Morphological Dependence of the Tully-Fisher Relation

The question of a possible dependence of the Tully-Fisher relation on morphology expressed as a Hubble type has been first raised by Roberts roberts75. It received drastic support from Rubin et al. rubin80,rubin82,rubin85 who measured H rotation velocities in a magnitude-limited sample of 21 Sc, 23 Sb and 16 Sa galaxies. At a fixed rotation velocity V, Rubin et al. rubin85 claimed that Sc galaxies were 2 magnitudes brighter in absolute B magnitude than Sa's and 0.5 magnitude brighter than Sb's and that the Tully-Fisher relation had a slope of 10 nearly independent of Hubble Type. This result deserves considerable attention as such a strong morphological dependence could be used to argue that any systematic deviation from the local Tully-Fisher relation seen at intermediate redshifts might be due to a wide range of local galaxy properties and strong selection effects.

De Vaucouleurs et al. devauc82 used a sample of 173 galaxies with revised morphological type T between 2 and 10 and heliocentric recession velocities between 500 and 2000 km s to find least-squares solutions to the equation

where is the total absolute magnitude corrected to face-on, = and is the revised morphological type. They obtained three types of solutions: (1) all free parameters, (2) fixed slope = 5 and (3) fixed slope = 10. The free parameter solutions yielded a value of 3 and a value of of zero (within the mean error). A fixed slope of 10 introduced a morphological type dependence with values of significantly different from zero and as high as 0.377 (with a mean error = 0.056).

Aaronson and Mould aaron83 in their study of 300 nearby ( < 3000 km s) galaxies raised a number of important issues. They pointed out that treating magnitude as the independent variable guarantees the same width at fixed magnitude for a volume- or magnitude-limited sample, but it does not guarantee the correct slope. For example, they found that regressing on magnitude for each individual type produced steeper slopes which caused an artificial spread in velocity width at a fixed magnitude i.e. a false type dependence. They obtained the best slope representation of their sample in a least squares sense by weighting the fit according to the distribution spread in each coordinate (in addition to measurement errors) and not by treating either absolute magnitude or velocity width as an independent variable. In the B band, they found no dependence of the slope on morphological type and a slight dependence of the zero point ( 0.4 mag going from Sa to Sc with the Sc zero point being fainter) on morphological type.

Many other studies [\protect\astroncitePierce and Tully1988][\protect\astronciteBothun et al.1984][\protect\astronciteIchikawa and Fukugita1992][\protect\astronciteFouqué et al.1990] argued against a strong morphological dependence of the Tully-Fisher relation. Bothun et al. bothun84 found that the mean line width of their sample of 20 Sc I galaxies ( = 467 20 km s) was virtually identical to the value obtained for the Sb sample of Rubin et al. rubin82. Pierce and Tully pierce88 found a slight type dependence of at most 0.4 mag in volume-limited samples in the Virgo and Ursa Major galaxy clusters.

How can one reconcile the Rubin et al. rubin85 result with the above works? The source of the discrepancy may reside in the internal absorption corrections applied to different samples. Internal absorption corrections are very important and poorly-understood in the B band, and this is why much recent work has been devoted to an infrared Tully-Fisher relation. Rubin et al. rubin85 applied the same internal absorption correction ( where a/b is the axial ratio of the galaxy image) to all morphological types. However, Kodaira and Watanabe kodaira88 conducted a statistical study of internal absorption in a sample of 184 disk galaxies and found that internal absorption was more important in late-type (Sc) galaxies because they contained more dust. Any error in the internal absorption correction should therefore have more of an impact on late-type galaxies. It is interesting to note that, in the sample of Rubin et al. rubin85, the dispersion in absolute B magnitude increases going from Sa to Sb galaxies and is thus correlated with the amount of internal absorption given by Kodaira and Watanabe kodaira88. Since the different subsamples have different intrinsic dispersions and the Rubin sample is magnitude-limited, it is conceivable that they may have been differentially affected by the Malmquist bias. Since Malmquist bias would lead to steeper slopes, one would expect the slope of the linear regression for the Sc subsample to be larger than for the Sa and Sb subsamples. The slopes obtained by Rubin et al. rubin85 for their Sa, Sb and Sc subsamples are 9.95, 10.2 and 11.0 respectively. As mentioned previously, an overestimate of the slope will introduce an artificial type dependence [\protect\astronciteAaronson and Mould1983][\protect\astronciteVaucouleurs et al.1982].

Small sample sizes preclude one from settling the issue of morphological dependence and the Tully-Fisher relation. The discrepancy between the Rubin result and other studies may only be symptomatic of large variations in local galaxy properties. These variations can be brought to light with a truly large local sample. Mathewson et al. mathew92 measured total I magnitude and H rotation velocity for a sample of 1355 local galaxies with morphological classification. Figure shows log versus absolute I magnitude for galaxies with morphological types between T = 1 (Sa) and T = 8 (Sdm). Although Mathewson et al. provided total I apparent magnitudes corrected for internal absorption, their uncorrected I magnitudes were used to derive absolute magnitudes. This is to insure that the Mathewson et al. data can be later compared with the CFHT internal kinematics data for which CNOC provided uncorrected apparent magnitudes. Figure is a beautiful illustration of the local Tully-Fisher locus and its dispersion. There are no systematic shifts between loci of different morphological types, but it is obvious that certain types (e.g. T = 6) display larger dispersions. Since galaxy heliocentric velocities were used to calculate absolute I band magnitudes, some of the scatter in Figure actually comes from galaxy peculiar velocities. It should therefore be kept in mind that this scatter will lead to conservative luminosity evolution estimates whenever the Mathewson et al. relation is used as a reference.

The absence of a type dependence in the I band does not necessarily apply to the B band since galaxies have different BI colors. We can convert Figure to the B band using the BI colors of local galaxies. Frei and Gunn frei94 have generated tables of colors and k-corrections for different Hubble types. Their local (z = 0) BI colors are 1.72 (Sbc, T = 34), 1.32 (Scd, T = 67) and 1.16 (Im, T = 89). These color correction were applied to the data shown in Figure to build Figure . Rubin et al. rubin85's Sa and Sb Tully-Fisher relations are also plotted for comparison. There are no systematic shifts between loci of different morphological types at the 2.0 mag level as claimed by Rubin et al. rubin85.

Figures and shed some much needed light on the Rubin et al. rubin85 result. First, the Rubin relations have the wrong slope. They are too steep (in the magnitude versus log V sense). The Rubin sample is biased against fast rotating Sc galaxies or alternatively, it is biased against faint Sc galaxies at a fixed rotation velocity. There is a significant number of Sb and Sc galaxies in the Mathewson et al. mathew92 sample lying right on the Rubin Sa relation even though there is only one Sa galaxy in the Mathewson et al. mathew92 sample. The key here is the larger dispersion displayed by Sc galaxies. This dispersion coupled to small sample size and selection effects probably lead to an artificial type dependence of the Tully-Fisher relation. However, the absence of type dependence should not obscure the fact that local late-type galaxies do show a large scatter and that this scatter should be carefully considered in claims of luminosity evolution at intermediate redshifts. This issue is the topic of the next section.