<p>When looking at say an elliptical galaxy, the line of sight velocity dispersion is usually measured from the spectra of very many unresolved stars. The line of sight velocity dispersion is calculated from the broadening of the summed stellar line profiles after taking account of the instrumental broadening. Crudely, the dispersion you are after is  <span class="math-container">$\sqrt{\sigma_L^2 - \sigma_I^2}$</span>, where <span class="math-container">$\sigma_L$</span> is the measured line standard deviation and <span class="math-container">$\sigma_I$</span> is that component due to the finite spectral resolution of the instrument. The <em>intrinsic</em> line widths of the individual stars contributing, of the order 1-2 km/s for most stars, can usually be ignored.</p>
<p>On the other hand, if looking at say stars in a Galactic globular cluster, you can measure individual (radial) velocities from their spectral. In which case, the line of sight velocity dispersion is the standard deviation of the individual velocities. The line widths due to instrumental and intrinsic broadening are not used in the calculation and are unimportant other than as a source of noise in the individual velocity measurements.</p>

CC BY-SA 4.0
2024-09-04T14:31:21.030Z
Thank you! Would there be a couple of such broadened lines (sums of multiple stars) in an elliptic galaxy? Would a broadened line typically be a Gaussian as it is the sum of multiple stars absorption lines? And finally - if you have a reference with real spectra and a corresponding calculation I would be happy to see.
Goose
CC BY-SA 4.0
2024-09-03T06:08:36.727Z
2024-09-03T06:08:36.727Z
ProfRob
<p>Velocity dispersion of stars in a galaxy could be useful for example in calculating the galaxy's mass, using the viral theorem.</p>
<p>I am wondering how this velocity dispersion (standard deviation in velocities) is usually calculated. Are doppler shifts of individual stars being measured (and is it the case that usually the spectral lines of individual stars can be resolved)?
If so, I imagine that each spectral line of a star would have some width. Would we take the doppler shift of each star to be the velocity corresponding to the center of the line? Or are the widths of individual stars taken into account?</p>

<observational-astronomy><galaxy><spectroscopy><radial-velocity><dispersion>
Velocity dispersion of stars in galaxies
Goose
# Velocity dispersion of stars in galaxies

(Asked by: username_0 on 2024-09-03)

Velocity dispersion of stars in a galaxy could be useful for example in calculating the galaxy's mass, using the viral theorem.

I am wondering how this velocity dispersion (standard deviation in velocities) is usually calculated. Are doppler shifts of individual stars being measured (and is it the case that usually the spectral lines of individual stars can be resolved)?
If so, I imagine that each spectral line of a star would have some width. Would we take the doppler shift of each star to be the velocity corresponding to the center of the line? Or are the widths of individual stars taken into account?

**Tags:** observational-astronomy, galaxy, spectroscopy, radial-velocity, dispersion

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## 1 answers

### Answer by username_1 on 2024-09-03. Score: 8 | Accepted answer

When looking at say an elliptical galaxy, the line of sight velocity dispersion is usually measured from the spectra of very many unresolved stars. The line of sight velocity dispersion is calculated from the broadening of the summed stellar line profiles after taking account of the instrumental broadening. Crudely, the dispersion you are after is $\sqrt{\sigma\_L^2 - \sigma\_I^2}$, where $\sigma\_L$ is the measured line standard deviation and $\sigma\_I$ is that component due to the finite spectral resolution of the instrument. The *intrinsic* line widths of the individual stars contributing, of the order 1-2 km/s for most stars, can usually be ignored.

On the other hand, if looking at say stars in a Galactic globular cluster, you can measure individual (radial) velocities from their spectral. In which case, the line of sight velocity dispersion is the standard deviation of the individual velocities. The line widths due to instrumental and intrinsic broadening are not used in the calculation and are unimportant other than as a source of noise in the individual velocity measurements.

> **Comments:**
> **username_0**: Thank you! Would there be a couple of such broadened lines (sums of multiple stars) in an elliptic galaxy? Would a broadened line typically be a Gaussian as it is the sum of multiple stars absorption lines? And finally - if you have a reference with real spectra and a corresponding calculation I would be happy to see.  

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