THE SMALL SCATTER OF THE BARYONIC TULLY–FISHER RELATION

This work is based on the Spitzer Photometry and Accurate Rotation Curves (SPARC) data set, which will be presented in detail in F. Lelli et al. (2016, in preparation). In short, we collected more than 200 high-quality ${\rm{H}}\;{\rm{I}}$ RCs of disk galaxies from previous compilations, large surveys, and individual studies. The major sources are Swaters et al. (2009, 52 objects), Sanders & Verheijen (1998, 30 objects), de Blok & McGaugh (1997, 24 objects), Sanders (1996, 22 objects), and Noordermeer et al. (2007, 17 objects). The RCs were derived using similar techniques: fitting a tilted ring model to the ${\rm{H}}\;{\rm{I}}$ velocity field (Begeman 1987) and/or using position–velocity diagrams along the disk major axis (de Blok & McGaugh 1997).

Subsequently, we searched the Spitzer archive for 3.6 μm images of these galaxies. We found 173 objects with useful [3.6] data. We derived surface brightness profiles and asymptotic magnitudes using the Archangel software (Schombert 2011), following the same procedures as Schombert & McGaugh (2014). The surface brightness profiles will be presented elsewhere; here we simply use asymptotic magnitudes at [3.6] to estimate stellar masses (M_*).

For the sake of this study, we exclude starburst dwarf galaxies (eight objects from Lelli et al. 2014a and Holmberg II from Swaters et al. 2009) because they have complex ${\rm{H}}\;{\rm{I}}$ kinematics and are likely involved in recent interactions (Lelli et al. 2014b). This reduces our starting sample to 164 objects.

Empirically, the velocity along the flat part of the RC minimizes the scatter in the BTFR (Verheijen 2001). The choice of velocities at some photometric radius, like 2.2 R_d (the disk scale length; Dutton et al. 2007) or R₈₀ (the radius encompassing 80% of the i-band light; Pizagno et al. 2007), lead to relations with larger scatter. This can be simply understood. For some low-mass, gas-dominated galaxies, these radii may occur along the rising part of the RC (Swaters et al. 2009). For high-mass, bulge-dominated galaxies, they may occur along the declining part of the RCs (Noordermeer & Verheijen 2007). These photometric radii do not necessarily imply a consistent measurement of the rotation velocity due to the different distribution of baryons within galaxies. Conversely, V_f appears insensitive to the detailed baryonic distribution, being measured at large enough radii to encompass the bulk of the baryonic mass. V_f represents our best proxy for the halo virial velocity.

We estimate V_f using a simple automated algorithm. We start by calculating the mean of the two outermost points of the RC:

$$\begin{eqnarray}&&\bar{V}=\displaystyle \frac{1}{2}({V}_{N}+{V}_{N-1}),\end{eqnarray} \tag{ 1 }$$

and require that

$$\begin{eqnarray}&&\displaystyle \frac{| {V}_{N-2}-\bar{V}| }{\bar{V}}\leqslant 0.05.\end{eqnarray} \tag{ 2 }$$

If the condition is fulfilled, the algorithm includes ${V}_{N-2}$ in the estimate of $\bar{V}$ and iterates to the next velocity point. When the condition is falsified, the algorithm returns ${V}_{{\rm{f}}}=\bar{V}$. This algorithm returns similar values of V_f as previously employed techniques based on the logarithmic slope ${\rm{\Delta }}\mathrm{log}(V)/{\rm{\Delta }}\mathrm{log}(R)$ (Stark et al. 2009), but is more stable against small radial variations in the RC. Galaxies that do not satisfy the condition at the first iteration are rejected, hence we only consider RCs that are flat within ∼5% over at least three velocity points. This excludes 36 galaxies with rising RCs (∼20%), reducing the sample to 129 objects.

The error on V_f is estimated as

$$\begin{eqnarray}&&{\delta }_{{V}_{{\rm{f}}}}=\sqrt{\displaystyle \frac{1}{N}\displaystyle \sum _{i}^{N}{\delta }_{{V}_{i}}^{2}+{\delta }_{\bar{V}}^{2}+{\left(\displaystyle \frac{{V}_{{\rm{f}}}}{\mathrm{tan}(i)}{\delta }_{i}\right)}^{2}},\end{eqnarray} \tag{ 3 }$$

where (i) ${\delta }_{{V}_{i}}$ is the random error on each velocity point along the flat part of the RC, quantifying non-circular motions and kinematic asymmetries between the two sides of the disk; (ii) ${\delta }_{\bar{V}}$ is the dispersion around $\bar{V}$, quantifying the degree of flatness of the RC; and (iii) δ_i is the error on the outer disk inclination i. High-quality ${\rm{H}}\;{\rm{I}}$ velocity fields can be used to obtain kinematic estimates of i and trace possible warps (e.g., Battaglia et al. 2006). This is another advantage of interferometric surveys over single-dish ones, since the latter must rely on photometric inclinations that depend on the assumed stellar-disk thickness and may be inappropriate for warped ${\rm{H}}\;{\rm{I}}$ disks. Inclination corrections become very large for face-on disks due to the $\mathrm{sin}(i)$ dependence, hence we exclude galaxies with i < 30°, reducing the sample to 118 galaxies. Clearly, this does not introduce any selection effects since galaxy disks are randomly oriented across the sky, but decreases the mean error on V_f to ∼0.03 dex.

We estimate the baryonic mass as

$$\begin{eqnarray}&&{M}_{{\rm{b}}}={M}_{{\rm{g}}}+{{\rm{\Upsilon }}}_{*}{L}_{[3.6]},\end{eqnarray} \tag{ 4 }$$

where M_g is the gas mass, L_[3.6] is the [3.6] luminosity (adopting a solar magnitude of 3.24, Oh et al. 2008), and ${{\rm{\Upsilon }}}_{*}$ is the stellar mass-to-light ratio. Several studies suggest that ${{\rm{\Upsilon }}}_{*}$ is almost constant in the near-infrared ([3.6] or K-band) over a range of galaxy types and masses. A small scatter of ∼0.1 dex is consistently found using different approaches: stellar population synthesis models (McGaugh & Schombert 2014; Meidt et al. 2014), resolved stellar populations (Eskew et al. 2012), and the vertical velocity dispersion of galaxy disks (Martinsson et al. 2013). Large inconsistencies persist in the overall normalization (McGaugh & Schombert 2015). In this Letter, we assume that ${{\rm{\Upsilon }}}_{*}$ is constant among galaxies and systematically explore the BTFR for different ${{\rm{\Upsilon }}}_{*}$.

In disk galaxies M_g is typically dominated by atomic gas (probed by ${\rm{H}}\;{\rm{I}}$ observations). The molecular gas content can be estimated from CO observations assuming a CO-to-H₂ conversion factor, which can vary from galaxy to galaxy depending on metallicity or other properties. CO emission is often undetected in low-mass, metal-poor galaxies (Schruba et al. 2012). Luckily, molecules generally are a minor dynamical component, contributing less than 10% to M_b (McGaugh & Schombert 2015). Similarly, warm/hot gas is negligible in the disk (McGaugh 2012). Hence we assume ${M}_{{\rm{g}}}=1.33{M}_{{\rm{H}}{\rm{I}}}$, where the factor 1.33 takes the contribution of helium into account. We note that any contribution from "dark gas" (molecular and/or ionized) is implicitly included in ${{\rm{\Upsilon }}}_{*}$ as long as this scales with M_*.

The error on M_b is estimated as

$$\begin{eqnarray}&&{\delta }_{{M}_{{\rm{b}}}}=\sqrt{{\delta }_{{M}_{{\rm{g}}}}^{2}+{({{\rm{\Upsilon }}}_{*}{\delta }_{L})}^{2}+{(L{\delta }_{{{\rm{\Upsilon }}}_{*}})}^{2}+{\left(2{M}_{{\rm{b}}}\displaystyle \frac{{\delta }_{D}}{D}\right)}^{2}},\end{eqnarray} \tag{ 5 }$$

where ${\delta }_{{M}_{g}}$ and δ_L are, respectively, the errors on M_g and L_[3.6] due to uncertainties on total fluxes (generally smaller than ∼10%). We assume ${\delta }_{{{\rm{\Upsilon }}}_{*}}=0.11$ dex, as suggested by stellar population models (McGaugh & Schombert 2014; Meidt et al. 2014). Galaxy distances (D) and corresponding errors (δ_D) deserve special attention. The 118 galaxies with accurate values of V_f have three different types of distance estimates (in order of preference):

• 1.  
  Thirty-two objects have accurate distances from the tip of the red giant branch (26), Cepheids (3), or supernovae (3). These are generally on the same zero-point scale and have errors ranging from ∼5% to ∼10% (see Tully et al. 2013). The majority of these distances (24/32) are drawn from the Extragalactic Distance Database (Jacobs et al. 2009).
• 2.  
  Twenty-six objects are in the Ursa Major cluster, which has an average distance of 18 ± 0.9 Mpc (Sorce et al. 2013). For individual galaxies, one should also consider the cluster depth (∼2.3 Mpc, Verheijen 2001), hence we adopt ${\delta }_{D}=\sqrt{{2.3}^{2}+{0.9}^{2}}=2.5$ Mpc, giving an error of ∼14%. We consider two galaxies from Verheijen (2001) as background/foreground objects: NGC 3992, having D ≃ 24 Mpc from a SN Ia (Parodi et al. 2000), and UGC 6446, lying near the cluster boundary in both space and velocity (D ≃ 12 Mpc from the Hubble flow).
• 3.  
  Sixty objects have Hubble-flow distances corrected for Virgocentric infall (as given by NED³ assuming ${H}_{0}=73$ km s⁻¹ Mpc⁻¹). These are very uncertain for nearby galaxies where peculiar velocities may constitute a large fraction of the systemic velocities, but become more accurate for distant objects. Considering that peculiar velocities may be as high as ∼500 km s⁻¹ and H₀ has an uncertainty of ∼7%, we adopt the following errors: 30% for D ≤ 20 Mpc; 25% for $20\lt D\leqslant 40$ Mpc; 20% for $40\lt D\leqslant 60$ Mpc; 15% for $60\lt D\leqslant 80$ Mpc; and 10% for $D\gt 80$ Mpc. The most distant galaxy in our sample is NGC 6195 at D ≃ 130 Mpc.

We perform separate analyses for the total sample (mean ${\delta }_{{M}_{{\rm{b}}}}\simeq 0.2$ dex) and for galaxies with accurate distances (58 objects in groups 1 and 2; mean ${\delta }_{{M}_{{\rm{b}}}}\simeq 0.1$ dex).

Figure 1 (top left) shows that ${\sigma }_{{\rm{obs}}}$ decreases with ${{\rm{\Upsilon }}}_{*}$ and reaches a plateau at ${{\rm{\Upsilon }}}_{*}\gtrsim 0.5{M}_{\odot }/{L}_{\odot }$. This plateau actually is a broad minimum that becomes evident by extending the ${{\rm{\Upsilon }}}_{*}$ range up to unphysical values of ∼10 ${M}_{\odot }/{L}_{\odot }$ (not shown). The observed scatter is systematically lower for the accurate-distance sample, indicating that a large portion of ${\sigma }_{{\rm{obs}}}$ in the full sample is driven by distance uncertainties.

Figure 1 (top right) shows that σ_int is below 0.15 dex for any realistic value of ${{\rm{\Upsilon }}}_{*}$. The similar intrinsic scatter between the two samples suggests that our errors on Hubble-flow distances are realistic. For a fiducial value of ${{\rm{\Upsilon }}}_{*}=0.5$ ${M}_{\odot }/{L}_{\odot }$, we find ${\sigma }_{{\rm{int}}}=0.10\pm 0.02$ for the full sample and ${\sigma }_{{\rm{int}}}=0.11\pm 0.03$ for the accurate-distance sample. As we discuss in Section 4, this represents a challenge for the ΛCDM cosmological model.

Figure 1 (bottom panels) shows that the BTFR slope (normalization) monotonically increases (decreases) with ${{\rm{\Upsilon }}}_{*}$. This is due to the systematic variation of the gas fraction (${f}_{{\rm{g}}}={M}_{{\rm{g}}}/{M}_{{\rm{b}}}$) with ${V}_{{\rm{f}}}$. Figure 2 (top panels) shows the BTFR for ${{\rm{\Upsilon }}}_{*}=0.5$ ${M}_{\odot }/{L}_{\odot }$, color-coding each galaxy by f_g. Low-mass galaxies tend to be gas-dominated (f_g ≳ 0.5) and their location on the BTFR does not strongly depend on the assumed ${{\rm{\Upsilon }}}_{*}$ (Stark et al. 2009; McGaugh 2012). Conversely, high-mass galaxies are star-dominated and their location on the BTFR strongly depends on ${{\rm{\Upsilon }}}_{*}$. By decreasing ${{\rm{\Upsilon }}}_{*}$, ${M}_{{\rm{b}}}$ decreases more significantly for high-mass galaxies than for low-mass ones, hence the slope decreases and the normalization increases.

Zoom In Zoom Out Reset image size

For any ${{\rm{\Upsilon }}}_{*}$, we find no correlation between BTFR residuals and galaxy effective radius: the Pearson's, Spearman's, and Kendall's coefficients are consistently between ± 0.4. Figure 2 (bottom panels) shows the case of ${{\rm{\Upsilon }}}_{*}=0.5$ ${M}_{\odot }/{L}_{\odot }$. Similarly, we find no trend with effective surface brightness. We have also fitted exponentials to the outer parts of the luminosity profiles and find no trend between residuals and central surface brightness or scale length. These results differ from those of Zaritsky et al. (2014) due to the use of ${V}_{{\rm{f}}}$ instead of ${\rm{H}}\;{\rm{I}}$ line widths (see also Verheijen 2001). The lack of any trend between BTFR residuals and galaxy structural parameters is an issue for galaxy formation models, which generally predict such correlations (Dutton et al. 2007; Desmond & Wechsler 2015).

Gas-dominated galaxies provide an absolute calibration of the BTFR. McGaugh (2012) used 45 galaxies that were selected for having ${f}_{{\rm{g}}}\gt 0.5$ (assuming ${{\rm{\Upsilon }}}_{*}$ based on stellar population models). Here we use a slightly different approach: we fit the BTFR weighting each point by $1/{f}_{{\rm{g}}}^{2}$. In this way, galaxies with f_g = 0.5 are 25 times more important in the fit than galaxies with f_g = 0.1. Our results are shown in Figure 2 for ${{\rm{\Upsilon }}}_{*}=0.5$ ${M}_{\odot }/{L}_{\odot }$, using both standard error-weighted fits and ${f}_{{\rm{g}}}^{2}$-weighted fits. The latter hints at a small, systematic increase of ${{\rm{\Upsilon }}}_{*}$ with $1/{f}_{{\rm{g}}}$ or ${V}_{{\rm{f}}}$ (by ∼0.1 dex) in order to have symmetric residuals.