1 Introduction

The Baryonic Tully-Fisher Relation of Gas Rich Galaxies as a Test of LCDM and MOND


The Baryonic Tully-Fisher Relation (BTFR) is an empirical relation between baryonic mass and rotation velocity in disk galaxies. It provides tests of galaxy formation models in CDM and of alternative theories like MOND. Observations of gas rich galaxies provide a measure of the slope and normalization of the BTFR that is more accurate (if less precise) than that provided by star dominated spirals, as their masses are insensitive to the details of stellar population modeling. Recent independent data for such galaxies are consistent with with . This is equivalent to MOND with . The scatter in the data is consistent with being due entirely to observational uncertainties. It is unclear why the physics of galaxy formation in CDM happens to pick out the relation predicted by MOND. We introduce a feedback efficacy parameter to relate halo properties to those of the galaxies they host. correlates with star formation rate and gas fraction in the sense that galaxies that have experienced the least star formation have been most impacted by feedback.

Subject headings:
galaxies: kinematics and dynamics — galaxies: dwarf — galaxies: irregular — galaxies: spiral

1. Introduction

The Tully-Fisher relation (Tully & Fisher, 1977) is one of the strongest empirical correlations in extragalactic astronomy. In addition to being a useful distance indicator (e.g., Sakai et al., 2000), the physical basis of this relation is of fundamental importance. In the context of CDM, the Tully-Fisher relation places tight constraints on galaxy formation models (e.g., Eisenstein & Loeb, 1996; McGaugh & de Blok, 1998a; Mo et al., 1998; Courteau & Rix, 1999; Steinmetz & Navarro, 1999; Courteau & Rix, 1999; van den Bosch, 2000; Bullock et al., 2001; Mo & Mao, 2004; Mayer & Moore, 2004; Gnedin et al., 2007; Governato et al., 2007; Trujillo-Gomez et al., 2010; de Rossi et al., 2010; Tonini et al., 2010). In a broader context, it provides a useful test of theories that seek to modify gravity in order to obviate the need for dark matter (e.g., Milgrom, 1983; Sanders, 1984; Mannheim & Kazanas, 1989; Moffat, 2006). These should make specific and testable predictions for the Tully-Fisher relation.

The Tully-Fisher relation was originally posed as an empirical relation between optical luminosity and the width of the 21 cm line. These are presumably proxies for more physical quantities. The 21 cm line-width is a proxy for rotation velocity (e.g., Verheijen, 2001; Yegorova & Salucci, 2007) which is widely presumed to be set by the dark matter halo. Luminosity is a proxy for stellar mass, which is presumably related to the total mass of the system. Thus a relation like Tully-Fisher seems natural, with the mass of the parent halo specifying both luminosity and rotation velocity.

Luminosity is not a perfect proxy for mass. The stellar mass-to-light ratio can, and presumably does, vary with galaxy type. This leads to Tully-Fisher relations with different slopes and scatter depending on the band-pass used to measure the luminosity (e.g., Tully et al., 1998; Verheijen, 2001; Courteau et al., 2007; Noordermeer & Verheijen, 2007; Masters et al., 2008). Moreover, low luminosity galaxies tend to fall below the extrapolation of the fit to bright galaxies (Persic & Salucci, 1991a; Matthews et al., 1998). Yet all these observed relations presumably stem from a single underlying physical relation.

Freeman (1999) and McGaugh et al. (2000) suggested that baryonic mass is a more fundamental quantity than luminosity. For any given stellar population, different mass-to-light ratios will be produced in different bands by the same mass of stars. More importantly, physics should not distinguish between mass in stars and mass in other forms. Adding the observed gas mass to the stellar mass results in a Baryonic Tully-Fisher Relation (BTFR) that is linear (in log space) over many decades in mass (Verheijen, 2001; Bell & de Jong, 2001; Gurovich et al., 2004; McGaugh, 2005b; Pfenniger & Revaz, 2005; Begum et al., 2008a; Stark et al., 2009; Trachternach et al., 2009; Gurovich et al., 2010). The BTFR appears to be the fundamental physical relation underpinning the empirical Tully-Fisher relation. As such, the true form of the BTFR is of obvious importance.

In this paper, I calibrate the BTFR with gas rich galaxies. The masses of gas rich galaxies can be more accurately measured than those of the star dominated galaxies that traditionally define the Tully-Fisher relation because they are less affected by the systematic uncertainty in the stellar mass-to-light ratio (McGaugh et al., 2000; Stark et al., 2009; McGaugh, 2011). This results in a purely empirical calibration with a minimum of assumptions.

In §2, I discuss the data required construct the BTFR, assemble samples of gas rich galaxies, explore the statistics of the data, and calibrate the relation. In §3 I examine the implications of the relation for CDM and MOND. I define a feedback efficacy parameter to quantify the mapping between observations and theoretical dark halo parameters. I estimate the value of the MOND acceleration constant from the empirical BTFR. A summary of the conclusions are given in §4.

2. The Baryonic Tully-Fisher Relation

In order to construct the BTFR, it is necessary to estimate the baryonic mass of a galaxy on the one hand, and its rotation velocity on the other. The baryonic mass is the sum of all observed components, stars and gas:


Combining this with measurements of the rotation velocity, one obtains the BTFR:


Remarkably, this simple formula suffices to describe the data for rotating galaxies. There is no indication of any need for a third parameter such as radius or surface brightness (Zwaan et al., 1995; Sprayberry et al., 1995; Courteau & Rix, 1999; McGaugh, 2005a): the Tully-Fisher relation is the optimal projection of the fundamental plane of disk galaxies.

2.1. Stellar Mass

The stellar mass of a galaxy is determined from its measured luminosity and an estimate of its stellar-mass-light ratio :


The mass-to-light ratio is commonly obtained from stellar population models (e.g., Bell et al., 2003; Portinari et al., 2004). These predict correlations between  and color so that the observed color is used to estimate .

The use of a stellar mass-to-light ratio estimated from stellar population models causes a significant systematic uncertainty in the stellar mass (Conroy et al., 2009), which is particularly sensitive to the IMF. Depending on the choice of model, one can derive BTFRs with slopes anywhere in the range of to 4 (McGaugh, 2005b). This is a basic limitation that is hard to avoid in samples of spiral galaxies where most of the baryonic mass is in the form of stars. This can be overcome by use of samples of galaxies whose baryonic mass is dominated by gas rather than stars (Stark et al., 2009). In such gas rich galaxies, the uncertainty in stellar mass is reduced to a minor contributor to the total error budget. Nevertheless, I am careful to employ a self-consistent population model to all galaxies employed in the analysis (Table 1).

2.2. Gas Mass

Galaxies with can be found among very late type (Sd and Sm) spirals and Irregulars with rotation velocities (e.g., Eder & Schombert, 2000; Walter et al., 2008; Begum et al., 2008b; Swaters et al., 2009). With enough such galaxies, it is possible to obtain an absolute calibration of the BTFR that is effectively independent of the stellar mass estimator (Stark et al., 2009). Here I describe gas mass estimates for galaxies used in the analysis.

Atomic Gas

Neutral atomic (HI) gas usually dominates the gas component of disk galaxies. The mass of HI follows directly from the physics of the spin-flip transition of hydrogen, the cosmic hydrogen fraction, and the distance to each galaxy. There is no uncertain mass-to-light ratio as with stars.

The gas mass is estimated by


where is the observed atomic gas mass. The factor accounts for the cosmic hydrogen fraction and forms of gas other than HI. Since the majority of objects of interest here are low luminosity late type galaxies, a primordial hydrogen fraction is assumed () rather than the commonly assumed solar value (). This difference is less than the typical uncertainty in , but does make the numbers here differ slightly from those in the source data.

Molecular Gas

After atomic hydrogen, molecular gas is the next significant gas component. Molecular gas persists in being difficult to detect in late type, low surface brightness galaxies. Where detected, it appears to contain relatively little of the interstellar gas mass (Matthews & Gao, 2001; O’Neil et al., 2000; O’Neil & Schinnerer, 2004; Matthews et al., 2005; Das et al., 2006, 2010). Such direct measurements of molecular gas are not available for most galaxies. To estimate the molecular gas mass, we might resort to scaling relations for such that


Young & Knezek (1989) showed that depends on morphological type, albeit with large scatter. From their data, McGaugh & de Blok (1997) derived the scaling relation . The vast majority of of gas rich galaxies are very late types (), so the typical correction is small ().

We can use the recent observations of THINGS galaxies (Walter et al., 2008) to provide an independent estimate of . Leroy et al. (2008) use the observed (low) star formation in dwarf Irregular galaxies to infer the amount of molecular gas present. This maps smoothly into the observed molecular content of brighter galaxies, with all disks following a relation between surface densities such that (their equation 33). This is a relation between local surface densities, while a relation between global masses is desired here. The atomic gas is typically spread over a much larger area than the stars, so we cannot simply assume that the ratio of surface densities is the same as the ratio of masses. However, the size of the stellar and molecular disk are usually similar (Helfer et al., 2003). Assuming that they are the same gives


In this case, the amount of molecular mass depends on the stellar mass rather than the atomic gas mass and morphological type as in equation (5).

In order to apply equation (6), we need an estimate of the average HI surface density. This is almost always in low surface brightness galaxies (de Blok et al., 1996). In the prototypical case DDO 154, at all radii (Leroy et al., 2008). Taking this value as typical implies that molecular gas is perhaps 9% of the stellar mass of these systems. For galaxies with less than half of their mass in the form of stars, we again infer that the molecular gas makes up of the baryonic mass.

For either of these estimators, the molecular gas content of HI-dominated late type galaxies is small. So small, in fact, that it is less than the uncertainty in the measurement of the HI mass in most cases. This would seem to be a good working definition of negligible, so rather than make an uncertain correction for the molecular gas mass, we neglect it entirely. In effect, we assume a constant .

Other Forms of Gas

We restrict our baryonic mass budget to the components that are actually detected, as discussed above. It is of course possible that other reservoirs of baryons exist in some undetected form. There is, however, no direct evidence for baryons in any form that is comparable in mass to the stars and atomic gas.

One conceivable reservoir of baryonic mass is hot gas in the halo. Such a component is expected in some galaxy formation scenarios. While some hot gas is certainly present, it appears that its density is too low to make a substantial contribution here (Anderson & Bregman, 2010).

Another suggestion is that some of the dark matter could be in the form of undetected cold molecular gas in galaxy disks (Pfenniger et al., 1994). Pfenniger & Revaz (2005) showed that a marginal improvement in the scatter of the BTFR of McGaugh et al. (2000) could be obtained by treating as a free parameter, and suggested that the best fit value () implied additional dark baryonic mass in the disk. In more accurate data, the scatter is consistent with being caused entirely by observational uncertainties (Verheijen, 2001; McGaugh, 2005b; Stark et al., 2009; Trachternach et al., 2009; McGaugh, 2011), nullifying this motivation for dark baryons.

In sum, there is no compelling evidence for substantial reservoirs of baryons in disk galaxies in any form other than those that are directly detected. I therefore consider only the physically motivated value of . Obviously, a different result would be obtained if we allowed to include hypothetical mass components (cf.  Pfenniger & Revaz, 2005; Begum et al., 2008a).

2.3. Rotation Velocity

The characteristic circular velocity of a disk galaxy can be measured with resolved rotation curves, or approximated by line-widths. The latter is a flux-weighted integral over the velocity field, and lacks the clear physical meaning of a rotation curve. It is, however, readily obtained from single disk 21 cm observations, whereas obtaining and analyzing interferometric data cubes is considerably more effort intensive (e.g., de Blok et al., 2008; Trachternach et al., 2009).

When a full rotation curve is available, one can make different choices about where to measure the circular velocity. Common choices include the maximum observed velocity (, e.g., Noordermeer & Verheijen, 2007), that in the outermost measured regions where rotation curves tend towards becoming approximately flat (, e.g., Verheijen, 2001; McGaugh, 2005b) or the velocity at some particular optical radius (e.g., Persic & Salucci, 1991b; McGaugh & de Blok, 1998a; Courteau & Rix, 1999). The gas rich galaxies of greatest interest here typically have gradually rising rotation curves that slowly bend over toward becoming flat without ever having an intermediate peak. For this common morphology, all these measures of velocity are very nearly equivalent. In particular, , and the velocity measured at a particular optical radius is also the same unless that radius is chosen to be very small (see Yegorova & Salucci, 2007). Indeed, it is not particularly meaningful to define an optical radius for gas rich galaxies, as the scale length of the baryonic mass can be considerably larger than that of the stars alone (McGaugh, 2005a; Sellwood & McGaugh, 2005).

For quantitative analysis, I restrict consideration to cases where is explicitly measured, simply adopting the measured velocity from each source. For comparison purposes, I also show line-width () data, but do not include these data in the calibration of the BTFR as line-widths and rotation velocities are comparable but not identical measures. In order to make comparisons, I assume for gas dominated galaxies with line-width measurements on the presumption that they would show rising rotation curves like those observed for comparable galaxies. For the star dominated galaxies, I assume . This qualitatively accounts for the fact that bright galaxies have rotation curves that peak before falling gradually toward flatness (Persic & Salucci, 1991b). The quantitative factor is chosen to reconcile the BTFR fit to (McGaugh et al., 2000, ) with that fit to (McGaugh, 2005b, ). This is only done to facilitate comparison between independent samples.

To account for non-circular motions, I accept whatever prescription was used by each source. In general, these corrections turn out to be rather small (cf. Tully & Fouque, 1985; Oh et al., 2008; Trachternach et al., 2009; Dalcanton & Stilp, 2010), as the typical velocity dispersion of low mass galaxies is (Kuzio de Naray et al., 2009). Since the correction to the circular velocity occurs in quadrature, it is only somewhat important only in the slowest rotators. There exist dynamically cold, rotating disks down to at least , and perhaps as low as (Tikhonov & Klypin, 2009).

2.4. Gas Rich Galaxy Samples

There is an enormous amount of data in the literature, but only a finite amount of it pertains to gas dominated galaxies with that also have all the required information for constructing the BTFR. I describe here the accumulated data to date. I also include associated samples of star dominated galaxies. This is done strictly for comparison as the uncertainties for star dominated galaxies are dominated by the choice of stellar IMF. This is the systematic uncertainty that we eliminate by making use of gas dominated galaxies. For gas rich galaxies, the major sources of uncertainty are the 21 cm flux measurement and the distance to each galaxy. The uncertainty in the measurement of is usually much smaller than that in mass (Table 1), provided that the inclination is reasonably well constrained. Cases where this condition might not be satisfied are discussed and, if suspect, excluded from fitting.

Stellar Mass Estimates

The data of Begum et al. (2008a), Stark et al. (2009), and Trachternach et al. (2009) constitute the sample of gas rich galaxies for which all the necessary information is available to construct the BTFR. For these galaxies, stellar mass is, by definition, a minority of the baryonic mass. Nevertheless, we take care to estimate stellar masses self-consistently. For specificity, we use their observed colors and the population model of Portinari et al. (2004) with a Kroupa IMF (Kroupa, 2002). Other plausible choices make no difference to the result because the gas mass dominates (Stark et al., 2009).

Several other samples are included for comparison (McGaugh et al., 2000; McGaugh, 2005b; Gurovich et al., 2010). Since these data are not used in the analysis, we make no effort to reconcile them to the same population model and simply adopt the stellar mass estimated by each source. Despite the minor differences in the methods used to estimate stellar mass, these data are remarkably consistent with one another.

Galaxies with Rotation Curve Data

Begum et al. (2008a): Data for remarkably slowly rotating galaxies are presented and analyzed by Begum et al. (2008a, b). The corrections for non-circular motions are substantial in some cases. However, the most significant difference in the analysis here is that we determine the gas mass as discussed in §2.2, and we select galaxies for consistency in their inclination determinations. It is not uncommon for the optical portions of gas rich galaxies to be much smaller than their dominant HI disks (e.g., Meurer et al., 1996; Gentile et al., 2007) and for the outer HI isophotes to have a different shape than the optical isophotes (Oh et al., 2008; Trachternach et al., 2009). If the stars suffer an oval distortion (bar), then an inclination based on optical isophotes will be systematically overestimated (Mihos et al., 1997; de Blok & McGaugh, 1998). Indeed, Begum et al. (2008b) observe that the optically inferred inclinations often exceed those inferred from HI. One must be wary that beam smearing might make the HI isophotes rounder than they should be, potentially causing an underestimate of inclination. To avoid either effect, I demand consistency between optical and HI inclination determinations. Specifically, I require that be within 12% of . The restriction is placed on because this is what matters to the accuracy with which can be measured. We are concerned with the velocity at large radii, so the HI inclination is adopted (as in Begum et al., 2008a); the difference is treated as an additional source of error. The particular threshold of 12% is determined by trial and error, balancing the desire for accuracy with the need to retain at least some objects. With this criterion, 16 objects are retained from Table 1 of Begum et al. (2008a) and 12 are rejected. Requiring any greater accuracy eliminates nearly the entire sample. UGC 5456 is excluded for being marginally star dominated. Three objects are also in the sample of Stark et al. (2009).

Stark et al. (2009): Data for gas rich galaxies are selected from the literature with an emphasis on data quality. Stark et al. (2009) required resolved rotation curves that were observed to flatten sufficiently that the logarithmic slope became in their outer parts (see Swaters et al., 2009, from which much of this sample originates). This requirement prevents the data from skewing to low due to the inclusion of galaxies whose rotation curves are still rising rapidly at the last measured point.

Galaxy D Ref.
Mpc M M
DDO 210 00.94 1,2
Cam B 03.34 1,2
UGC 8215 04.5 1
DDO 183 03.24 1
UGC 8833 03.2 1
D564-8 06.5 3
DDO 181 03.1 1
P51659 03.6 1
KK98 246 07.83 1
UGCA 92 03.01 1
D512-2 14.1 3
UGCA 444 00.95 2
KK98 251 05.6 1
UGC 7242 05.4 1
UGC 6145 07.4 1
NGC 3741 03.0 1,2
D500-3 18.5 3
D631-7 05.5 3
DDO 168 04.3 2
KKH 11 03.0 1
UGC 8550 05.1 2
D575-2 12.2 3
UGC 4115 07.5 1
UGC 3851 03.2 2
UGC 9211 12.6 2
NGC 3109 01.3 2
UGC 8055 17.4 1
D500-2 17.9 3
IC 2574 04.0 2
UGC 6818 18.6 2
UGC 4499 13.0 2
NGC 1560 03.45 2
UGC 8490 04.65 2
UGC 5721 06.5 2
F565-V2 48. 2
F571-V1 79. 2
IC 2233 10.4 2
NGC 2915 03.78 2
NGC 5585 05.7 2
UGC 3711 07.9 2
UGC 6983 18.6 2
F563-V2 61. 2
F568-1 85. 2
F568-3 77. 2
F568-V1 80. 2
NGC 2403 03.18 2
NGC 3198 14.5 2

References. – 1. Begum et al. (2008a); 2. Stark et al. (2009); 3. Trachternach et al. (2009). Note. – An electronic version of this table is available at http://www.astro.umd.edu/~ssm/data/gasrichdatatable.txt

Table 1Data for Selected Gas Rich Galaxies

Trachternach et al. (2009): New WSRT observations of 11 very gas rich galaxies selected from the sample of Schombert et al. (1997) and Eder & Schombert (2000) are used to measure both and . Of the 11 galaxies observed, 6 show clear evidence of rotation. Interestingly, these all have HI profiles that are double-horned or at least show some structure. Galaxies for which rotation is not positively detected tend to have single-horned profiles. Trachternach et al. (2009) discuss two methods for measuring the outer circular velocity that are largely consistent. I adopt for their values measured from the position-velocity diagram ( from their Table 3). For D575-2, I estimate based on the outermost detected 21 cm emission in their data.

McGaugh (2005b): The data for star dominated galaxies with measured from extended HI rotation curves are shown for comparison. They are not used in the analysis here. For illustration, stellar masses adopt the MOND mass-to-light ratios. The BTFR fit to these data [] can be tested by extrapolation to the much lower circular velocities of the new gas rich galaxies.

Figure 1.— Baryonic mass as a function of line-width (left) and circular velocity as measured from resolved rotation curves (right). Left panel: data from McGaugh et al. (2000) are shown as light gray points with uncertainties suppressed for clarity. Gas dominated galaxies (with ) are shown as small circles and star dominated galaxies () are shown as small squares. More recent data are shown with larger symbols. The data of Trachternach et al. (2009) are shown as dark blue stars in cases where rotation is detected and as open stars when it is not. The data of Begum et al. (2008a) are shown as light blue squares for cases with consistent optical and HI inclinations and as crosses in cases where these differ substantially. The HI data of Gurovich et al. (2010) are shown as orange left-pointing triangles. The data of the star dominated galaxies of Sakai et al. (2000) are shown as red right-pointing triangles with stellar mass from the -band data of Gurovich et al. (2010). The two samples of star dominated galaxies are compared in the inset. This also serves to illustrate the range covered by the bright spirals that traditionally define the Tully-Fisher relation — a range that is vastly expanded by the inclusion of gas rich dwarfs. In the right panel, data for galaxies with measured from resolved rotation curves include the rotating cases of Trachternach et al. (2009), the data of Begum et al. (2008a) with consistent inclinations, the gas dominated galaxies compiled by Stark et al. (2009, green circles), and the star dominated galaxies compiled by McGaugh (2005b, dark gray squares).

The three data sets of Begum et al. (2008a), Stark et al. (2009),and Trachternach et al. (2009) are the most interesting for our purposes here. They compose a sample of gas dominated galaxies with resolved rotation curve measurements. These data are listed in Table 1 and are used in the analysis. The data for these galaxies is less precise than those for bright star dominated galaxies. However, they are more accurate, as their masses are not sensitive to systematic uncertainties in the IMF and stellar population mass-to-light ratios.

Galaxies with Line-width Data

Samples of galaxies with line-width data are used for comparison purposes only. We shall see that line-width data give a systematically shallower slope and greater scatter for the BTFR than do rotation curve data. This appears to be a limitation of the line-width as a proxy for the rotation velocity. Aside from this detail, the data present a consistent overall picture.

McGaugh et al. (2000): Gas rich galaxies were selected from the sample of Schombert et al. (1997) and Eder & Schombert (2000) to have inclinations . Stellar masses were estimated from -band data assuming a constant mass-to-light ratio (). We retain this original prescription, making only the correction to stellar mass discussed by Gurovich et al. (2004). Data for star dominated galaxies with -band photometry were taken from Bothun et al. (1985). A constant -band mass-to-light ratio of was assumed. These data are included here for comparison to more recent data but are not used in the analysis.

Gurovich et al. (2010): New data are presented for galaxies with . Detailed stellar population modeling has been performed to determine the stellar mass (Table 11 of Gurovich et al., 2010). The same modeling procedure has also been applied to the star dominated sample of Sakai et al. (2000). We adopt the stellar mass determinations based on -band photometry1 for these galaxies (Table 4 of Gurovich et al., 2010) to facilitate direct comparison with the data of Bothun et al. (1985) employed by (McGaugh et al., 2000). All of the bright galaxies of Sakai et al. (2000) have nicely double horned line profiles with well measured line-widths. This is not the case for the new gas rich galaxies presented by Gurovich et al. (2010), for which the HI profiles are all narrow and single-horned (Gurovich, private communication). Such profiles can be poor indicators of (Trachternach et al., 2009).

2.5. Comparison of the Data

Fig. 1 shows the data. Over all, the agreement between the various data is good. This is true even for the star dominated galaxies, despite the differences in the population modeling. The spiral galaxies of Sakai et al. (2000) fall in the midst of the data of Bothun et al. (1985) (inset of Fig. 1) with indistinguishable slope and normalization. The chief difference is the lower scatter in the data of Sakai et al. (2000). This presumably results from the more accurate distances to the individual galaxies based on HST observations (Freedman et al., 2001). This is expected: as the data improve, the scatter is reduced. Indeed, essentially all of the scatter can be accounted for by observational uncertainty and the expected variation in stellar mass-to-light ratios (Verheijen, 2001; McGaugh, 2005b; Stark et al., 2009).

Figure 2.— The data in a coordinate system rotated (equations 7 and 8) so that the abscissa is the position of a galaxy along the BTFR and the ordinate is its perpendicular distance from the BTFR. Symbols as per Fig. 1 with the exception that the star dominated galaxies of Sakai et al. (2000) are shown twice: once with the -band stellar masses of Gurovich et al. (2010) as in Fig. 1 and again for the hybrid and band stellar masses of Gurovich et al. (2010, open right-pointing triangles). This illustrates the systematic uncertainty that plagues star dominated galaxies but from which gas dominated galaxies are largely immune. The top panel shows the BTFR with slope 3 fit to the data of (Gurovich et al., 2010). The bottom panel shows the BTFR fit of McGaugh (2005b) with slope 4. The data shown in the bottom panel are independent of those to which the fit was made.

The scatter is larger for lower luminosity galaxies. This is also expected: it is more challenging observationally to obtain precise data for faint dwarfs than it is for bright spirals. Nevertheless, the data for the gas rich dwarfs appear to be largely consistent. Moreover, the scatter is reduced as we progress from line-widths to resolved measures of the rotation velocity (left vs. right panels in Fig. 1). The improved accuracy with which the rotation velocity is measured is reflected in reduced scatter.

There are some outliers among the gas rich dwarfs, especially when is used as the velocity estimator. The two most dramatic outliers from the sample of Begum et al. (2008a) are UGC 3755 and KK98-65, which fall far to the left of the main correlation. These are rejected by the requirement of consistent optical and HI inclinations as are a number of galaxies that do lie along the BTFR in Fig. 1 when the HI inclination is used. The outliers from Trachternach et al. (2009) are those objects for which rotation is not clearly detected. (D575-5 is the most discrepant case). Three objects from the sample of Gurovich et al. (2010) stand out from the rest of the data: ESO 085-G 088, AM 0433-654, and HIPASS J1424-16b. ESO 085-G 088 has a large velocity error, so its status as an outlier is only marginal. The other two objects clearly deviate from the rest of the data, including the bulk of the data of Gurovich et al. (2010). Since we do not have resolved measures of rotation for these objects, it is possible that they, like the single-peaked HI profiles of Trachternach et al. (2009), would not show clear evidence of rotation if so observed. Another possibility is that the optical inclinations used by Gurovich et al. (2010) are systematically in error as discussed in §2.4.2. Lacking resolved HI observations, we can not make the same consistency check that is possible with the data of Begum et al. (2008a). The inclination of AM 0433-654 is and that of HIPASS J1424-16b is . A shift of dex in would reconcile these galaxies with the other data. If inclination is responsible, the corresponding inclinations would be for AM 0433-654 and for HIPASS J1424-16b. From experience with such objects (e.g., de Blok & McGaugh, 1998; Trachternach et al., 2009), this is well within the realm of possibility. Indeed, it would be surprising if something like this did not happen with only optical inclinations to work with.

Over all, the data paint a consistent picture (Fig. 1) despite the inevitable concern over the accuracy of a few individual measurements. The BTFR is continuous and linear in log space over five decades in mass. There is a hint of curvature when is used as the rotation velocity measure, but it is not statistically significant. There is not even a hint of curvature when resolved rotation curves are available to measure .

We expect the accuracy of the data to improve as we move from integral measures like the line-width to physically meaningful rotation velocities measured from resolved HI data cubes. This expectation is realized in the reduced scatter seen in the right panel of Fig. 1. We can quantify this with ordinary least-square fits to the data.

If we apply the maximum likelihood method of Weiner et al. (2006) to fit the new line-width data (i.e., all the data in the left panel of Fig. 1 except those of McGaugh et al., 2000), we obtain a slope and an intercept . If we perform the same exercise with the rotation curve data in the right panel of Fig. 1, we obtain and . These formal fits have rather different slopes even though the chief difference is in the scatter. The method of Weiner et al. (2006) accounts for uncertainties in both velocity and mass, and can optionally estimate the amount of intrinsic scatter in a sample. Accounting for the stated observational errors (e.g., Weiner et al., 2006) implies an intrinsic scatter in the line-width data of dex. That is, there is 0.2 dex of scatter not explained by observational uncertainty. In contrast, this number is zero for the rotation curve data in the right hand panel: observational errors suffice to explain all the scatter, leaving no room for intrinsic scatter. This difference in scatter implies some source of variance in line-width measures that is perhaps not accounted for in the stated uncertainties, and not likely to be meaningful to the physics we are interested in here.

At least part of the reason for the difference in fitted slopes is physical: is not a perfect proxy for . Bright galaxies typically have peaked rotation curves with . Line-width measurements do not resolve where emission originates, and are thus sensitive to the maximum velocity rather than that at larger radius. This immediately skews the slope to shallower values. A similar if less pronounced effect can be seen in optical rotation curves that are resolved but not particularly extended (Torres-Flores et al., 2011).

For the physical interpretation of the BTFR, we wish to have as precise a probe as possible of the quantity of interest. For CDM, this would be the virial velocity of the dark matter halo. In the case of MOND, we need a measure that probes the low acceleration regime. In both cases, provides a better measure than .

2.6. Statistics of the Gas Rich Galaxy Data


The slope of the BTFR has been addressed by many authors (e.g., McGaugh et al., 2000; Bell & de Jong, 2001; Gurovich et al., 2004; McGaugh, 2005b; Begum et al., 2008a; Stark et al., 2009; Gurovich et al., 2010), and is typically found to be between and 4. Most of this variation is due to the prescription adopted for estimating the mass-to-light ratio (McGaugh, 2005b), which is by far the strongest systematic effect. This can be remedied by selecting gas dominated galaxies as described in §2.4.1.

Stark et al. (2009) found a slope with a bisector fit while Gurovich et al. (2010) found from their gas dominated galaxies (their Table 12). These results appear to be contradictory. Repeating the analysis of Gurovich et al. (2010), I obtain the same result. However, simply looking at the data, it is not obvious that they are in conflict. Part of the difference is simply the difference in velocity estimators, as line-widths give shallower slopes than resolved rotation velocities. Their sample is also small, so is easily influenced by a few outliers as discussed above (§2.5).

This difference between and 4 is precisely the difference between the nominally anticipated slopes in CDM and MOND. The correct answer is therefore of considerable importance and lends itself to controversy. Bear in mind that the data themselves are quite consistent (Fig. 1), so any controversy is in the fitting and choice of velocity estimator, not in the data. The line-width is a convenient proxy for the rotation velocity when resolved rotation curves are unavailable, but not a substitute for them when predictive theories are being tested.

Figure 3.— Fits to the BTFR of gas dominated galaxies. Symbols as per Fig. 1. The solid lines show the best fit to the whole sample (with slope ) and the bisector (slope 3.98) of the forward and reverse fits (dashed lines) limited to the samples of Stark et al. (2009) and Trachternach et al. (2009) — that is, excluding the less accurate points (squares) from Begum et al. (2008a). The various fits are consistent within the uncertainties, and yield the same basic result.

To illustrate the difference between slopes of 3 and 4, we rotate the BTFR into a new coordinate system (,) parallel to it such that measures the position of a galaxy along the BTFR and its deviation perpendicular to it. That is,


where is the slope of the BTFR. In Fig. 2 we show the data rotated into the frames defined by the fits of Gurovich et al. (2010) with slope and McGaugh (2005b) with .

A slope of 4 does a good job of describing the bulk of the data: there is no systematic deviation from the dotted line in the lower panel of Fig. 2. In contrast, there is an obvious residual from a slope of 3 in the the top panel. The data show a clear preference for the steeper slope.

Fig. 2 also illustrates the systematic caused by different mass-to-light ratio estimates for star dominated galaxies. The galaxies from Sakai et al. (2000) are shown twice, once with masses based on -band data (Table 4 of Gurovich et al., 2010) and again with the average of and band stellar masses (Table 5 of Gurovich et al., 2010). The hybrid - masses are nicely consistent with a slope of 3, while alone prefers a slope of 4. While one must choose which band to trust for bright galaxies, this choice is rendered irrelevant for gas dominated galaxies. We therefore concentrate the rest of our analysis on the gas rich galaxy sample described in §§2.4.2 and 2.4.1.

Taking the 47 gas rich galaxies with reasonably trustworthy data (Table 1), the best fit slope is with intercept . Various methods (forward, reverse, and maximum likelihood) all give the same result. Of these data, the velocity measurements of Begum et al. (2008a) are the most challenging: only three of their objects meet the quality criteria of Stark et al. (2009). Excluding the data of Begum et al. (2008a) but retaining those of Stark et al. (2009) and Trachternach et al. (2009) gives from a forward fit, from a reverse fit, and a bisector slope (Isobe et al., 1990) of . The corresponding intercepts are , 1.82, and 1.71, respectively, with uncertainty . The slight difference that occurs when also fitting the data of Begum et al. (2008a) may result from a slight skew to low velocities, and testifies to the difficulty in obtaining quality data for very slow rotators (see also Trachternach et al., 2009). The skew effect becomes more pronounced if we relax the selection criterion that requires consistent optical and HI inclination determinations. Irrespective of these details, these fits are all consistent with one another (Fig. 3), and none yield a slope that is meaningfully different from 4.


An interesting question we can pose to the gas rich galaxy data is what the normalization of the BTFR is with slope fixed to 4. The gas rich galaxies provide an absolute calibration of the BTFR that is very nearly independent of the estimator we use for the stellar mass-to-light ratio. By fixing the slope, we can sharpen our estimate of the intercept, and also examine higher order moments of the distribution. In addition to the scatter perpendicular to the BTFR, , we can also examine the skew and kurtosis .

Table 2 shows these statistics for several combinations of the rotation curve samples. We start with the sample of Stark et al. (2009) as their sample was constructed for this exercise. Adding the galaxies of Trachternach et al. (2009) decreases the intercept slightly. Adding the data of Begum et al. (2008a) raises again, back near to the value found by McGaugh (2005b, ).

Table 2 gives quantitative estimates for the intercept. For each combination of samples both a weighted and unweighted estimate is given. The weighted estimate comes from minimizing by adjusting with fixed slope: we minimize the orthogonal deviations. The unweighted estimate is made by finding the centroid of the distribution in the orthogonal deviations of the rotated data and requiring the centroid to lie at . The latter method gives no heed to the accuracy of the data, but is less sensitive to the influence of outliers whose formal uncertainty may understate systematic effects. These methods give indistinguishable results.

Sample N
weighted unweighted skew kurtosis weighted unweighted
S 28 1.04 1.71 1.65 0.07 0.14 1.18 1.36
ST 34 0.92 1.67 1.64 0.01 +0.01 1.30 1.39
STB 47 0.99 1.69 1.71 0.13 0.10 1.24   1.18

Note. – Sapmles: S: Stark et al. (2009); T: Trachternach et al. (2009); B: Begum et al. (2008a). The units of are and those of are . The skew and kurtosis are the statistics of the orthogonal deviations (equation 8).

Table 2Statistics of Gas Rich Galaxy Rotation Curve Samples

All combinations of the various samples give consistent estimates for the intercept. To be specific, we adopt the weighted estimate from the combination of the Stark-Trachternach sample:


This is the midpoint of the various determinations and has the smallest . This value is of course very similar to the unweighted value () obtained from the Stark et al. (2009) sample alone (McGaugh et al., 2010; McGaugh & Wolf, 2010). It is also similar to the value () found by McGaugh (2005b) from star-dominated galaxies, but has the virtue of being completely empirical with little sensitivity to how stellar masses are estimated.

It appears that at this juncture, systematic effects and the necessary assumptions are as limiting as data quality. It is traditional to assume which is appropriate for solar metallicity atomic gas, but is slightly larger than the primordial 1.33 taken here for these little evolved, gas rich, low mass galaxies. The choice of is not important to the star dominated galaxies in McGaugh (2005b), but we note that adopting it here would cause a change in mass of at most 5% (for purely gaseous galaxies) that would drive . This is at the level of our best guess in the mass of molecular mass. At this level the systematic uncertainty in the IMF that plagues stellar mass estimates also begins to play a role even in gas dominated galaxies. Given these considerations, any value in the range 45 —  is plausible, with values much outside this range quickly becoming less plausible. If we use the BTFR to estimate stellar masses (Stark et al., 2009), then more massive star dominated galaxies are consistent with a Kroupa IMF. Grossly different stellar masses would imply a break in the slope of the relation around where the galaxy population transitions from frequent to infrequent gas domination.


The scatter in our rotated coordinates is with only the third digit varying from sample to sample. In terms of mass this is 0.24 dex. The bulk of this uncertainty (0.20 dex) is provided by the error in the baryonic mass. Velocity uncertainties are less important in most samples, and only rival the uncertainty in mass in the lowest mass systems. Together, the uncertainties in mass and rotation velocity suffice to account for essentially all of the observed scatter.

A robust estimator of the dispersion of a sample is the median absolute deviation (MAD). The dispersion of the various samples about the BTFR as measured by their MAD is shown in Fig. 4. Also shown is the median measurement uncertainty . For data that scatter about the relation simply because of observational uncertainty, 1.48MAD as N becomes large presuming there are no systematic uncertainties that have not been accounted for. Fig. 4 shows that indeed 1.48MAD in most cases. This leaves little, if any, room for intrinsic scatter in the BTFR. Indeed, the only times when scatter measured by the MAD significantly exceeds that expected from measurement errors are in the cases where we have identified systematic uncertainties that are not included in the error estimate: in the subset of the sample of Trachternach et al. (2009) where no signature of rotation has been detected, and in the sample of Begum et al. (2008a) where the optical and HI inclination estimates differ greatly. Had we been unaware of these sources of systematic error, we might infer some finite intrinsic scatter. Instead, it appears that including less accurate data merely increases the scatter as one would expect.

The BTFR of gas rich galaxies appears to have essentially zero intrinsic scatter. The same conclusion follows from noting that is close to unity (Table 2). A finite intrinsic width to the relation would cause (Weiner et al., 2006). We can place an upper limit on the scatter by assuming the data are Gaussian (a dubious but conservative assumption for this purpose) and subtracting the measurement error in quadrature from 1.48MAD. This gives a limit on the intrinsic scatter of dex in mass for the Stark et al. (2009) sample and for the trimmed sample of Begum et al. (2008a). Essentially zero intrinsic scatter is inferred by this method for the gas rich data of Gurovich et al. (2010) or the star dominated galaxies of Sakai et al. (2000). For the data of Trachternach et al. (2009), the expected error exceeds the observed scatter. This might indicate that the uncertainties (especially in distance) have been overestimated, but the sample is small so we should not expect the assumption of a normal distribution to hold. Indeed, that is generally true here, which mitigates against there being much intrinsic scatter.

Figure 4.— The median error (triangles) and median absolute deviations (MAD) of the various samples. Open symbols refer to line-width measurements; solid symbols represent rotation curve measures. Lines connect samples that have been trimmed for quality as discussed in the text, showing both trimmed and untrimmed samples. Samples are labeled by first author with the MAD of gas dominated galaxies represented by circles and that of star dominated galaxies by squares. The MAD is a robust estimator of the dispersion of a sample; for a normal distribution MAD. The similarity of the observed scatter to the typical uncertainty is an indication that the BTFR has very little intrinsic scatter.

It is common in extragalactic astronomy for correlations between physical quantities to exhibit lots of scatter. In the BTFR we seem to have a relation with little or no intrinsic scatter. There are some possible couple of caveats to this. First, it is conceivable that the uncertainties in the data have been overestimated. Second, selection effects might limit the sample to only those objects that obey the BTFR.

Overestimation of the uncertainties seems unlikely. That is a sign not only that the model is a reasonable description of the data, but also that the uncertainties have been neither over nor underestimated. Even if the data were arbitrarily accurate, so that all of the observed scatter is intrinsic, the intrinsic width of the relation is still small: the observed 0.24 dex. If anything, this would appear to confirm that the disks of the selected galaxies are well behaved dynamically cold rotators. If it were otherwise, e.g., if there were substantial non-circular motions in the disks of these galaxies, then this would be a substantial source of intrinsic scatter and the observed scatter would be much larger (Franx & de Zeeuw, 1992; Trachternach et al., 2009).

It is more difficult to address the issue of potential selection effects. However, in the time since Zwaan et al. (1995), the persistent surprise has been how tight the Tully-Fisher relation remains when challenged with data for new types of galaxies that we would naively expect to deviate from it. Since this exercise has now been performed by many different observers for many different types of galaxies, it seems unlikely that selection effects play a significant role in suppressing the scatter of the BTFR. We select gas rich galaxies here, but it is already well established that star dominated galaxies obey the BTFR with little scatter (e.g., Verheijen, 2001; McGaugh, 2005b; Noordermeer & Verheijen, 2007). The only further selection made here is on data quality. When lower quality data are admitted, the scatter increases (Fig. 1 and 4), just as one would expect. There is no clear evidence for intrinsic scatter beyond that induced by observational uncertainty and the systematic effects inevitable in astronomical data.

It is impossible to estimate the scatter that might result from including objects that are not already included. We have restricted our discussion to rotating galaxies, since these are the objects that define the Tully-Fisher relation. It is not obvious that it should apply to non-rotating galaxies, though the Faber-Jackson relation for elliptical galaxies provides an obvious analog. Dwarf spheroidal galaxies fall remarkably close to the BTFR with a simple estimate of the equivalent rotation velocity (: Wolf et al., 2010) though the ultrafaint dwarfs fall systematically below it (McGaugh & Wolf, 2010). Even objects that have no apparent business adhering to to the BTFR either do so, or come surprisingly close to doing so (e.g., the HI ring in Leo). Among rotating disk galaxies one would expect differences in the baryonic mass distribution to have some impact on the BTFR, just because . And yet disk dominated spirals, bulge dominated early types, diffuse low surface brightness galaxies, and dwarf irregulars all adhere to the same BTFR in spite of their palpably different mass distributions (Zwaan et al., 1995; Sprayberry et al., 1995; Tully & Verheijen, 1997; McGaugh & de Blok, 1998a; Courteau & Rix, 1999; Verheijen, 2001; Noordermeer & Verheijen, 2007). Rather than finding greater scatter as we expand our samples beyond the Sc I galaxies traditionally utilized in distance scale studies, we find that rotating galaxies of all types adhere to the same BTFR. It therefore seems unlikely that there remains some large, unknown population of galaxies in rotational equilibrium that would induce substantial scatter in it.

Skew and Kurtosis

A persistent problem with interpreting astronomical data is whether gaussian statistics actually apply. One reason for this is that systematic errors often dominate random ones. We can check whether this might be an issue here by examining higher order statistics like the skew (lopsidedness) and kurtosis (pointiness) of the distribution in the deviations perpendicular to the BTFR. If the data are well behaved (i.e., distributed normally), the skew and kurtosis should be small.

For all three combinations of samples in Table 2, both the skew and the kurtosis are small: and . These samples are consistent with normal distributions as they should be if random errors dominate over systematic effects. This is especially true for the combination of the Stark et al. (2009) and Trachternach et al. (2009) samples, which have .

For comparison, the sample of Gurovich et al. (2010) has and around their best fit. If we trim the three outlying galaxies discussed in §2.5, the skew is reduced () but the kurtosis grows (). The negative kurtosis means that the distribution is flat-topped: it does not have a well-defined central peak as it should if the data simply scatter around the correct relation. These data are not distributed normally. This undermines the basic assumption of the least square fitting method, so the apparently straightforward fit of Gurovich et al. (2010) yields a misleading constraint on the slope.

3. Discussion

Understanding the physical basis of the BTFR is central to one of the most fundamental issues presently confronting extragalactic astronomy. The BTFR provides a simple test that in principle can distinguish between CDM and MOND (Milgrom, 1983). Here we discuss the interpretation of the BTRF in terms of each paradigm in turn.

3.1. Cdm

The interpretation of the BTFR in CDM depends on how we relate dark and baryonic mass. Different assumptions can plausibly be made, leading to rather different interpretations. Here we start from general considerations, later proceeding to specific models.

Figure 5.— The BTFR in CDM. Symbols as per Fig. 1. The dashed line illustrates the baryons that are available within the virial radii of dark matter halos with . Solid lines in the top left panel illustrate the model of Mo et al. (1998) for two choices of their disk fraction parameter . In the top right panel, the curved solid line is the model of Trujillo-Gomez et al. (2010) while the box illustrates the range of the model of Tonini et al. (2010). The bottom left panel shows the model of de Rossi et al. (2010). The bottom right panel plots the model galaxies of Piontek & Steinmetz (2011) and Governato et al. (2010) along with the data. Here the peak rotation velocity is used, as this is more directly comparable to the models of Piontek & Steinmetz (2011). The inset shows that these agree quite well with the data for star dominated galaxies.

It is conventional in cosmology to refer to structures by the density contrast they represent with respect to the critical density of the universe. The mass enclosed within a radius encompassing the over-density is


With , this becomes


By the same token, the circular velocity of a tracer particle at is . Consequently,


In CDM, the density contrast marks the virial extent of a dark matter halo (Eke et al., 2001). For ,


This includes all mass, dark and baryonic, that reside within the radius .

Dark matter halos thus obey a scaling relation reminiscent of the Tully-Fisher relation (e.g., Steinmetz & Navarro, 1999). This is not the same as the observed Tully-Fisher relation, as both mass and velocity refer to a radius that is much larger than what is observed. In order to map these theoretical quantities into the observed baryonic mass and circular velocity , we need to introduce some factors:




The total mass of baryons available within is , where is the cosmic baryon fraction (: Komatsu et al., 2009). The factor represents the fraction of these available baryons that contribute to the observed baryonic mass. Similarly, the observed velocity need not be identical to ; hence the factor .

General Constraints

The BTFR would follow from equation (13) with a slope of and a normalization if . This is shown as the dashed line in Fig. 5. Had this matched the data, it would provide a very satisfactory interpretation (Steinmetz & Navarro, 1999). Instead, it exceeds the data, leading us to consider models with and .

Equations 13-15 to provide a generic constraint on CDM galaxy formation models. These typically invoke feedback to suppress the cold baryon content of disks so that . We define a feedback efficacy parameter


When , is just the inverse of , so a galaxy with has lost 90% of its baryons, and one with has lost 99% of them. By “lost” we mean that they are not part of the disk of stars and cold gas that contribute to in the BTFR. It is irrelevant whether these baryons have been physically expelled as envisioned in many feedback models, or remain in the halo in some dark, unobserved form. merely tells us how effective feedback (or any other mechanism) has been in suppressing the inclusion of the available baryons into the observed disk.

In addition to variations in, the mapping of the observed to virial velocity can also play a role. This ratio can easily vary with mass as more massive disks boost by engendering more adiabatic contraction in their host dark matter halos than low mass disks (e.g., Bullock et al., 2001). My own experience with modeling adiabatic contraction (Sellwood & McGaugh, 2005) leads me to expect this to be a modest effect, most likely in the range . Very recently, Reyes et al. (2011) have combined dynamical and gravitational lensing measurements to estimate for galaxies with . This represents the upper end of the BTFR (roughly ). In order to help with the observed variation in , would need to be substantially less in lower mass galaxies. While these often have rising rotation curves (so possibly ), flatness of the rotation curve is an explicit selection criterion of the Stark et al. (2009) sample. For these galaxies, seems the best guess, as their disks are not massive enough to substantially alter their dark matter halos to yield , nor does it seem likely that the rotation curve will resume rising after being observed to flatten out so as to yield . For the very low mass galaxies of Begum et al. (2008a), is certainly possible, but it seems unlikely to be tiny. If at and 1.3 at , the total variation in is a factor of 3. This does not, by itself, suffice to explain the data.

Equation 16 maps between the parameters of the dark matter halo and those of the disk galaxy it hosts. It is completely general, encompassing what needs to happen to connect theory with observational reality. In order to be consistent with the data presented here, models must meet the requirement


This constraint applies to models of disk galaxies over the range . In applying this constraint, care should be taken to use consistent definitions of total mass ( with ) and rotation velocity. The term involving is a simple shift in normalization that translates to other common definitions of halo mass (e.g., instead of 100). The constraint holds irrespective of the particular choice of definition for halo mass.

Equation 17 is a scaling between and rotation velocity. That the efficacy of feedback scales with velocity is not surprising, as the shallower potential wells of low mass halos should be less effective at retaining baryons. The efficacy of feedback should also depend on the physics driving it. For low mass systems like those under consideration here, feedback is often imagined to be driven by star formation and subsequent supernova events. In this case, we might expect some correlation between and measures of star formation.

Fig. 6 shows as a function of past average star formation rate and gas mass fraction. The gas fraction while to obtain the past average star formation rate we assume all galaxies have an age of 12 Gyr so that /(12 Gyr). Both of these quantities correlate with , albeit with substantial scatter at low and high . The curious thing about these correlations is that feedback has been most effective in removing or suppressing baryons in galaxies that have experienced the least star formation and have the most cold gas remaining. This would seem to be the opposite of what one would expect from a picture in which intense star formation results in winds that drive out baryons.

The effect is not subtle. Galaxies with have ; those with have . It appears that the vast majority of baryons associated with the halos of these galaxies have been ejected or otherwise suppressed.

A more remarkable thing is that scales so precisely with . In order not to impart scatter to the BTFR, there must be very little scatter in equation 17. I would expect feedback from supernovae to be a chaotic process: the lack of scatter in seems unnatural. A halo of a given circular velocity could easily host galaxies of very different . Apparently, they do not — all halos of the same have basically the same .

The tight relation between and provides strong constraints on models of galaxy formation. Indeed, it seems to me that it constitutes a serious fine-tuning problem. One can readily envision models that obtain equation 17 as a mean scaling relation (e.g., Stringer et al., 2011). However, doing so with so little scatter is a much tougher requirement. I would expect and to fill out some distribution of values (e.g., Kauffmann et al., 1999; Bullock et al., 2001), even for halos of fixed mass. There is no obvious reason why these distributions should be either narrow or closely coupled. Yet they must be in order to avoid introducing substantial scatter into the BTFR.

Figure 6.— The efficacy of feedback (equation 16) as a function of past average star formation rate (left) and gas fraction (right). Symbols as per Fig. 1. Feedback solutions reconciling CDM with the BTFR require galaxies that have experienced the least star formation to have been most effective in driving out baryons. This holds even in galaxies where there are almost no stars ().

Stochastic feedback is not the only expected source of scatter in the BTFR. The scatter in the concentration–circular velocity relation for dark matter halos is another. Without some tuning (Bullock et al., 2001), this alone exceeds the scatter allowed by the error budget.

Another expected source of scatter is the triaxiality of dark matter halos (e.g., Kazantzidis et al., 2010). If the gravitational potential is non-symmetric, then the observed velocity depends on the viewing angle with respect to the principle axis of the potential. Otherwise identical galaxies will appear to have difference depending on how they happen to be oriented on the sky. A significant degree of non-axisymmetry is clearly excluded (Andersen et al., 2001; Trachternach et al., 2009; Kuzio de Naray et al., 2009). This constraint might be lessened by the rounding of dark matter halos by the formation of the baryonic disk, but the disk needs to be heavy enough to have an impact. Kazantzidis et al. (2010) find that disks need to be at least half maximal to have a significant impact on their halos. The low surface brightness disks that make up the majority of the gas rich galaxy sample typically fall a factor of two below this level McGaugh (2005b).

There are other factors though could, and probably should, contribute to the scatter (Eisenstein & Loeb, 1996; McGaugh & de Blok, 1998a). It is difficult to accommodate all the expected sources of scatter in the error budget. This holds even if we have overestimated the observational uncertainties, simply because the observed scatter is small.

The small intrinsic scatter in the BTFR appears to be a serious challenge for CDM. This is hardly the only problem faced by the theory (e.g., Kroupa et al., 2010; Kuzio de Naray & Spekkens, 2011; Bovill & Ricotti, 2011), so the persistent failure to detect WIMPs is worrisome. The XENON100 Collaboration (2011) has excluded much of the mass–cross section parameter space where WIMPs were expected to reside (Trotta et al., 2008). Given the persistence of dynamical puzzles (e.g., McGaugh & de Blok, 1998a; Sellwood & Kosowsky, 2001), one might worry that non-baryonic dark matter does not exist after all.

Some Specific Models

The generic constraints discussed in the previous section may or may not be adequately addressed by specific models. There are very many relevant models in the literature. It is well beyond the scope of this paper to review all of them. Nevertheless, it is useful to examine several illustrative cases.

Mo et al. (1998) built a model following the same logic outlined above. They define a parameter that is the fraction of the total mass in the disk. This is analogous to , such that when . The resulting Tully-Fisher relation (their equation 4) is shown as lines in the first panel of Fig. 5 for (as they suggest) and for (as suggested by Gurovich et al., 2010). The higher value gives a tolerable approximation to high mass spirals while the lower value is appropriate to gas dominated dwarfs. A single universal value, as often assumed in galaxy formation modeling, does not work simply because the slope of the BTFR is steeper than the CDM halo mass—rotation velocity relation.

More recent models include semi-analytic treatments (e.g., Tonini et al., 2010; Trujillo-Gomez et al., 2010) and fully numerical results (e.g., Mayer & Moore, 2004; Governato et al., 2007; de Rossi et al., 2010; Governato et al., 2010; Piontek & Steinmetz, 2011). These are shown in the remaining panels of Fig. 5. All invoke feedback in some way, but the prescription for feedback is specific to each so some come closer to matching the constraint on than others.

The model of Tonini et al. (2010) exhibits a substantial scatter and dependence on the distribution of stellar mass. This is natural: the distribution of baryonic mass should matter (McGaugh & de Blok, 1998a). This model matches the high mass end of the observed BTFR reasonably well, but over-predicts the mass in galaxies with . Nevertheless, this model, like that of Mo et al. (1998), provides an honest example of what we most naturally expect from CDM: a substantial amount of scatter about a relation with a slope of approximately 3.

The model of Tonini et al. (2010) also indicates a shift in normalization of the relation for galaxies residing in sub-halos relative to that of primary galaxies (their Fig. 6). This shift is already apparent at relatively large . There is no evidence of this in the data, which includes both primary galaxies and dwarfs that are sometimes isolated and sometimes group members (and hence sub-halos). For example, KK98-251 is a dwarf Irregular satellite of NGC 6946, but both lie on the same BTFR with no indication of a shift in normalization. Current data for the ultrafaint dwarf satellites of the Milky Way do place them below the BTFR, but this shift occurs at much smaller scales ( Walker et al., 2009; Wolf et al., 2010; McGaugh & Wolf, 2010).

The model of Trujillo-Gomez et al. (2010) exhibits significant curvature. It matches the data well at but falls below it at both high and low . There is no evidence in the data for curvature in the BTFR of rotating galaxies. It is of course conceivable that curvature may appear beyond the bounds of the current data, and may already be observed if one considers dwarf Spheroidals along with dwarf Irregulars (McGaugh & Wolf, 2010). This occurs at a significantly lower velocity scale () than in the model of Trujillo-Gomez et al. (2010, ).

The model of de Rossi et al. (2010) fares better, being in rather good agreement with the bulk of the data for gas rich galaxies. Like most CDM models, its slope is relatively shallow (; their Table 3), but this is only problematic at the high mass end where the model under-predicts the mass of star dominated galaxies. This might be remedied by adopting a different IMF, but the change would have to be substantial: a factor of 3 or so in stellar mass. The scatter is respectably small.

Most recently, detailed numerical simulations have produced model galaxies consistent with the BTFR (Governato et al., 2010; Piontek & Steinmetz, 2011). These are shown in the final panel of Fig. 5. Here we plot the velocity measured at the peak of the baryonic rotation curve rather than because this is closer to what is reported by Piontek & Steinmetz (2011). Even though there is not a great deal of difference between these two measures of rotation velocity, the already good agreement between the models and the data shown by Piontek & Steinmetz (2011) improves even further when is utilized.

Numerical simulations of this sort remain computationally expensive. Piontek & Steinmetz (2011) present eight model galaxies, which at present is a large number for such models Governato et al. (2007). Of these eight, seven are very nicely consistent with the observed BTFR. As they discuss, the other case is very discrepant. Obviously, no conclusions can be drawn from one case in eight. But it does raise some obvious and important questions. What is the intrinsic scatter of the BTFR? How often we should expect to see grossly discrepant cases? Would we recognize these objects as normal disk galaxies?

Resolution also remains an issue in numerical modeling. The model galaxies of Piontek & Steinmetz (2011) are all in the star dominated regime. Only the one model galaxy of Governato et al. (2010) resides in the regime of gas dominated galaxies where the comparison is most clean. Comparison with the Millenium simulation suggests that simulated galaxies might well follow the observed slope and normalization of the BTFR down to low rotation velocities (White & Guo 2011, private communication). The critical issue of the intrinsic scatter that should be exhibited by low mass model galaxies in such simulations remains open.

Detailed numerical simulations appear to have finally reached the point where they can produce quantitatively realistic model galaxies. This is clearly a great success. It remains to be seen if they can provide a satisfactory explanation for the small intrinsic scatter in the BTFR. If they can, then the question becomes whether feedback prescriptions that accomplish this feat correspond to how feedback works in nature.

3.2. Mond

Among alternatives to dark matter, one, MOND (Milgrom, 1983), makes strong a priori predictions about the BTFR, which is a consequence of the form of the force law in MOND. This deviates from purely Newtonian at small accelerations, , where is the one new parameter introduced by the theory. The value of must be constrained by observation (: Sanders & McGaugh, 2002) but once specified is constant.

In the deep modified regime, , the effective acceleration , where is the Newtonian acceleration calculated for the observed baryonic mass in the usual way. For circular motion around a point source, we can equate the centripetal acceleration to this effective force to obtain


Since both and are constants, and all mass is baryonic, one recognizes the BTFR, .

There are several consequences of equation 18 (Milgrom, 1983). The Tully-Fisher relation is absolute, being a direct consequence of the force law. There should be no intrinsic scatter. The normalization is set by constants of nature. Contrary to the case in Newtonian dynamics, the circular velocity does not depend on the radial extent of the mass distribution, so there is no dependence on surface brightness or scale length. This naturally explains the lack of observed size or surface brightness residuals from the Tully-Fisher relation (Zwaan et al., 1995; Sprayberry et al., 1995; Courteau & Rix, 1999; McGaugh, 2005a).

Indeed, Milgrom (1983) specifically predicted that low surface brightness galaxies would share the same Tully-Fisher relation, with the same normalization, as high surface brightness spirals. This prediction was subsequently confirmed (McGaugh & de Blok, 1998b). The data for gas rich galaxies now provides the opportunity to test another a priori prediction.

A Test with No Free Parameters

The circular velocities of the gas rich dwarfs considered here are measured. Their baryonic masses are observed. Their location on the BTFR is fixed. Consequently, the gas rich dwarfs provide a test of the MOND prediction with zero free parameters.

The results of this test is shown in Fig. 7. Even though the value of was set by early fits to bright spirals (: Begeman et al., 1991), equation 18 fits the data for much lower mass galaxies well. The gas rich dwarfs follow the prediction of MOND with no fitting whatsoever (McGaugh, 2011).

There was no guarantee that MOND would pass the test posed by gas rich galaxies. A Tully-Fisher type relation is built into the MOND force law, but its slope is fixed and its normalization is constrained by star dominated galaxies. The data for gas rich galaxies could have fallen anywhere in the BTFR plane. They might have exhibited the slope of 3 that is more natural to CDM. Instead, they fall along the one and only line permitted in MOND.

The Value of

The data for gas rich galaxies are consistent with existing estimates of the acceleration constant (Sanders & McGaugh, 2002). We can reverse the question, and ask what the best value of is according to these data. The best fit intercept of the BTFR already contains this information, and the corresponding value of is given in Table 2.

There are some subtle issues to consider in calibrating . This has previously been done by fits to full rotation curves (e.g., Begeman et al., 1991; Swaters et al., 2010). These contain more information than simply the outer circular velocity, but require an additional assumption. In particular, a form for the function that interpolates between the Newtonian and MOND regimes must be assumed. For an assumed form, the best fit mass-to-light ratio must be found simultaneously with the best fit value of . There is not much range allowed in either of these parameters, but there can be covariance between them, so it is challenging to disentangle them uniquely. Here, the data are not so precise, and we make no use of the information in the shape of the rotation curve, only its outer value. However, this approach has the virtue of making a minimum of assumptions and using mass estimates that are completely independent of the theory.

Strictly speaking, equation 18 applies at infinite distance from an isolated point mass. Real galaxies are not point masses, and is measured at a finite radius. Consequently, the intercept of the empirical BTFR is related to the acceleration constant of MOND by


where is a factor of order unity that accounts for the finite size and non-spherical geometry of galaxies.

McGaugh & de Blok (1998b) estimated from purely geometrical considerations — thin disks rotate faster than the equivalent spherical mass distribution. The sample discussed by McGaugh (2005b) has MOND fits performed for and a separate fit for the BTFR yielding . This gives an empirical estimate of which I adopt here. The gas rich galaxy data thus give


The uncertainty here includes the formal uncertainty in (equation 9) plus an allowance for 20% uncertainty in . It is conceivable that there could be a small systematic offset in the mean value of between gas and star dominated galaxies.

Figure 7.— The BTFR in MOND. Symbols as per Fig. 1. The band shows the expectation of MOND (Milgrom, 1983). The width of the band represents the uncertainty in based on detailed fits to the rotation curves of a subset (Begeman et al., 1991) of the star dominated galaxies (gray squares). The gas rich galaxies test the MOND prediction in a new regime with zero free parameters.

Detailed rotation curve fits to some of the same galaxies suggest (Swaters et al., 2010). This becomes if we exercise the same quality control on the data as applied here. This gives some idea of the systematic uncertainty and the impact of the best-fit stellar mass-to-light ratio when it is treated as a fitting parameter along with . However, data quality remains the paramount issue.

The BTFR is clearly a success of MOND, despite its many other drawbacks. Further tests are possible, as adherence to the BTFR does not guarantee that it is possible to obtain an acceptable fit to the entire rotation curve of each galaxy. While MOND is known to do quite well in some gas rich galaxies (e.g., Gentile et al., 2010), it may have problems in others (e.g., Sanchez-Salcedo & Hidalgo-Gamez, 2011, where again the issue appears to be with the inclination). It is particularly challenging to fit the full rotation curve of NGC 3198 (Bottema et al., 2002; Gentile et al., 2011), the most massive galaxy to meet the gas rich criterion imposed here. Perhaps this suffices to falsify MOND, but we should not be so eager to disbelieve MOND as a theory that we ignore the remarkable empirical virtues of the simple formula suggested by Milgrom (1983).

There are other suggestions for modifying gravitational theory besides MOND. Conformal Weyl gravity (Mannheim & Kazanas, 1989) and MoG (Moffat, 2006) offer two examples. If these theories can fit rotation curves (Brownstein & Moffat, 2006; Mannheim & O’Brien, 2010), then they will also fit the BTFR (Moffat & Toth, 2011). However, I am not aware of a clear a priori prediction of this phenomenon by any theory other than MOND.

4. Conclusions

The baryonic mass–velocity relation provides an important test of CDM and alternatives like MOND. I have assembled recent data for gas rich rotating galaxies in order to test these theories. With , the location of these objects in the baryonic mass–rotation velocity plane is effectively measured directly from observations without the usual systematic uncertainty in the stellar mass-to-light ratio.

A long standing prediction (Milgrom, 1983) of MOND is that rotating galaxies will fall on a single mass–velocity relation with slope 4: . This prediction is realized in multiple independent data sets. Gas rich galaxies fall where predicted by MOND with no free parameters. There are not many predictions in extragalactic astronomy that fare so well over a quarter century after their publication.

In CDM, the naive expectation of a of Tully-Fisher type relation with a slope around 3 fails to predict the location of gas rich galaxies in the plane of the BTFR. Only the most recent detailed numerical models (e.g., Governato et al., 2010; Piontek & Steinmetz, 2011) succeed in reproducing the observed phenomenology. This is certainly progress. However, a physical understanding for why galaxy formation in the context of CDM should pick out the particular phenomenology predicted a priori by MOND remains wanting.

The work of S.S.M. is supported in part by NSF grant AST 0908370.


  1. Whether optical or -band data are used to estimate the stellar mass for the bright galaxies of Sakai et al. (2000) makes a difference to the slope of the BTFR: in the optical and in the -band. This exemplifies the systematic uncertainty that gas dominated galaxies save us from.


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