A FIRST ATTEMPT TO CALIBRATE THE BARYONIC TULLY–FISHER RELATION WITH GAS-DOMINATED GALAXIES

To be included in the sample, each galaxy must satisfy certain quality criteria. The first criterion for a galaxy to be included in the sample was that it had to have rotation curve data (line widths are not accurate enough) and its rotation curve had to reach a constant velocity. This flat velocity (v_f) is the velocity³ used in the BTF relation. Many apparently gas-rich galaxies fail to meet this criterion. It is common for the observed region of the rotation curve to rise continuously with no discernable flatness to be found. This may simply be because the measurements do not extend far enough in radius to see v_f. This is a common issue with dim dwarf galaxies, for which obtaining reliable data is challenging.

A rotation curve is considered to meet the flatness criterion if the measured velocity changes by a small amount over a defined range in radius. Quantitatively, the difference in velocity between that at three disk scale lengths and the last measured point had to be less than 15%. If measurements extended past four disk scale lengths, then the difference between the velocity at four disk scale lengths and the last measured point had to be less than 10%. Figure 1 provides examples of usable versus unusable rotation curves.

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We adopt the inclination given with the source data where possible. This is usually from a tilted ring fit, but is sometimes based on observed axis ratios. In eight cases, the inclination was not given, so we estimated it with

$$\begin{equation} \sin {i}=\sqrt{\frac{1-(b/a)^2}{1-0.15^2}}. \end{equation} \tag{ 1 }$$

The values a and b are the semimajor and semiminor axes of the galaxy. We adopt 0.15 as the intrinsic thickness of a disk galaxy seen edge-on (Kregel et al. 2004).

We select galaxies with inclinations i > 45°. Since v_f = v_obs/sin i, the uncertainty becomes large for smaller inclinations. As is often the case in the literature, inclination uncertainties were not reported. Given the adopted lower limit on the inclination, a reasonable and conservative estimate of the uncertainty on the inclination is 5°. Choosing the minimum inclination of 45° ensures the relative error in v_f from the inclination stays below 10%. This way, the uncertainty in the inclination contributes to the error budget no more than most other sources of uncertainty. Another option would have been to limit the inclination to >60°, reducing the relative error to less than 5%. However, this reduced the size of the galaxy sample significantly, and the number of available galaxies is already constrained heavily by the M_g > M_s requirement. Limiting the inclination to this degree would have made the results excessively vulnerable to the inadequacies of small number statistics (see Section 4.3). As a third option, we could have allowed all galaxies with inclinations >30°, but this allowed the relative uncertainty to be as large as 30%, making it a major contributor to the uncertainty in v_f.

No upper limit on inclination was adopted. For galaxies with very high inclinations, the shape of the rotation curve can be affected by internal extinction. However, the outer, flat portion, which is the key quantity used in the BTF relation, is usually unaffected (Spekkens & Giovanelli 2006). Removing galaxies with i > 80° from our sample has a negligible effect on the results (<1%). We discuss the implications of this inclination requirement in Section 4.3.

The primary criterion for inclusion in our fit to the BTF relation is gas dominance: M_g > M_s. Some galaxies always satisfy this criterion while many never do. A few skirt the boundary, depending on the stellar mass estimator employed (Section 3).

Table 1 provides basic information about the galaxies used in this study. Both star- and gas-dominated galaxies are tabulated in order of increasing gas mass. Though the former are not used in determining the BTF relation, we do use them to test the derived relation and stellar population models, so they are included here. We also include several low-inclination galaxies. Again, these are not used in the determination of the BTF relation, but are used to test the implications of the i > 45° limit (see Section 4.3).

Many distances to galaxies were already provided in the papers in which these galaxies were found, but a search was always made for more modern or reliable distance measurements. The most common ways the distances were determined were with a Hubble flow model (H₀), the red giant branch method (TRGB), the Cepheid method (CEPH), and the brightest star method (BS). Distances found with Cepheids or the red giant branch were always used if they were available, since they have been found to be more reliable indicators of a galaxy's distance (Karachentsev et al. 2004). Unless otherwise noted, the uncertainties for the red giant and Cepheid distance measurements were taken to be 10%. Likewise, the uncertainty for the brightest star method was taken as 25%. When redshifts are used as the distance indicator, we assume H₀ = 75 km⁻¹ s⁻¹ Mpc⁻¹. The uncertainty in this case is taken from the spread in various flow models:

$$\begin{equation} \sigma _D = \frac{D_{\rm max}-D_{\rm min}}{2}. \end{equation} \tag{ 2 }$$

Here, D_max and D_min are the maximum and minimum estimates of the distance from the Hubble flow models provided by NED.⁴ It should also be noted that a number of galaxies are part of the Ursa Major cluster (Verheijen 2001). In this case, the uncertainty is that in the cluster distance (Tully & Pierce 2000). Only a few of these galaxies are gas rich, so most do not contribute to the derivation of the BTF relation.

The uncertainty in v_f varies from case to case. It depends on how well the flat velocity could be distinguished from the rest of the rotation curve, along with the uncertainty in the velocity measurements themselves. The errors predominantly fell from 1% to 8%, but got as high as 25% for some cases, and 60% for the most extreme (CamB).

The gas mass estimates were derived from the mass of neutral hydrogen (H i), which follows the flux from the 21 cm hydrogen emission line. We estimate the gas mass as $M_g = \frac{4}{3}M_{{\rm H}\,\hbox{\scriptsize {\sc i}}}$. The factor of 4/3 comes from the fact that hydrogen gas makes up roughly 75% of the universe. We ignore molecular gas, which is not measured in most of the galaxies of concern here. In the cases where it is known (Helfer et al. 2003), it is always outweighed by either the stars or the atomic gas, and almost always by both. The work of Young & Knezek (1989) suggests that this should be generally true for galaxies in our sample. The uncertainty in the value of M_g is dominated by the distance uncertainty, but there is also some error from source noise in the measurement.

We adopt the H i masses reported by the original observers. In most cases these are synthesis observations (VLA⁵ or WSRT) or single dish maps (e.g., with the Giant Metrewave Radio Telescope (GMRT)). It is common for H i fluxes to be reported without uncertainties. We adopt a typical error of 30% based on the variance between the synthesis observations and single dish estimates. This value, coupled with the uncertainty due to distance, yields the total uncertainty in M_g,

$$\begin{equation} \sigma _{M_g}=\sqrt{\left(\frac{2\sigma _D}{D}M_g\right)^2+(0.3\,M_g)^2}. \end{equation} \tag{ 3 }$$

There is no clear indication in the data we have collected that the synthesis observations detect systematically less flux than single dish observations.

As with the gas mass, the uncertainty in luminosity is dominated by that in the distance, and again the uncertainties are seldom reported. We adopt a 10% uncertainty in the luminosity to reflect the likely disparity arising from many different measurements, instruments, and filters used in data collected from diverse sources in the literature. Therefore, the total uncertainty including the affect of distance is given by

$$\begin{equation} \sigma _{L}=\sqrt{\left(\frac{2\sigma _D}{D}L\right)^2+(0.1\,L)^2}. \end{equation} \tag{ 4 }$$

For the color, the B – V band was used if available. Otherwise, B – R was used. In a vast number of cases, color uncertainties are not reported. We assume them to be negligible. The consequences of this assumption are discussed in Section 5.2. All sources of uncertainty are added in quadrature in the analysis.

We have intentionally limited our sample to gas-dominated galaxies with reasonably high quality data. These are predominantly, though not exclusively, dwarfs. Ideally, one would like a sample that is not biased toward or against any particular type of galaxy. This is an impossible ideal already broken by the distinct Tully–Fisher and fundamental plane relations for disk and elliptical galaxies. Among rotating disk galaxies, typical Tully–Fisher samples are numerically dominated by bright galaxies with v_f > 100 km s⁻¹ simply because they are easier to see. This type of bias is unavoidable.

Perhaps the most remarkable aspect of the BTF relation is the extent to which galaxies of very different types appear to obey it. The relation is continuous over five decades in stellar mass (McGaugh 2005a), from the fastest rotators (∼300 km s⁻¹) to the slowest (∼20 km s⁻¹). There is no persuasive evidence for a second parameter in the relation, as might be expected for surface brightness (Aaronson et al. 1979; Zwaan et al. 1995; McGaugh & de Blok 1998) or scale length (Courteau & Rix 1999; McGaugh 2005b). If disk galaxies do indeed all fall on a single BTF relation, as appears to be the case, then it hardly matters what portion of the relation we sample in order to calibrate it. We do of course wish to sample as broadly in velocity as possible in order to constrain the slope. Nonetheless, if the relation is both valid and universal, calibrating it with lower velocity gas-rich dwarfs is no worse than calibrating it with high velocity star-dominated giants as is usually done. Indeed, it has the advantage of providing an absolute normalization to the mass scale.

We have made six separate stellar mass estimates per galaxy. These are hereafter named for the population model-IMF combination utilized (Table 4). The idea is to only use the gas-dominated galaxies to determine the BTF relation. For each subsample, only galaxies with gas mass greater than stellar mass (M_g > M_s: Tables 2 and 3) are used to make fits.

Table 4. BTF Fit to Gas-Dominated Galaxies

Subsample	N	xv|M	$\sigma _{x_{v|M}}$	Av|M	$\sigma _{A_{v|M}}$	χ2ν,v|M	xM|v	$\sigma _{x_{M|v}}$	AM|v	$\sigma _{A_{M|v}}$	χ2ν,M|v	xbis	Abis	χ2v,bis
Portinari-Kroupa	23	3.77	0.22	2.08	0.42	1.28	4.11	0.25	1.43	0.47	1.18	3.93	1.78	1.25
Portinari-Salpeter	14	3.59	0.40	2.44	0.73	1.42	4.37	0.50	1.02	0.91	1.46	3.94	1.79	1.46
Portinari-Kennicutt	26	3.74	0.21	2.14	0.41	2.01	4.33	0.26	0.99	0.50	1.85	4.01	1.62	1.99
Bell-Scaled Salpeter	23	3.77	0.20	2.09	0.39	1.41	4.09	0.23	1.47	0.45	1.31	3.93	1.80	1.37
Bell-Kroupa	26	3.72	2.17	0.20	2.30	0.40	4.36	0.94	0.25	2.10	0.49	4.01	1.61	2.27
Bell-Bottema	36	3.55	2.45	0.14	2.02	0.29	3.96	1.63	0.16	2.06	0.32	3.74	2.06	2.04

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The number of galaxies in each subsample differs because of the different stellar mass estimators. Under one population model, a galaxy might have more gas mass than stellar mass, while under another model, the opposite could be the case. The final number of galaxies in each subsample is given in Table 4.

For comparison, the median mass-to-light ratios (in M_☉/L_☉) of the subsamples are 0.61 for Portinari-Kroupa, 0.87 for Portinari-Salpeter, 0.51 for Portinari-Kennicutt, 0.67 for Bell-Scaled Salpeter, 0.47 for Bell-Kroupa, and 0.28 for Bell-Bottema. Note that in all cases, the typical mass-to-light ratio is less than unity. This is perhaps not surprising in that galaxies are more likely to be found to be gas dominated if they have low ϒ. However, we also note that some prescriptions seem to give implausibly low mass-to-light ratios. This is particularly true of the Bell–Bottema case, which leads to the inclusion of many galaxies that would not otherwise be considered gas dominated (e.g., NGC 2998).

We assume the BTF relation is linear in the logarithm. If there is any curvature in the intrinsic relation, it is not apparent in our data. Therefore, fits of the form

$$\begin{equation} \log {M_b}=x\log {v_f}+A \end{equation} \tag{ 8 }$$

were made to each of the six gas-dominated subsamples. We used the OLS bisector linear fit method to determine the BTF relation coefficients. Forward and reverse fits were first conducted following the standard χ² minimization routine outlined in Press et al (1992, Chapter 15.3), taking into account uncertainties in both M_b and v_f. Using the coefficients from these fits and their respective uncertainties, the final OLS bisector slope and intercept were determined following the technique in Isobe et al. (1990).

The hope and intent was that the choice of stellar population model would have a little effect on the total baryonic mass, and in turn, on the derived BTF relation. Table 4 shows the results of the fit for each gas-rich subsample. One can immediately see that indeed the population model does not have a drastic effect on the BTF relation.

As a further test, we considered the selection criterion M_g > 2 M_s to further suppress any dependence on the choice of the population model. The results are indistinguishable. Rather few galaxies survive this very restrictive selection criterion, so we do not consider this case further.

The choice of stellar mass estimator has an impact on the quality of the fits. Three cases (the Portinari-Kroupa, Portinari-Salpeter, and Bell-Scaled Salpeter subsamples) have reasonable reduced χ² values (Table 4). These are shown in Figure 2. The other cases look similar, albeit with worse formal χ². In order to determine a single calibration for the BTF relation, a weighted average is made of the coefficients from the three illustrated subsamples, with weights given by

$$\begin{equation} w=\frac{1}{\mid \chi ^2-1 \mid }. \end{equation} \tag{ 9 }$$

The resulting BTF relation is

$$\begin{equation} \log M_b = 3.94 \log v_f + 1.79. \end{equation} \tag{ 10 }$$

The formal random uncertainty in this OLS bisector fit is ±0.07 in the slope and ±0.26 in the intercept. In addition to the random errors, we attempt to estimate the systematic uncertainties. The choice of the population model still plays some role in constraining the intercept of the BTF relation, as the stellar mass is minimized but not zero. Likewise, there is some systematic uncertainty in the slope that seems to depend on the inclination cutoff. This will be discussed in further detail in Section 4.3. For now, the estimated magnitude of the systematic uncertainties are ±0.08 in the slope and ±0.25 in the intercept.

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We have investigated how our sample selection criteria affect the result. The choice of the stellar mass estimator, the line between star and gas domination, and the velocity measure employed (v_f vs. v_max) all make little difference. The limit on inclination, i > 45°, has a mild effect on the slope, discussed further below. While variations in the sample selection makes no net systematic difference to the intercept, the range of variation does give some handle on the possible amplitude of the systematic error.

The formal error in the intercept has been determined by bootstrap resampling of the data. Approximately 95% of the data fall within 2σ of the best-fit value. As is usually the case for astronomical data, it is rather harder to estimate the systematic uncertainty. By examining the variation in the zero points from the fits to the various subsamples, forward and reverse OLS as well as the bisector method, we estimate a systematic uncertainty in the intercept of ±0.25. This covers the full range of variation, so is rather conservative.

For the uncertainty in slope, there does seem to be a mild but systematic dependence on the imposed inclination limit. Some cutoff on the inclination of galaxies included in the sample is necessary, as the velocity depends on 1/sin(i). Faces on galaxies have very large uncertainties due to inclination. Inclusion of such data will bias the determination of the slope Jefferys (1980), so we impose a limit to combat this effect. This inclination limit is chosen to provide a middle ground between data quality and sample size (see Section 2.2). As a check, we vary the inclination limit imposed on the gas-dominated subsamples, letting the minimum range from 0° to 70°. The BTF relation is derived for each inclination cut. The bisector slope and sample size of each fit are shown in Figure 3.

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The slope is relatively stable from no cut to an inclination limit of i > 55°, varying between 3.76 and 3.95. For the most part, the magnitude of these variations is comparable to the uncertainty in the fits. For higher inclination cuts, the slope becomes much larger. The cause of this is the drastic reduction in sample size, as seen in Figure 4. For the last inclination cut, only 15% of the sample remains, leaving only 3–6 galaxies depending on the population model used. Therefore, we reach a point where restricting the inclination too much simply leads to small number statistics. This effect sets in around i_min ≈ 60°, where the slope strongly deviates from the mean, and the uncertainty drastically increases. This is most clearly seen in the transition from i_min = 60° to i_min = 65°. Over this range, the sample size is effectively cut in half. This leaves one galaxy with a low relative velocity, but it has a very large uncertainty, effectively leaving nothing to constrain the low end of the fit. Choosing i > 45° keeps us well away from this region where small number statistics issues like this one take over. Accounting for the variation in best-fit slope with the choice of inclination limit gives a systematic uncertainty of 0.08, discounting the sharp change due to small numbers at very large inclination limits.

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The stellar mass-to-light ratios determined from the BTF relation provide a means of testing the population synthesis models. These models utilize our knowledge of stellar evolution and prescriptions for galactic star formation histories to predict the color-ϒ relation. We utilize dynamical information in the form of the BTF relation to estimate the mass-to-light ratios of star-dominated galaxies.

The methods and data involved in this process are distinct and very nearly independent. They are not completely independent as population synthesis models have been used in the BTF calibration. However, as presented in Table 4, the differences in stellar mass given by the different population models have only a negligible impact on the BTF. By construction, the gas mass dominates the calibration. Consequently, this may be as close as it is possible to get to an independent test of the predictions of population synthesis models.

We detect a clear correlation between mass-to-light ratio and color. This is expected in the models of Portinari et al. (2004) and Bell et al. (2003), which are consistent in predicting a slope of s_B = 1.7 for the color-ϒ relation. Our data are consistent with this slope (Figure 4). This appears to be a remarkable confirmation of a basic prediction of population synthesis models.

The main uncertainty in population models is from the IMF. Here we apply our calibration of the BTF relation to derive our own color-ϒ relation. This provides an independent constraint on the IMF.

We fit the IMF normalization q_B of Equation (5) to the data in Figure 4. The slope of the data is consistent with the expectation of the models, within the uncertainties. To provide a direct comparison to the IMF of the models, we fix the slope and fit only for the normalization q_B. In this process, we utilize only those star-dominated galaxies with B – V colors that were not part of the three subsamples used to calibrate the BTF relation (Section 2.4).

The BTF-derived IMF coefficient provides a constraint on the IMF. We find q_B = −0.94 ± 0.20 such that, in the mean,

$$\begin{equation} \log {\Upsilon }=1.7({ B-V})-0.94. \end{equation} \tag{ 12 }$$

The BTF-derived values of ϒ and the fit (Equation (12)) are shown in Figure 4.

The constraint derived here on the IMF agrees most closely with the Bell-Scaled Salpeter or Portinari-Kroupa models. This is not a strong statement, as the uncertainties allow many of the IMFs considered by both Bell et al. (2003) and Portinari et al. (2004). However, it does suggest that the IMF is normal, and not outrageously different from that assumed in population synthesis models. Indeed, the more extreme IMFs, both heavy (Portinari-Salpeter) and light (Bell-Bottema), are disfavored.

The scatter in ϒ is 0.20 dex. Roughly half of this can be attributed to observational uncertainties, leaving an intrinsic scatter of ∼0.14 in the mass-to-light ratio. This is consistent with the intrinsic scatter expected by Bell et al. (2003) and Portinari et al. (2004).

As mentioned in Section 2.3, we take the color uncertainties to be negligible as they are widely unreported. We test the implication of this assumption by repeating the linear fits to find q_B but giving the B – V values uniform uncertainties. The IMF normalization q_B goes down by 0.02 for σ_B−V = 0.05, and then drops by 0.04 for every addition of 0.05 to the color uncertainty up to σ_B−V = 0.2, the largest value considered. It seems unlikely that the average uncertainty in the colors would be larger than this, so the effect is likely to be modest. However, we cannot exclude the possibility of a larger effect if the galaxies with extremal colors also have large uncertainties.