(Almost) Dark Galaxies in the ALFALFA Survey: Isolated H i-bearing Ultra-diffuse Galaxies

The ALFALFA observations, data reduction, and catalog products are detailed elsewhere (e.g., Giovanelli et al. 2005; Saintonge 2007; Haynes et al. 2011). Columns 1–8 of Table 1 give the H i data from the ALFALFA 70% catalog for sources in the HUDS-R and HUDS-B samples. In brief, ID numbers (column 1) are taken from the Arecibo General Catalog (AGC), and the J2000 coordinates (columns 2 and 3) are those of the identified very low surface brightness optical source. Recessional velocities (column 4) are measured at the center of the H i line at the 50% flux level. W₅₀ (column 5) is the width of the H i line measured at the 50% flux level, corrected for channel broadening. H i line fluxes (column 6) are calculated from fits to the spatially integrated line spectrum, Distances (column 7) are calculated using the ALFALFA flow model, which is simply Hubble Flow at ${\rm{c}}z\gt 6000$ km s⁻¹; for sources in this velocity range (∼2000–8000 km s⁻¹) distance uncertainties due to proper motions are ≲15%. H i masses (column 8) are calculated from the given integrated fluxes and distances, assuming that the gas is optically thin.

Table 1.  Properties of HUDS

AGC ID	OC R.A.	OC Decl.	cz	W50	$\int {SdV}$	Dista	log(${M}_{{\rm{H}}{\rm{I}}}$)	${\mu }_{g,0}$	re	Mg	$g-r$	Sampleb
	J2000	J2000	(km s−1)	(km s−1)	(Jy km s−1)	(Mpc)	log ${M}_{\odot }$	(mag/''2)	(kpc)	(mag)	(mag)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)
322019	344.6121	1.8497	4819	33 ± 14	0.54 ± 0.07	72	8.81 ± 0.08	24.62 ± 0.18	3.9 ± 1.0	−15.8 ± 0.7	0.36 ± 0.25	R
103796	5.1650	6.9658	5647	31 ± 7	0.48 ± 0.04	80	8.86 ± 0.06	24.23 ± 0.14	3.9 ± 0.7	−16.2 ± 0.5	0.47 ± 0.21	R
113790	18.2587	27.6369	4952	31 ± 10	0.33 ± 0.03	69	8.57 ± 0.07	24.31 ± 0.14	2.9 ± 0.6	−15.4 ± 0.5	0.42 ± 0.19	R
114905	21.3271	7.3603	5435	27 ± 3	0.96 ± 0.04	76	9.11 ± 0.06	24.85 ± 0.19	5.1 ± 1.5	−16.2 ± 0.8	0.53 ± 0.25	R
114943	26.7775	7.3311	8416	32 ± 7	0.40 ± 0.04	116	9.10 ± 0.06	24.48 ± 0.18	4.8 ± 1.2	−16.4 ± 0.7	0.36 ± 0.28	R
113949	27.4108	30.6808	7380	44 ± 7	0.44 ± 0.05	102	9.03 ± 0.07	24.29 ± 0.18	3.4 ± 0.8	−15.8 ± 0.7	0.53 ± 0.28	R
122966	32.3708	31.8528	6518	35 ± 6	0.53 ± 0.04	90	9.00 ± 0.06	25.37 ± 0.23	7.4 ± 3.3	−16.4 ± 1.1	0.38 ± 0.41	R

Notes. H i and optical table parameters come from ALFALFA and SDSS, and are described in Sections 3.1.1 and 3.2.1 of the text, respectively.

^a Distances from the ALFALFA flow model (see Haynes et al. 2011); distance-dependent quantities include an error of 5 Mpc due to peculiar velocities. ^b R—HUGS-R, B—HUGS-B.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Of particular relevance to this paper, optical identification is done by eye, matching sources in SDSS or DSS2 images with the ALFALFA position. We emphasize this visual identification because extended nearby sources, including low surface brightness sources, are often shredded into multiple sources, and classified as more distant objects in automated catalogs. Further, we are able to identify a likely counterpart even in cases where the catalog does not include an entry due to a failure in the fit or proximity to a star. Without a corresponding optical redshift, identification necessarily relies on a small spatial offset between the optical source and the ALFALFA position. Though rare sources have been identified at large offsets from the ALFALFA centroid (Cannon et al. 2015), the average ALFALFA H i centroid accuracy is ≲20'', and confirmation observations have found the identifications to be quite reliable in almost all cases with an identified optical counterpart.

We observed AGC122966 and AGC 334315 with 2 × 12h pointings with WSRT as part of exploratory observations of (almost) dark sources in the ALFALFA survey (program R13B/001; PI Adams). The observations were centered on the central H i velocity measured in ALFALFA, with a 10 MHz bandpass with 1024 channels and 2 polarizations, leaving ample line-free channels for continuum subtraction, but still sufficient velocity resolution of 4.1 km s⁻¹ after Hanning smoothing.

We observed AGC 219533 under a separate program (14B-243; P.I. Leisman) with the Karl G. Jansky Very Large Array (VLA) in 2014. We observed the source for two there-hour observing blocks in C-configuration, using the WIDAR correlator in dual polarization mode with a single sub-band 8 MHz wide with 1024 channels, giving a native channel width of 1.7 km s⁻¹.

We reduced the WSRT data following the same process described in Janowiecki et al. (2015) and Leisman et al. (2016), using the automated pipeline of MIRIAD (Sault et al. 1995) data software wrapped with a Python script (see Serra et al. 2012; Wang et al. 2013). The pipeline automatically flags the data for RFI and iteratively deconvolves the data with the CLEAN algorithm after primary bandpass calibration, in order to apply a self-calibration. The calibration solution and continuum subtraction are applied in the visibility domain before inverting the data. The resulting data cubes are iteratively cleaned down to their rms noise, using CLEAN masks determined by clipping after filtering with Gaussian kernels. This process produces cubes with three different robustness weightings, r = 0.0, r = 0.4, and r = 6.0, and bins the data to a velocity resolution of 6.0 km s⁻¹ after Hanning smoothing.

We reduced the VLA data using standard procedures in the CASA package (Common Astronomy Software Applications; McMullin et al. 2007), including flagging of the visibilities, calibration, and continuum subtraction. We imaged the calibrated uv data following standard procedures, producing data cubes using the CLEAN task in CASA, with a Briggs robust weighting of 0.5. Since we expected the source to be extended, we used the multiscale clean option, which improves localization of extended flux (Cornwell 2008). It models the source as a collection of point sources and Gaussians of the beam width and several times the beam width.

For each source we created H i total intensity maps by creating a 3σ mask on images smoothed to two times the beam size, applying the mask to the unsmoothed cubes, and then summing along the velocity axis. We convert these maps to H i column densities assuming optically thin H i gas that fills the beam, and also produce H i moment-one maps (representing velocity fields) from the masked cubes. The resulting H i images and velocity maps are shown in Figure 3.

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We obtained calibrated, background-subtracted SDSS optical images in the g, r, and i bands for the full sample from the SDSS mosaic server described in Blanton et al. (2011). They estimate that the uncertainty in the background contributes a systematic uncertainty of up to ∼10%.

Since the inclination is poorly constrained for the low surface brightness sources in question, we measure the average flux in concentric circular annuli to approximate the surface brightness profile of the sources, using Python code we developed based on Astropy (Astropy Collaboration et al. 2013), and its affiliated package Photutils. We note that we chose the galaxy center to be the center of the extended optical flux, which, for sources with clumpy morphologies and significant evidence of star formation, may not be the location of the peak flux. We then fitted exponential functions to the surface brightness profiles, including a term to estimate a constant offset due to the background.

We correct our measured profiles for galactic extinction, but do not correct for cosmological surface brightness dimming for consistency with other local universe studies, and since the dimming corrections are small at the distances of our sample. We model the effect of the PSF on our fitted values by simulating model 1D profiles convolved with a 1D approximation of the SDSS PSF, and then applied our fitting method to both the true and convolved profiles to calculate analytic approximations of its effect. We then correct our measured parameters accordingly. We note that the sources in this sample are very extended relative to the SDSS PSF, so these corrections tend to be small.

We report the results of these fits in columns 9–12 of Table 1. Specifically, column 9 gives the measured central surface brightness from the exponential fit to the g-band data, and errors that are the quadrature sum of uncertainties from the fit and an assumed 10% uncertainty in the absolute background calibration. They do not account for additional systematic errors introduced from uncertainty in the inclination or galaxy centroid. Column 10 gives the effective radius, which is 1.68 times the disk scale length from the exponential fit, and contains half the light from the galaxy. Column 11 gives the estimated absolute magnitude derived from integration of the exponential fit and the assumed distance. We note that this total magnitude has not been truncated and should be used with caution: it is significantly brighter than measurements derived from aperture magnitudes (the average offset is ∼0.5 mag), especially given the large estimated disk scale length for these extended sources. Column 12 gives the $g-r$ color derived from the offset between the exponential fits, which is a better measurement than from using the absolute magnitude of these sources.

We observed AGC 122966 and AGC 334315 in October 2013 with the WIYN¹⁴ 3.5 m telescope at Kitt Peak National Observatory¹⁵ using the partially populated One Degree Imager (pODI). At that time, pODI had a field of view of 24' × 24', and we used a dithering sequence to eliminate chip gaps. We obtained nine 300 s exposures in SDSS g and r filters (Gunn et al. 1998; Doi et al. 2010) for both targets. Due to unfavorable weather conditions, we did not observe AGC 219533. While faint, it is significantly detected in SDSS images and we use those in this analysis.

We reduced our observations using the QuickReduce (QR, Kotulla 2014) data reduction pipeline via the ODI Pipeline, Portal, and Archive (ODI-PPA; Gopu et al. 2014; Young et al. 2014) at Indiana University. The QR pipeline removes instrumental signatures including bias, dark, flat, pupil ghost, nonlinearity, cosmic rays, and fringes. We also applied an iterative dark-sky illumination correction to produce very flat final stacked images, following the methods of Janesh et al. (2015) and Janowiecki et al. (2015), using the "odi-tools" package.¹⁶ Our final images are calibrated using catalog fluxes of SDSS stars in the frames (Alam et al. 2015) and our photometric zeropoints typically have rms errors ≤0.05 mag.

While these HUDS appear to have large H i masses relative to their stellar mass, UDGs are optically defined by their large radii for their stellar masses. Thus, for the three sources with existing H i-synthesis observations we analyze their H i radii and distribution. Estimates of their properties are limited by the low physical resolution of the data (6–14 kpc), but are still sufficient to constrain their extended nature.

All three sources are resolved with multiple beams in H i, which allows us to estimate the radii of the sources, albeit with a fairly large uncertainty. As a first-order estimate, we place an upper limit on the size of each source by measuring the largest projected extent on the sky, uncorrected for the effects of beam smearing. Specifically we measure largest extents of 44 ± 7, 38 ± 3, and 38 ± 6 kpc for AGC 122966, 219533, and 334315, respectively, assuming uncertainties of half the beam width along the major axis. We next estimate radii by fitting the observed H i profile with tilted rings every half beam width using the GIPSY task ELLINT, assuming a constant position angle and inclination, and then estimating radii at a surface density of 1 ${M}_{\odot }$ pc⁻² to compare to measurements from The H i In Nearby Galaxies Survey (THINGS; Walter et al. 2008) described below. We choose the major axis to be the kinematic major axis (see Section 4.2.2), which approximately corresponds to the morphological major axis, except in the case of AGC 122966, where the morphological major axis is poorly determined due to the elongated WSRT beam.

However, fitting flux in rings is limited by the minor axis resolution, so we additionally estimate the surface density profile using Lucy–Richardson deconvolution (Lucy 1974; Warmels 1988), which collapses the flux to a 1D profile along the major axis, and then models that profile as a disk of uniform coplanar rings. This method has the advantage of not requiring an estimate of the inclination of the sources, and is insensitive to low resolution along the minor axis, but is still limited by the resolution along the major axis. Outside of the central beam width, Warmels (1988) estimate the uncertainty in the modeling as ∼25%. The surface density profiles estimated from the Lucy method are consistently higher than those from the 2D modeling with ELLINT by ∼25%, but the measured radii are consistent within half the beam width, our estimated error.

We then use these radial models to estimate the radii at 1 ${M}_{\odot }$ pc⁻² to roughly compare to the expected radii from standard scaling relations. Using the H i-mass–radius relation from Broeils & Rhee (1997; which is similar to the relation from Wang et al. 2016) we compute expected H i diameters from the measured H i masses. The ratios of the measured diameters to the predicted diameters are given in Table 2. All of the sources lie above the relation, i.e., their H i radius is extended for their H i mass. However, this comparison is still limited by the effect of the beam, which tends to push flux to larger radii, exaggerating the size of a galaxy, while reducing the measured column density, which can underestimate the size of the galaxy for low-density systems. Thus, given the uncertainties in the radius measurements, all three sources are consistent with the scatter in the relation. We note that while the H i radii appear to be consistent with the expected radii for their H i mass, as noted above, all three sources have large H i masses relative to their stellar populations. Thus, these sources are significantly more extended than typical H i-rich sources with comparable stellar mass.

As a suggestive exercise, we compute the median H i profile of four dwarf galaxies and $4\sim {L}_{\star }$ galaxies from the THINGS sample (Walter et al. 2008), and compare the results to the results for HUDS. We used profiles fitted using the tilted rings method from Leroy et al. (2008), and smoothed them to a physical resolution of 7 kpc to approximately compare with the physical resolution of the measured HUDS. The results of this exercise are shown in Figure 5. The beam smeared profiles of the HUDS are shown as thick colored lines, and the smoothed median profiles and their spread are shown as gray shaded regions. Importantly, the H i disks are more extended than typical dwarf galaxies, which are point sources at 7 kpc resolution, and more consistent with H i disks of ${L}_{\star }$ spirals. The average surface density seems to be somewhat lower than the typical THINGS galaxies, suggesting that these sources may be somewhat more diffuse in H i than typical H i sources. However, we emphasize that this result is at best suggestive due to the small number statistics and low resolution; higher resolution observations of a larger sample will be required to confirm this suggestion.

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The bottom panels of Figure 3 shows the H i velocity fields for the three sources with resolved synthesis observations. All the sources show signs of ordered motions, and evidence of a significant velocity gradient, though the gradient is only over a relatively narrow range. Indeed, AGC 219533 has the largest velocity width of 66 km s⁻¹. The relatively narrow velocity widths of the three resolved sources, however, are consistent with the ALFALFA velocity widths for the entire H i-bearing UDG sample. The center panel of Figure 4 shows the velocity distribution of HUDS compared with a similarly selected ALFALFA sample. The mean velocity width of the HUDS-B (HUDS-R) sample is 59 (44) km s⁻¹, compared to 194 km s⁻¹ for all ALFALFA galaxies, and 119 km s⁻¹ for ALFALFA galaxies with similar selection criteria. Specifically, we expect lower velocity widths for HUDS given their lower baryonic masses, and also due to the fact that surface brightness is a function of inclination. However, even after removing sources with H i mass (log (M_{H i} /${M}_{\odot }$) > 9.5) and with inclinations >66° (approximating the inclination distribution of HUDS), the HUDS still populate the low-velocity-width part of the distribution. This result is not entirely unexpected, due to their lower mass and a surface brightness selection against edge-on galaxies. We return to this in Section 5.2.2.

Since the sources are only resolved with a few beams along the major and minor axes, traditional fitting of tilted ring models to the 2D velocity profile tend to overestimate the dispersion and underestimate the rotational velocities due to the many velocities along the lines of sight contained within the beam. Thus we instead estimate the rotation curve of the sources by fitting the "envelope" of velocities observed at each position in a position–velocity (PV) diagram (e.g., Sancisi & Allen 1979). We follow the methods of Hallenbeck et al. (2014), using the GIPSY task ROTCUR to estimate the position angle of the velocity field (using tilted ring models), and then extract a position–velocity field using a slice two beams wide along the major rotational axis. We then extract the spectrum at each position and fit a third-order Gauss–Hermite polynomial, and estimate the final rotation curve as the velocity where the integrated area under the curve is 3.3% from the approaching or receding edge. We then average the rotation values from the approaching and receding envelope, and correct for inclination by dividing by sin(i).

We note that the inclination uncertainty contributes significantly to the rotational uncertainty (Section 3.4), and that our estimated rotation velocity could be biased by gas inflow or outflow, enhanced velocity dispersion, and the assumptions of a disk like structure with a negligible disk scale height. In spite of these limitations, however, the data are still sufficient to constrain the allowed parameter space, as we discuss below.

The right panel of Figure 7 shows the dynamical mass of the HUDS as inferred from the H i rotation curves estimated in Section 4.2.2 (black squares, pentagons, and hexagons), compared with the dynamical mass estimates from globular cluster spectroscopy for UDGs from van Dokkum et al. (2016) (DF 44; gray triangle) and Beasley et al. (2016) (VCC 1287; gray circles), and predicted models from Di Cintio et al. (2017; colored lines). The uncertainties in geometry dominate the uncertainties in estimating the rotation velocity. However, even accounting for these, the H i data provides significant constraints on the dark matter masses. All three sources have measured dynamical masses consistent with or slightly larger than the measurements from Beasley et al. (2016) of VCC 1287, and slightly smaller than the measurement from van Dokkum et al. (2016). Specifically, Beasley et al. (2016) find a dynamical mass of 4.5 ± 2.8 × 10⁹ ${M}_{\odot }$ within 8.1 kpc, while the HUDS give values ranging from 5–10 × 10⁹ ${M}_{\odot }$ within a similar radius (and with similar errors—see Table 2). These dynamical estimates yield dynamical-to-stellar-mass ratios for these sources consistent with those reported in Beasley et al. (2016), though the error bars are large. The dynamical-to-baryonic mass ratios are significantly smaller, however, since the H i mass is large compared with the stellar mass.

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The HUDS also match reasonably well with the predicted profiles from Di Cintio et al. (2017), though for the best resolved source AGC 219533, the measured rotation curve appears somewhat flatter than those predicted. We estimate a halo mass from the best fitting profiles from Di Cintio et al. (2017) of $\sim {10}^{10.7}{M}_{\odot }$ for AGC 219533, and somewhat smaller masses of ${10}^{10.4}$ and ${10}^{10.3}{M}_{\odot }$ for AGC 334315 and 122966, respectively. However, the extrapolation from dynamical mass to total halo mass necessarily relies on the type of model fit. We note that abundance matching (using the data from Papastergis et al. 2012) implies that galaxies with the baryonic masses of these sources should live in halos with log M_halo/${M}_{\odot }$ ≳11.1. However, this estimate seems unreasonably large given the dark matter mass estimated within r = 20 kpc.

As an instructive exercise, we attempt to model the rotation curve as composed of a gaseous disk, stellar disk, and dark matter halo using the GIPSY task ROTMAS for AGC 219533. The left panel of Figure 7 shows the best-fitting model stellar and gas disk contributions to the rotation, assuming the mass surface density distributions shown in Figure 5, multiplied by 1.3 to account for the presence of helium (and assuming an infinitely thin disk, an assumption that has little effect on the analysis given the size of the errors). The remaining rotation is modeled as the result of either an pseudo-isothermal or Navarro–Frenk–White (NFW; Navarro et al. 1997) halo (shown as dash-dotted and long-dashed lines, respectively), such that ${V}_{\mathrm{obs}}^{2}={V}_{\mathrm{gas}}^{2}+{V}_{* }^{2}+{V}_{\mathrm{DM}}^{2}$.

We also note that while low surface brightness galaxies usually exhibit steadily rising rotation curves that are not well fit by NFW profiles (e.g., McGaugh & de Blok 1998), AGC 219533 appears to flatten out, potentially suggesting that is is more consistent with the NFW profile, similar to the extreme gas-rich, high spin parameter, low surface brightness galaxy UGC 12506 (Hallenbeck et al. 2014). However, while these results are suggestive, the limited resolution of our current observations cautions against overinterpreting these results.

While the analysis in Section 5.2.1 indicates that HUDS are likely to occupy dwarf halos, it does not explain the mechanism for the extended radii. Here we attempt to estimate the spin parameters of the dark matter halos of HUDS, to test the prediction that "ultra-diffuse" sources are spatially extended due to large halo spin parameters (Amorisco & Loeb 2016).

The spin parameter is a dimensionless quantity that describes the angular momentum in the halo:

$$\begin{eqnarray*}&&\lambda =\displaystyle \frac{J\ast | E{| }^{1/2}}{G\ast {M}^{5/2}}\end{eqnarray*}$$

where J is the halo angular momentum, E is the energy, and M is the halo mass. The halo spin parameter is difficult to constrain observationally, since almost any λ can fit a halo with given parameters from rotation curve fitting. Instead, we employ the common practice of simplifying the calculations by assuming that the dark and baryonic matter are coupled such that their angular momentum per unit mass (j and j_b respectively) are equal (e.g., Mo et al. 1998). Thus we are technically calculating the modified spin parameter $\lambda ^{\prime} ={j}_{b}/j\,\times \lambda$ (henceforth, we will drop the prime).

Under this assumption, we approach the calculation of λ two different ways. We first follow Huang et al. (2012), measuring the exponential disk scale length R_d from SDSS and the rotation velocity V_rot from the ALFALFA H i line width (${V}_{\mathrm{rot}}={W}_{50}/2/\sin (i)$), and adopting the λ estimator from Hernandez et al. (2007):

$$\begin{eqnarray*}&&\lambda =21.8\displaystyle \frac{{R}_{d}[\mathrm{kpc}]}{{V}_{\mathrm{rot}}{[\mathrm{km}{{\rm{s}}}^{-1}]}^{3/2}}.\end{eqnarray*}$$

This estimator further assumes self-gravitating, virialized, isothermal dark matter halos that dominate the galaxy's potential energy, flat rotation curves, and, importantly, a constant disk mass fraction M_*/${M}_{\mathrm{total}}$ of 0.04. We assume that all sources are inclined at 45°, as discussed in Section 3.4.

The left panel of Figure 8 shows the results of this analysis. The distribution for all galaxies in the ALFALFA 70% sample is shown in dark pink. The distribution for the HUDS-R and HUDS-B samples are shown in yellow and orange, respectively, and appear to be significantly elevated relative to the rest of the ALFALFA distribution, i.e., the the radii of these sources are large given the rotation velocities of their disks. A Kolmogorov–Smirnov test estimates a probability that they are drawn from the same distribution as 10⁻¹⁸ and 10⁻³⁴.

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To better understand the potential impact of selection effects and our other assumptions, the left panel also shows the lognormal fit to the spin parameter distribution of H I-selected sources derived by Huang et al. (2012) through similar analysis (blue dashed curve). Like the spin parameters derived by Huang et al. (2012), the ALFALFA 70% spin parameters follow a lognormal distribution, but have a lower mean and wider dispersion, demonstrating the impact of source selection. The distribution shown in pink includes the entire ALFALFA sample, whereas the blue dashed curve is restricted to a volume limited sample from the ALFALFA 40% catalog, and only includes sources over the absolute magnitude range −20 > ${M}_{r}\gt -23$  mag. The difference between the distributions is also in part due to the assumption of constant inclination.

Further, the left panel also shows the lognormal distribution derived by Huang et al. (2012) under the assumption the sources do not have a constant halo mass fraction, but instead have mass fractions derived from abundance matching (dark blue solid curve). As discussed in Huang et al. (2012), allowing the mass fraction to vary can have a large impact on the distribution. Still, the calculated spin parameters for the HUDS are large compared to the overall sample in all three cases, though we note that if the HUDS reside in large halos for their stellar masses as predicted through abundance matching, their estimated spin parameters would be significantly lower.

The trend to high spin parameters is perhaps not entirely unexpected given our selection of extended sources, and the observation that they have high gas fractions relative to the ALFALFA sample (Section 4.2). Indeed, Huang et al. (2012) showed that sources with high gas fractions tend to have high spin parameters for a given stellar mass, and Obreschkow et al. (2016) suggest that gas fraction depends on a global stability parameter which scales linearly with the angular momentum of the disk. Thus, the high estimated spin parameters make sense in light of the sample's observed gas fractions.

For resolved sources we can do a somewhat more detailed estimate of the modified halo spin parameter. We follow the procedure detailed in Hallenbeck et al. (2014), which, in brief, estimates the angular momentum of the halo by summing the product of the H i disk mass, velocity, and radius at each point on the rotation curve, and assuming the angular momentum of the halo scales with that of the baryons. The energy is calculated from the maximum velocity from the isothermal halo fit, and the halo mass is estimated from abundance matching. We note that, as discussed in Section 5.2.1, abundance matching poorly estimates the total halo mass for these sources. Thus we also estimate spin parameters assuming the halo masses derived from comparison to the models from Di Cintio et al. (2017), and discuss the effect on the results below.

The right panel of Figure 8 shows the distribution of spin parameters for resolved THINGS galaxies computed by Hallenbeck et al. (2014), which approximately follow the lognormal distribution for the 40% ALFALFA sample (also plotted in the left panel) from Huang et al. (2012). The three HUDS with synthesis observations, assuming the halo masses from Di Cintio et al. (2017), are overplotted as yellow bars. Though this analysis uses resolved rotation curve fitting rather than SDSS radii and ALFALFA line widths, the three sources again appear to have higher spin parameters than most of the THINGS galaxies.

However, it is important to note that if one instead assumes the halo masses from abundance matching, which are larger by a factor of ∼4, the spin parameters are reduced by the same factor, and fall squarely within the THINGS distribution. Also, for two of the three sources, the values estimated from our resolved analysis are significantly lower than those estimated from the velocity width and optical radius. This is likely due to the low physical resolution relative to the sources from Hallenbeck et al. (2014), which tends to suppress the j value, and the comparatively large ${r}_{\mathrm{optical}}$/${r}_{{\rm{H}}{\rm{I}}}$, which increases the spin parameter measurement in the unresolved method (which relies on optical radii) relative to the resolved method (which relies on H i radii measurements). With only three sources, we hesitate to read more into this.

With these caveats in mind, the results in this section still suggest that if the assumptions used to calculate the spin parameters are valid and the halo masses are indeed more consistent with dwarfs, then HUDS may reside in high spin parameter halos.