## 1. IntroductionThe morphological structure of galaxies contains very valuable information about the history of their formation and subsequent interaction with their environments. In particular, the surface brightness distribution has long been recognized as an essential clue to describe quantitatively the morphological features of galaxies, as well as to investigate possible relationships to other galactic properties. The luminosity profile of disk galaxies is often considered as the sum of two overlapping components: the bulge and the disk. These components are parameterized by simple empirical fitting functions with free parameters which are determined by a non-linear fit to the observed profiles (see, e.g., the reviews by Simien 1989, and Capaccioli and Caon 1992). The most frequently used function describing the radial surface brightness profile of the disks of spiral galaxies is an exponential function (de Vaucouleurs 1959, Freeman 1970): where is the central surface luminosity and
In that concerning the bulge, the early papers used a de Vaucouleurs law to describe its brightness profile. However, since strong deviations from this law have been frequently observed in bulges of spirals, an exponential fitting function has been more recently proposed for this component (Frankston & Schild 1976, Kent el al. 1991, and Andreakis & Sanders 1994) where and are the bulge central surface luminosity and scale length, respectively. Although the use of an exponential bulge improves the quality of fits, we note however that a single bulge/disk decomposition (whatever the bulge law is) still fails frequently for the most advanced types of the Hubble sequence, while it generally gives very good results for S0 galaxies (see, e.g., Kent 1985). Inspection of the observed profiles of Sa-Sc galaxies shows that a failure in the fitting of data is often produced by the existence of sinusoidal-like oscillations in the radial brightness distribution. This strongly suggests that such a failure originates from having neglected a third component whose importance could be crucial in many Sa-Sc galaxies: the spiral arms. The aim of this paper is to deduce and to apply a luminosity law for the contribution of spiral arms. Unlike the purely empirical functions traditionally used for the bulge and unperturbed disk components, we will obtain the luminosity law of spiral arms by means of dynamical arguments based on the density wave theory. An advantage of this dynamical approach is that it provides the relationship between the fitting parameters and certain dynamical properties of the galaxy under consideration. This allows one to establish a connection between theory and observations much clearer than that obtained through purely empirical expressions. The density wave theory is an extensive formalism describing the dynamics of differentially rotating disks (see, e.g., the reviews by Rohlfs 1977, Athanassoula 1984, and Binney and Tremaine 1987). This theory assumes that spiral arms are the excess matter associated with a wavelike oscillation that propagates through the galactic disk. This spiral pattern generates a non-axisymmetric component of the gravitational field which, in principle, can help to produce long-lived spiral arms. The main difficulty in giving a complete treatment of the problem lies in the long-range nature of gravitational forces: all parts of a galaxy are strongly coupled together and, in general, the wave features can only be determined numerically. Fortunately, there exists one important limit in which the analysis is much simpler: if the radial wavelength of waves is much smaller than the radius, the long-range coupling is negligible, the response is determined locally, and the relevant solutions are analytic. In addition to this limit, known as the WKB approximation, the simplest version of the density wave theory also assumes that waves are quasi-stationary (Lin and Shu 1966). Although the Lin-Shu theory yields a broad range of successful predictions, the validity of its hypotheses is still the subject of intense debate and it cannot be considered as a final solution to all spiral structure problems. In general, the WKB theory must be augmented with new physical concepts, as feedback loops and the swing amplifier (Toomre 1981), which require numerical programs to follow the non-linear evolution of wave packets. The paper is arranged as follows. The basic equations of the density wave theory are briefly described in Sect. 2.1, as well as the approximations assumed in this paper. These expressions are then used in Sect. 2.2 to obtain an analytical function for the spiral arm contribution to the surface brightness profiles of galaxies. The fitting procedure using our bulge-disk-arms composite model and its application to some observed profiles is then presented in Sect. 3. Finally, Sect. 4 summarizes our main results and conclusions. © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |