3 Dynamical mass models

Above and in Paper I, we used the rotation curves to construct simple integrated dynamical mass estimates, based on the formula given by Lequeux (1983). In this section we will briefly discuss more detailed mass models involving a dark matter halo, and which also take the radial behaviour of the luminosity profile and the rotation curve into account.

Given the luminosity profiles for the burst and disk components, and their associated M/L values (see Table 2), its is possible to predict what the rotation curve should look like, under the assumption that there is no dark matter. We call these photometric rotation curves. These photometric rotation curves are the minimum allowed velocities for rotational support. In Figs. 3 to 5, these are shown as full drawn lines with a hatched region for the allowed interval (given by the M/L intervals in Table 2). We show only the disk component, since the burst component in all cases gives a neglible contribution.

Dynamical mass models, based on the observed rotation curve and the observed shape of the disk and burst luminosity profiles, were also constructed. Again, the burst component gives a neglible contribution and for reasons of clarity we do not include it in the presented mass models.

The technique used to construct the mass models is described in Carignan & Freeman (1985) and the code we used was kindly provided to us by Claude Carignan. Both "maximum disk" and "best fit" approaches were undertaken. A $\chi^2$ minimization technique is used in the three-parameter space of the model. These parameters are: The mass-to-light ratio of the stellar disc ( $M/L_{\rm disk}^\star$), the core radius ( $r_{\rm halo}$) and the one-dimensional velocity dispersion ( $\sigma_{\rm halo}$) of the dark isothermal halo. The mass models are presented in Figs. 3 to 5: the dash-dot lines gives the disk component, the crosses give the halo component and the long dashed lines give the total disk+halo component. The latter should be compared with the observed rotation curve which is shown as dots with errorbars (here the error bars reflect the discrepancy between the approaching and receeding sides). In general we show the maximum disk models since they provide upper limits to the mass contained in stars.

When viewing Figs. 3 to 5, the galaxies fall in two categories: four of the galaxies (ESO350-38, ESO 400-43, ESO185-13 and ESO338-04) have observed rotation curves that are well below the predicted photometric rotation curves. This was expected in view of the mass dicrepancies discussed in Sect. 2.5. In ESO350-38 and ESO185-13 the shape of the rotation curve and the disk component of the mass model agree reasonably well when using the primary component of the decomposed velocity fields, but the observed rotation is to slow (marginally in ESO185-13, significantly in ESO350-38). ESO185-13 could possibly be saved by adopting an IMF with a flat low mass part, if the dynamical mass has also been underestimated (e.g. by overestimating the inclination), but not ESO350-38. For ESO400-43 and ESO338-04 it was even impossible to construct a mass model, due to the divergent shapes of the observed and photometrical rotation curves.

In the other galaxies, there is no apparent deficiency of rotation. In ESO480-12 and ESO338-04B we see a significant contribution from a dark matter halo to the dynamics. In ESO400-43B and Tololo0341-407E there is marginal evidence for dark matter: the fit is improved by including a dark halo, but the luminous matter still dominates within the extent of the rotation curve. In Tololo0341-407W we see no signature of a dark halo. In these 5 cases the disk components agree rather well with the photometric rotation curves.

The galaxy ESO338-IG04B has both well behaved dynamics and photometric structure. For this galaxy, the maximum disk and best fit models give the same result: $M/L_{\rm disk}^\star=4$, core radius of the dark halo $r_{\rm halo}=9$ kpc, and a halo velocity dispersion $\sigma_{\rm halo}=100$ kms^-1. The 99.9% confidence interval is $M/L_{\rm disk}^\star = 3.2$ to 4.8. The photometric estimate is $M/L_{\rm disk}=3.4$ with a 3 sigma confidence interval 2.5 to 6.0. A photometric mass slightly lower than the dynamical one is expected with the presence of an H I disk. Thus there is a very good agreement between the purely dynamical and the purely photometric M/L estimates, which give us confidence in our photometric M/L estimating procedure.