Unless otherwise stated, in this Chapter we will make the following assumptions. For sake of simplicity we will consider galaxies with purely circular, concentric isophotes. More complex and more realistic shapes will be introduced later. Thus, the only coordinate is the radius , in units of seconds of arc * [how do you translate this angular into linear units?]*.

Note that any galaxy can be circularized by building circular models where, at the same surface brightness level, the isophotes in the real object and in the galaxy have the same area. This is simply done by measuring the area of an isophote of the real galaxy from where the radius of the corresponding circular isophote in the model is computed as: .

The star added to the symbol for the radius serves to remember that this quantity is relative to the circular model, that is that it is an “equivalent” radius.

Given the small angular dimension of galaxy images (usually not larger the a few minutes of arc), they are taken to place on a plane rather than on the sphere, as it is the sky * [why?]*. This prevents us from wild extrapolations of

We shall first comment on the data reduction. We will then present the empirical formulae developed to represent the light profiles of early-type galaxies, then those for spiral disks. The rest of the lecture is devoted to present some parameters used nowadays in studies of very large sample of galaxies.

The tremendous developments of observational equipment has produced, since the last few decades of 1900, extraordinary data-bases with millions of galaxies which must be treated and analyzed by automatic tools and statistical methods. This flow of data has been a genuine revolution in a field where, no more than 30 years ago, a paper would deal usually with no more than a few tens of galaxies at the time.

Two papers with the photometric techniques and the results of a study of the standard E galaxy NGC 3379. Credit: *The Astrophysical Journal*.

The first empirical law describing the trend of the surface brightness in spheroidal galaxies was devised by the British astronomer John H. Reynolds (1874-1949) and then by Hubble (*Ap.J.*, 71, 231, 1930). The normalized Reynolds law writes as:

There are no free parameters, since and are just scale parameters * [explain]*. is the central brightness and is the radius at which the surface brightness is diminished by a factor of . The surface intensity is almost constant for thus well mimicking the behavior of the atmospheric blurring (seeing) as we shall see later, and it runs as for

The very center * [but does a very center make sense if one relaxes the assumption that a galaxy is a continuous system?]* is discontinuous (spiky):

The luminosity at radius is:

As we can see, it diverges as increases. This does not mean, though, that tends to infinity with , since the radius cannot grow indefinitely, as we have said in listing our assumptions.

In 1955 (*PASP*, 83, 199) the Reynolds-Hubble formula was reviewed by William A. Baum in order to match more spiky nuclei. The normalized Baum law writes as:

Again there are no free parameters, since and are scale parameters.

Now the surface intensity raises steadily as decreases:

.

Outwards it continues to run as .

At the very center the surface brightness is discontinuous. Actually it diverges for , as the luminosity does* [solve the integral analytically]* when tends to infinity (with the standard caveat on the limits for the extrapolation of the radius).

In 1976 G. Oemler *(Ap.J.*, 209, 693) proposed an implementation to the Reynolds-Hubble’s formula to account for the occurrence of the so-called tidal truncation of the light profile. To this end he introduced an exponential factor and a free parameter, , where stays for tidal:

In this way, for it is:

,

while for the surface brightness vanishes: .

For one gets the standard Reynolds-Hubble formula.

The breakthrough in the field came with the thesis work of Gérard de Vaucouleurs (*Ann. d’Ap*, 11, 247, 1948), who introduced the following formula, often referred to as the *R*^{1/4 }law:

It has two scale factors, and , and, as we shall see, no free parameters.

In fact, can be fixed by properly choosing the scales for and . To this end, let us define an “effective radius” as the isophote radius that encircling half of the total light of the galaxy.

We can do this since, as it will be clear later, a de Vaucouleurs model has a finite luminosity, and since we are considering circular (circularized) galaxies only.

Remember that, when dealing with real galaxies with elongated or irregular isophotes, the effective radius of the circular model just introduced would be written as to remind us that it is not only effective but also equivalent.

By definition:

Placing:

,

we obtain:

,

from where:

.

is the effective surface brightness, ~3.33 times fainter than the peak brightness.

Left: the R1/4 profile of NGC 3379. The (O-C) residuals (right-top) show the presence of a nuclear component in excess of the R1/4 law.

In terms of *R*_{e}, the de Vaucouleurs formula writes: The (formal) total luminosity (again we extrapolate to infinity) converges. It is:

From: , we calculate the mean surface intensity within the effective radius:

* [Translate this result in units of mag/arcsec ^{2}.]*

The relation between effective and peak intensities is: which, in magnitudes, reads:

In conclusion, the magnitude difference between peak and effective intensities is constant:

Assume that you have obtained the light profile of a (circular) galaxy by sampling the surface brightness at a set of radial distances from the center of the object. How do you verify whether the profile follows a de Vaucouleurs law and, in any case, how do you determine the values of the scale parameters? The answer to the first question is simple. You check if the surface brightness data versus (no matter if not zero-point calibrated yet) define a straight line. You then extrapolate the best-fitting linear profile to * [how do you do the best fit?]*, so to find and . The abscissa of is .

**[As you see, you have measured the fourth root of the quantity that you want. Does this affect the accuracy of your final result? Think about all sources of errors in the above procedure.]**

The photometric profile of the normal E NGC 3379, plotted in the de Vaucouleurs plane, is well fitted by the straight line. Its extrapolation to zero gives the starting point to derive the scale parameters.

In 1968, while preparing his Atlas de Galaxias Australes (Cordoba Obs.), José Luis Sersic (1933-1993) realized that the de Vaucouleurs law could be profitably generalized by letting the exponent be a free parameter :

The idea came to him from the fact that with just one parametric formula he was able to represent the trends of the light profiles of ellipticals and spiral disks , as we shall see soon. So, Sersic speculated that the parameter , now called *Sersic parameter *after his name, could typically assume any value between 1 and 4.

According to Occam’s principle: “*entia non sunt moltiplicanda praeter necessitatem*“; that is, never add a parameter without a very good reason * [explain the validity of this statement by thinking at polynomial fittings of observational data]*. At Sersic’s time there were no reasons to accept his formula. The reason was found 25 years later by Caon et al. (see next two pages), who discovered a correlation between the size, or the luminosity, of a galaxy (early types and spiral bulges) and the value of the Sersic parameter given by the best fit of its luminosity profile with the Sersic formula.

Discovery paper of the correlation between Sersic parameter and the size/luminosity of galaxies. Credit: *MNRAS*.

In 1968 (*A.J.,* 71, 276), inspired by the results of his studies of quasi-isothermal stellar systems, Ivan King wrote the following empirical formula for the light profiles of early-type galaxies:

, named *core radius*, is a scale factor, while the so-called tidal radius is a free parameter; the smaller it is, the steeper is the surface brightness decay in the outer envelope. Note that, but for , the profiles are truncated. Note also that, for very large tidal radii, the inner profile mimics the behavior of seeing convolution * [see lecture no. 23 about seeing]*: the inner profile is in excess of the inward extrapolation of the outer trend.

Let us now derive the space density of light, *j*(*r*), from the project light profile, *I*(*R*), in the case of a spherically symmetric, transparent galaxy.

From the geometrical relation: one gets:

The integral of the light density along the line of sight is:

assuming simmetry of .

Being an integral equation with Abelian kernel, it can be inverted:

Applying this result to Reynolds-Hubble’s profile:

we get:

Asymptotically, *i.e.* for , it is: and . As a general rule, if at large radii the light density decreases as a power law with exponent , also the surface brightness profile does, with an exponent .

In dynamical modeling it is much more convenient to use the space density profile rather than the projected light distribution. For this purpose quite a few formulae have been proposed.

Some of them are generated by the following general formula:

where is the total luminosity and is a characteristic length.

By varying the parameter we obtain the different formulae:

- gives the Hernquist (1990) formula;
- gives a trend very similar to the deprojection of a de Vaucouleurs surface brightness profile;
- gives the Jaffe (1983) formula.

While reducing high resolution HST images, Tod Lauer et al. (*A.J*., 110, 2622, 1995) speculated on whether the excess light noted above King’s flat cores, there where light profiles have a break, had to be considered as independent subsystems. Thus they developed the so called “nuker” formula which fits well the profiles over a wide range including the center, keeping the notion of the break as physical feature characterized by its radius and the corresponding surface brightness :

For matching the outer power law;

for describing the cusp;

controls the smoothness of the transition between the cuspy core and the outer profile.

The Nuker formula fails convergence into a finite total luminosity for .

The Nuker formula gives excellent fits; the price to pay is the introduction of 3 free parameters (but remember Hoccam’s rasor:* entia non sunt moltiplicanda praeter necessitate*, and the need in this case must be physical). It accounts for the various classes of cores recently discussed by Kormendy *et al. *(2009) for a complete sample of Virgo Cluster galaxies:

- cores tend to flatten up towards the center in more luminous Es

They are explained by “dry mergers” (no gas), where the accreted gas is rubbed out by a binary BH; - they rise steeply toward the center in midsize Es The excess light may come from nuclear starburst resulting from “wet mergers” (that is with gas).

Correlation of the Nuker formula parameters and of the central velocity dispersion with the total magnitude of ETs and bulges of S0-Sb. HST data (Faber et al., *Ap.J.*, 114, 1771, 1999).

The Petrosian radius (*Ap.J.*, 209, L1, 1976) is defined through the ratio of the mean luminosity within the ring of radii and , and the mean luminosity within the aperture of radius :

If we assign a value to the ratio and to a constant , the Petrosian radius is the value of at which . For instance, for the SDSS: , and . Why the Petrosian radius is so popular in modern galaxy surveys? The reason is that does not depend on the photometric scale parameter of the light profile: *i.e.* it is the same for two galaxies differing only for the mean surface brightness, while it depends on the shape of the light profile. Note that for , for .

On average early-type galaxies are red: the rest-frame (B-V) color ranges from 0.8 to 0.9. The color is due to a population of aged stars (young stars have already completed their cycle). The color gradients are negative, more and more as the mean color reddens (a trend which is weaker in dense environments, *e.g.* rich clusters). In true ellipticals the gradient slowly flattens up with increasing total luminosity. Why it is so, it is still matter of debate since several concurring causes are in play: plain stellar evolution, metallicity gradients, merging (but when?) and (consequent) AGN activity. Bulges are bluer than E galaxies (even at the same luminosity), indicating that they are younger and or more metal poor than ellipticals. Most bulges do not reach solar metallicity. Bulges also show negative color gradients (bluer outwards). Rest-frame colors of ellipticals change with redshift (distance) intrinsically (evolution). This property, as well as the new classification of galaxies based on SDSS and GALEX data, which is driven by the color versus luminosity (mass) diagram (the red and blue sequences) will be examined later.

*1*. Introduction to the realm of nebulae

*3*. Photometry of early-type galaxies

*4*. Photometry of late-type galaxies

*5*. Apparent and true flattening of galaxies

*6*. Properties of elliptical galaxies

*8*. Spiral arms

*9*. Origin and stability of spiral arms

*10*. Scale relations

*12*. Cosmic distance scale - Part II

*13*. Cosmic distance scale - Part III

*14*. Galaxy dynamics

*16*. Galaxy dynamics - Part III

*18*. Stellar populations in galaxies

*19*. Galaxy clusters

M. Capaccioli, "The de Vaucouleurs and the r^{1/4} law", 1989, article in *The de Vaucouleurs: a Life for Astronomy*, M. Capaccioli and H.G. Corwin Jr. (eds.), Adv. Series in *Astrophysics and Cosmology*, Vol. 4, World Scient. Press: Singapore, p. 173.

M. Capaccioli and G. de Vaucouleurs and , "Luminosity distribution in galaxies. II. A study of accidental and systematic errors with applications to NGC 3389", 1983, *Ap.J.Supp.*, 52, 465 (with G.de Vaucouleurs).

M. Capaccioli, "Photometric properties of galaxies", 1989,* Second Extragalactic Astronomy Regional Meeting* (Cordoba, Argentina), Academia de Ciencias: Cordoba, 317.

G. de Vaucouleurs and M. Capaccioli, "Luminosity distribution in galaxies. I. The elliptical galaxy NGC 3379 as a luminosity distribution standard", 1979, *Ap.J.Supp.*, 40, 699 (with G.de Vaucouleurs).

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