WEIGHING GALAXY DISKS WITH THE BARYONIC TULLY–FISHER RELATION

The term "Tully–Fisher relation" has been used to mean a variety of subtly different things. Most commonly, the Tully–Fisher relation is posed as a relation between absolute magnitude and line-width (Tully & Fisher 1977) and employed as a distance indicator (e.g., Sorce et al. 2012). Here we construct several distinct flavors of Tully–Fisher relations from the assembled data in hopes of identifying the underlying physical basis of the relation.

We start with the relation between luminosity and asymptotic rotation velocity in each band. This is analogous to the common Tully–Fisher relation, but differs from it in that V_f from extended, resolved rotation curves is used rather than line-widths. While similar, these are not identical measures of the characteristic rotation velocity of a galaxy. The primary consequence is a systematic difference in the slopes of the fitted relation. Line-widths are sensitive to the presence of inner humps in rotation curves, which are prominent in luminous galaxies but largely absent in faint ones (see, e.g., Verheijen 2001; Noordermeer & Verheijen 2007). This causes the L–W relation to be shallower than the L–V_f relation, which we will refer to as the LTFR.

Next, we construct the STFR. We employ stellar population models to estimate the stellar mass of each galaxy based on their observed luminosities and the color–mas-to-light ratio relation (CMLR) of McGaugh & Schombert (2014). We then construct the STFR implied independently by the luminosity observed in each band pass. If the our semi-empirical CMLR are correct, then each band's LTFR should transform into the same STFR.

Finally, we construct the BTFR. This is a relation between rotation velocity and baryonic mass (McGaugh et al. 2000; Bell & de Jong 2001; Verheijen 2001; McGaugh 2005; Pfenniger & Revaz 2005), where now the mass includes all observed forms of baryonic mass, stars, and gas: ${{M}_{b}}={{M}_{*}}+{{M}_{g}}$. Again, if all is well, the same relation should be obtained independently from each band. We can also compare the result to that obtained independently from fitting gas-rich galaxies, which is very nearly independent of the stellar mass estimate (McGaugh 2011, 2012). The LTFR, STFR, and BTFR are shown in Figure 1.

Zoom In Zoom Out Reset image size

The use of V_f  as the velocity measure rather than line-widths leads to systematically steeper slopes for all flavors of the relation. For the BTFR, McGaugh (2012) found a slope x = 3.41 ± 0.08 when line-widths are employed as the velocity measure, consistent with the 3.5 ± 0.2 reported independently by Zaritsky et al. (2014). In contrast, the slope of the BTFR is very close to 4 when V_f is employed as the measure of rotation velocity (Stark et al. 2009). It also consistently yields relations with lower scatter. For example, McGaugh (2012) estimates an irreducible scatter of ∼0.2 dex in mass in the BTFR when line-widths are employed, but negligible intrinsic scatter with V_f.

For the various possible ordinates of the Tully–Fisher relation, we fit relations of the form

$$\begin{eqnarray}&&{\rm log} (\ell )={\rm log} A+x{\rm log} ({{V}_{f}}).\end{eqnarray} \tag{ 2 }$$

Here A is the normalization of the relation (${\rm log} A$ being the intercept in the log–log plane) and x is the slope. The symbol ℓ represents any of the luminosity, stellar mass, gas mass, or baryonic mass, while V_f  is of course the rotation velocity measured in the outer portion of the resolved rotation curve.

To fit the data here, we use the maximum likelihood method of Weiner et al. (2006). This allows for a finite intrinsic scatter in the ordinate in addition to random errors in both dimensions. The uncertainty that we estimate in the abscissa V_f is given in Table 1. The error in luminosity is dominated by uncertainty in distance rather than in apparent magnitudes. The uncertainty in the distance to each galaxy is heterogenous and hard to quantify with confidence. We therefore treat it as a contributor to the scatter. The best-fit scatter found in this way is a combination of random errors and intrinsic scatter, placing an upper limit on the latter. The results of the fits are reported in Table 2.

Table 2.  Tully–Fisher Relations and Their Scatter

Band	ℓ	x	${\rm log} A$	${{\sigma }_{{\rm log} \ell }}$
Luminous Tully–Fisher Relation
V	LV	4.27 ± 0.19	0.68 ± 0.41	0.18
I	LI	4.77 ± 0.30	−0.37 ± 0.62	0.19
3.6	L[3.6]	4.93 ± 0.17	−0.28 ± 0.36	0.15
Stellar Mass Tully–Fisher Relation
V	M*	4.70 ± 0.22	−0.20 ± 0.48	0.21
I	M*	5.01 ± 0.31	−0.84 ± 0.64	0.20
3.6	M*	4.93 ± 0.17	−0.61 ± 0.36	0.15
Baryonic Tully–Fisher Relation
V	Mb	3.92 ± 0.18	1.81 ± 0.39	0.17
I	Mb	4.02 ± 0.24	1.56 ± 0.51	0.15
3.6	Mb	4.09 ± 0.15	1.49 ± 0.32	0.13
Gas-only Tully–Fisher Relation
...	Mg	3.24 ± 0.28	2.89 ± 0.59	0.29

Note. Fits to the data in Table 1 of the form ${\rm log} \ell ={\rm log} A+x{\rm log} {{V}_{f}}$.

Download table as:  ASCIITypeset image

The slope of the LTFR becomes progressively steeper as we go to progressively redder bands (Table 2), consistent with previous findings (Tully et al. 1998; Verheijen 2001). As one might expect, the [3.6] LTFR has less scatter and better determined fit coefficients than that in V or I. Nevertheless, we caution against overinterpreting the specifics of these LTFR fits given the high gas content of the slower rotators.

The LTFR found here are steeper than many published relations. This is expected, as the particular value of the fitted LTFR slope depends on the dynamic range of the data. As one goes to lower velocities, one finds more gas-rich galaxies. The luminosity progressively underestimates the mass, so these galaxies fall too low, steepening the fitted slope. The scatter in luminosity at a given velocity also goes up (Matthews et al. 1998). The precise value of the fitted slope therefore depends on both the dynamic range of the sample, and the happenstance of the gas fractions of the galaxies therein. Magnitude limited samples will be biased to lower slopes, as they will preferentially include brighter, lower gas fraction galaxies at a given V_f. This can be a strong effect, since it occurs at one end of the relation where relatively few points can have considerable leverage on the best-fit slope. In our sample, exclusion of the three slowest rotators (with ${{V}_{f}}\approx 50\ {\rm km}\;{{{\rm s}}^{-1}}$) flattens the fitted slope perceptibly (if not significantly).

An important consequence of sample dependence is that there probably is no such thing as a universally correct LTFR. The "right" answer is unique to each sample. This effect is difficult to discern in most samples, which tend to lack galaxies with ${{V}_{f}}\lesssim 90\ {\rm km}\;{{{\rm s}}^{-1}}$ where the effect becomes pronounced. This may be enough to explain subtle differences in published calibrations. It may also introduce systematic errors in distance determinations, as one is sampling different parts of the relation at different distances. Fortunately, this effect will be small as long as only bright galaxies are utilized in both calibration and distance determination.

Stellar mass is a more fundamental physical quantity than luminosity. The LTFR can and does vary from band to band. These all presumably originate from the same STFR, with differences in the LTFR reflecting systematic variations in the stellar populations composing fast and slow rotators. From the color–magnitude relation, one would expect progressively steeper LTFR as one goes from blue to red bands, but these should all originate from one STFR.

The stellar mass is estimated from ${{M}_{*}}={{\Upsilon }_{*}}L$ for each band with the mass-to-light ratio ${{\Upsilon }_{*}}$ estimated by population synthesis models. Many models currently in wide use do not pass a straightforward self-consistency test, in that stellar masses estimated from the luminosity in different pass bands for the same galaxy return wildly different results (McGaugh & Schombert 2014). By comparing data over a broad range, McGaugh & Schombert (2014) constructed empirically self-consistent color–${{\Upsilon }_{*}}$ relations.

We employ the CMLR of McGaugh & Schombert (2014) to estimate ${{\Upsilon }_{*}}$ with B − V colors, which we found to be the most sensitive to variations in stellar mass-to-light ratio among the available colors. To be specific, we adopt the model of Bell et al. (2003) modified to give self-consistent photometric results as quantified in Table 7 of McGaugh & Schombert (2014):

$$\begin{eqnarray}\begin{array}{rcl} {\rm log} \Upsilon _{*}^{V} & = & -0.628+1.305(B-V) \\ {\rm log} \Upsilon _{*}^{I} & = & -0.275+0.612(B-V) \\ \Upsilon _{*}^{[3.6]} & = & 0.47 \\ \end{array}\end{eqnarray} \tag{ 3 }$$

As expected, the dependence of ${{\Upsilon }_{*}}$ on color becomes weaker as one goes to redder bands. It effectively disappears in the NIR, so we neglect the meaninglessly tiny color slope (−0.007) at [3.6].

We make this particular choice of model because the model of Bell et al. (2003) required the least modification of the various models considered. By construction, the V-band ${{\Upsilon }_{*}}$indicator is identical to the original one of Bell et al. (2003), and is subject to the usual normalization uncertainty due to the IMF. Both the I-band and [3.6] ${{\Upsilon }_{*}}$ are modified to obtain photometric self-consistency. Requiring self-consistent stellar masses implies $0.4\lt \Upsilon _{*}^{[3.6]}\lt 0.5$ for all models, including those that initially predicted rather different values. A constant NIR ${{\Upsilon }_{*}}$ is therefore a good approximation.

The dependence of the NIR ${{\Upsilon }_{*}}$ on color in Equation (3) is considerably weaker than the already weak relation anticipated in many models. This is required for self-consistency: the photometric data are not consistent with NIR mass-to-light ratios that depend much on color (see McGaugh & Schombert 2014). While there can certainly be some color dependence, the variation of $\Upsilon _{*}^{[3.6]}$ with color appears to be rather less than the intrinsic scatter in $\Upsilon _{*}^{[3.6]}$ from galaxy to galaxy, which itself is also fairly small.

In practice, the weak relation between NIR ${{\Upsilon }_{*}}$ and color means that NIR luminosity is a good proxy for stellar mass. One might hope to do better by fitting the entire spectral energy distribution (SED) of a galaxy, and this no doubt matters for extreme objects like those dominated by the light of very young stellar populations. However, for most galaxies, there is no value added from this procedure. Most of the information about ${{\Upsilon }_{*}}$ is contained in the first color (B − V here), which is already negligible in the NIR. Further corrections from other colors carry progressively less information. In the case of [3.6] and the K_s-band, attempts at SED fitting are more likely to add noise than improve the estimate of ${{\Upsilon }_{*}}$. Worse, they are prone to induce systematic errors from imperfections in the population models. For example, many models assume a single metallicity for the stellar population. This appears to be one driver for the predicted dependence of the NIR mass-to-light ratio on color being stronger than observed (McGaugh & Schombert 2014). Models with realistic metallicity distributions have a much weaker dependence of NIR ${{\Upsilon }_{*}}$ on color (Schombert & McGaugh 2014a).

If the stellar mass-to-light ratios estimated with Equation (3) are correct, the ${{\Upsilon }_{*}}$ for each band should succeed in converting the different LTFR observed for each independent band into the same STFR. This procedure is indeed successful. The STFR in Table 2 are equivalent within the errors.

Since the adopted NIR mass-to-light ratio is constant, the [3.6] STFR is identical to the [3.6] LTFR except for the change in normalization due to the non-unity value of the mass-to-light ratio. Consequently, it is the slope of the V and I LTFR that change to result in an STFR that matches that from [3.6]. This goes in the expected sense from the color–magnitude relation: fainter galaxies are bluer and have lower ${{\Upsilon }_{*}}$ on average. Use of incorrect CMLR would lead to discordant STFR in the various bands. That we find consistent SFTR provides independent confirmation that the CMLR in Equation (3) determined photometrically by McGaugh & Schombert (2014) are very nearly correct.

The conversion from the LTFR to the STFR can only work to the extent that $B-V$ is predictive of ${{\Upsilon }_{*}}$. That it does work suggests that B − V is an adequate ${{\Upsilon }_{*}}$ indicator. It certainly is not perfect: one expects a fair amount of scatter about the mean CMLR. The expected consequence is an increase in the scatter in the STFR relative to the LTFR. Even though the mean slope of the CMLR succeeds in converting the LTFR into the STFR, the scatter in the latter increases due to misestimates of the mass-to-light ratios of individual galaxies. That is, the scatter in the STFR should include whatever scatter is already in the LTFR, plus an extra component for the scatter in the CMLR. This effect does appear to be present in the data, as seen by the amount of scatter in Table 2.

From a population perspective, one expects a greater amount of scatter in ${{\Upsilon }_{*}}$ at bluer wavelengths. The scatter fit in Table 2 is a combination of intrinsic scatter and that from random errors like those in distance. While there is considerable uncertainty in these uncertainties, we estimate that random errors contribute ∼0.08 dex to the scatter in the ordinate of the various Tully–Fisher relations. Subtracting this in quadrature from the fitted value for each band gives an estimate of the intrinsic scatter in ${{\Upsilon }_{*}}$: 0.16 dex in V, 0.13 in I, and 0.12 in [0.36]. For comparison, Bell & de Jong (2001) anticipate 0.1 dex of scatter in K from variations in the star formation history alone, and Portinari et al. (2004) anticipate 0.15 dex.

It thus appears that the observed scatter is entirely explained by random errors plus the expected scatter in mass-to-light ratios. The scatter in ${{\Upsilon }_{*}}$ is consistent with that expected from the inevitable variations in star formation histories. This leaves essentially no room for variations in the IMF from galaxy to galaxy. Apparently, the IMF in disk galaxies is universal, at least averaged over the many star formation events that build a galactic disk over a Hubble time.

There is also precious little room for intrinsic scatter in whatever physics underlies the Tully–Fisher relation. Given the small size of the sample, we should not overinterpret the precise values of the scatter. However, it is not obvious that larger samples will inevitably lead to larger scatter, as they bring with them the larger risk of unaccounted systematic errors (see discussion in McGaugh 2012). Bear in mind that in order to increase the intrinsic scatter of this sample, we must have overestimated random errors. Even in the limit of zero experimental error and no scatter in ${{\Upsilon }_{*}}$, the intrinsic scatter is still small ($\lt 0.15$ dex; Table 2).

The atomic and molecular gas masses given in Table 1 are used to compute the total gas mass

$$\begin{eqnarray}&&{{M}_{g}}=1.33[M({\rm HI})+M({{{\rm H}}_{2}})].\end{eqnarray} \tag{ 4 }$$

The factor 1.33 accounts for the mass in helium. One can debate the appropriate value of the second digit past the decimal, but not at a level that matters here. We adopt 1.33 for consistency with McGaugh (2012), whose BTFR calibration will be utilized in Section 4. The contribution of other gas phases is generally negligible, at least within the visible disk where all relevant quantities are measured.

Luminosity is the original ordinate of the Tully–Fisher relation. Stars are the dominant baryonic mass component of bright galaxies. We therefore do not expect there to be a meaningful gas-only Tully–Fisher relation. For completeness, we do construct one. It is shown in Figure 2.

Zoom In Zoom Out Reset image size

At first glance, the relation in Figure 2 appears perfectly reasonable. It has considerably greater scatter than any of the STFR, but it is certainly possible to fit a line through the data (left panel of Figure 2). This is not particularly meaningful, as the scatter here is real. Galaxies of the same V_f can differ in gas mass by an order of magnitude.

Galaxies segregate by gas fraction (right panel of Figure 2). To estimate the gas fraction, we use $\Upsilon _{*}^{[3.6]}=0.47$ from Equation (3) to estimate the stellar mass, and bin the data by the ratio ${{M}_{*}}/{{M}_{g}}$. Galaxies with ${{M}_{*}}\lt {{M}_{g}}$ hover near the BTFR previously fit to gas-rich galaxies (McGaugh 2012). Since stellar mass is ignored here, these should all fall slightly below the BTFR line. Most do, though a few fall a little too high, albeit within the errors. In contrast, galaxies with ${{M}_{*}}\gt {{M}_{g}}$ all fall well below the line. The obvious inference is that we are missing an important component of the mass in these galaxies: the stars. Indeed, the amount of the shift is what we expect for the typical gas fractions of bright spirals.

Just as we should not ignore the gas in faint galaxies, we should not ignore the stars in bright ones. Neither the LTFR nor the gas-only Tully–Fisher relation are fundamental. Instead, they must stem from a more basic relation that does not distinguish between mass in the form of stars or gas.

In order to construct the BTFR, we take the sum of stellar and gas mass to obtain the baryonic mass

$$\begin{eqnarray}&&{{M}_{b}}={{M}_{*}}+{{M}_{g}}.\end{eqnarray} \tag{ 5 }$$

The stellar mass is computed with Equation (3) and the gas mass is from Equation (4). The BTFR resulting from the stellar mass as computed from each of V, I, and [3.6] is shown in the bottom row of Figure 1.

If all has gone well, this BTFR should be consistent from band to band. This should necessarily follow if the STFR has been computed correctly for each band. Indeed, this is the case: the relations computed separately for each band are mutually consistent. Both the slopes and the intercepts are indistinguishable within the errors. This convergence is readily seen in Figure 1. These relations are also consistent with the BTFR previously fit to independent data for gas-rich, predominantly slow rotators by McGaugh (2012).

Another encouraging point is that scatter in the BTFR is less than that in either the LTFR or the STFR. Including the gas mass as well as the stellar mass has made a better relation in this sense. Indeed, the relation is near-perfect in the sense that their appears to be very little intrinsic scatter. In Section 3.3 we estimated the intrinsic scatter in stellar mass-to-light ratios for this sample to be 0.16 dex in V, 0.13 in I, and 0.12 in [0.36]. The scatter in the BTFR is 0.17 dex in V, 0.15 in I, and 0.13 in [0.36] (Table 2). These are effectively identical given the uncertainties, leaving precious little room for intrinsic scatter in the relation itself.

The fundamental, underlying physical relation appears to be one between baryonic mass and asymptotic rotation velocity. The data are consistent with a single power law (a single linear slope in log–log space) over four decades in baryonic mass. There is no evidence for a bend or break in the relation as sometimes anticipated by models (e.g., Trujillo-Gomez et al. 2011).

The luminosity of the traditional LTFR is merely a proxy for baryonic mass. This works well for bright spirals that are gas-poor, but starts to break down at lower velocity where galaxies tend to be more gas-rich. As a consequence, the LTFR suffers increased scatter at low rotation velocity, and may appear to bend. This simply illustrates the shortcomings of luminosity as a mass indicator, not any break in the underlying relation.

Since luminosity is an imperfect indicator of baryonic mass, calibrations of the LTFR are inevitably inaccurate at some level. Bright spirals do contain gas, and the gas fraction varies from galaxy to galaxy. The LTFR will inevitably deviate from the underlying BTFR in a way that depends both on the band pass and the gas fractions of the sample of calibration galaxies. Fortunately, this appears to be a minor effect for bright galaxies.

The small intrinsic scatter in the BTFR suggests that there is no large reservoir of missing baryons remaining to be discovered in the disks of spiral galaxies (McGaugh et al. 2000). It is obvious if we miss either the stars or the gas (Figures 1 and 2). If we were still missing a comparable number of baryons, then the sum in Equation (5) would be incomplete. This would show up as a systematic deviation from the BTFR, which is not apparent. One can always have some mass in an undetected component (e.g., in the form of very cold molecular gas—Pfenniger & Revaz 2005), but its mass must be $\ll {{M}_{b}}$ from Equation (5) so that the data stay within the scatter. Consequently, the stars and cold gas that we detect appear to compose the bulk of the baryonic mass in disk galaxies.

One of the most important works constraining the stellar masses of disk galaxies is the DiskMass survey (Bershady et al. 2010, 2011). DiskMass observes velocity dispersions across the disks of face-on galaxies. It then assumes a disk thickness that is statistically consistent with observations of edge-on galaxies (Kregel et al. 2005) in order to estimate the stellar mass profile. This method measures the vertical restoring force to the disk, and is independent of the IMF.

The results of the DiskMass survey are summarized in the second and third rows. Martinsson et al. (2013a) found a near-constant NIR mass-to-light ratio of $\langle \Upsilon _{*}^{K}\rangle =0.31\pm 0.07\ {{M}_{}}/{{L}_{}}$. The small scatter and near-constancy of the mass-to-light ratio is consistent with our findings here. The normalization differs by nearly a factor of two.

The rather sub-maximal disks found by the DiskMass survey imply that the dark matter halo contributes a non-negligible amount to the observed velocity dispersions. This will inevitably drive the implied mass of the disk down further. Swaters et al. (2014) make a first estimate of this effect, revising the mean mass-to-light ratio down to $\langle \Upsilon _{*}^{K}\rangle =0.24\ {{M}_{}}/{{L}_{}}$. This is a factor of ∼2.4 times lower than what we find here. The effect of the dark matter halos may be compensated somewhat by the presence of super-thin disks in some galaxies (Schechtman-Rook & Bershady 2014). We will return to the factor of ∼2 discrepancy between the DiskMass and BTFR results after discussing other constraints.

Our own Milky Way provides a unique environment in which we can hope to both count stars directly and constrain their mass from kinematics. This is a rich field that is rapidly becoming richer; we consider here only a few relevant results. Flynn et al. (2006) use star counts from two independent surveys to constrain Galactic structure. Among other things, they estimate the mass-to-light ratio of the stellar disk of the Milky Way to be $\Upsilon _{*}^{V}=1.5\pm 0.2$ and $\Upsilon _{*}^{I}=1.2\pm 0.2\ {{M}_{}}/{{L}_{}}$.

Starting from an initially exponential disk, McGaugh (2008) deformed the radial mass distribution of the Galactic disk in order to fit the terminal velocity curve in the fourth quadrant (Luna et al. 2006; McClure-Griffiths & Dickey 2007). The result is a mass model with bumps and wiggles reminiscent of those observed in other spirals. Indeed, the Centaurus arm is prominent in the inferred Milky Way mass profile. Matching the details of the terminal velocity curve requires the disk to be nearly maximal. The stellar mass obtained in this way is $\sim 2\%$ greater than estimated by Flynn et al. (2006): ${{M}_{*}}=5.48\times {{10}^{10}}\ {{M}_{}}$ for R₀ = 8 kpc. To estimate the mass-to-light ratio, we adopt the total I-band luminosity estimated by Flynn et al. (2006), ${{L}_{I}}=4.5\times {{10}^{10}}\ {{L}_{}}$, and the K-band luminosity from Drimmel & Spergel (2001), ${{L}_{K}}=8.6\times {{10}^{10}}\ {{L}_{}}$, which includes a somewhat uncertain correction for the bulge fraction. The result is $\Upsilon _{*}^{I}=1.22\ {{M}_{}}/{{L}_{}}$ and $\Upsilon _{*}^{K}=0.63\ {{M}_{}}/{{L}_{}}$.

Bovy & Rix (2013) have measured the vertical restoring force to the disk ${{K}_{z}}(|Z|\lt 1.1\ {\rm kpc})$ over a wide range of Galactocentric radii. They obtain a mass for the stellar disk of ${{M}_{*}}=4.6\pm 0.3\times {{10}^{10}}\ {{M}_{}}$. Combining this with the disk-only luminosity estimated by Flynn et al. (2006), $L_{I}^{{\rm disk}}=3.5\times {{10}^{10}}\ {{L}_{}}$, and $L_{K}^{{\rm disk}}=6.9\times {{10}^{10}}\ {{L}_{}}$ from Drimmel & Spergel (2001) yields $\Upsilon _{*}^{I}=1.31\pm 0.09\ {{M}_{}}/{{L}_{}}$ and $\Upsilon _{*}^{K}=0.66\pm 0.04\ {{M}_{}}/{{L}_{}}$. The bulge component is excluded from this calculation, as the measurement of Bovy & Rix (2013) is specific to the Galactic disk. The uncertainty in the mass-to-light ratio here is only that from the error in stellar mass quoted by Bovy & Rix (2013), and does not include the uncertainty in the luminosity. The latter translates to roughly ±0.2 in ${{\Upsilon }_{*}}$ in absolute terms.

The three independent methods just discussed, star counts, the radial force in the disk, and the vertical force from the disk, all find mass-to-light ratios for the Milky Way that are consistent within the errors. They are also consistent with the result from the BTFR. They are not consistent with the low mass-to-light ratios implied by DiskMass, despite the similarity of the DiskMass approach to that of Bovy & Rix (2013).

Eskew et al. (2012) perform star counts in the LMC with NIR data. They sum up the mass of the stars observed to find $\Upsilon _{*}^{[3.6]}=0.5\ {{M}_{}}/{{L}_{}}$. This method depends some on the IMF, as one does have to extrapolate to unseen, low-mass stars. As in the Milky Way, the stars are counted directly, albeit to a brighter limit, so the assumption on the IMF is less strong than in population synthesis models.

Stellar population synthesis is a well-developed field. It is possible to use our knowledge of stellar evolution to construct detailed models for the composite stellar populations of spiral galaxies for various assumed star formation histories and metallicities (e.g., Bruzual & Charlot 2003; Le Borgne et al. 2004). For an assumed IMF, such models can be used to estimate the mass-to-light ratio of a population.

One expects that as stars age, the population will redden and the mass-to-light ratio will increase. Many models have been constructed to quantify this behavior. A few relevant ones are noted in Table 4.

The models of Bell et al. (2003), Portinari et al. (2004), Zibetti et al. (2009), and Into & Portinari (2013) have been explored in detail by McGaugh & Schombert (2014). These all give comparable mass-to-light ratios in the optical portion of the spectrum: $1\lt \Upsilon _{*}^{V}\lt 1.5\ {{M}_{}}/{{L}_{}}$ evaluated at B − V = 0.6. Small differences there appear to be attributable largely to modest differences in the assumed IMF. However, the models have very different SEDs in the NIR, where agreement breaks down. In terms of ${{\Upsilon }_{*}}$, Bell et al. (2003) estimate $\Upsilon _{*}^{K}=0.73\ {{M}_{}}/{{L}_{}}$ for B − V = 0.6 while Zibetti et al. (2009) give $\Upsilon _{*}^{K}=0.21\ {{M}_{}}/{{L}_{}}$. This large discrepancy persists if we correct for difference in the IMF: scaling up the Zibetti et al. (2009) estimate to match that of Bell et al. (2003) in the V-band has little impact in the NIR: $\Upsilon _{*}^{K}=0.21\to 0.28\ {{M}_{}}/{{L}_{}}$. This is still a factor of 2.6 lower than the $0.73\ {{M}_{}}/{{L}_{}}$ of Bell et al. (2003). The models do not produce consistent SEDs (Kriek et al. 2010).

McGaugh & Schombert (2014) used multi-wavelength photometric data to test these various models. Requiring self-consistency—i.e., that the same mass of stars produce the observed luminosity in each band for every galaxy—leads to Equation (3). This stipulation obliges all models to fall in the range $0.4\lt \Upsilon _{*}^{[3.6]}\lt 0.5\ {{M}_{}}/{{L}_{}}$.

The reasons for the difference between models are many, but the primary issue appears to be the treatment of thermally pulsing AGB (TP-AGB) stars. The luminosity of this component is grossly exaggerated in the models of Zibetti et al. (2009). The TP-AGB stars, as modeled, produce a lot of NIR photons without contributing much in the optical nor representing much stellar mass. Consequently, the NIR mass-to-light ratios are underestimated. A similar conclusion was reached by Zibetti et al. (2013), who sought but did not find the predicted spectral features of TP-AGB stars.

We constructed our own population models in Schombert & McGaugh (2014a). These models have an empirically motivated distribution of stellar metallicities as well as different ages corresponding to a variety of star formation histories. The metallicity distribution weakens the dependence of the NIR mass-to-light ratio on color relative to models built with a single metallicity. Most of model parameter space is consistent with $\Upsilon _{*}^{[3.6]}=0.5\ {{M}_{}}/{{L}_{}}$. Independently, Meidt et al. (2014) obtain $\Upsilon _{*}^{[3.6]}=0.6\ {{M}_{}}/{{L}_{}}$ albeit with a stronger color dependence. These values are consistent with the majority of methods. The exception to this is the DiskMass result, which we examine further in the next section.

Examination of Table 4 suggests that nearly all methods are consistent with a NIR mass-to-light ratio $\Upsilon _{*}^{K}\approx 0.57$ and $\Upsilon _{*}^{[3.6]}\approx 0.45\ {{M}_{}}/{{L}_{}}$. Since the population synthesis result for a Kroupa IMF is consistent with the normalization independently required by the gas-rich BTFR, it is tempting to conclude that the problem of stellar mass in disk galaxies is essentially solved. Unfortunately, the method of vertical velocity dispersions, as employed by the DiskMass project (Bershady et al. 2010, 2011), does not yield a consistent result.

Since DiskMass provides a result that is independent of the IMF, it deserves more consideration than the mutual consistency of the various population synthesis models. We can maintain that consistency while rescaling the IMF to match the DiskMass scale with the translation Δa = −0.29. So conceivably we simply need to adopt a different IMF in the population modeling. While this may be acceptable form a population synthesis perspective, it is harder to reconcile with Milky Way constraints. We check here whether we can reconcile it with the BTFR.

We construct the BTFR from the DiskMass data (Martinsson et al. 2013a, 2013b) for two choices of mass-to-light ratio. In one case, we use that measured by DiskMass (Martinsson et al. 2013a). In the other, we adopt a constant mass-to-light ratio $\Upsilon _{*}^{K}=0.6\ {{M}_{}}/{{L}_{}}$ as suggested by population synthesis models (McGaugh & Schombert 2014). The result is shown in Figure 5. Unsurprisingly, the two BTFR are shifted with respect to each other due to the difference in the adopted mass-to-light ratios.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The BTFR constructed using the population synthesis value $\Upsilon _{*}^{K}=0.6\ {{M}_{}}/{{L}_{}}$ places the DiskMass galaxies directly on the BTFR fit independently to gas-rich galaxies by McGaugh (2012). This is not guaranteed to happen. The gas-rich galaxies are, by and large, much slower rotators than the bright spirals of the DiskMass sample. The extrapolation of that fit will not work if the mass-to-light ratio is discordant.

Indeed, the data for the mass-to-light ratios determined by DiskMass are systematically shifted off the gas-rich BTFR. Most of the data parallel it, falling $\sim 1\sigma$ below the fit line. The DiskMass data are consistent with a slope four BTFR, but with a lower normalization. One possibility is thus that the DiskMass ${{\Upsilon }_{*}}$ are too low due to some unknown systematic error, and simply need to be shifted upwards to be consistent with other results.

Looking closely at the DiskMass BTFR (right panel of Figure 5), one may note that the data almost bifurcate into two parallel relations. Most of the data fall low, but a half-dozen galaxies stand to one side, consistent with the gas-rich BTFR. We do not know why the data might split like this, and suspect it may be indicative of a systematic error. This bifurcation disappears when a higher, constant K-band mass-to-light ratio is adopted. The scatter also decreases, but this is an inevitable result of setting the inclinations of the DiskMass galaxies with the K-band LTFR in the first place (Martinsson et al. 2013a). As ${{\Upsilon }_{*}}$ grows, the gas becomes less important to M_b, and one eventually just recovers a scaled version of the LTFR.

So far, we have kept distinct the various samples: gas-rich galaxies, the [3.6] sample of this paper, and the DiskMass data. Another approach is to make a fit of the BTFR with the combined data. The [3.6] and DiskMass data are consistent with each other for the same choice of mass-to-light ratio, so we make two combined fits. In one, we simply combine the [3.6] data as presented here with the gas-rich data from McGaugh (2012). In the other, we combine the DiskMass data with the gas-rich data, scaling the stellar masses of the gas-rich galaxies down by a factor of two to be consistent with the DiskMass scale. These fits are shown in Figure 6 with fit parameters given in Table 5.

Table 5.  BTFR of Combined Samples

Sample	x	${\rm log} A$
3.6 + gas-rich	4.04 ± 0.09	1.61 ± 0.18
DiskMass + gas	3.87 ± 0.10	1.80 ± 0.21

Note. Fits of the form ${\rm log} {{M}_{b}}={\rm log} A+x{\rm log} {{V}_{f}}$.

Download table as:  ASCIITypeset image

In combining the [3.6] and gas-rich samples, we have been careful to exclude duplicate galaxies. The gas-rich galaxies from McGaugh (2012) are largely independent of the sample here, but there is some overlap. Of the 47 galaxies considered in McGaugh (2012) and the 26 in Table 1, there are 7 objects in common: D631-7, DDO 168, IC 2574, F568-V1, F568-1, NGC 2403, and NGC 3198. These seven are plotted (and fit) only once in Figure 6, as part of the [3.6] sample.

The combined fit obtained in this fashion is consistent with the previous fit to the gas-rich data alone. It is slightly steeper, as we have retained the stellar population estimate of $\Upsilon _{*}^{[3.6]}=0.47\ {{M}_{}}/{{L}_{}}$, which is ever so slightly higher than the 0.45 of a straight extrapolation from the gas-rich BTFR. The fit lines are hardly distinguishable.

Indeed, the [3.6] data follow the extrapolation of the fit to the gas-rich galaxies remarkably well. This happens with zero effort: the location of the galaxies from Table 1 in the BTFR plane is determined entirely by the data and the mass-to-light ratio provided by population modeling. It is completely independent of the gas-rich calibration.

The gas-rich BTFR calibration also extrapolates well downwards to lower masses. This holds for both star-dominated dwarf Spheroidals (McGaugh & Wolf 2010) and for the recently discovered gas-rich dwarf Leo P (Giovanelli et al. 2013). In Figure 6 we adopt the distance of McQuinn et al. (2013) when plotting the stellar (Rhode et al. 2013) and gas (Giovanelli et al. 2013) mass of Leo P so that the datum is independent of the fitted relation. Leo P falls along the extrapolation of the BTFR to lower rotation velocity just as the [3.6] data fall along the extrapolation to higher V_f. The BTFR appears to be well described as a single power law over nearly six decades in mass.

In the combined DiskMass plus gas-rich BTFR fit, the relation again appears to be well described as a single power law. The chief difference is a slightly lower slope. Instead of $x\approx 4.0$, we now find $x\approx 3.9$.

This change of slope is driven by the lower mass-to-light ratios favored by DiskMass. The lower stellar masses adopted for the gas-rich galaxies themselves make no perceptible difference. The baryonic masses of these objects were already dominated by gas; for the new assumption this is slightly more true. However, the gas-rich galaxies are predominantly slow rotators, while the DiskMass sample is composed entirely of fast rotators. In order to fit the two data sets simultaneously, the BTFR simply pivots around the low-mass end, with the fitting routine selecting the slope necessary to hit the data at the high-mass end. The two lines illustrated in the right panel of Figure 6 are indistinguishable over the range constrained by the gas-rich data, but the slight difference in their slopes amounts to a factor of ∼2 in ${{\Upsilon }_{*}}$ for the fast rotators.

It therefore appears that a factor of ∼2 systematic uncertainty in stellar mass persists. The natural conclusion of our work is that the population synthesis models are essentially correct for a reasonable (e.g., Kroupa) IMF. However, we cannot exclude the lower mass scale indicated by the DiskMass data. The BTFR fit to DiskMass is within the uncertainties of the original gas-rich BTFR fit.

The situation could be improved with better data. Direct distance measurements of many gas-rich galaxies would considerably improve the calibration. If the small intrinsic scatter of the BTFR persists, then the scatter will come down as the data improve. The gas-rich BTFR provides a promising new method to constrain the absolute stellar mass scale.