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Astron. Astrophys. 339, 409-422 (1998)

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3. Results

3.1. Rotation curve fitting

With the exception of NGC 2841 and perhaps NGC 4698, our rotation curves do not extend far enough to constrain effectively both the stellar M/L's and the halo parameters of Eq. 10, and result in high uncertainties on the parameters (of order 50%) when all of them are fit simultaneously. Moreover, a global fit of the three components, at least in the case of the isothermal halos, tends to attribute most of the mass to the halo at all radii. These difficulties were also noted by Kent (1988), who considered further assumptions and simplifications to better constrain the parameter space. In particular he suggested three possible kinds of fits.

a) Maximum bulge+disk solutions (MBD hereafter), in which the maximum amount of mass compatible with the rotation observed is ascribed to the visible matter. This can be achieved by fitting the RC with only the two stellar components for radii within two disk scale lengths, or up to the disk velocity peak for the np case. When the observed RC is constant or rising beyond this limit the dark halo contribution is added. In our case the MBD approach can be justified by noting that these are all relatively high-luminosity objects with [FORMULA], so that DM should not be important in the inner regions (Kormendy 1990; Salucci et al. 1991); in general the rising part of the RC is well reproduced, confirming that this hypothesis is probably correct. Moreover, in the case of NGC 2841 the simultaneous fit of the three components does not differ appreciably from the MBD one. For NGC 4698 it yields M/L ratios for disk and bulge lower by about a factor of two, although with higher uncertainties which make the result again consistent with the MBD results.

b) Fits with fixed asympotic halo velocity, [FORMULA]. If we assume that this parameter can be determined from the flat portion of the RC, the number of free parameters is reduced to three. Only a few of our galaxies have extended RC's, so that an independent estimate of [FORMULA] is possible only for NGC 2841 and NGC 4698. In these two cases, however, the fitting routine is able to estimate [FORMULA] with reasonable accuracy (the uncertainty is [FORMULA]), so that we did not consider this kind of solutions.

c) Fits with constant-density halos. Adopting the halo distribution described by Eq. 11, again we have only three free parameters. We find that in general such a halo does not affect appreciably the contribution of bulge and disk to the rising part of the RC, thus yielding M/L's consistent with the MBD values.

On the basis of the above statements, and to define a homogeneous set of parameters, we decided to consider only the MBD solutions, assuming that the associated errors are realistic estimates of the true parameter uncertainties. The dark halo contribution was added for six galaxies out of 14; five of these have RC's measured at 21 cm, while for the sixth, NGC 2639, the optical RC was sufficient to constrain the halo contribution. In the case of NGC 3593, the radio measurements, extending to about 1.5 [FORMULA], could be fit reasonably well without any dark component. For each galaxy we chose the model halo (the constant density or the pseudo-isothermal sphere) which yielded the best-quality RC fit; the constant-density halo was adopted for NGC 2639, NGC 4450, and NGC 5879. Comparing quantities which can be derived in both cases, such as the central density or the mass within the optical radius, we find no systematic difference between the two models.

The resulting M/L's and halo parameters are given in Table 2, and Table 3 reports the masses of the various components. The values presented in the tables are the mean of the parametric and np results; the error is the largest of the values for the semi-difference (parametric and np) and the formal error on the fitted parameter. Fig. 1 illustrates the fits to the observed rotation curves for both the parametric and np surface brightness decompositions. We note that the best results in most cases are obtained with the np fits, which in general match more closely the features of the observed curves. Quantitatively, comparing the [FORMULA] obtained in the two cases, we find a median ratio of the parametric to the np value of 1.3.


Table 2. Fits to the rotation curves: mass-to-light ratios and halo parameters.
Columns 2-5 : M/L's are in solar units


Table 3. Masses of the components.
Column 2 : bulge mass; Column 3 : disk mass;
Column 4 : bulge + disk mass;
Column 5 : halo mass;
Column 6 : total mass (all values are in units of [FORMULA] [FORMULA]).
Column 7 : dark-to-visible mass ratio.

[FIGURE] Fig. 1. Observed and fitted rotation curves. Upper and lower panels show respectively the results from parametric and np decompositions (K band). The dots are the observed curve; the dashed, dotted, and dot-dashed lines are respectively the contributions of bulge, disk, and dark halo to the model RC (continuous line). The mark on the abscissa corresponds to [FORMULA].

[FIGURE] Fig. 1. (continued)

[FIGURE] Fig. 1. (continued)

[FIGURE] Fig. 1. (continued)

3.2. M/L ratios

Disks have a mean M/L ratio of 1.6 and 1.0 (solar units) in J and K, with a dispersion of 0.6 and 0.4, respectively; the mean bulge M/L is [FORMULA] in J and [FORMULA] in K. We note that five galaxies (namely NGC 2639, NGC 3593, NGC 3898, NGC 4419 and NGC 5879) yield low values of bulge M/L; these were excluded from the mean calculation (see Sect. 4.1). For NGC 4698, this happens only in the np case. Note that NGC 3593 and NGC 4419 are actively forming stars (Paper I), so a low M/L might be expected. Kent (1988) also found low M/L ratios for these five galaxies, and interpreted this as an inconsistency between photometric and kinematical data, possibly due to non-circular motions of the ionized gas in the inner part of the galaxies. The same conclusion was attained by Fillmore et al. (1986), who have modeled both rotation and velocity dispersions from emission and absorption-line data for six galaxies, three of which are also in our sample (see Table 1). Their models suggest that in most cases even the observed emission-line velocity underestimates the actual RC of the galaxy in the inner regions ([FORMULA] kpc). In the case of NGC 3898 and NGC 4450, for instance, a rough estimate of the bulge M/L from their model RC turns out to be respectively six and two times higher than ours. Even these `corrected' values, however, remain lower by at least 15% than the disk M/L's estimated from the observed rotation.

3.3. Dark halos

The dark halo parameters we obtain are similar to the ones derived for other samples, in particular for later type spirals using photometry in the optical passbands (e.g., Kent 1986; Kent 1987). Scale lengths [FORMULA] are comparable to the optical size of the galaxies and range from 18 to 30 kpc; the average halo mass is about [FORMULA] [FORMULA] within [FORMULA]. The central halo densities show a rather narrow distribution, peaked at [FORMULA] g cm-3, with a dispersion of about 20%. The average ratio of luminous to dark matter within the optical radius is around 2, a typical value for bright spirals (PSS).

3.4. MOND

For the six objects selected in Sect. 2.4, we first performed a fit of the RC with [FORMULA] and the bulge and disk M/L as free parameters. This yielded for [FORMULA] a weighted mean of 1.3 in units of [FORMULA] cm s-2. We then performed two sets of fits keeping [FORMULA] fixed respectively to our value of 1.3 and to the value of 0.8, corresponding to the [FORMULA] obtained by BBS rescaled to [FORMULA] km s-1.

A visual inspection of the observed RC's and the models does not clearly favor one paradigm over the other. This qualitatitve statement is confirmed by the similar values of [FORMULA]. Two representative cases are shown in Fig. 2: NGC 2841, for which MOND yields a particularly good fit, and NGC 5879 for which a dark halo seems to produce a significantly better result. The choice of [FORMULA] turns out to be crucial only for NGC 2841, where only [FORMULA] yields a good fit. BBS also found that this galaxy required a higher critical acceleration, probably because of an error in the estimated distance, and did not include it in their average. After removing NGC 2841 from the sample, the remaining five galaxies yield [FORMULA], fully consistent with the BBS value.

[FIGURE] Fig. 2. Fit with MOND to the RC's of NGC 2841 and NGC 5879 from parametric (upper panels ) and np decompositions (lower panels ). Dots represent the observed RC, whereas the continuous line is the model one. In both cases we plot the results for [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: October 21, 1998