# Massimo Capaccioli » 7.Galaxy kinematics

### Gauging the kinematics

All the ingredients of a galaxy (stars, gas, dust, DM) must move in order to oppose the self-attraction and prevent the overall collapse of the system. For sake of simplicity, for now we model the galaxy as an ensemble of $N$ particles with mass $m$, velocity $\overlive {\bf v}$ (relative to the barycenter), and rest-frame spectral energy distribution $f(\lambda)$. In the pixel $px(x,y)$ of the galaxy image, they project $n$ particles $\mbox{P}_i,\ (i=1,n)$, all those within the column along the line of sight. Each one appears to possess a radial velocity $v_{0,i}$, or simply $v_i$ once the systemic velocity $v_0$ is subtracted. The latter is the velocity of the galaxy barycenter relative to the observer, usually assumed to coincide with the velocity of the photometric center of the galaxy.

### Rotation and velocity dispersion

Let us call rotational velocity, at the given galaxy image pixel, the mean of all the radial velocities $v_i$ along the line of sight: $v_r=\frac{\sum_{i=1}^n m_iv_i }{M},$

where: $M=\sum_{i=1}^n m_i ,$

and let’s call  velocity dispersion the standard deviation: $\sigma_r=\sqrt{\frac{\sum_{i=1}^n m_i\left(v_i-v_r\right)^2}{M}}.$

These quantities, though having a direct dynamical meaning, cannot be measured in what the galaxy particles are usually not resolved.

### Gauging kinematics

Normally (exceptions are particular dynamical tracers as, e.g., Planetary Nebulae, Globular Clusters, and satellite galaxies), all we can do is to extract rotational velocity and velocity dispersion from the integrated spectrum of each pixel $px(x,y)$ (its size will be likely defined by the resolution of the spectroscopic setup [how?]). Ignoring internal extinction, if any, the assumption is that we can evaluate $v_r$ and $\sigma_r$ weighting by the luminosity instead of the mass [Isn't it the same thing?]: $v_r=\frac{\sum_{i=1}^n f_i(\lambda^*, v_i+v_0)v_i}{L},$ where: $L=\sum_{i=1}^n f_i(\lambda^*, v_i+v_0) ,$ and $\lambda_i^*$  is the center of the (narrow) spectral band (e.g. a spectral line) where we make the observation (we shall see later why it depends on the velocity), and: $\sigma_r=\sqrt{\frac{\sum_{i=1}^n f_i(\lambda^*,v_i+v_0)\left(v_i-v_r\right)^2}{L}}.$

### Gauging the kinematics

How do we measure the above kinematical quantities? By measuring the integrated spectrum $f_t(\lambda)$. This is the sum of the spectra of all the particles properly shifted in wavelength by the Doppler term: $\frac{\Delta\lambda_i}{\lambda_0}= \Delta\ln(\lambda_i)=\frac{v_0+v_i}{c}=z_i ,$ $f_t(\lambda)=\sum_{i=0}^n \frac{1}{1+z_i}f_i\left(\frac{\lambda}{1+z_i}\right).$ Imagine that the rest-frame spectra of all particles have one line narrower than the resolution, thus infinitely narrow. How will it appear to an observer? Shifted according to $v_0+v_r$, broader depending on $\sigma_r$, and more or less deep according to the way it is in the individual spectra.

### Spectroscopy of extended images

Since spectroscopy adds one further dimension, that for the wavelength, and since optical detectors are two-dimensional [what about radio detectors?], it is clear that one exposure can map at maximum one spatial dimension of the two of a projected galaxy image. This is achieved with a (long) slit, which maps one strip of the galaxy image. The slit width is measured in wavelengths (or equivalently km/s [how?]), the length in arc seconds [Note the difference between the spectra of point like sources and extended sources. They may seen alike but the extension of stellar spectra is obtained by trailing the image along the slit]. A galaxy spectrum is then a sequence of monocromatic [but how much such?] images cut by the slit. Modern 3D spectrographs such as the panoramic integral field spectrograph Sauron provide at once a complete datacube, that is 3D matrix made piling up image slices at increasing/decreasing wavelength (velocity). The trick of redistributing a datacube in two dimensions is achieved by sampling the image with lenslet arrays or fiber bundles, or by slicing it with ad hoc optics.

### Basic spectral lines

On long slit spectra the radial velocity $v_0+v_r ,$ can be measured directly in the presence of narrow spectral lines (small $\sigma_r$) as they are  emission lines of spiral galaxies (outside the nucleus, if this is active). [It is instructive to get acquainted with the classical procedure of measuring spectra by eye, following all the steps of wavelength calibration, distortion correction and so on, down to the derivation of the radial velocities].Typical (strong) emission lines in the optical spectra of galaxies, from blue to red, are:

• $\lambda = 3727-3729$ : triplet of [OII], which is a forbidden transition of the single ionization Oxygen [do these lines require a high or a low density environment?]; 1Å at [OII] is $\Delta v \simeq 80\ km/s$.
• $\lambda = 6548-6584$ : two forbidden lines of [NII].
• $\lambda = 6562.8$ : the $H\alpha\$ line, the most intense of the Balmer series (transition from n=3 to 2).

Other Balmer lines may be in emission, down to $H\beta\$; 1Å at [NII] is $\Delta v \simeq 46\ {\it km/s}$. Another very important emission line is in the radio domain, at $\lambda\sim 2\ cm$ or $\nu = 1420.4\ MHz$. The $21\ cm$ line is the spontaneous spin transition of HI, the Hydrogen atom at the ground level. Because of magnetic interactions, the configuration with the electron and proton aligned in the same direction (parallel) is slightly more energetic than the other one (anti-parallel). This transition between these two states is highly forbidden, with an extremely small probability of $2.9\times 10^{-15} s^{-1} \sim 10^{-7}\ yr^{-1}$. The corresponding, extremely long lifetime can be considerably shortened by collisions. At astronomical level the rarity of the event is won by the huge number of potential emitters [estimate the number of events per seconds in a $1\ M_\odot$  cloud of HI].

### Rotation of a flat disk: projection

What would we see by long slit spectroscopy on a thin disk forming an angle $i$ with the line of sight and rotating about its center with a differential velocity ${\bf v}(r)$ ? Being $\bf n$ and $\bf s$ two unit vectors orthogonal to the orbital plane and along the line of sight (see figure), the rotational velocity vector is: ${\bf v}({\bf r})=\Omega(r)\left({\bf n}\times {\bf r}\right) ,$ and the (observed) radial velocity (relative to the barycenter): $v_r(r,\phi) = {\bf v}\cdot{\bf s}= \Omega(r)\left({\bf n}\times{\bf r}\right)\cdot {\bf s}= \Omega(r)\left({\bf s}\times {\bf n}\right)\cdot {\bf r}=$ $=\Omega(r)\left({\bf k}\cdot{\bf r}\right)\sin i= \Omega(r)r\cos\phi\sin i=V(r)\cos\phi\sin i.$ [Repeat the exercise for a disk having a velocity which still depends on radius (axial symmetry) but which is not purely tangential (expanding/contracting disk).]

### Velocity field of a rotating flat disk Application of the formula giving the radial velocity field of a purely rotating disk seen at an angle i. The rotation curve (formula reported in the figure) mimics well enough a real galaxy. The inclination of the example at the lower right is 45°.

### Toy models of rotation velocity curves of thin disks Observed rotation curve vr(R) of a thin disk in rigin (top) and keplerian (bottom) rotation. R is the apparent distance from the center of the galaxy image along the kinematical major axis.

### Rotation of a flat disk: expectation

What is the shape of the rotation curves of real spirals? Up to the ’70s, the expectation, apparently supported by the observations, was a combination of an inner rigid rotation with an outer Keplerian rotation. The curve $v_r(R)$ was expected to raise up to a maximum and then descend steeply as $R^{-1/2}$. The turning point would demark the range where the mass of the galaxy was no longer increasing significantly with $R$ [try to understand why this is so, why the rotation curve is then called keplerian, and ask yourself if the explanation you found for this expectation bears no assumptions of the relation between surface brightness and surface density of matter]. Actually, since 1975 ca. the deep and accurate rotation curves produced by radio astronomers using the 21 cm hydrogen at the fundamental state (spin transition) have disclosed a different scenario. There is no Keplerian decline; the rotation curve raises up and than remains roughly flat. We shall see later why this is an indication of the existence of the Dark Matter.

### Flat rotation curves of spiral galaxies Rotation curves of 25 galaxies of various types, including our own Galaxy, from optical measurements of emission lines and radio measurements of the HI line at 21 cm (from A. Bosma, Ph. Thesis, Gröningen Univ., 1978).

### Shape of a spectral line in an unresolved disk

What is the shape of a spectral line from an unresolved galaxy disk? The question is far from being academic since lack of spatial resolution, typical of former radio observations, still affects very distant objects.

Let us consider the velocity field of a tilted disk with a rotation curve $v_r(R)$ mimicking a real spiral; for instance, a flat rotation curve (see figure).

Now, compute the histogram of the velocities. It will be clearly symmetric, with two peaks separated by twice the maximum rotational velocity (see figure).

In wavelengths: $\Delta\lambda(\mbox{two peaks})\simeq 2\lambda\times v_r(\mbox{max})/c.$

To make a more realistic line shape, one shall weight the velocities by their surface brightness and add some noise. The symmetry is less perfect and the peaks less pronounced, but the conclusion remains the same.

### Shape of a spectral line in an unresolved disk

Let us compute the shape of a spectral line of an unresolved exponential disk using a rotation model: $\displaystyle{ V_{\rm rot} = \frac{100 r}{r+r_0}},$

shown in the figure for two values of $r_0$: black is for $r_0=1$ and gray is for $r_0=10$, in arbitrary units. The models flatten to $V_{\rm rot} = 200$ km/s for $r \gg r_0$.

In the next page we reproduce the simulated velocity distributions for our exponential thin disk seen edge-on, chosen to have a disk scale length $h=10$ (arbitrary units as above) and characteristic radius of the rotation law $r_0=1$ (top-left in the figure) and $r_0=10$ (top-right).

At the bottom there are the same velocity distributions as they would be measured with an instrumental error of $20\ km/s$. The measurement errors broaden the whole velocity distribution, an effect which is more evident for the two peak which are used as a rough estimate of the maximum velocity of the disk $(V_{\rm max}=\Delta V_{\rm peak}/2)$. In both cases the last measured data-point is at $r=100$.

### Gauging kinematics form broad lines

If the spectral lines are broad (larger than the instrumental widening), it is hard and inaccurate to measure directly the rotational velocity. In any case it is necessary to estimate also the velocity dispersion. Both quantities are obtained simultaneously by comparing the galaxy spectrum with a template spectrum of a stellar (like) object with narrow [how narrow?] lines and known redshift, properly broadened to match the dispersion, shifted to mach the radial velocity, and rescaled to match the depth of the spectral features in the band where the comparison is made. Let us assume that the galaxy spectrum $G(\ln\lambda) = f_t$ [notice the use of the logarithm and explain why] is the convolution of a template spectrum $S(\ln\lambda)$( for instance that of a star matching the average spectral type of the galaxy) with a broadening function $B(\ln\lambda)$: $G(\ln\lambda) = S(\ln\lambda) \otimes B(\ln\lambda) = \int_{-\infty}^{+\infty}S(\ln\lambda^\prime)B(\ln\lambda-\ln\lambda^\prime)d\ln\lambda^\prime. \hspace{2cm}(1)$

Reasons of convenience induce us to model the broadening function with a Gaussian with a velocity dispersion $\sigma_\lambda,$ centered at $v_0^\prime$ (mean velocity of the galaxy pixel relatively to the velocity of the template), and with an intensity $\gamma$ (which is the ratio of the line strengths in the galaxy and in the star). Replacing the wavelength with the velocity thought the Doppler formula $\ln\lambda=v/c,$  the broadening function writes: $B(v) = \frac{\gamma}{\sqrt{2\pi\sigma^2}}\mbox{exp}\left[\frac{\left(v-v_0^\prime\right)^2}{2\sigma^2}\right] .$

### Gauging kinematics form broad lines

You can solve equation (1) by chasing the values of $v_0, \sigma,$ and $\gamma$ which provide the best value of some optimization estimator, e.g. $\chi^2$. For instance, one might explore the entire parameters volume, searching for the minimum of the estimator [why not?]. Actually you can do better by the procedure described by Rix and White. Alternatively, you may exploit the Convolution theorem [revisit the properties of the Fourier transform]: $G= S\otimes B \hspace{1cm}\rightarrow\hspace{1cm} \widetilde{G} =\widetilde{S}\cdot\widetilde{B},$

where the upper tilde indicates the Fourier transformations of the corresponding functions.

The Fourier Quotient method is based then on the assumptions that: $\frac{\widetilde{G}(k)}{\widetilde{S}(k)}= \widetilde{B}(k),$ $B(k) = \frac{\gamma}{\sqrt{2\pi\sigma^2}} \mbox{exp}\left(-2\pi\sigma^2k^2+i2\pi v_0 k \right).$ This method appears far simpler than the one above, but it is not so.

The ratio $\widetilde{G}/\widetilde{S}$ is easy to compute but the result is useless because of the noise [discuss why quotients are very sensitive to noise].

### Anomalies

Usually rotation curves are symmetrically placed about the center. You may fold them by changing the sign to both radius and velocity on one of the two sides opposite to the center (this is a good way to evaluate the systemic velocity).

Apart from the effects of the noise, the folded rotation curves are expected to possess only non-negative velocities. [Why? Think to the angular momentum vector.]

Actually this is not always true, and the deviation we are going to describe calls for an explanation. In some galaxies it has been found that the overall spin vector may change its direction in the central regions, up to the point of being oriented in the opposite direction. These conter-rotating cores manifest a relevant kinematical decoupling with the rest of the galaxy. As we shall see later, they may be the remnants of a merging event where the angular momentum of the victim has not yet been digested with an increase of the velocity dispersion [why so?].

Other kinematical anomalies concern the presence of a nuclear black hole causing the formation of an Active Galactic Nucleus (AGN) or simply causing a very steep turn up of the velocity dispersion: a phenomenon discovered before the launching of HST but tackled mostly by the very high spatial resolution spectra produced by this telescope in space.

### Counter-rotating cores An example of a kinematically decoupled core (NGC 3593) compared to a rather normal S0 (NGC 4111), characterized by a steep raise of the central velocity. Credits: spectra from K. Kuijken, images from SDSS.

### Hidden counter-rotation in the core of NGC 4374

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