Some Musings on Galaxy Image Decomposition

(Or… things you must really know!)

As a way to clarify my own thoughts I decided to create this page, but I am sure it is going to be useful to many… This is organized in different parts. First, I recall the main functions that describe the surface brightness profiles of the galaxy components that are present, e.g., in BUDDA, i.e., their mathematical and graphical description. After that, I recall what boxy and disky isophotes mean in reality, as well as other interesting configurations that can be studied using BUDDA, like, for instance: barred galaxies, double-exponential disks, edge-on galaxies…I will end some day with some other hints specifically to someone who wants to run BUDDA, although many are already described in the “How to use it” section (but not again here).

THE EXPONENTIAL FUNCTION. Used to describe galactic disks, the simplest of all formulae here. Written in linear units (e.g., counts in the detector), it looks like:

r is the galactocentric radius [putting it more accurately, the length of the semi-major axis of the (elliptical!) isophote], h the length scale of the disk and 0 refers to central values. If the plot below at left (in linear units) was in magnitudes it would only contain a straight line (see below). Thus, to find a first guess for h you can make a plot like that (in magnitudes) and see where the brightness drops from the center by approximately 1 mag, using the outer (exponential!) part of the brightness profile of a disk galaxy.

Things get a bit more complicated when dealing with edge-on disks (we have more about that below). In this case one must consider the brightness vertical dependence. Including that, then one has (assuming that the disk may be described as isothermal):

This assumes (fair enough!) that the vertical length scale z0 does not have any dependence on r. The plot below at right shows how the sech² function looks like in linear units.

THE SÉRSIC FUNCTION. A very versatile and useful expression!

Where e refers to effective values and n is the famous Sérsic index. When n equals 4 the Sérsic function becomes the de Vaucouleurs function; when n equals 1 the Sérsic function becomes an exponential, and when n equals 0.5 it becomes a Gaussian! For values in the range 1-4 (see plot below at left, from left to right), approximately, it is well suitable to describe from bulges in late-type spiral galaxies (or pseudo-bulges) to bulges in early-type spirals (or classical bulges) and elliptical galaxies. It is easy to realize that the larger the value of n the more concentrated is the light (and mass!) of the bulge (or elliptical) in the center.

The effective radius of a galaxy is the one that contains half of the light emitted by the galaxy. The numerical constants b_n and c_n are chosen so that the brightness at the effective radius is the effective brightness, and they depend only on n.

You may find the same function written in other ways, like:

Don’t worry, they are all indeed the same… that only depends on whether you use the effective or the central value for the brightness parameter, and whether you choose the natural base (e) or the decimal one.

Note that, for n=4, the difference between the central surface brightness and the effective one is about 8 magnitudes, so this is useful also to estimate a first guess for the effective parameters (see case above for disks).

The Sérsic function can also be suitable to describe bars, considering values in the range 0.4 to 1. (Values smaller than that may be unphysical as the brightness would then drop in the center!) A bar in a late-type galaxy can be well fitted by an exponential (n=1), whereas bars in early-type galaxies have a flatter luminosity profile (n=0.6, say). The plot below at right shows the Sérsic function for n=0.4, 0.6, 0.8 and 1 (upwards).

Bars are hard to fit… and on top of the Sérsic function you will certainly need boxy isophotes, generally with a somewhat high ellipticity. The effective radius of the Sérsic function describing the bar must also be very well chosen, or better, carefully fitted. In many cases too, you will need an outer cutoff radius. Below I describe in more details how I fit bars with BUDDA, including very nice plots!

THE MOFFAT FUNCTION. It looks pretty much like a Gaussian, but the Moffat function fits better the effects of seeing caused by atmospheric turbulence because of its longer tails. Hence, BUDDA uses it to describe central bright point sources like an AGN or a stellar cluster. Of course the seeing effects on the whole model fitted by the code are also taken into account using the Moffat function. OK, so there it is:

Here, r_d is related to the seeing FWHM and n is normally set to 4.765. For n→∞ it becomes a Gaussian.

BOXY & DISKY ISOPHOTES (GENERALIZED ELLIPSES). A generalized ellipse is described by the following expression:

Where q is the axial ratio (related to the ellipticity) and a pure ellipse (top left panel of figure above) has c=0. For c>0 one has a boxy ellipse, or isophote (top right panel). In this case there is a deficit of light in the directions of the major and minor axes. For c<0 one gets a disky isophote (bottom left panel), where there is an excess of light in the directions of the major and minor axes.

The ability to fit boxy or disky isophotes may be very important in studies of elliptical galaxies, since it seems to exist a dichotomy between the properties of boxy and disky ellipticals. The former are radio-loud, pressure-supported, and in high density environments, in contrast with the latter.

Boxy isophotes are also essential to fit bars. The bottom right panel shows what could be a bar in an early-type galaxy: eccentric, square and with a flat luminosity profile.

THE FOUR COMPONENTS. The figure below at left shows examples of the four components BUDDA fits in a galaxy: the central bright source (top left), the bulge (top right), the bar (an appropriate example of a bar in a late-type galaxy, with a steep brightness profile — bottom left), and the disk (bottom right). Of course, these are only some examples! The variety one can have in the structure of a galaxy is enormous, as is the parameter space of the properties of the four structural components included in the code.

The figure below at right represents how BUDDA sees a galaxy… (Note that the central bright source is there but you can not distinguish it since the brightness and contrast settings of this figure make it immersed in the bulge light. But then now you can better see the outer disk.)

DOUBLE EXPONENTIAL DISKS. In the new version of BUDDA the user has the choice to fit the galactic disk with a single exponential function, or a double exponential, i.e., like two different disks (inner and outer disks, say) that share the same position angle and ellipticity, but have different central brightness and length scales. This ability tends to become very useful since galaxies with double exponential disks are being now frequently found with ever deeper and finer images. In the figures below I present the two distinct sorts of disks described by different inner and outer parts. In the left figure below one can see what is traditionally called disk truncation, i.e., the outer disk shows a steeper profile. In the figure at right one sees what is being called anti-truncation, i.e., the outer disk profile is shallower than that of the inner disk. These physical properties may be related to a number of different processes, like bar evolution, the stellar halo, late gas accretion, and galaxy harassment. BUDDA also is able to fit edge-on disks (bottom figure below), also with a single or a double exponential function.

TWO DIFFERENT BARS. Bars come in two distinct types: the ones in early-type disk galaxies show a flat brightness profile (figure below at left), whereas those harbored by late-type spirals show a steep (exponential) profile (below at right). Both can be modeled by a Sérsic function like BUDDA does. This different property may also be related to different physical processes, like the disk building, star formation and the age of the bar itself. These figures below show how BUDDA can fit these different sorts of bars and also the brightness cutoff at the bar ends.