## 3. Applications## 3.1. Fitting procedure and determination of physical parametersThe expression (29) for the contribution of the spiral arms to the surface brightness profile of a galaxy introduces five new fitting parameters. In principle, this is not very satisfactory since a so high number of parameters can introduce a considerable degree of arbitrariness. Furthermore, as in the usual bulge/disk decomposition, very different sets of the free parameters can lead to very close values of the statistical measures (as the r.m.s.) for the fit quality. In order to choose the best approximating model, a criterion just based on such statistical measures is often inappropriate because other very different models could lead to fits with just a very slightly smaller goodness. We remark however that Eq. (29) has been obtained through physical arguments and, consequently, it is not just a mathematical fitting function. As a matter of fact, the fitting parameters are related with measurable physical properties through (Eqs. 32): where is expressed in km/s and
in
/arcsec Hence, the freedom in the choice of such parameters is limited by the requirement that they must imply reasonable physical properties for the galaxy under study. If this requirement cannot be satisfied for a given galaxy, we must consider that Eq. (29) does not introduce a reliable improvement with respect to the usual bulge/disk decomposition. In the present study, we have used a fitting procedure in which reproducibility of reasonable physical properties, as well as the fit goodness, is also taken as a criterion to identify the best model. Furthermore, since the result of a non-linear fitting can depend on the value initially assigned to the free parameters, our fitting procedure consists of two stages where the best model is identified after exploring a grid of input parameters. The first stage of this procedure is conceived to explore all the range of reasonable parameters and thus find the better choice of their input values. Toward this end, the bulge and disk parameters, as well as the arbitrary spiral arm constants and , have input values generated through the grids: where represent their input central values, chosen after performing some tests. The input dynamical parameters (, , and ) are instead generated so that they cover a wide interval of values considered as physically reasonable. In other words, they are generated so that Eqs. (32) imply a grid of , and values covering all their corresponding allowed intervals. For example, and similarly for and . The allowed interval considered for the pitch angle is (Kennicut 1981) , while the interval considered for covers from to (see, e.g., Kent 1985, Forbes 1992). The values are instead more poorly known. In the case of our Galaxy, Lin, Yuan and Shu (1969) obtained that km/s/kpc fits well various observed data, while Marochnik, Mishurov and Shuchkov (1972) obtained km/s/kpc. We have considered here a rather wide allowed interval km/s/kpc. Starting from these input grids, we have then performed a standard non-linear fitting (based on the Levenberg-Marquard method). The selection of the best model is carried out by taking, among all those implying reasonable physical properties, that which leads to the smallest r.m.s. For reasonable physical properties we mean, not only that the corresponding , , and values are within the above described intervals, but also that the resulting profile for the spiral arms does not violate one of the basic hypothesis of the density wave theory: . In fact, we see from Eq. (29) that density perturbations have a very large amplitude when . The possibility of obtaining large density amplitudes has been previously noted by Toomre (1969) and Shu (1970). According to Eq. (24), such values appear when the LWM and SWM solutions coincide (). In that case, the wave number is marginally real and the disk is marginally stable. Since the density wave theory assumes that is small as compared to the density of the unperturbed disk, such models are not acceptable. In our numerical computation, we have rejected those models with . The second stage of our procedure takes the values of obtained in the previous stage as input central values to generate a new set of grids, but now using Eq. (35) for any parameter. From the same criteria and non-linear fitting algorithm as before, we select a new best model. The values of this model are then used as input central values to generate new grids. This iterative procedure is followed until the output values converge. Obviously, this procedure consumes a considerable amount of computing time ( hr), but it reduces notably the possible arbitrariness of results. The obtained model is in fact rather insensitive to the initial central values adopted at the beginning of the first stage. ## 3.2. ResultsIn order to illustrate the application of Eq. (29) to some observed profiles, we have analyzed the surface brightness distribution of two galaxies in the Kent (1984, 1986) sample: UGC 2885 and IC 467. UGC 2885 has been selected as being a typical Sc galaxy for which the usual bulge/disk decomposition provided rather bad results. Using the rotation curve measured by Rubin et al. (1985) ( km/s and km/s/kpc), and applying the above procedure, we find that the surface brightness profile of UGC 2885 is very well fitted by our triple decomposition (see Fig. 1). The physical parameters implied by this procedure are , km/s/kpc and . This value coincides within a very reasonable margin with those obtained from different techniques. For example, the maximum-disk and the constant-density halo solutions of the UGC 2885 rotation curve (Kent 1986) imply . In the same way, Roelfsema & Allen (1985) found the values and km/s/kpc, which are just slightly smaller than ours. Comparison with Canzian's (1985) results is however much more difficult because he considered the existence of two superposed spiral patterns in UGC 2885, while our approach assumes a single two-armed spiral structure. Nevertheless, we note that the outer pattern in Canzian (1985) has km/s/kpc, very close to our results (the inner pattern has instead a considerably higher speed, km/s/kpc).
IC 467 has been selected as another Sc galaxy but, now, with a rising rotation curve ( km/s, km/s/kpc) also measured by Rubin et al. (1985). The fitting of its surface brightness distribution is again much better than that found from the usual double decomposition (see Fig. 2). The resulting dynamical parameters (, km/s/kpc and ) are also reasonably close to those found from other procedures. The disk mass-to-luminosity ratio is in fact intermediate between that found by Forbes (1992), , and that obtained by Kent (1986), . It must be mentioned that, although IC 467 is considered as a galaxy with a normal spiral structure, it probably has a companion (NGC 2336) at a projected separation of about 135 kpc (van Moorsel 1987). The good modeling of its surface brightness by our WKB approximation suggests that possible tidal effects on IC 467 have a very small dynamical influence.
Another important consequence of the incorporation of spiral arms in this kind of analysis is that the relative importance attributed to the bulge with respect to the other components can be different from that obtained through the usual approach. In the examples shown above, our calculations imply that the luminosity of the bulge is greater by a factor of than that obtained by Andreakis and Sanders (1994), who did not consider the presence of spiral arms perturbing the disk component. © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |