A Generalist, Automated ALFALFA Baryonic Tully–Fisher Relation

The ALFALFA survey is a drift-scan, sensitivity-limited survey using the seven-beam Arecibo L-band Feed Array (ALFA) receiver on the Arecibo 305 m telescope. This survey detected more than 31,000 extragalactic sources (Haynes et al. 2018), with the goal of sampling the H i mass function to ≃100 Mpc. The typical sensitivity of the ALFALFA survey (5σ detection of 0.75 Jy km s⁻¹ for W₅₀ = 200 km s⁻¹) allows for the detection of low-mass (log(M_{H I}/M_⊙) < 8) sources out to ∼35 Mpc and higher-mass sources to ∼260 Mpc (heliocentric velocity = 17,912 km s⁻¹). When calibrating the template relation for this study, we use the distance estimates from the ALFALFA catalog, which are a combination of flow-model distances and distances from cluster catalogs (Haynes et al. 2018). The sample used in this analysis is drawn from the ALFALFA extragalactic catalog with the goal of being representative of the same population. The H i masses and velocity widths used in this analysis are calculated using the α.100 catalog and ALFALFA spectra, respectively. The α.100 catalog also contains measures of profile velocity widths including W_50,P (defined as the width of the profile at 50% of the peak flux), which is calculated with some manual intervention. The new widths presented here are calculated fully automatically and thus can also be homogeneously calculated for samples where there is interest in applying this analysis.

Assuming optically thin emission, H i mass is computed using ALFALFA catalog flux values and the following standard formula:

$\begin{eqnarray}&&{M}_{{\rm{H}}\,{\rm\small{I}}}=2.356\times {10}^{5}{D}_{{\rm{H}}\,{\rm\small{I}}}^{2}{S}_{21}{M}_{\odot },\end{eqnarray} \tag{ 1 }$

where D_{H i} is the distance to the source taken from the ALFALFA catalog (as described above) in megaparsecs, and S₂₁ is the integrated H i line flux in Jansky kilometer per second. Previous works find that the inclination-dependent self-absorption correction is likely only ∼10% of the H i mass in most ALFALFA galaxies, and reaches ∼35% only in the most edge-on sources (Jones et al. 2018), so no correction for self-absorption is made in the fiducial relation. To account for other contributions to the gas mass, the fractions of metals and molecular gas are incorporated via the flat conversion M_gas = 1.33M_{H I}, which is commonly used in BTFR studies.

The stellar masses used in the fiducial relation are based on Sloan Digital Sky Survey (SDSS; Blanton et al. 2017) optical photometry and are tabulated using Equation (7) of Taylor et al. (2011) and additional considerations discussed in Durbala et al. (2020). These masses are recommended among those discussed in Durbala et al. (2020) based on their agreement with rigorous spectral-energy-distribution-based mass estimates and the availability of data for a large fraction of sources in the full ALFALFA catalog. Additionally, near-infrared (NIR) masses tabulated using unWISE (Lang et al. 2016) photometry are subject to shredding effects, leading to underestimated masses in the most extended ALFALFA sources; SDSS-based stellar masses are not subject to these effects. Section 5.2.1 presents a BTFR using unWISE NIR photometry-based stellar masses, which may be used to extend this work beyond sources with available optical observations.

The Doppler shift of the 21 cm line is a useful tracer of the motion of galaxies' baryons, and thus the rotational velocity at the extreme extent of galaxies. When resolved rotation curves are unavailable, a galaxy's rotational velocity can be approximated by measuring the width of the spectral-line profile. To ensure uniformity, specific definitions of velocity width are used across the literature. Commonly used velocity width definitions measure the width across the global H i line profile at some percentage of the peak flux (e.g., W_50,P ), or similarly of the mean flux (e.g., W_50,M ). Some velocity definitions are, in general, larger and in better agreement with the flat velocities at the extreme edge of a galaxy's rotation curve, e.g., W_20,P (Lelli et al. 2019) and W_50,M (Courtois et al. 2009); others may be more robust at low signal-to-noise (S/N), e.g., W_50,P (Springob et al. 2005). Additional velocity width measurements come from parametric fits to the full shape of a line profile (e.g., busy function; Westmeier et al. 2014), or fitting line widths using physical source modeling (Peng et al. 2022).

The velocity width used in this analysis is based on the curve-of-growth (COG) width-finding method proposed in Yu et al. (2020). As discussed there, this algorithm is straightforward to implement automatically. Automatically measuring profile line widths allows for efficient and uniform reduction of large data sets, which is our aim in this study. The program used here fits a piecewise function to a pair of lines; the cost surface of this problem is generally simple, and relevant parameters (the pivot channel, the slope of the line on either side of the pivot) are straightforward to initialize with rigorous guesses. ⁷ Because of this, this method is computationally faster than a nonlinear least-squares fitting algorithm that includes more parameters (e.g., the busy function; Westmeier et al. 2014; Stewart et al. 2014), and is currently more reliable than recent attempts using trained neural networks (e.g., Hallenbeck et al. 2019). Additionally, its dependence on the integrated flux of a line profile rather than its peaks and boundaries means its performance at low S/N is significantly more reliable than attempts at automating W_50,P and W_20,P .

The implementation of a COG width-finding algorithm presented here is slightly different from the original presentation in Yu et al. (2020). Three examples of the fitting procedure used in this analysis are presented graphically in Figure 1 for galaxies with different profile shapes and typical S/N values, and a general sketch of the procedure is as follows:

• 1.  
  Find the heliocentric velocity of a line profile using the hermite polynomial matched filtering described in Saintonge (2007). This determines the velocity channel where the two black curves of growth meet and are centered in the figure.
• 2.  
  Moving outward from the central channel, add the flux in subsequent channels to both the left and right in order to create a COG—the integrated flux of the profile as a function of channel from profile center. The result of this integration is the black curve in Figure 1. Errors for this curve are determined from the sum of the per-channel rms.
• 3.  
  Fit a piecewise function consisting of two straight lines to the resulting COG, allowing for a coarse determination of the integrated flux of the profile; we use the pivot of the piecewise as this integrated flux. This best-fit piecewise is plotted on each profile in Figure 1 using a dashed–dotted line, and the pivot of the piecewise intersects the dotted red line, indicating this integrated flux. The red region in each figure is centered on the best-fit integrated flux value, with the region extending to ±1× the uncertainty on that integrated flux value.
• 4.  
  To measure the COG velocity width, which is defined as the width of the profile at some percent of the integrated flux of the profile, the COG is renormalized, and the profile width is determined by the velocity width where the COG summed from the data intersects the desired percentage of the total flux. In Figure 1, this occurs when the black COG intersects the dotted horizontal black line at 0.75 of the integrated flux, with the profile width indicated using solid black vertical lines on the figure. As discussed below, this threshold of 0.75× the integrated flux is chosen to balance the increased uncertainty near the edge of the integrated curve with the desire to reach as large of a velocity width as possible, in attempt to define a velocity proxy more likely to describe the flat part of a galaxy's rotation curve. In this, V₇₅ is designated to balance two desires: first, defining a velocity width that can be reliably measured automatically, even in low-S/N profiles; and, second, defining a velocity which is a rough proxy for the rotational velocity on the edge of a galaxy's baryons.

In theory, the summing of the COG and subsequent piecewise fitting could be done separately for each half of the profile in order to more flexibly measure the widths of very asymmetric profiles. Testing this procedure on a subsample of sources finds that the widths are more reliable (i.e., follow the same general trends between V₇₅ versus W_50,P confirmed with a high-S/N sample and shown in Figure 2) when both sides are summed simultaneously; in the typical S/N range of this sample, the increased S/N of the summed COG is more important than carefully treating the profile shape.

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Occasionally, this width-fitting program outputs widths that are clearly discrepant from ALFALFA width measures (where the fit V₇₅ is more than twice, or less than half, the rotation velocity measured using W_50,P /2). The most robust indicator that the width fitting has failed is if the output V_helio measurement greatly differs from that of the ALFALFA catalog; we use ΔV_hel = V_{helio,ALFALFA} − V_helio,WF, where V_{helio,ALFALFA} is the catalog measurement and V_helio,WF is the measurement made by the width-fitting program. When ΔV_hel is large, the matched filtering step finds some other noise peak, or uses a single peak of a strongly double-horned profile as its center. We find that taking a cutoff of ∣ΔV_hel∣ < 50 km s⁻¹ removes the majority of such sources from the sample, as illustrated in the first panel of Figure 2. In this panel, the blue points are the sources selected with ∣ΔV_hel∣ > 50 km s⁻¹, superimposed on the remaining sources which have ∣ΔV_hel∣ ≤ 50 km s⁻¹ criteria (gray). As demonstrated in the following panels, two additional cuts further reduce the number of outliers created by the width-fitting program: (i) unphysical negative uncertainties, which are indicative of an algorithmic issue normalizing the COG, and (ii) a long-term slope of the COG above an empirically determined value, which indicates that the piecewise curve fit did not find the flat part of the COG successfully. These three exclusion criteria remove many of the most clearly incorrect V₇₅ values for the α.100 sample, minimizing the number of good fits that are removed as we control for the performance of the width-fitting program. In total, these criteria flag 2539/31,502 (∼8%) ALFALFA sources, reducing the number of sources with V₇₅/W_50,P ratios vastly different from the average range of 0.5 <2V₇₅/W_50,P < 2 (bottom-right panel, Figure 2).

In order to have confidence that the reported uncertainties represent the width-finding program's success in finding the true width of a line profile, we test the program on a sample of 66 real profiles with S/N > 300 by degrading them to lower S/N values. Comparing the high-S/N reliable values of the profile widths with those measured in the noisy profiles, ≳68% of these noisy profile widths agree with the "base truth" value within 1σ for S/N > 7. The number within 2 and 3σ is nearly consistent with Gaussian distributions above this S/N threshold, as well, although for some classes of profiles there is an outlier fraction of ∼a few percent. This suggests that, although the reported uncertainties are reasonably good at describing the difference between true and noisy profile widths, ∼2% of widths for profiles with S/N > 9 may be inconsistent with expectations because of poor performance of the width-finding program. The outlier fraction increases in the lower-S/N regime of the sample, which have 4.5, 6, and 10% sources with ΔV > 3σ for S/Ns of 9, 7, and 5, respectively. Given the distribution of profile S/N values for the full ALFALFA sample and the sample used to fit the template relation (5.8, 8.4, 17, and 6.8, 10.2, 21 for the 16th, 50th, 84th percentiles, respectively), we expect ≲6.6% of sources to have widths significantly distorted by noise in an ALFALFA BTFR and ≲4% of the sample used to fit the template relation (α.R) to have distorted widths attributable to the performance of the width-finding program. We run similar tests using the approximate profile shapes used as the matched filtering template bank, finding comparable results above S/N = 5.

An additional factor that makes it difficult to reliably measure the width of low-S/N profiles is variation in the baseline of the spectrum. To determine how the width-fitting program is affected by baseline artefacts, we simulate noise and simple baseline variations over noiseless profiles simulated from input model rotation curves using the schema discussed in Obreschkow et al. (2009). Again, we find similar consistency between the true widths and degraded widths as in the ALFALFA data experiment, as the piecewise fit in the width-fitting program is appropriately insensitive to artefacts beyond the bulk of the line (as can also be seen in Figure 1). In the case of higher-order baselines (sloped, curved baselines), the method for calculating the uncertainty of the pivot flux takes these variations into account in determining appropriate uncertainties for measured widths.

Though V₇₅ is a smaller width than W_50,P , the two have a relatively constant proportionality for most double-horned profiles, with V₇₅/W_50,P increasing at low widths where profiles are more Gaussian (see Figure 2). This suggests that V₇₅ weights the shape of the line profile differently in determining the line profile width than W_50,P , as one would expect given that one looks at the integrated flux of the profile and the other looks at the extrema. Though in general smaller velocity measurements represent less extreme velocities in a galaxy's rotation curve, a straightforward flat multiplier correction (V₇₅ = 0.41W_50,p ; see Figure 2) could be applied to recover W_50,P from V₇₅ across the majority of the ALFALFA mass range (for sources with log (W_50,P ) > 1.8, true for ∼26,000/31,000 ALFALFA galaxies). In fact, the increase in V₇₅/W_50,P at low mass suggests that V₇₅ may in fact be more effective at tracing the extreme velocities at the edges of dwarf galaxies' slow-rising rotation curves (e.g., McQuinn et al. 2022). Given the difficulty tying the width of a profile to the internal dynamics of a source, the differences between this velocity width and others does not preclude the use of V₇₅ in BTFR studies; rather, it provides a new avenue for automatically fitting widths for use in large data sets.

The observed Doppler-broadened line width measures only the radial component of the galaxy's rotation and must be corrected for disk inclination. We estimate the inclination of these sources using their observed axial ratios from SDSS photometry and the relation

$\begin{eqnarray}&&\sin (i)=\sqrt{\displaystyle \frac{1-{\tfrac{b}{a}}^{2}}{1-{q}_{0}^{2}}},\end{eqnarray} \tag{ 2 }$

where $\tfrac{b}{a}$ is the observed axial ratio of the source, and q₀ is the intrinsic ratio (Hubble 1926). These inclinations allow us to correct the sources' rotational velocities for projection effects:

$\begin{eqnarray}&&{V}_{\mathrm{rot},75}=\displaystyle \frac{{V}_{75}}{\sin (i)(1+z)}.\end{eqnarray} \tag{ 3 }$

V_rot,75 is thus a measure of source rotational velocity as it includes corrections for inclination effects and is converted to a rest-frame velocity (via multiplication by a 1/(1 + z) factor). We use q₀ = 0.15; most studies use values between q₀ ∼ 0.1–0.2 for blue spirals, which we expect to dominate the ALFALFA sample (e.g., Karachentsev et al. 2017, their Figure 2; Yuan & Zhu 2004). To investigate the reliability of these inclination estimates, we compare them with estimates tabulated in the HYPERLEDA catalog, ⁸ which derives morphology-dependent inclination values using

$\begin{eqnarray}&&\sin (i)=\sqrt{\displaystyle \frac{1\mbox{--}{10}^{-2\mathrm{log}({r}_{25})}}{1\mbox{--}{10}^{-2\mathrm{log}({r}_{0})}}},\end{eqnarray} \tag{ 4 }$

where r₀ is a dimensionless parameter that varies with morphological Hubble type and r₂₅ is the axial ratio of the B-band 25 mag arcsec⁻² isophotes for the source (Paturel et al. 2003). We plot these inclination estimates against one another in Figure 3 and find general agreement, consistent with Bradford et al. (2016)'s finding that a morphology-dependent inclination does not appear to significantly affect the best-fit BTFR.

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As illustrated by the dashed lines around the 1:1 ratio in Figure 3, there is an average spread of ∼10° difference in the inclinations measured via these different methods for the sample of crossmatched HYPERLEDA-ALFALFA sources. This is incorporated as a systematic term in the uncertainty of the inclination correction. The statistical term of this uncertainty comes from the propagation of the axial ratio uncertainty in the photometry of each source, and is in most cases much smaller than the systematic term.

As can be seen in Figure 3, there are a number of sources with inclinations >10° disparate between measurement methods; the axial ratios for these sources are different between these data sets and thus have different inclinations and inclination-corrected velocity measurements. We find consistent relations fit using SDSS and HYPERLEDA inclinations when using an outlier-downweighting fitting method. Sources with δ i > 10° between estimation methods are often outliers from only one of the two BTFRs, which we take as evidence that these inclinations are not reliable. Of the 31/842 sources which are only outliers in the SDSS sample, 28/31 sources have δ i > 10°. Similarly, 31 of the 35 outliers from the HYPERLEDA relation have δ i > 10°. Corrections based on an over/underestimated inclination will move points away from the central relation, making them more likely to be outliers from the BTFR. When we find outliers where the inclination is overestimated, this overestimated inclination will lead to smaller sin(i) uncertainties, causing these unreliable inclinations to have disproportionately large influence when using a simple fitting likelihood that does not account for an outlier fraction as described in Section 3.1. These sources are dealt with through the downweighting of outliers using a Gaussian mixture model and, in the sample of sources used to fit the template relation, imposing an inclination restriction designed to reduce the fraction of sources with unreliable SDSS inclinations in the HYPERLEDA-ALFALFA crossmatch sample (i > 60°, red vertical line in Figure 3). In the sample with i >60°, 5/353 sources are identified as SDSS-only outliers, and these sources all have HYPERLEDA i ≪ SDSS i. We note that sources with understimated inclinations from SDSS axial ratios have previously been noted as outliers in other studies of the BTFR (Catinella et al. 2012).

Note that the fiducial relation does not include a turbulence correction for the rotational velocities as is occasionally done in BTFR studies, although we do discuss a BTFR with such a correction applied in Section 5.2.2. As this is the first attempt to use and interpret a V₇₅-based BTFR, applying a uniform correction to a velocity measure likely to be more sensitive to V/σ than other width measures (see, for example, the different relation between shape and width demonstrated by the V₇₅/W_50,P relation) adds unnecessary complication for our specific application of these velocities.

We fit the relations discussed here using two likelihood functions. The first is a simple linear model in log–log space, much like the ones used in Lelli et al. (2019) and Papastergis et al. (2016). As these observations have significant uncertainties in both the x and y variables, using straightforward linear regression is known to introduce a bias to the slope of a fit relation (Willick et al. 1995). Generative likelihood models rigorously handle this while maintaining model simplicity. This model is an intrinsic Gaussian distribution perpendicular to a best-fit linear relation in log–log space. This model assumes that the uncertainties, here approximated as Gaussian, describe the probability of an observation given a true underlying point within this distribution (see Appendix appendix for more details.) The other model used to produce the presented fiducial relation, which we call the mixture model, fits for a second "outlier" component parallel to this best-fit relation. As discussed in Hogg et al. (2010), this class of model can be used to reduce the influence of outlier sources on a population's characteristics. Sources have an intrinsic binary flag placing them in the "outlier" or "bulk" components of the model. The best-fit likelihood includes the probability of each source belonging to each component, meaning that sources that have a low probability of belonging to the "bulk" component have less influence over the best-fit parameters for that component. The likelihood for this model is discussed further in Appendix appendix, where there is also discussion of additional tests of more complicated outlier models.

We can identify sources as confident members of the outlier component by looking at the posterior probability for each data point. These sources are difficult to exclude from the sample using the parametric cuts described in Section 3.2. They may include sources with underestimated uncertainties, confused sources in the α.100 sample (Jones et al. 2015) or sources with unreliable or misidentified optical photometric objects (Durbala et al. 2020), or sources with intrinsic properties which make them distinctly different from the regular rotators most adherent to the BTFR (e.g., blended sources, mergers, tidally disturbed galaxies, ultra-diffuse galaxies; Mancera Piña et al. 2019). Each of these types of sources appears multiple times in the most discrepant sample of outliers, discussed in more detail in Section 4.2.

Restricted samples of high-quality, targeted observations are important for use in the BTFR in cosmological analyses, as they reduce observational uncertainties and probe the fundamental relation while minimizing competing effects. As discussed in Bradford et al. (2016), the properties of a sample used to define a template are important for determining where that template is applicable. If restrictive cuts are not appropriately accounted for in the fitting process, they can limit the applicability of, and in some cases bias, the resulting relation (see demonstrations of this in Section 5.2).

To define the template sample here, we take a different approach: we make minimal cuts to preserve the properties of the underlying α.100 sample of sources. The sample α.R (R for "restricted"), which defines the fiducial template relation, includes some cuts which make it more selective than the full ALFALFA catalog. However, these cuts are agnostic to the demographics of the galaxy population (e.g., gas fraction, stellar mass, morphology), and are instead motivated by a desire to reduce the number of outliers caused by improper fitting, low-quality data, or sources with disturbed morphologies/that unlikely to be regularly rotating. We also run fits which relax the α.R-defining restrictions, showing that broadly the full α.100 sample is consistent with this template (see Section 5.1). The criteria used to define the α.R sample are as follows:

• 1.  
  Sources recommended for use by the output of the width-fitting program. As described in Section 2.3, we determine a number of criteria that prune the most severe failures of the width-finding program.
• 2.  
  Sources that have stellar masses derived from SDSS photometry in the Durbala et al. ( 2020 ) catalog.
• 3.  
  Sources far from large overdensities. One intended application of this relation is to measure signatures of infall onto the PPS. We want to fit a template that is unlikely to contain infall signatures, necessitating the removal of certain regions of the sky from the sample. To this end, we also exclude Virgo, the other large overdensity in the ALFALFA survey range. Within the ALFALFA footprint, we exclude all sources that are at 22^h < R.A. <3^h to exclude PPS and all sources with R.As. 12^h08^m−12^h56^m, decls. 2°–18°, and velocities 350–3000 km s⁻¹, to exclude Virgo from the sample.
• 4.  
  Sources with inclinations i > 60°. As discussed in Section 2.4, we use an inclination cut of SDSS i > 60°, as shown in Figure 3, in order to reduce the number of sources with unreliable SDSS inclinations. This does not remove all of the outliers with SDSS i > HYPERLEDA i but instead minimizes the fraction of sources with misidentified inclinations while maximizing the size of the remaining, nonrestricted sample.
• 5.  
  Sources with uncertainties <20% profile widths. This restriction is essentially a width measurement S/N threshold. We test a number of uncertainty thresholds and find, as expected, that the slope of the relation increases as we impose a stricter uncertainty threshold and the noisy scatter of the distribution of widths is reduced. Above this imposed 20% uncertainty threshold the slope of the fit BTFR stays relatively constant. This threshold therefore appropriately limits the additional scatter and uncertainty caused by low-S/N widths and is the least restrictive cut we can make to this effect.
• 6.  
  Sources which do not appear highly asymmetric to the width-fitting program. Sources may appear highly asymmetric for a number of reasons. First, this selects for many of the remaining failures of the width-fitting program; if the program finds an inaccurate V_helio, it will center the curve on a noise spike or on one horn of a double-horned profile, causing different "width" measurements for the left and right sides of the profiles when measured separately. Second, interacting sources may also have highly asymmetric line profiles (Espada et al. 2011; Bok et al. 2019). As these sources are disturbed, they are more likely to be offset from the BTFR defined by regularly rotating sources. Confusion (e.g., Jones et al. 2015) may also contaminate the sample; this emission may appear lopsided if the two sources in the beam are at different distances from the H i centroid.

The number of sources in the α.100 sample which satisfy each selection criteria are listed in Table 1. The imposition of all six of these criteria produce a reliable sample of ∼4500 sources broadly representative of the α.100 population, which we use to define a template relation that can be used as a distance indicator with minimal uncertainty and contamination.

The fiducial "inclusive ALFALFA relation" presented here is based on the α.R sample discussed in Section 3.2. The fiducial relation, as well as a best-fit relation fit without an outlier component, is plotted over both the α.R and α.100 samples in Figure 4.

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The template α.R relation,

$\begin{eqnarray}&&\mathrm{log}({M}_{b}/{M}_{\odot })=m\times (\mathrm{log}({V}_{\mathrm{rot},75}/\mathrm{km}\,{{\rm{s}}}^{-1})-2)+b,\end{eqnarray} \tag{ 5 }$

(where V_rot,75 is V₇₅ corrected for inclination and redshift broadening) has best-fit slope (hereafter m), intercept (b), and intrinsic perpendicular scatter (σ_⊥) values of m = 3.30 ± 0.06, b = 9.9 ± 0.014, and σ_⊥ = 0.025 ± 0.004, respectively. This model includes an additional component that describes the distribution of outliers—a Gaussian distribution which runs parallel to the best-fit relation (Equation (5)) with a perpendicular offset (−0.07 ± 0.01, σ = 0.13 ± 0.01), and a weighting of (0.2 ± 0.03) relative to the main component. This relation is fit using 3000 galaxies randomly drawn from the 4525 galaxies in α.R to leave a significant sample of α.R sources available for validation. We note that this relation is stable within the uncertainties regardless of which random sample of 3000 sources is drawn. This template relation, along with a relation fit without a Gaussian outlier component, is presented in Figure 4 and the first row (Ia) of Table 2, which summarizes all of the fits run in this investigation. To the eye, the fiducial relation may appear steeper than the distribution of data; this is an expected result of rigorously fitting a relation to data with significant uncertainties in the independent variable: fits by eye or using methods which do not appropriately treat x uncertainties will tend to fit relations that are inappropriately shallow.

Table 2. Summary of Fits Discussed in Section 5

ID	Sample Characteristics	Sample Size	Velocity Width	Simple Slope	Simple Intercept	Simple Scatter	Mix Slope	Mix Intercept	Mix Scatter
Ia	Fiducial	4525	V75	3.36 ± 0.04	9.94 ± 0.01	0.065 ± 0.0025	3.30 ± 0.03	9.9 ± 0.01	0.024 ± 0.008
Ib	Restrictions w/o 3 (No PPS/Virgo removal)	6811	V75	3.4 ± 0.04	9.96 ± 0.01	0.07 ± 0.002	3.34 ± 0.03	9.9 ± 0.01	0.026 ± 0.004
Ic	Restrictions w/o 4 (No inclination restriction)	6970	V75	3.43 ± 0.04	9.94 ± 0.006	0.072 ± 0.002	3.34 ± 0.03	9.89 ± 0.01	0.027 ± 0.003
Id	Restrictions w/o 5 (No uncertainty restriction)	5160	V75	3.32 ± 0.03	9.96 ± 0.005	0.0677 ± 0.015	3.28 ± 0.03	9.91 ± 0.07	0.026 ± 0.005
Ie	Restrictions w/o 6 (No asymmetry restriction)	5818	V75	3.33 ± 0.03	9.96 ± 0.005	0.071 ± 0.002	3.29 ± 0.03	9.9 ± 0.01	0.025 ± 0.0045
If	Forced same mass sampling	2931	V75	3.37 ± 0.035	9.949 ± 0.005	0.064 ± 0.001	3.326 ± 0.03	9.91 ± 0.01	0.029 ± 0.003
IIa	Fiducial, McGaugh mass, $\delta ({log}(M))\lt 0.3$	4253	V75	3.48 ± 0.03	9.99 ± 0.01	0.004 ± 0.003	${3.52}_{-0.05}^{+0.06}$	9.96 ± 0.01	0.004 ± 0.003
IIb	Fiducial, Ponomareva mass, $\delta ({log}(M))\lt 0.3$	4253	V75	3.34 ± 0.03	9.77 ± 0.01	${0.023}_{-0.007}^{+0.003}$	${3.62}_{-0.05}^{+0.07}$	9.69 ± 0.01	0.000 ± 0.00
IIc	Fiducial, recovered Taylor mass, $\delta ({log}(M))\lt 0.3$	4253	V75	3.54 ± 0.04	9.93 ± 0.01	0.069 ± 0.002	${3.51}_{-0.05}^{+0.06}$	9.91 ± 0.01	0.03 ± 0.01
IIIa	Fiducial	4525	W50,P	3.35 ± 0.04	9.72 ± 0.01	0.08 ± 0.01	3.34 ± 0.03	9.66 ± 0.01	0.045 ± 0.002
IIIb	Fiducial, turbulence corrected V75	4525	V75,TC	${3.22}_{-0.03}^{+0.04}$	9.95 ± 0.01	0.07 ± 0.002	3.19 ± 0.03	9.90 ± 0.01	0.026 ± 0.003
IIIc	Fiducial/Springob crossmatch	705	V75	3.33 ± 0.07	9.89 ± 0.01	0.032 ± 0.003	${3.35}_{-0.07}^{+0.08}$	9.87 ± 0.02	0.01 ± 0.01
IIId	Fiducial/Springob crossmatch	705	W50,M	${3.43}_{-0.06}^{+0.07}$	9.52 ± 0.02	0.026 ± 0.004	3.44 ± 0.06	9.52 ± 0.02	${0.02}_{-0.008}^{+0.004}$
IIIe	Fiducial/Springob crossmatch	705	W20,P	${3.54}_{-0.07}^{+0.08}$	9.41 ± 0.02	0.033 ± 0.003	3.56 ± 0.07	9.41 ± 0.02	${0.02}_{-0.008}^{+0.005}$
IIIf	Fiducial/Springob crossmatch	705	WC	3.35 ± 0.07	9.69 ± 0.02	0.033 ± 0.003	3.37 ± 0.06	9.68 ± 0.02	0.007 ± 0.005
IVa	Fiducial/Springob crossmatch, McGaugh masses	705	WC	${3.66}_{-0.07}^{+0.08}$	9.78 ± 0.02	0.029 ± 0.004	3.68 ± 0.07	9.78 ± 0.02	0.01 ± 0.01
IVb	Fiducial/Springob crossmatch, Ponomareva masses	705	WC	3.45 ± 0.07	9.73 ± 0.02	0.035 ± 0.003	3.45 ± 0.07	9.72 ± 0.02	0.02 ± 0.01
IVc	Gas fraction > 2	1836	W50,P	3.80 ± 0.08	9.81 ± 0.02	0.089 ± 0.002	${3.92}_{-0.10}^{+0.08}$	9.77 ± 0.02	0.05 ± 0.005
IVd	$\mathrm{log}({M}_{b})\lt 10.75$	4096	W50,P	3.37 ± 0.04	9.74 ± 0.01	0.079 ± 0.002	3.39 ± 0.03	9.68 ± 0.01	0.046 ± 0.001
IVe	b/a < 0.25	1178	W50,P	${3.34}_{-0.06}^{+0.07}$	9.71 ± 0.01	0.066 ± 0.002	3.30 ± 0.05	9.70 ± 0.01	0.042 ± 0.006
Va	$\mathrm{log}({M}_{b})\gt 10$	2666	V75	2.82 ± 0.06	10.02 ± 0.02	0.059 ± 0.002	${2.88}_{-0.05}^{+0.06}$	9.97 ± 0.01	${0.011}_{-0.007}^{+0.007}$
Vb	$\mathrm{log}({M}_{b})\gt 10$	2666	V75,TC	${2.78}_{-0.04}^{+0.05}$	10.02 ± 0.01	0.060 ± 0.02	2.85 ± 0.04	9.97 ± 0.01	${0.02}_{-0.01}^{+0.005}$
Vc	$\mathrm{log}({M}_{b})\gt 10$	2666	W50,P	2.83 ± 0.07	9.83 ± 0.02	0.071 ± 0.002	${2.93}_{-0.05}^{+0.05}$	9.75 ± 0.02	0.036 ± 0.003
Vd	$\mathrm{log}({M}_{b})\leqslant 10$	1859	V75	${4.13}_{-0.11}^{+0.12}$	10.01 ± 0.02	0.066 ± 0.002	${4.16}_{-0.11}^{+0.12}$	10.00 ± 0.01	0.036 ± 0.004
Ve	$\mathrm{log}({M}_{b})\leqslant 10$	1859	V75,TC	3.68 ± 0.10	10.00 ± 0.01	0.073 ± 0.002	${3.77}_{-0.09}^{+0.10}$	9.98 ± 0.01	0.040 ± 0.004
Vf	$\mathrm{log}({M}_{b})\leqslant 10$	1859	W50,P	${4.00}_{-0.11}^{+0.12}$	9.73 ± 0.01	0.084 ± 0.002	${4.08}_{-0.11}^{+0.12}$	9.69 ± 0.01	0.055 ± 0.002
Vg	DALF ≤ 40 Mpc	299	V75	${3.83}_{-0.15}^{+0.16}$	9.92 ± 0.04	0.063 ± 0.007	${3.86}_{-0.15}^{+0.17}$	9.88 ± 0.04	${0.03}_{-0.02}^{+0.01}$
Vh	DALF ≤ 70 Mpc	786	V75	${3.76}_{-0.09}^{+0.10}$	9.94 ± 0.02	0.072 ± 0.005	3.67 ± 0.08	9.89 ± 0.01	0.03 ± 0.01
Vi	DALF ≤ 100 Mpc	1663	V75	3.48 ± 0.05	9.92 ± 0.01	0.063 ± 0.003	${3.43}_{-0.04}^{+0.05}$	9.88 ± 0.01	0.025 ± 0.05
VIa	Fiducial, scatter = 0	4525	V75	3.15 ± 0.02	9.958 ± 0.003	0	${3.30}_{-0.03}^{+0.07}$	9.90 ± 0.01	0
VIb	Fiducial, scatter = 0	4525	W50,P	2.70 ± 0.01	9.81 ± 0.002	0	${3.28}_{-0.03}^{+0.05}$	9.66 ± 0.01	0

Notes. Horizontal bars demarcate different portions of the analysis; the first section demonstrates that relaxing the criteria which define α.R results in a relatively stable fiducial relation generally applicable to the full α.100 sample. Note that we do not present results for a relation without relaxing restriction 2, which is that the sources must have stellar masses in our catalog, as it is not possible to place sources on our relation without a usable stellar mass estimate. The second region illustrates the differences between fits to the full fiducial sample using different stellar mass estimates, the third illustrates the difference induced by using different velocity definitions, the fourth provides relations with data methods/velocity definitions/sample criteria more closely matched to literature relations, and the fifth demonstrates the slope discrepancy between the high- and low-mass ends of the sample.

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The uncertainties reported here include both the statistical uncertainties from the fitting procedure (reported in Table 2) and systematic uncertainties (0.05 in m and 0.01 in b), estimated using the spread in best-fit relations when we relax the imposed restriction criteria in Section 3.2. Importantly, the slope and intercept of this template relation (Equation (5)) are consistent with these relations (see Section 5.1). Though defining a restricted sample is useful for ensuring that the relation is fit with a high-confidence sample, the consistency with fits to less restricted samples suggests that this α.R relation can be used to estimate distances to the broader α.100 sample, including sources which do not fit all listed selection criteria. As discussed later in this section, the corresponding uncertainties and outlier fractions will correspondingly be larger in this case.

This fiducial relation is drawn very generally from the full α.100 catalog—itself complicated by issues of varying coverage, the impact of radio frequency interference, confusion, and uncertainties in the identification of optical counterparts—meaning that there is a possibility of distorted data which may affect the relation and be difficult to remove parametrically. This motivates use of the mixture model to de facto downweight the most anomalous data. We investigate the 80 sources with the lowest probability of membership in the bulk relation model, ⁹ finding the following common classes:

• 1.  
  Sources with close companions. Thirty sources (∼38% of the 80 most extreme outliers) have nearby companions, identified in SDSS imaging by small sky (≲6', ≲120 kpc at 70 Mpc) and velocity (∼few hundreds of kilometers per second) separations. Seven of these sources are deblended from nearby sources in the ALFALFA data-reduction process. Nearby companions may impact sources' positions on the BTFR, as confusion with gas-rich sources within the ALFALFA beam or distortion of H i disks via interaction add significant nonrotational velocity to these source profile widths. Other studies find evidence that line profiles are affected by the presence of a close companion, including twice as many kinematic anomalies in pairs versus field galaxies (Kannappan & Barton 2004), significantly more asymmetry in H i line profiles for sources in close pairs versus isolated sources (Bok et al. 2019), and more flat-topped or widened H i profiles for sources in close pairs (Zuo et al. 2022). The outlying sources with close companions sit both above and below the bulk relation in Figure 5, demonstrating that sources with close companions add scatter to the bulk relation but do not preferentially displace these sources to larger velocity widths.
• 2.  
  Sources with misidentified inclinations. Eight (10%) of the 80 extreme outliers can be confidently identified as nearly face-on in SDSS imaging. All eight sources have external evidence suggesting SDSS inclinations may be unreliable, including HYPERLEDA inclinations >10° different for 7/8 and detailed modeling in Meert et al. (2015) and Galaxy Zoo classification as "not edge-on" (Willett et al. 2013) for the eighth. All eight of these sources sit to the left of the bulk relation in Figure 5, suggesting that their inclination-corrected velocity is underestimated, consistent with the interpretation that the 1/sin(i) factor is too small for these sources. Though the inclination selection criteria in Section 3.2 are designed to account for most outliers due to misattributed inclinations, a small fraction of overestimated SDSS-based inclinations in the above HYPERLEDA-ALFALFA comparison persists even with this selection criteria. The persistence of a (albeit small, ∼0.25% of the total sample) fraction of misattributed inclination outliers is unsurprising.
• 3.  
  Sources with inconsistent stellar mass estimates. Nine (∼11%) additional outliers have stellar masses that are likely to be unreliable. Four such sources have bright foreground stars in the same field as the galaxy in SDSS imaging. These sources all sit significantly above (large, positive ${\rm{\Delta }}\mathrm{log}({M}_{b})$) the general relation, suggesting that their masses are overestimated, potentially appearing brighter than expected because of leaking flux from the foreground star. The remaining five sources without foreground stars have inconsistent >5× SDSS and unWISE stellar masses. These sources all have an SDSS photometric object identifier (PhotoObjID) which is offset from the center of the source and instead centered on a compact region of brightness ≪the size of the galaxy, meaning that a misassigned PhotoObjID may contribute to these sources' separation from the bulk relation.
• 4.  
  Sources with inconsistent velocity measures. Several sources may be outliers because of unreliable V₇₅ measurements. Twenty-one sources identified as outliers when fitting the V₇₅ relation are not identified as outliers when fitting the W_50,P relation. Ten of these sources overlap with the above categories; 9/21 have close companions, 1/21 has an overestimated inclination. The remaining 11 sources represent a fraction of the α.R 3000-source sample (0.3%) significantly smaller than the estimated percentage of ≲4% incorrect widths described in Section 2.3. Even if the majority of these 21 sources are primarily outliers because their V₇₅ values are incorrect, this still indicates an outlier percentage 0.7% ≪ 4%. This closely analyzed sample of 80 outliers represents only the most discrepant sources from the fiducial relation; this underrepresentation of width-based outliers in this analysis may indicate that inconsistent widths increase the scatter of the relation, but it is not significant enough to dramatically increase the number of outliers in the ALFALFA BTFR.

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The above four classes account for 58/80 of the sources identified as outliers by their low probability of membership in the bulk component of the Gaussian mixture model. The remaining ∼quarter (22/80) of outliers are somewhat more complicated to interpret. Three are additional blue sources with irregular morphologies, three are the remaining highly inclined sources which may be affected by unaccounted for extinction/absorption corrections, but most (16/22) are too faint in SDSS imaging to interpret and are not particularly anomalous in other one- and two-parameter descriptions of the full ALFALFA data set (e.g., gas fraction versus mass, V₇₅ versus W_50,P , S/N, etc.). Regardless, we are able to account for the majority of outliers from the fiducial relation, finding tractable reasons for their offsets.

We use all ALFALFA extragalactic sources with recommended widths and Durbala et al. (2020) SDSS-based stellar masses to estimate the average uncertainty of a BTFR-derived distance measurement. We propagate these errors using the observational uncertainties on photometry, width measurement, inclination, and mass-to-light ratio, plus the uncertainties of the best-fit relation and intrinsic scatter. For this full ALFALFA sample, we find a median uncertainty on log(distance) estimates of 0.17 dex, with 16th and 84th percentile uncertainty estimates of 0.1 and 0.28 dex. The cumulative distribution of uncertainties for BTFR-derived distances is shown as the gray curve in Figure 6, with vertical bars at each of these reported percentiles. The bump in the α.R curve at 0.2 dex is caused by the systematic uncertainty on ALFALFA H i flux measurements, which becomes the dominant uncertainty on log(M_b ) measurements at high S/N.

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When we compare the predicted distance measurements to ALFALFA distances, which are calculated using flow models as described in Haynes et al. (2018), we find that the distribution has a significantly larger standard deviation than would be expected if the departures were pure Gaussian noise (e.g., μ = 0, σ = 1). The gray distribution in Figure 7 is the difference between the ALFALFA distances and the BTFR-based distances for the full ALFALFA sample. The distribution is more sharply peaked than the best-fit Gaussian, indicating the heavy tails of this distribution and the significant number of sources with discrepancies between these distance measurements: ∼6% of these distances are >3σ different. This is larger than expected if the differences are solely attributable to Gaussian noise; however, given the nonzero outlier fraction from Gaussian mixture modeling, the numerous causes of additional noise around the relation discussed above, and the corresponding increase in noise from looking at a less restricted sample, we expect a larger-than-Gaussian number of outliers. This distribution is also asymmetric, with more sources at positive values of D_BTFR − D_ALFALFA than negative values. When plotted using their ALFALFA distance estimates, such sources sit above the template BTFR; these sources are more likely to belong to the outlier component of the best-fit likelihood than the bulk component, and thus may be a byproduct of the average data quality of the sample (in fact, a number of the outlier classes identified in Section 4.2 preferentially set sources above the best-fit relation, including overestimated inclinations and overestimated stellar masses either from foreground objects or off-center photometry). Indeed, when we consider this distribution for the more restricted, higher-quality α.R sample, we see a smaller (albeit still present) degree of asymmetry in the distribution of distance residuals, with the skewness reduced from −4.2 to −1.7.

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Using the more restricted α.R sample of sources, the median, 16th, and 84th percentile uncertainties decrease to 0.092, 0.07, and 0.12, respectively (shown as the green curve in Figure 7, which reaches larger cumulative fractions at a given uncertainty than the gray curve). Overall, these distances are in better agreement with the ALFALFA distances, demonstrating, as expected, that pruning for data quality can reduce the average uncertainty and outlier occurrence in an application sample. Despite this reduction, there are still about ∼5% sources with distance disagreements larger than 3σ, and the heavy tails demonstrate the presence of outliers even in the restricted sample.

Additionally, comparing the residuals from the template relation with various sample properties allows us to search for systematics that may be important when using this relation to measure source distances. We find that sources are generally distributed symmetrically about the relation, as shown in the top-left panel of Figure 8, although there are hints of a falloff at the lowest masses/velocities, suggesting galaxies have larger velocities at a given mass. A similar trend appears in the residuals of the turbulence-corrected relation, suggesting that if the fractionally larger contribution of turbulence to velocity widths at low velocity causes this departure, the single turbulence correction as a function of velocity proposed in Yu et al. (2020) may not fully account for the variation of this correction across the galaxy population. The strongest correlation we find is between rotational velocity and perpendicular distance (left panel, second row of Figure 8). This moderate correlation is likely attributable to the distribution of data points rather than to a slope mismatch for a number of reasons. First, a large number of sources with the largest positive offsets do not sit at the lowest masses and rather at log(M_b ) > 9.5, suggesting that these sources have large perpendicular offsets specifically because they have low rotational velocities for sources of their mass. We further test this by varying the slope of the relation and recalculating the correlation coefficients between the residuals and velocity/mass, finding that as the magnitude of the V_rot,75 coefficient decreases, the magnitude of the log(M_b ) coefficient increases—if the correlation was due to a mismatch in the slope, both coefficients would decrease as we reached the correct value. In addition to this distribution-based correlation, we find additional, weak correlations between the residuals and the heliocentric velocity, a selection effect due to the flux-limited strategy of the ALFALFA survey (Kourkchi et al. 2022), and with gas fraction, consistent with results from (Ponomareva et al. 2018).

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To explore structure in this relation beyond a simple power-law fit, we include constrained B-spline quantile fits (implemented in R as COBS; Ng & Maechler 2015) overlaid in Figure 4. The overplotted gold lines are spline fits to the central 20, 50, and 80% percentiles of the log(M_bary) values for the α.R sample, where the blue lines are the same for a more inclusive subsample of α.100. In this subsample, sources with velocity uncertainties of >0.2 dex have been removed, as COBS accounts for dependent variable uncertainty and not independent variable uncertainty. When we fit to the full α.100 sample, including sources with still larger uncertainties, the relation appears shallower still, exactly as expected given the impact of these uncertainties. We note that this likely indicates that the "true" relation is steeper than the fit splines. Beyond this, these fits highlight the additional structure in the sample beyond a simple power law, as both lines curve away from power-law fits in the high- and low-mass range. We note that the fit to the less restricted data set has increased spread and more significant deviation, demonstrating that some part of this deviation is related to data quality. These effects persist in the spline fits to the α.R sample, suggesting that some portion of this structure may be intrinsic. Previous studies have discussed the expectation of curvature in the BTFR as predicted by semi-analytic ΛCDM modeling (Trujillo-Gomez et al. 2011), as well as the observational difficulties of conclusively finding such deviations (Iorio et al. 2017). Generally, the expected curvature from semi-analytic models curves to faster rotation for lower galaxy mass, leading to higher slopes at lower galaxy masses and lower slopes at higher galaxy masses. Though we see some downturn in the shape of the relation at high rotational velocity, the relation curves upward at high velocity; further dedicated study is necessary to determine the origin and interpretation of this additional apparent structure.

We confirm the consistency of the α.R fit with the broader ALFALFA sample by running fits relaxing each of the criteria in Section 3.2 in sequence, collecting the best-fit parameters for these relations in section I of Table 2. As demonstrated in Figure 9, the lines fit to less restricted versions of α.100 are consistent with the fiducial relation. In all cases, the relations fit with an outlier component have higher intercepts than those fit without such a component, consistent with the fact that most of the outliers observed sit above and to the left of the bulk distribution of sources (see Figure 5).

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5.2.1. Stellar Mass Measurement Method

5.2.2. Velocity Definition

5.2.3. Mass Cutoff

5.2.4. Gas Fraction

5.2.5. Fitting Routine