take a typical star like our Sun and put it at the distance to our nearest neighbor star, Alpha Centauri, which we will just say is 4 light years. The angle subtended by this star is roughly (in radians) $\theta\approx R_{sun}/(4 ly)\approx2\times 10^{-8}$ this is ~.002 arc seconds. A typical (non-dwarf) galaxy is ~100,000 light years across, and let's place it at a distance of perhaps 100 million light years (this is ~10 times farther out than our local group). This gives an angle subtended of $\theta\approx 100000ly/100000000ly\approx 10^{-3}$ which is ~4 arc minutes.

As you can see, it is FAR easier to resolve a nearby galaxy than it is to resolve a nearby star.

In addition, even if the object appears at a dot (unresolved), we can STILL figure out something about its rotation by the spread of the emission/absorption lines that we see in the spectra. Because part of the object will be going towards you, and part of the object will be going away from you, this will produce a spread in the usually quite sharp emission and absorption lines. So, by using spectroscopy, we can in fact figure out rotation of an object, even if we can not resolve that object. We do this quite often for stars actually.

Of course, this will depend on the angle at which we are viewing these objects! So if you are unable to resolve the object to confirm that you are looking at it perpendicular to the axis of rotation, you can only use this as a lower bound on the rotation rate.

 

[connorp]

Doppler measurements relating red/blue shift of stars moving near the galactic plane. Naturally, your instrument must have enough resolution to perform such measurements.

Yes, this. I've actually done it before with radio telescopes. My team collected Doppler shifts from neutral hydrogen clouds from a section of the galactic plane, calculated distances and orbital speeds, and plotted those. Fascinating stuff.