In order to describe the 1D distributions we need simple model profiles for the different components. The bulge and disk components of spiral galaxies have different profile shapes. The bulges have profiles similar to ellipticals while the disk component is exponential in nature. In this section we will discuss a few simple model profiles that are in use to describe the galaxy components.
The so called Hubble's Law is really due to Reynolds ( 1913). It bears the name of Hubble because it was he (Hubble, 1930) who made extensive use of it. According to this law the surface brightness is given by
where r is the major axis. stands for the surface brightness and is the surface brightness at r=0. The scale length is the distance at which the surface brightness falls to a quarter of its central value.
Hubble's law describes very well the radial brightness distribution of most ellipticals within certain bounds of r. However, the light contained within a given r diverges logarithmically with increasing r. This certainly cannot be true for real galaxies.
King ( 1966) introduced two dynamical models of self-gravitating stellar systems. These were successfully applied to globular clusters and later extended to elliptical galaxies ( King, 1982). A King profile can be described in terms of the following parameters:
King's profiles are based on specific dynamical models, and provide good fits to globular cluster surface brightness. However, recent studies ( Lauer et al., 1995) have shown that these are not really suitable for elliptical galaxies.
de Vaucouleurs' law is perhaps the most widely used empirical law to describe the surface brightness profile of an elliptical galaxy. Also known as the law ( de Vaucouleurs, 1953), it is given by
The scale length, , known as the effective radius, contains half of the total light of the galaxy. The total light B of the bulge is given by
is the surface brightness at . de Vaucouleurs' law gives . A plot of the logarithm of surface brightness versus gives a straight line if de Vaucouleurs' law is valid (see Figure ).
de Vaucouleurs' law is empirical in nature. It does not necessarily fit all elliptical galaxies over all ranges of r. Burkert ( 1993) has shown that de Vaucouleurs' law describes very well the brightness distribution in ellipticals for .
Studies have shown that the law could be the end result of the relaxation of merged galaxies. For example, NGC 7252 is a peculiar galaxy with two large luminous tails. Schweizer (1982) has shown that the central region of the radial luminosity profile of this galaxy, which is made of two disk galaxies, obeys the law. However, two purely stellar disks cannot give rise to an elliptical galaxy. This is because stars in an elliptical galaxy are more tightly packed than in a spiral. Hence, to form an elliptical, the participation of gas which can pack during a merger is necessary.
For de Vaucouleurs' law the deprojected mass density , corresponding to an intensity distribution, is not analytically tractable ( Young, 1976). As a result, de Vaucouleurs' law is not very useful for theoretical analyses. Attempts have hence been made to obtain density profiles resembling that of the law and having simple analytic forms
(e.g. Binney, 1982; Bertin and Stiavelli, 1984, 1989 and Hernquist, 1990). However, these analytic forms are not as convenient to use as de Vaucouleurs' law. Moreover, though empirical in nature, de Vaucouleurs' law is seen to provide adequately good fits to many ellipticals. The approach that has been taken in this thesis is to fit de Vaucouleurs' law to the observed profiles, and to analyze the distribution of the fitted parameters as well as the deviations from the best-fit law. As we shall see, the latter provide important physical information.
de Vaucouleurs' law can be generalized by replacing the 1/4 by 1/n, where n is a variable. The law is then given by
with chosen so that half of the luminosity is contained within . By fitting a large number of galaxy surface brightness profiles with a generalized de Vaucouleurs' law, it has been shown that dwarf galaxies have while luminous galaxies have . (Caon et al., 1988; Graham et al., 1996)
Petrosian ( 1976) introduced another useful way of understanding the bulge. He introduced a dimensionless monotonic function , which is the ratio of the average surface brightness up to a radius r to the surface brightness at r. It is scale-less and does not assume any particular cosmological model as it involves only the structural parameters of the galaxy. It is useful for studies involving, say, the dependence of surface brightness as a function of redshift without reference to any particular cosmological model.
Jaffe ( 1983) proposed that elliptical galaxies have a density distribution given by
Quantities like surface brightness, gravitational potential, projected velocity dispersion etc. can be calculated easily from this law. Hence this profile is widely used in numerical studies of spherical galaxies.
Dehnen ( 1993) proposed a family of density profiles for which the total luminosity within the projected radius r does not diverge as :
Here L is the total luminosity of the model and is the scale length. This form of j(r) yields an analytically tractable projected density when is an integer or a half-integer. The models of Jaffe (1983) and Hernquist (1990) described above correspond to and respectively. The closest approximation to the law is given by . The Dehnen models will be discussed further when we consider the 3-D distribution of luminosity in elliptical galaxies.
It has been found (Freeman, 1970; Kormendy, 1977) that the surface brightness of the disk component of a spiral galaxy has an exponential form:
where is the scale length. The total light in the disk is given by
The disk-to-bulge ratio,
is also the ratio of the luminosities of the disk and the bulge. The ratio is sometimes expressed in terms of the bulge-to-total light ratio,
The disk becomes less and less prominent as one goes from spirals to lenticulars to ellipticals ( Kent, 1985). A galaxy classified as Sc can easily have a D/B ratio of 4, while a typical S0 galaxy will have D/B;SPMgt;0.3. An elliptical galaxy, by extension, would be expected to have a D/B;SPMlt;0.3. The disk in a disk galaxy is likely to be made of an old disk, a young disk and a dust layer ( e.g. Wainscoat, 1989). Burstein ( 1979) showed for a sample of S0 galaxies that they possess a thick disk and a thin disk in addition to a bulge. The orientation of the disk and the presence of dust complicates the accurate determination of the components. The subtleties of the disk-bulge decomposition will be described in the next section. Not much is known about the faint disks being detected in galaxies originally classified as ellipticals.
A small fraction of galaxies ( ) are known to possess an Active Galactic Nucleus (AGN). An AGN is a compact object at the center of the galaxy and is believed to be powered by accretion of gas onto a supermassive black hole at the center of the galaxy. AGN luminosities can reach and can outshine the host galaxy by over a factor of . This factor is highest for quasars and least for normal galaxies.
When present in a galaxy, an AGN corresponds to a sharp peak in the surface brightness profile at r=0. The extent of the AGN is small compared to the PSF. However, the light from the AGN gets distributed upto a few arcsec during convolution by the PSF. As a result, if the AGN is weak, it is very difficult to reliably detect it from ground-based observations. AGNs are readily detectable in Seyferts, but we find that in radio galaxies they are generally too weak to be extracted from our data. If such a source is present it will be weak and indistinguishable from other compact features like a small scale disk. Therefore we have not considered point sources during the profile decomposition for the sample. In cases where there seem to be strong reasons for believing that an AGN is present, additional processing was done to look for the point source.