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Astron. Astrophys. 335, 449-462 (1998)

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4. Best-approximation process

The stellar populations and mass distribution in M 81 have been studied by means of modelling. For all models the best-approximation parameter set has been found using the least-squares algorithm. The algorithm minimizes the sum of squares of relative deviations of the model from all observations.

The basic formulae and general steps of model construction were described in a previous paper (Tenjes et al. 1994) where the stellar populations in the Andromeda galaxy were studied. In addition, the basic formulae in a compact form will be given in Appendix. Due to the nonlinear nature of these relations and the composite structure of the galaxies, fitting of the model to observations is not a straightforward procedure, and mathematically correct solution may be completely unphysical. (For example, a six-component model, where photometrical profile is approximated by some appropriate three components with nearly zero M/L ratios and rotational curve by other three components with very large M/L ratios may have nice fit with observations, but it is unphysical.) For this reason, the approximation process must be done in several steps.

First, the number of model components was fixed. The components we decided to include were described and emphasized in Sect. 3.

For three components - the nucleus, the extreme flat subsystem and the metal-poor halo subsystem - several parameters were determined independently of other subsystems (Sects. 3.1, 3.3, 3.4). In subsequent fitting processes they were kept fixed. This step allows to reduce the number of free parameters in the approximation process.

In the next stage crude estimates for remaining population parameters were made. The purpose of this step is only to exclude obviously unphysical values of parameters. During this stage we studied also the sensitivity of population parameters to different observations.

In the last stage the final model was constructed by the subsequent least-square approximation process.

4.1. Parameters of the approximation process

The set of initial data in final approximation process consists of:

  1. photometrical data (surface photometry in UBVR colours along the major and minor axes, in I colour along the major axis);
  2. the rotational curve in the plane of the galaxy;
  3. the mean line-of-sight stellar velocity dispersion for the nucleus, the core and the bulge;
  4. the mean line-of-sight velocity dispersion of globular clusters and satellite galaxies of M 81.

The total number of combined observational data points was 258, 216 data points of them are the surface brightness data, 39 data points describe the rotation curve and 3 points are the mean velocity dispersions of components. We assume that the rotation curve has the same total weight as the surface photometry. Thus, because the number of rotation data points is smaller than the number of photometry data points, the weight ascribed to each of them was correspondingly larger. The velocity dispersions were used only for mass determination.

In principle, in the seven-component model the maximum number of the degrees of freedom in the fitting process is 58 (6 visible populations with 9 parameters each ([FORMULA], [FORMULA] [FORMULA] [FORMULA] [FORMULA] and 4 mass-to-light ratios in UBVR) and an invisible corona with 4 parameters ([FORMULA], [FORMULA] [FORMULA] M)). Some of these parameters can be fixed earlier. First, the parameter [FORMULA] indicating the depth of the central density depression, was taken zero for all components except the disk and the flat component. Further, the parameters of the nucleus were determined independently of others and fixed thereafter (Sect. 3.1). In Sect. 3.3 we derived the density distribution parameters [FORMULA] [FORMULA] [FORMULA] the colour indices (B-V), (V-R) and the mass-to-light ratio [FORMULA] of the halo from the distribution of globular clusters. In Sect. 3.4 from the distribution of the young stellar component all the parameters for the flat population were derived. These parameters of the halo and the flat subsystems were also unchanged. The parameters [FORMULA], [FORMULA], and [FORMULA] of the corona were also fixed earlier (Sect. 3.6). When taking into account all the fixed parameters the number of free parameters is 28.

During the preliminary model construction the space of the remaining 28 parameters was divided into separate regions and analysed separately. The axial ratios of all subsystems form a nearly independent subspace, as they depend mainly on the light distribution along the minor axis, i.e. the projected isophotal eccentricities. Also the colour indices form an independent subspace, reducing the number of parameters to 15. Further, the masses and the luminosities of several subsystems depend also on different sets of observations. As it was in the case of modelling of the galaxy M 31 (paper IV), the most mixed are only seven parameters: the radii and the structure parameters of the core, the bulge and the disk, and the central depression of the disk. Coupling of these parameters is the most controversial part of the modelling process. We will analyse these problems in the following subsection.

4.2. Coupling of parameters

The core and the bulge subsystems are mixed in photometry, i.e. light profiles allow variation of their structural parameters in a quite large interval. When limiting to the light profile in V only and neglecting kinematics, the presence of the core is not even necessary. More strict limits to the parameters of these components result from kinematics: specific first maximum at about [FORMULA] 1 kpc and minimum at about [FORMULA] 2-3 kpc. To represent adequately both the inner part of the rotation curve and the surface photometry, a two-component inner spheroid with different [FORMULA] ratios is needed. From Fig. 8 it is seen that the model with the bulge component only and giving best fit with the rotation curve and surface photometry cannot be handled as a good one. Main characteristics of the inner parts of the rotation curve and surface brightness distribution are poorly fitted. Hence, further we study core + bulge models. From the modelling of the Milky Way rotation velocities Rohlfs & Kreitschmann (1988) also concluded that the bulge alone does not give the first maximum in the rotation curve and hence an additional inner component was needed.

[FIGURE] Fig. 8. The inner part of the rotation curve and the surface brightness distribution in V (cf. Figs. 1 and 4) for the best fit model without the inner metal-rich core. The mean calculated line-of-sight velocity dispersion for the bulge was fixed to be 145 km/s.

In Fig. 9 we demonstrate the sensitivity of the rotation curve and the surface brightness distribution to the core radius. In these calculations the structure parameter [FORMULA] 3 and the mass of the core corresponds to the mean velocity dispersion of [FORMULA] 160 km/s. The radii of the core are 0.07, 0.14, and 0.30 kpc, the masses of the core are 0.3, 0.6, and 1.0 (in units of [FORMULA] for models (a), (b), and (c), respectively. Fig. 10 illustrates models with a different structure parameter N of the core [FORMULA] 0.35, [FORMULA] 0.14 kpc). In model (a) [FORMULA] 1.5 and in model (b) [FORMULA] 5. As the radius of the core is more than seven times smaller than the radius of the bulge, the latter is quite insensitive to the changes of the core parameters and also to the parameters of all other components.

[FIGURE] Fig. 9. The inner part of the rotation curve and the surface brightness distribution in V (cf. Fig. 1) for three different models. The core radii are 0.07 kpc, 0.14 kpc, 0.30 kpc for the lower, middle and upper panels, respectively.

[FIGURE] Fig. 10. The inner part of the rotation curve and the surface brightness distribution in V for two core models. In lower model [FORMULA] 1.5, in upper model [FORMULA] 5.

Since [FORMULA] and N for the halo parameters were fixed on the basis of the distribution of globular clusters, all the remaining parameters for the core and the halo, and all parameters for the bulge can be determined with sufficient accuracy.

4.3. The disk and the massive corona

The parameters of the disk depend both on photometrical and kinematical data. Increasing eccentricity of the isophotes beyond [FORMULA] 2 kpc (Fig. 3) indicates roughly the region where the disk becomes dominating in the photometry. In addition, the specific second maximum at [FORMULA] 6.5 kpc in the rotation curve quite firmly determines the disk radius and the structure parameter N.

Fig. 11 illustrates the sensitivity of the rotation curve and the major axis brightness profile to the disk parameter [FORMULA]. Models (a) and (b) correspond to the models with fixed parameters of the central depression [FORMULA] 0.2 and [FORMULA] 0.9, respectively. The radii of the disk were [FORMULA] 3.7 kpc and 4.2 kpc, the mass-to-light ratios 12 and 24 in B. In the case of [FORMULA] 0.2 also the bulge mass decreases to [FORMULA].

[FIGURE] Fig. 11. The rotation curve and brightness profile for different central depression of the disk. In lower model [FORMULA] 0.2, in upper model [FORMULA] 0.9.

As we mentioned in Sect. 3.6 from observations the parameter of the corona [FORMULA] is quite uncertain. This means also that the central density of the corona may vary considerably causing variation of the disk mass. Hence we must study the coupling of disk mass and dark matter density. In Fig. 12 two models for different disk masses are given. Model (a) has [FORMULA] 5.7, model (b) has [FORMULA] 11.3. Other parameters of the disk as well as the radius of the corona were fixed. In both models the velocity dispersion of the corona was 114 km/s. It can be seen that in the case of M 81 the rotation curve allows quite firmly to discriminate between the visible and dark matter.

[FIGURE] Fig. 12. The rotation curve of M 81 for different disk masses. In lower model [FORMULA] 5.7, in upper model [FORMULA] 11.3 (in units of [FORMULA]).

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998