# Massimo Capaccioli » 10.Scale relations

### Applying the Virial Theorem

The Virial Theorem [recap!] states that a virialized galaxy satisfies (on average) the equation: $2T+U = 0$,

where: $T = \frac{1}{2} \sum_{i=1}^N m_i v_i^2 = \frac{1}{2} M \langle v^2\rangle,$

is the kinetic energy, and $U = \displaystyle{-\frac{1}{2}G \sum_{i=1}^N \sum_{j=1}^N \frac{m_i m_j}{r_{ij}}= G\frac{M^2}{\langle r\rangle}},$

is the potential energy. $M$ is the total mass, and $\langle r \rangle$  and $\langle v^2 \rangle$ are some average radius and squared velocity.

### Applying the Virial Theorem

In order to replace with observables the unknown average quantities, we make the following positions:

• $R= k_R\langle r \rangle,$  where $R$ is some sort of photometric radius (e.g. de Vaucouleurs’ effective radius) and the parameter $k_R$ reflects the structural properties of the object (e.g. the trend of the density with the radius);
• $V^2= k_V\langle v^2 \rangle,$  where $V^2$ is some sort of operationally defined squared velocity (for instance, the square of the maximum rotational velocity in a spiral galaxy, or of the velocity dispersion in an early type) and the parameter $k_V$ reflects the kinematical structure of the object (relative weight of disordered over ordered motions, anisotropies, etc.).

We further place: $L= k_L I R^2$, where $L$ is the total luminosity, $I$ is an operationally defined surface brightness.

The parameter $k_L$ reflects the luminosity structure of the object [not necessarily that in density] in the color in which $L$ is given ( $k_L$ is the same for galaxies with the same photometric trend).

For instance, $k_L= 22.68$ for a $R^{1/4}$ galaxy if radius and surface brightness are those at the effective isophote.

### Applying the Virial Theorem: scaling laws

Replacing the above positions in: $\displaystyle{ \langle V^2\rangle =G\frac{M}{\langle R\rangle}}$,  one gets: $\displaystyle{ \frac{V^2}{k_V}=Gk_R\frac{M}{R}=Gk_R\frac{\left(M/L \right)L}{R}=Gk_R\frac{\left(M/L \right)k_LIR^2}{R}=Gk_R\left(\frac{M}{L}\right)k_LIR},$

or: $\displaystyle{ R=k_{SR}V^2\left( \frac{M}{L}\right)^{-1}I^{-1}},$

where: $\displaystyle{ k_{SR}=\frac{1}{Gk_Rk_Vk_L}},$

or, by squaring: $\displaystyle{ \frac{V^4}{k_V^2}=G^2k_R^2\frac{M^2}{R^2} =G^2k_R^2\frac{\left(M/L\right)^2L^2}{R^2}= G^2k_R^2\frac{\left(M/L\right)^2k_LIR^2L}{R^2}= G^2k_R^2k_L\left(\frac{M}{L}\right)^2 IL},$

that is: $\displaystyle{ L=k_{SL}V^4\left(\frac{M}{L}\right)^{-2}I^{-1},$  where: $\displaystyle{ k_{SL}=\frac{1}{G^2k_R^2k_V^2k_L}}}.$

### The luminosity-velocity relation

By the only assumption of virialization, we have obtained two relations which can be used to gauge distances. In fact, they relate distance independent observable (a surface brightness and a velocity) with other observables which depend on distance (a characteristic radius or the total luminosity[remember that we measure angles rather than distances, and fluxes].

We may attempt to rewrite the relation between luminosity and velocity as: $L\propto V^{\beta},$ assuming (to be demonstrated, though) that: $\displaystyle{k_{SL} \left(\frac{M}{L}\right)^{-2}I^{-1} \propto L^\gamma.$

If so: $L\propto V^4 L^\gamma$, from where: $\beta=4/\left (1-\gamma\right).$

In magnitudes it is: $-2.5\log L= M=-2.5\times\beta\log V.$

Note that, if this relation holds for absolute magnitudes $M,$ it holds as well for apparent magnitudes $m$ provided that, in the second case, it is applied to galaxies all at the same distance, as it is for objects belonging to the same cluster [but remember the problem of the membership]. In this case, in fact, the distance modulus, $m-M$, is constant.

### The luminosity-velocity relation

The above considerations show that the relation between luminosity and velocity may be an excellent distance indicator, provided that it is applied to clusters of objects obeying the same relation (same slope and same zero point), in the way shown in the figure. There we have plotted the luminosity-velocity relation for two clusters with distance moduli differing by $\Delta m.$

### The Tully-Fisher relation

The simplest assumptions relative to the luminosity-velocity relations were originally used by the Estonian pioneer astronomer E. Öpik to derive, as early as 1922 (Ap.J., 55, 406), a distance of $450\ kpc$ for the Andromeda nebula (a measure preceding the seminal work by Hubble and closer to the modern value of $780\ kpc$).

In 1977 (AA, 54, 661) B. Tully and R. Fisher applied the relation to Local Group spirals (with known distances and thus absolute luminosities).
To estimate the velocity $V$ they used global HI profile widths $W$ (corrected to edge-on view [how?]) and observed that these velocities obey a relation $L_{pg}\propto W^{\beta}$ with $\beta = 2.5 \pm 0.3.$
Comparison with similar measurements for objects of the M81 and M101 groups [remember that galaxies in groups or clusters share about the same distance from the observer] and in the Virgo cluster allowed them to estimate the distances of these systems and the value of the Hubble constant.
They found $H_0 = 80\ {km\ s}^{-1} {Mpc}^{-1}.$ Not bad!

### The Tully-Fisher relation (TFR)

Let us consider the simplest form for the Tully-Fisher relation (TFR): $-2.5\log L_\lambda =M_\lambda = a+b\log W.$

Here the subscript to the absolute magnitude  M reminds us the dependence on the observing band. The relation applies well to spiral galaxies. As we said, $W$ (from “width”, since the first measurements were integrated HI line widths) is a kinematical estimator representing the velocity $V$. Since Doppler effect measures only the radial component of the velocity, $W$ must be corrected for the inclination $i$ of the galaxy main plane to the line of sight. Other corrections [which?] are in order for the photometric measurements. Clearly, both $a$ and $b$ depend on the adopted photometric band. It is found that the slope increases with the wavelength while the dispersion decreases. Also, we may speculate (correctly) that $b$ depends on the morphological class (indeed S0s and early spirals have a smaller slope of the TFR than later types [guess why]). The most intriguing question is whether it epends from redshift (that is, on look-back time). It is so: galaxies at $z\simeq 0.5$ are 1.5 mag brighter that objects with the same morphology and the same $V$. This observation shows that the TFR can be used to gauge distances or to test cosmic evolution.

At the RHS the modern TF relation for 5 galaxy clusters, all reduced at the same distance.

### The Faber-Jackson relation (FJR)

The same scaling law (Tully-Fisher) applying to galaxies with ordered motions (rotation) holds for those dominated by disordered motions (velocity dispersion $\sigma$) as they are early-type galaxies: $-2.5\log L_\lambda =M_\lambda = a^\prime +b^\prime\log\sigma$.

The validity of this relation was first proved by S. Faber and Jackson (Ap.J., 204, 668, 1976) using the central velocity dispersion. Recent studies on SDSS data have shown that the parameters of the FJ relation depend on the interval of magnitudes considered for the sample: both size and mean value.

The figure shows the FJ relation for a sample of elliptical galaxies and bulges of spirals covering a large range in luminosity.

### The 7 Samurai

Dressler et al. (1987, Ap.J., 313, 42), self-nicknamed the “Seven Samurai”, used a modified version of the FJR to derive a powerful distance indicator for elliptical galaxies, which allowed them to estimate in 150 to 350 km/s the infall velocity of the Local Group towards the Virgo Cluster.

To this end they defined a new photometric parameter $D_n,$ as that of the circular aperture within which the mean total surface brightness assumes an assigned value $\Sigma$. In particular, for the blue band they adopted: $\Sigma = 20.75\ \mbox{B-}mag\ arcsec^{-2}$.

The reason is that, for nearby galaxies, the value corresponds to apertures large enough to absorb seeing blurring and off-centering errors [expand], situated within a range rich of photoelectric aperture photometry (photoelectrical fluxes measured through centered circular diagrams). With this choice $D_n$ is calculated by interpolation.

The Seven Samurai showed that $D_n$ correlates well with just $\log\sigma$.

### The Fundamental Plane

Let us reconsider the relation: $\displaystyle{ R=k_{SR}V^2\left( \frac{M}{L}\right)^{-1}I^{-1}},$

where: $\displaystyle{ k_{SR}=\frac{1}{Gk_Rk_Vk_L}}$

is a constant. For ellipticals (and bulges of spirals), using the central velocity dispersion $\sigma_0$, the de Vaucouleurs’ effective radius $R_e$, and the average surface brightness within $R_e$: $\langle I\rangle_e=L/2\pi R_e^2,$ it writes as: $\displaystyle{ R_e \propto \sigma_0^2\left( \frac{M}{L}\right)^{-1}\langle I\rangle_e^{-1}}.$

### The Fundamental Plane

That is, if E galaxies obey the prescriptions of the model, i.e. they are in Virial equilibrium, they form an homologous photometric family, e.g. they all follow $R^{1/4}$ profiles, and $M/L$ is a regular function of $L$, and if the mean value of the mass-to-light ratio does not vary from galaxy to galaxy, the effective radius, the mean effective surface brightness, and the central velocity dispersion define a surface, which we shall call Fundamental Plane (Djorgovski and Davis, 1987).

### The Tilt of the Fundamental Plane

Actually real galaxies do follow this kind of behaviors. For instance, Virgo galaxies obey the relation (Dressler et al., 1987): $R_e \propto \sigma_0^{1.4}\langle I\rangle_e^{-0.85},$

within a small dispersion. The exponents, however, differ from those predicted by the theory. The departure is called “tilt“. The theoretical prediction is completely rescued if one assumes that the mass-to-light ratio varies with the total luminosity as: $M/L\propto L^{0.25}$, as one can easily verify by simply substituting this expression into the relation: $\displaystyle{ R_e \propto \sigma_0^2\left( \frac{M}{L}\right)^{-1}\langle I\rangle_e^{-1}}.$

It is apparent that the Faber-Jackson relation is just the projection of the Fundamental plane onto the dynamical versus the photometric axes. The other projection, onto the $\langle I\rangle_e\ (\equiv\mu_e)$ versus $R_e$ plane, that we have already seen, is not as straight, having a different behavior for normal and bright galaxies.

Is the Fundamental Plane really fundamental? The question is still open.

### The k space

Bender et al. (Ap.J., 399, 380, 1992) have introduced the $k$ coordinate system obtained by combining linearly the observables originally defining the Fundamental Plane: $k_1 = \left(\log\sigma_0^2 + \log R_e\right)/\sqrt{2}\propto \log M ,$ $k_2 = \left(\log\sigma_0^2 + 2\log\langle I\rangle_e - \log R_e\right)/\sqrt{6} \propto \log\left[(M/L)\langle I\rangle_e^3\right],$ $k_3 = \left(\log\sigma_0^2 - \log\langle I\rangle_e - \log R_e\right)/\sqrt{3} \propto \log(M/L).$

Since, as we saw, the dependence on $M/L$ is weak, this transformation sets the Fundamental Plane almost perpendicular to the $k_3$ axis.

Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion

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