ARE DWARF GALAXIES DOMINATED BY DARK MATTER?

The R-band luminosity profiles presented in Paper II have been used to calculate the contribution of the stellar disk to the observed rotation curve, assuming the mass-to-light ratio ϒ^R_* is independent of radius. Because most of the galaxies in this sample have light profiles that are well represented by an exponential disk, and generally have little or no bulge, the light profiles were not decomposed into a disk and a bulge component. The contribution of the stellar disk to the rotation curve was calculated using the prescription given in Casertano (1983), assuming that the galaxies are optically thin and assuming an intrinsic thickness of q₀ = 0.2. This value was chosen because it appears to be a suitable value for the average of the intrinsic thickness over the range of galaxy types included in this sample. Late-type disk galaxies, of morphological types around Sd, are reported to have intrinsic thicknesses closer to q₀ = 0.1 (e.g., de Grijs 1997; Schwarzkopf & Dettmar 2000; Bizyaev & Kajsin 2004), but for late-type dwarf galaxies higher values may be found, up to q₀ = 0.4, especially toward the lowest luminosity systems (van den Bergh 1988; Binggeli & Popescu 1995; Sung et al. 1998). We estimate the uncertainty in the amplitude of the rotation curve introduced by the variations in q₀ from galaxy to galaxy to be ∼5% (see S99 for more details).

The major uncertainty in determining a mass model for a galaxy is the mass-to-light ratio ϒ_*, which is not known a priori and cannot be derived from the rotation curve fits alone. Equally well-fitting mass models can be obtained with a range in ϒ_* (e.g., van Albada et al. 1985; Swaters et al. 2000, 2003; Dutton et al. 2005; see also Section 4). However, it is possible to obtain limits on ϒ_* by scaling up the contribution of the stellar disk to explain most of the observed rotation curve in the inner parts (the so-called maximum disk hypothesis), and by reducing the contribution of the stellar disk to its minimum while still obtaining a good fit to the rotation curve.

The procedure for deriving the contribution of the gas to the rotation curve is very similar to that of the stellar disk. The H i radial profiles presented in Paper I were used to calculate this contribution. We assumed that the gas layer has an intrinsic thickness of q₀ = 0.2, the same value that was assumed for the thickness of the stellar disk. Little is known about the vertical distribution of the gas in late-type dwarf galaxies. There is some evidence that the gas and the stars have the same vertical distribution (Bottema et al. 1986). Fortunately, the precise choice of the thickness of the H i layer has very little influence on the shape or amplitude of the rotation curve and the uncertainty is smaller than that introduced by assuming a constant thickness of q₀ = 0.2 for the optical disk.

In order to derive the contribution of the gas to the rotation curve, the H i was assumed to be optically thin. To correct for the mass fraction of helium, the H i mass was scaled by a factor η = 1.32, derived assuming a primordial Helium fraction of 24% (Steigman 2007).

Other gas components that might contribute to the rotation curve, such as molecular hydrogen, have been ignored. To date, most evidence suggests that there is much less molecular hydrogen in late-type dwarf galaxies than there is H i, even when taking into account that the conversion factor to convert the observed CO flux into a molecular gas column density may be substantially higher in late-type dwarf galaxies (e.g., Israel et al. 1995; Boselli et al. 2002; Buyle et al. 2006). Moreover, if molecular gas were to have the same radial distribution as the stellar disk, its contribution is implicitly included in the maximum disk fits.

Several different halo radial mass profiles have been proposed over the years, most of which produce good fits to the observed rotation curves. Halo mass–density profiles as predicted by cosmological simulations have received a great deal of attention, because a confrontation of these profiles against the observed rotation curves provides a powerful test of these cosmological models. For most of the sample presented here, such tests have been presented in van den Bosch & Swaters (2001) and Swaters et al. (2003).

In this paper, we focus on an empirical description of the properties of the dark matter halos. For this purpose, we have used the density profile of an isothermal sphere that has often been used in the literature to represent the dark halo. The advantage is that it will be easier to compare our results with those of other studies (e.g., Broeils 1992a; Verheijen 1997; Noordermeer 2006; see also Section 5.2).

The radial density profile of an isothermal sphere can be approximated by

$$\begin{equation} \rho (r)=\rho _0\left[1+\left(\!\frac{r}{\,r_c}\right)^2\:\right]^{-1}, \end{equation} \tag{ 4 }$$

where ρ₀ is the central density of the halo, and r_c is the core radius. This density profile gives rise to a rotation curve of the following form:

$$\begin{equation} v_h(r)=\sqrt{4\pi \mathrm{G}\rho _0r_c^{\,2}\left[1-\frac{r_c}{\!r} \mathrm{arctan}\left(\!\frac{r}{\,r_c}\right)\right]}. \end{equation} \tag{ 5 }$$

The relative contribution of each of the three components described above is determined by a simultaneous fit of the right-hand side of Equation (3) to the observed rotation curve. For an isothermal halo, there are four free parameters in Equation (3). However, the contribution of the atomic gas to the rotation curve is known, and therefore, with the correction for the mass fraction of helium, η is fixed to 1.32. On the other hand, the value of ϒ_* is not known beforehand, and a range of ϒ_* and halo parameters exists which produce a good fit to the data (van Albada et al. 1985). Mathematically, it is possible to derive the parameters for the isothermal halo and the value of ϒ_* from a best fit to the observed rotation curve. However, the physical meaning of such a best fit is unclear, as the values found from such a fit depend on the small-scale variations and uncertainties in the rotation curves. Because of these uncertainties in ϒ_*, two limiting cases are presented. In one case, the contribution of the stellar disk is scaled up to explain most of the rotation curve out to two or three disk scale lengths. This is the maximum disk solution. In the other limiting case, the contribution of the stellar disk is completely ignored, and the rotation curve is fitted with the contributions of dark halo and gas only.

The maximum disk fits were done by eye, ensuring that the rotation curve of the stellar disk contributes maximally to the observed rotation curve within its observational uncertainties, following, e.g., Broeils (1992a), Verheijen (1997), and Noordermeer (2006). We have refit several mass models from the literature using our methods. We were able to reproduce the original results to better than 10%. A mass model is only considered to be compatible with a maximum disk if the stellar disk can be scaled up to explain at least ∼80% of the rotation velocity at 2.2 disk scale lengths (e.g., Sackett 1997; see also Section 5.1).

There has been considerable effort over the past decade to relate the broadband colors of galaxies to their ϒ_* through stellar population synthesis modeling (e.g., Bell & de Jong 2001; Bell et al. 2003; Zibetti et al. 2009). The advantage of using such models is that it provides an independent estimate of ϒ_* and possibly a more realistic value of ϒ_*. However, there is some uncertainty in ϒ_* as derived from population synthesis modeling, because of, for example, uncertainties in the star formation history, the initial stellar mass function, metallicity, and the details of the late phases of stellar evolution. The combined uncertainty may be a factor of four or even larger if some of uncertainties are systematic (see, e.g., Bershady et al. 2010).

The six galaxies in our sample with both B- and R-band observations presented in Paper II have an average B−R color of 0.8 mag. Following Bell et al. (2003), this corresponds in ϒ^R_* ∼ 0.8, although this value of ϒ^R_* is uncertain, as discussed above. However, for any ϒ^R_* near unity, all but one of the galaxies in our sample are completely dominated by dark matter, so the exact choice of ϒ^R_* has little influence on the derived dark halo parameters (see also Section 4). Given these considerations, and because we had already calculated the contribution of the stellar disk for ϒ^R_* = 1 for the input of the mass modeling, we used these to determine the dark halo properties in these population synthesis-based restricted-ϒ_* fits.

The rotation curve fits are shown in Figures 1 and 2. The minimum disk fits are not shown separately because in most cases the mass models are very similar to those for ϒ^R_* = 1. The fit parameters are given in Table 1. The uncertainties on the fitted parameters have been derived from the 68% confidence levels and include the covariance between ρ₀ and r_c. The uncertainties on the derived rotation velocities are non-Gaussian, as the points in the rotation curves are correlated, and the rotation curves and their uncertainties can be affected by systematic effects. The confidence levels and the corresponding uncertainties should be considered estimates.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The most striking result from the maximum disk fits, presented in Figure 1, is that in all 18 galaxies the contribution of the stellar disk can be scaled up to explain most of the inner parts of the rotation curves out to two or three disk scale lengths, with the exception of UGC 4499. Thus, it seems that the fact that the stellar disk can be scaled to explain most of the inner parts of the rotation curves of late-type dwarf galaxies is rule rather than exception. This is in contrast with the results from several previous studies of dwarf galaxies, in which the stellar disk only can explain a small fraction of the inner rise; this is examined in more detail in Section 5.2.

To illustrate how the maximum disk mass models depend on the optical galaxy properties, we have plotted the mass models against surface brightness and absolute magnitude in Figure 3. In this figure, the origin of each model indicates the μ_R and M_R for each galaxy. The rotation curves are represented by filled circles for radii smaller than three disk scale lengths and by open circles for larger radii. From an inspection of Figure 3 it appears that there is no clear trend between the relative contribution of the stellar disk to the rotation curve on the one hand, and rotation curve shape, luminosity, or surface brightness on the other. Maximum disks are seen at all surface brightnesses and all absolute magnitudes.

Zoom In Zoom Out Reset image size

This point is demonstrated in more detail in Figures 4 and 5. Figure 4 shows that there is no clear correlation between the rotation curve shape as determined by the logarithmic slope between two and three disk scale lengths S^h_(2,3), and the relative contribution of the stellar disk to the rotation curve at 2.2 disk scale lengths (see Paper III for details). Independent of the rotation curve shape, the contribution of the stellar disk to the rotation curve is about 90% at 2.2 disk scale lengths. Figure 4 also shows the correlation between rotation curve shape and the relative contribution of the stellar disk at four disk scale lengths. The contribution of the stellar disk to the rotation curve has dropped to about 70%. There is one galaxy, UGC 12060, where even at four disk scale lengths the stellar disk can still explain most of the observed rotation curve. This galaxy has excess light between three and four disk scale lengths, when compared with an exponential falloff.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

From Figure 5, it is clear that the maximum disk hypothesis works equally well for all surface brightnesses.

From the results presented here, it is apparent that for the vast majority of these late-type dwarf galaxies the contributions of their stellar disks to the rotation curves can be scaled to explain most of the inner parts of the observed rotation curves. Therefore, these dwarf galaxies do not necessarily have insignificant mass in their stellar disks and dominant dark halos, as concluded in earlier work. For the maximum disk fits, dark matter only becomes important at radii larger than three or four disk scale lengths. These maximum disk mass models for late-type dwarf galaxies are similar to those seen in spiral galaxies (e.g., Begeman 1987; Broeils 1992a; Verheijen 1997).

4.1.1. Stellar Mass-to-light Ratios

Although for the majority of the late-type dwarf galaxies in our sample the contribution of the stellar disk can explain most of the rotation curve in the inner parts, the stellar mass-to-light ratios ϒ^R_* may be high. In Figure 6, a histogram is presented of the derived values of ϒ^R_* for the maximum disk fits. The average value for ϒ^R_* is 7.7, but ranges from 1 to 15, with the high end being well outside the range predicted by current population synthesis models. These high values of ϒ^R_* will be discussed in more detail in Section 5.

Zoom In Zoom Out Reset image size

Figure 7 shows ϒ^R_* plotted against the central surface brightness μ_R and the absolute magnitude M_R. Galaxies with lower surface brightnesses have higher values for ϒ^R_* under the assumption of maximum disk (see also Section 5.3).

Zoom In Zoom Out Reset image size

4.1.2. Isothermal Halo Properties

In three cases (UGC 4325, UGC 5414, and UGC 7577), the rotation curve can be explained by the stellar disk and the H i alone (see Figure 1). Hence, the fitted dark halos will have central densities close to zero, and the core radii are not constrained. For several galaxies with more extended rotation curves (see Figure 2), the rotation curves of the dark halos have a solid body appearance, indicating that the dark matter halos have constant density cores. In these cases, the core radius is also unconstrained and only the halo density can be determined. Only for eight galaxies with sufficiently extended rotation curves both the core radius and the central density can be determined reliably.

Figure 8 shows that the core radius r_c and the disk scale length h are correlated. The halo core radius is on average 3.5 ± 0.5 times as large as the disk scale length. A similar ratio of 3.0 ± 0.5 is found for the spiral galaxies presented in Broeils (1992a), demonstrating the similarity between the maximum disk fits of the late-type dwarf galaxies in our sample and those of spiral galaxies. Such a ratio of core radius to disk scale length of about three is expected for galaxies with a more or less flat rotation curve, as was also mentioned by van Albada & Sancisi (1986), who found that the expected ratio is about 3.5.

Zoom In Zoom Out Reset image size

For scaling relations between the optical properties of the galaxies in our sample and the relative fractions of mass in stars, gas, and dark matter, we refer to S99.

Note from Figure 1 that in the case of maximum disk models the shapes of the halo rotation curves often are similar to the shapes of the gas rotation curves. This is remarkable and indicates that the ratio of dark matter surface density to gas surface density is constant in the outer parts. This was noted for spiral galaxies by Bosma (1981) and further investigated by Hoekstra et al. (2001). A detailed study of the apparent link between the H i and dark matter densities will be the subject of an upcoming paper.

A lower limit to the contribution of the stellar disk to the rotation curve can be obtained by decreasing ϒ^R_* to its lowest possible value, while the mass model still produces a good fit to the rotation curve. For the late-type dwarf galaxies in this sample, the rotation curves can be well fitted using only a dark halo and the H i (i.e., ϒ^R_* = 0). A minimum disk fit thus does not provide a useful lower limit on ϒ^R_*, but it does provide an upper limit on the halo mass within a given radius and on the central density, and a lower limit on the halo core radius. The results for the minimum disk fits are given in Table 1. The mass models themselves are not shown because the dark halos are very similar to those for ϒ^R_* = 1, as shown in Figure 2. Details of the correlations between the minimum-disk dark halo properties and optical properties were presented in S99.

As was shown above, and as will be discussed in more detail in Section 5, the maximum-disk ϒ^R_* values are higher than expected from population synthesis models. To investigate the dark halo properties for values of ϒ^R_* that are in agreement with population synthesis models (as was described in Section 3.4), we also fit mass models for ϒ^R_* = 1. The results of the fits are presented in Table 1, and the fits themselves are shown in Figure 2.

With ϒ^R_* = 1, the relative contribution of the stellar disk to the rotation curve is tightly correlated with surface brightness, as can be seen in Figure 5. Such a tight correlation is expected for models with a fixed ϒ^R_*, because in that case the surface brightness is directly linked to the surface mass density and hence to the relative contribution of the stellar disk.

The main result of the present study is that the contribution of the stellar disk can be scaled up ("maximum disk") to explain the inner parts of the rotation curve of all but one of the 18 late-type galaxies in the present sample. These maximum disk models are similar to those for late-type spiral galaxies in which the stellar disks can also be scaled to explain the inner rotation curves (e.g., Kalnajs 1983; Kent 1987; Begeman 1987; Broeils 1992a; Verheijen 1997) and also those of the LSB galaxies presented in Swaters et al. (2000, 2003).

How well the maximum disk works is described by ξ the fractional contribution of the maximum stellar disk to the rotation curve at two disk scale lengths. A low value for ξ means that even when scaled as high as possible, the stellar disk can only explain a small fraction of the observed rotation curve. A high value of ξ (near unity) means that the stellar disk can be scaled to explain most or all of the observed rotation curves.

For the galaxies in our sample ξ = 0.88, with a dispersion around the mean of 0.09. This is nearly identical to the average for the spiral galaxies in the sample of Broeils (1992a), for which ξ = 0.86 ± 0.07, and to the average for spiral galaxies in the Verheijen (1997) sample, with ξ = 0.81 ± 0.12 (see Figure 9).

Zoom In Zoom Out Reset image size

If we assume that the values for ξ near unity mean that these dwarf galaxies physically have maximal disks (despite the high required mass-to-light ratios), then the stellar disk can explain the mass distribution over the optical parts of the galaxy, and dark matter only becomes relevant at large radii.

Under the same assumption, the average ratio of stellar-to-dark mass within four disk scale lengths is 1.0 ± 0.6 for the dwarf galaxies in our sample. For the spiral galaxies in the sample presented in Broeils (1992a), the average stellar-to-dark mass ratio within four disk scale lengths is 0.9 ± 0.4. Thus, the maximum disk fits for dwarf galaxies and spiral galaxies are very similar, with similar relative amounts of mass in the dark and luminous components within similar numbers of disk scale lengths. In these maximum disk fits, dwarf galaxies are not dominated by dark matter at all radii. Dark matter is only needed to explain the outer part of the rotation curve.

The conclusion reached here contrasts with the general notion from previous work that in late-type dwarf galaxies dark matter dominated everywhere. The average value of ξ in the comparison samples of Broeils (1992a) and Verheijen (1997) when including only late-type dwarf galaxies is lower than what we find here. For the late-type dwarf galaxies in the sample of Broeils (1992a), ξ = 0.61 ± 0.20, and for the Verheijen (1997) sample, ξ = 0.58 ± 0.09. Such lower values of ξ are found when a rotation curve rises slowly.

Indeed, the conclusion that the stellar disk cannot be scaled to explain the inner rise of the observed rotation curve is reached in many of the dark matter studies of late-type dwarf galaxies, e.g., DDO 154 (Carignan & Freeman 1988; Carignan & Beaulieu 1989), NGC 300 (Puche et al. 1990), NGC 55 (Puche et al. 1991), NGC 3109 (Jobin & Carignan 1990), NGC 5585 (Côté et al. 1991), DDO 168 (Broeils 1992a), IC 2574 (Martimbeau et al. 1994), NGC 2915 (Meurer et al. 1996), five dwarfs in the Sculptor and Centaurus A groups (Côté et al. 1997; see also Côté 1995), NGC 5204 (Sicotte & Carignan 1997), NGC 2976 (Simon et al. 2003), and NGC 6822 (Weldrake et al. 2003). Many of these studies have led to the notion that dwarf galaxies are dominated by dark matter at all radii.

There are, however, also studies that provide a different picture. Some early mass models for late-dwarf galaxies showed that the stellar disk could be scaled to explain all of the inner rise of the rotation curves, suggesting that the dark matter properties of these galaxies were similar to those of spiral galaxies (Carignan 1985; Carignan et al. 1988). Similar results were found for the four late-type dwarf galaxies in the Virgo clusters studied by Skillman et al. (1987), DDO 170 (Lake et al. 1990), DDO 105 (Broeils 1992a), NGC 1560 (Broeils 1992b), most of the late-type dwarf galaxies presented in van Zee et al. (1997), the Large Magellanic Cloud (Kim et al. 1998; Bekki & Stanimirović 2009), and most of the galaxies presented in S99 and Swaters et al. (2003).

The spread in results seen in earlier studies raises the question whether the spread reflects an intrinsic spread in the rotation curve shapes of dwarf galaxies, which then results in a range of different mass models, or whether this observed spread is due to other factors that affect the derived rotation curves.

There are well-known systematic effects that can affect the shape of the derived rotation curve and can make it appear shallower. For example: (1) beam smearing can wash out steep gradients and make the inner rotation curve appear more shallow (e.g., Begeman 1987; Paper III); (2) rotation curves derived from high-inclination galaxies are susceptible to underestimated velocities in the inner parts (e.g., Sancisi & Allen 1979; Swaters et al. 2003); (3) some methods of deriving the velocity fields (e.g., adopting the intensity-weighted mean velocity of a profile) can lead to an underestimate of the derived rotation curve (e.g., Bosma 1978; Paper I; de Blok et al. 2008). In addition, in some cases derived rotation curves may have been affected by noncircular motions (e.g., Rhee et al. 2004; Spekkens & Sellwood 2007; Valenzuela et al. 2007).

Given that beam smearing has been ignored in many of the previous studies, that some of the galaxies in those studies have high inclinations, and that the majority of those studies used intensity-weighted mean velocity fields, it is likely that the inner slopes of the rotation curves in those previous studies have been underestimated. Because such an underestimate of the inner slope limits how much the contribution of the stellar disk can be scaled in a maximum disk fit, an underestimate of the inner slope directly results in an underestimate of ξ.

It is beyond the scope of this paper to review each rotation curve in the literature in detail and assess whether systematic effects may have played a role. However, it is useful to consider as an example the well-studied prototype dwarf galaxy DDO 154. From H i synthesis observations of this galaxy a slowly rising rotation curve was derived, and mass modeling indicated that DDO 154 was almost entirely dominated by dark matter (Carignan & Freeman 1988; Carignan & Beaulieu 1989). Based on the published mass model, we estimate that ξ = 0.52, much lower than the values found for our sample. Those early observations had a resolution of 45'', and the derivation of the rotation curve was certainly affected by beam smearing, especially given the galaxy's inclination of approximately 66° (de Blok et al. 2008).

Recently, de Blok et al. (2008) presented higher resolution H i data for this galaxy. Because of the 12'' resolution, and the fact that the velocity field was derived using skewed Gaussians, which can account for asymmetries often observed in line profiles, the derived rotation curve is less prone to the systematic effects discussed here. The rotation curve derived by de Blok et al. (2008) rises somewhat more steeply than the one derived earlier by Carignan & Beaulieu (1989), and de Blok et al. (2008) find ξ = 0.62. However, the position–velocity diagram shown in Figure 81 of de Blok et al. (2008) suggests that the rotation curve in the inner 2' could be even steeper. Indeed, a new study of the kinematics of DDO 154 in the central regions based on long-slit spectroscopy and integral-field spectroscopy (R. A. Swaters et al. 2011, in preparation) shows a rotation curve that rises somewhat more steeply than the one of de Blok et al. (2008), resulting in a value of ξ = 0.80.

A key feature of the sample presented in this paper is that the systematic effects mentioned above were avoided as much as possible. The sample galaxies have inclinations between 39° and 80° and galaxies with large asymmetries were excluded. The rotation curves were derived with an interactive technique using modeling of the observed data cube to correct for the effects of beam smearing (Paper III).

For the maximum disk models, the derived mass-to-light ratio ϒ^R_* is tightly coupled to surface brightness, with lower surface brightness galaxies having higher ϒ^R_*. This is expected in order to explain the high-mass density needed for the maximum disk despite the LSB nature of the stellar disks. The values for ϒ^R_* required in the maximum disk models, reach as high as ϒ^R_* = 15, well outside the range of what population synthesis models predict (e.g., Bell et al. 2003; Zibetti et al. 2009). The precise value of ϒ^R_* as derived from such population synthesis may be uncertain by a factor of 2–4 (Portinari et al. 2004), and possibly even higher (Bershady et al. 2010). However, even considering these uncertainties, values as high as ϒ^R_* = 15 appear unlikely, as they would require a strong bias in the initial mass function toward low-mass stars, or a large fraction of the stellar mass being locked up in stellar remnants.

The presence of a nonstellar component of mass with a distribution similar to that of the stars, e.g., in the form of cold molecular gas as suggested by Pfenniger & Combes (1994) and Revaz et al. (2009), could provide an explanation.

Alternatively, the stellar disks may be somewhat less than maximal. Independent measurements of ϒ_* from stellar velocity dispersions in spiral galaxies suggest that stellar disks have a submaximum contribution to the rotation curve, with a peak contribution of the stellar disk to the rotation curve of around 65% (Bottema 1993; K. B. Westfall et al. 2011, in preparation). At this level, the stellar disk still determines the inner shape of the rotation curve, i.e., one would still expect to find ξ ∼ 1. If the stellar disks would also contribute around 65% in late-type dwarf galaxies, this would reduce ϒ^R_* by almost a factor of 2.5. This would bring the mean and maximum ϒ^R_* of the galaxies in our sample to 3 and 6, respectively, still above what population synthesis models predict.

If ϒ^R_* is assumed to be near unity, as predicted by stellar population synthesis models (see Section 3.4), then well-fitting mass models are found as well. In this case, all galaxies except UGC 7577 are dominated by dark matter. Moreover, the relative contribution of dark matter increases toward lower surface brightness, and the contribution of the disk decreases correspondingly. Thus, for models with ϒ_* ∼ 1, we reach the same conclusions as are usually found for dwarf and LSB galaxies (e.g., Carignan & Beaulieu 1989; Broeils 1992a; de Blok & McGaugh 1997).

There is, however, a critical difference. In many of those previous studies, as described above, low values for ξ were derived. For our sample, however, we find a value for ξ that is near unity. This not only means that the stellar disk can be scaled to explain most of the inner parts of the observed rotation curve. If ϒ^R_* ∼ 1, this also means that the rotation curve of the stellar disk alone must have the same shape as that of the dark matter that dominates the potential. Otherwise, ξ could not be near unity.

In spite of the uncertainties in ϒ^R_*, one result from this analysis is firm: the fact that the contribution of the stellar disk can be scaled up to explain the inner parts of the rotation curve indicates that the shapes of the rotation curve of the stellar disk and of the observed rotation curve are similar. There appears to be a strong coupling between the distribution of the stars and that of the total mass (see also Sancisi 2004; Paper III).