Van Albada and Sancisi (1986) pointed out that mass modelling based on the assumption that the M / L ratio is constant as function of radius in the disk contains a degeneracy: it is a priori not clear whether a maximum disk, i.e., a disk so massive that its rotation curve fits the inner parts of the observed one without overshooting it, is the correct answer. Already it is not entirely justified to assume that the M / L ratio is constant throughout the disk, even though this is customary, since if there is a colour gradient in bright spirals, it is usually in the sense that the outer parts are bluer. Yet another problem is that the assumption of maximum disk generally leads to haloes which are cored, and thus do not follow a Navarro-Frenk-White (NFW) model (e.g., Navarro 1998).
This disk-halo degeneracy is a serious problem which is even now under debate. Relating a value of the M / L ratio to a disk colour, and working out whether “reasonable” stellar populations can then be assumed, does not appear entirely satisfactory, due to possible variations in the initial mass function (IMF). Various groups have thus tried to marshall dynamical arguments to break the degeneracy, but the answers are mixed. Mechanisms of spiral structure generation, in particular the swing amplifier mechanism (Toomre 1981), depend on the ratio of disk mass to halo mass. Athanassoula et al. (1987) used this in detail to set a range of allowed values of the disk/halo mass ratio, by remarking that most spirals are dominated by an m = 2 component, and thus requiring that an m = 2 spiral be allowed to amplify, while at the upper end suppressing an m = 1 component. This is further discussed in Bosma (1999) and illustrated in Fig. 6 (top right panel).
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Figure 6. Top left: disk mass fraction as function of the maximum velocity of the rotation curve, determined with several methods. “BR13” (Bovy and Rix 2013) indicates the value for our Galaxy. “Bars” concern the determination using gas flow models in barred spirals (see text). “KSR03” concern five galaxies studied by Kranz et al. (2003). Black filled stars concern the results from Athanassoula et al. (1987) for their “maximum disk with no m = 1” models, except for two vertical lines at Vmax = 114.0 and 280.0 km/s which indicate also the “no m = 2” models. For the DiskMass project, the results are taken as in Courteau and Dutton (2015), but the error bars are replaced by the area spanned by them. A similar representation has been done for the SWELLS survey (Barnabè et al. 2012; Dutton et al. 2013). Top right: mass models for NGC 3198 according to the method described in Athanassoula et al. (1987), and shown in detail in Bosma (1999); Bottom left: colour-magnitude diagrams calculated with IAC-STAR (Aparicio and Gallart 2004) in a manner similar to that used in Aniyan et al. (2016). Bottom right: spectra obtained by Westfall et al. (2011) in the Mgi region, and by Bershady et al. (2005) in the Caii region (reproduced by permission). |
Fuller dynamical modelling has been done for a number of galaxies. Kranz et al. (2003) calculated spiral structure models based on potentials derived from K′-band photometry and compared the gas flow in these with the observed velocity fields for a sample of five galaxies. Similarly, for bars, the gas flow can be calculated in a potential derived from imaging in the near- or mid-infrared. Seeking to fit the amplitude of the jump in radial velocity across a dust lane, which outlines the location of a shock in the flow, will constrain the M / L ratio of the disk (Lindblad et al. 1996; Weiner et al. 2001; Weiner 2004; Zánmar Sánchez et al. 2008). For the face-on barred spiral NGC 1291, Fragkoudi et al. (2016) calculated a range of models trying to fit the shape of the dust lane. Most of the barred spiral models require close to maximum disk, while for the spiral models a range of values depending on the maximum rotational velocity has been found, as shown in Fig. 6 (top left panel).
The study of the lensing galaxy associated with the quad-lens Q2237+0305 done by Trott et al. (2010) shows that at least the central part of that galaxy needs a maximum M / L value. Note that that galaxy is barred, and its bulge thus presumably of the boxy/peanut type. Barnabè et al. (2012) and Dutton et al. (2013) also find maximum bulges in the SWELLS survey. Dutton et al. (2013) suggest that the IMF in bulges is more like the Salpeter one, and in disks is closer to the Chabrier one.
The major method favouring non-maximum disks is based on the analysis of stellar velocity dispersions. This is not straightforward, since one has to assume either a disk thickness when the galaxy is face-on, or a ratio of radial to vertical velocity dispersion when the galaxy is edge-on. The results of Bottema (1993, 1997), Kregel et al. (2005) and the more recent DiskMass project (Bershady et al. 2011; Martinsson et al. 2013), and references therein) all point to sub-maximum disks. Yet doubts have been expressed, as in Bosma (1999), from which we quote
... as argued by Kormendy (private communication, see also Fuchs's contribution — Fuchs 1999), the influence of younger stellar populations could result in lower measured velocity dispersions.
Aniyan et al. (2016) recently quantified this concern, and I show in Fig. 6 (lower left panel) a variation on their principal result. As argued in Aniyan et al., the stellar populations in the blue-visible region, where the Mgi velocity dispersion is measured, are heavily contaminated by the presence of young stars, which might have lower velocity dispersions than the old stars. On the other hand, in the infrared (e.g., K-band), the light of the red giant branch stars dominate. Hence there is a mismatch between the stellar populations used to measure the velocity dispersion and those used to estimate the disk scale height, with as a result that the M / L ratio of the disk is underestimated. Calculations by Aniyan et al. (2016) show that this can roughly explain the difference between the results of the DiskMass project (sub-maximum disks), and those from several other dynamical estimators (maximum disks).
Although the authors of the DiskMass project were aware of a stellar population effect, witness the extensive discussions of this in, e.g., Bershady et al. (2005, 2010a, b) and Westfall et al. (2011), they argue clearly for the use of the Mgi region as the preferred region for doing velocity dispersion work, since the Caii region has 1) larger intrinsic line widths, 2) a higher background, and 3) more scattered light (cf. Bershady et al. 2005, and the spectra shown in Fig. 6, lower right panel). Most galaxies they discuss have been observed only in the Mgi region, and, in the few cases where data at both wavelength regions were available, no difference was noticed.
It should be possible, however, to investigate a possible systematic effect of stellar population differences on the velocity dispersions further, since even in the i-band the older stellar populations are substantially more prominent than in the g-band (see Fig. 6, lower row, second panel from left). There are now several spectrographs being built which will have a setup allowing the simultaneous measurement of the velocity dispersions of the same galaxy in both the Mgi region and the Caii region. In particular, the WEAVE spectrograph (Dalton 2016) has high spectral resolution, and can thus suitably be used to test whether Caii-derived velocity dispersion measurements are systematically larger than Mgi-derived ones in face-on spiral galaxies.