#  Herschel Photometric Observations of Little Things Dwarf Galaxies

Phil Cigan George Mason University, 4400 University Dr, Fairfax, VA
22030-4444, USA  Lisa M. Young New Mexico Institute of Mining and Technology,
801 Leroy Place, Socorro, NM 87801, USA  Haley L. Gomez School of Physics and
Astronomy, Cardiff University, Cardiff CF24 3AA, UK  Suzanne C. Madden  CEA
Saclay, Laboratoire AIM, CEA/DSM-CNRS-Université Paris Diderot DAPNIA/Service
d’Astrophysique, Bât. 709, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France
Pieter De Vis  School of Physics and Astronomy, Cardiff University, Cardiff
CF24 3AA, UK  Deidre A. Hunter Lowell Observatory, 1400 West Mars Hill Road,
Flagstaff, AZ 86001, USA  Bruce G. Elmegreen IBM T. J. Watson Research Center,
P.O. Box 218, Yorktown Heights, New York, USA  Elias Brinks Centre for
Astrophysics Research, University of Hertfordshire, College Lane AL10 9AB, UK
Phil Cigan<EMAIL_ADDRESS>
###### Abstract

We present here far-infrared photometry of galaxies in a sample that is
relatively unexplored at these wavelengths: low-metallicity dwarf galaxies
with moderate star formation rates. Four dwarf irregular galaxies from the
Little Things survey are considered, with deep Herschel PACS and SPIRE
observations at 100 $\mu$m, 160 $\mu$m, 250 $\mu$m, 350 $\mu$m, and 500
$\mu$m. Results from modified-blackbody fits indicate that these galaxies have
low dust masses and cooler dust temperatures than more actively star-forming
dwarfs, occupying the lowest $L_{\mathrm{TIR}}$ and $M_{\mathrm{dust}}$
regimes seen among these samples. Dust-to-gas mass ratios of $\sim$10-5 are
lower, overall, than in more massive and active galaxies, but are roughly
consistent with the broken power law relation between the dust-to-gas ratio
and metallicity found for other low-metallicity systems. Chemical evolution
modeling suggests that these dwarf galaxies are likely forming very little
dust via stars or grain growth, and have very high dust destruction rates.

††journal: AJ††facilities: Herschel Space Telescope, Very Large
Array††software:  astropy (Astropy Collaboration et al., 2018), ipython (Pérez
& Granger, 2007), lmfit (Newville et al., 2016), matplotlib (Hunter, 2007),
numpy (van der Walt et al., 2011), SAOImage DS9 (Joye & Mandel, 2003),
Scanamorphos (Roussel, 2013), scipy (Jones et al., 2001–).

##  1 INTRODUCTION

Dust plays an important role in the process of star formation and, therefore,
evolution of galaxies. Combined with metallicity measurements, the dust
content can provide a probe of the evolutionary status of a galaxy (e.g.,
Clark et al., 2015; Schneider et al., 2016). Historically, studies have
suggested that the dust-to-gas ratio linearly increases with the content of
metals (Dwek, 1998; Edmunds, 2001; Draine et al., 2007) as a galaxy uses up
its gas to form stars. Far infrared (FIR) to sub-millimeter (sub-mm)
photometry can directly probe the emission from dust in dwarf galaxies to help
answer a fundamental question: how do the bulk dust properties (mass,
temperature, emissivity) vary with metallicity ($Z$) and what are the dust
scaling relations for galaxies of different morphological type and in
different evolutionary phases? Constraining dust properties requires robust
sampling of the thermal emission profile – with sensitive measurements in the
infrared to sub-mm, extending beyond the peak to longer wavelengths – as well
as reliable dust and spectral energy distribution (SED) models. IR wavelengths
are also difficult to observe from ground-based instruments due to water vapor
absorption features in the atmosphere, so they can only detect the brightest
(highest metallicity or extreme star forming) dwarfs. The Herschel Space
Observatory (hereafter Herschel, Pilbratt et al., 2010) (and its sister
satellite Planck, Planck Collaboration et al., 2011) has made it possible to
study the FIR–sub-mm properties of local galaxies in large samples (e.g.,
Skibba et al., 2011; Boselli et al., 2010; Leroy et al., 2011; Dunne et al.,
2011; Negrello et al., 2013; Clemens et al., 2013; Agius et al., 2013; Rémy-
Ruyer et al., 2014; Roman-Duval et al., 2014; Clark et al., 2015; Beeston et
al., 2018) allowing for the first derivations of dust scaling relations and
comparison of dust properties across the Hubble Sequence with coverage over
the full FIR spectrum (Cortese et al., 2012; Smith et al., 2012; De Vis et
al., 2017a), as well as resolved dust studies of nearby galaxies (Smith et
al., 2012; Galametz et al., 2011; Gordon et al., 2014; Draine et al., 2014).
Many of these studies sample the more FIR-bright spirals or more massive early
type galaxies, but in the low stellar mass and low metallicity regime, dwarf
galaxies can provide a fundamental probe of the dust content in sources that
are chemically young and potentially representative of galaxies in the early
universe.

The largest Herschel sample of dwarfs to date comes from the Dwarf Galaxy
Sample (hereafter DGS) of Madden et al. (2013), where Rémy-Ruyer et al. (2014)
indicated that the dust-to-gas ratio (DGR) is linearly proportional to
metallicity from the solar level down to $Z\sim 0.2\,Z_{\odot}$, but the slope
steepens for lower metallicities such that these galaxies are predicted to
have less dust for a given gas mass compared to higher metallicity sources; at
$Z=0.025\,Z_{\odot}$ the dust can be lower by an order of magnitude or more
(see also Draine et al., 2007; Galliano et al., 2008). This result is also
suggested by De Vis et al. (2017b), who combined the DGS with galaxies from
the Herschel-ATLAS survey (Eales et al., 2010), roughly doubling the number of
low metallicity galaxies (though many were not detected in the sub-mm with
Herschel). Difficulty in determining the origin of the differences in dust
mass and/or physical properties between low and high metallicity galaxies is
further exacerbated due to the enormous scatter in the DGR at low $Z$, with
Hunt et al. (2014) demonstrating that even with the same metallicity, the
observed DGR in two low-$Z$ dwarf galaxies can vary by two orders of magnitude
(see also Schneider et al., 2016). In Rémy-Ruyer et al. (2014) and De Vis et
al. (2017b), chemical evolution models are used to probe the observed
differences in dust content with metallicity and the deviation from the linear
trend seen in the DGR for higher metallicity galaxies. Those works attribute
the variation to different sources of dust (i.e., different amounts of star-
dust from evolved stellar winds and/or supernovae, and grain growth in the
interstellar medium), whereas it is instead attributed to different star
formation rates in Zhukovska (2014) and the balance between inflows and
outflows in Feldmann (2015). Schneider et al. (2016) convincingly make the
case that the higher dust masses observed in the low $Z$ dwarf SBS 0335-052,
as compared to I Zw18 which has the same metallicity, is due to increased dust
grain growth in the ISM in SBS 0335-052 because of higher interstellar gas
densities.

Understanding what underpins the relationship between the dust content of
galaxies and metallicity and gas properties is therefore important, and the
low metallicity regime seems crucial for revealing trends that are different
compared to normal spirals. In this paper, we provide a sample of deep FIR
photometric measurements using Herschel for four low metallicity, low star
forming galaxies from the Little Things survey (Hunter et al., 2012). This
provides additional sources to the dwarf studies in the literature, crucially
sampling a regime that still lacks significant numbers: that of low mass,
moderate star formation rate, and extremely low metal content. We use this
information to analyze the physical properties of the ISM in these galaxies,
such as dust temperature, mass, emissivity, and infrared luminosity, and
compare to other galaxy samples. Information about the sample, the
observations, and data reduction are discussed in §2. The photometry methods,
including selection of regions and extraction of flux densities, are described
in §3. In §4, we compare the Herschel observations with other IR data. §5
introduces the SED model used to derive the properties of these sources, with
comparison to the literature discussed in §6.1.

##  2 DATA

###  2.1 The Sample

Four dwarf irregular galaxies with properties typical of normal dwarfs were
selected for observations with Herschel for this study: DDO 69, DDO 70, DDO
75, and DDO 210. The photometry sample is slightly different from the
spectroscopy sample of Cigan et al. (2016) – DDO 69, DDO 70, and DDO 75 are
common between the two, but DDO 210 has no Herschel emission line data. They
are all nearby (less than 1.3 Mpc) and small (RD between 0.17 and 0.48 kpc,
$M_{\mathrm{HI}}$ of 7.2$\times 10^{7}$ $M_{\odot}$ or less), with modest star
formation rates of 0.012 $M_{\odot}$ yr-1 or less. The most metal rich galaxy
in the sample is DDO 75, at 12+log(O/H) = 7.54 (7% $Z_{\odot}$), while the
poorest is DDO 210 at 7.2 (3.2% $Z_{\odot}$). A summary of the parameters for
the sample is given in Table 2.1.

For comparison with other local dwarf galaxies, the DGS metallicities span a
range of 7.14–8.43 (Madden et al., 2013) and star formation rate (SFR) values
range several orders of magnitude from 0.0008–43 $M_{\odot}$ yr-1 De Looze et
al. (2014).

Table 1:  Sample Galaxy Parameters

Galaxy | Other Names | D | log10 $M_{\mathrm{HI}}$ | RD | ${\mu_{0}}^{V}$ | log10 SFRFUV | log10 SFR${}_{D}^{FUV}$ | 12+log10(O/H)
---|---|---|---|---|---|---|---|---
|  | (Mpc) | ($M_{\odot}$) | (kpc) | (mag arcsec-2) | ($M_{\sun}\;\textrm{yr}^{-1}$) | ($M_{\sun}\;\textrm{yr}^{-1}\;\textrm{kpc}^{-2}$) |
DDO 69 | PGC 28868 | 0.8 | 6.84 | 0.19 $\pm$ 0.01 | 23.01 | -3.17 | -2.22 $\pm$ 0.01 | 7.38 $\pm$ 0.10
| UGC 5364 |  |  |  |  |  |  |
| Leo A |  |  |  |  |  |  |
DDO 70 | PGC 28913 | 1.3 | 7.61 | 0.48 $\pm$ 0.01 | 23.81 | -2.30 | -2.16 $\pm$ 0.00 | 7.53 $\pm$ 0.06
| UGC 5373 |  |  |  |  |  |  |
| Sextans B |  |  |  |  |  |  |
DDO 75 | PGC 29653 | 1.3 | 7.86 | 0.22 $\pm$ 0.01 | 20.40 | -1.89 | -1.07 $\pm$ 0.01 | 7.54 $\pm$ 0.06
| UGCA 205 |  |  |  |  |  |  |
| Sextans A |  |  |  |  |  |  |
DDO 210 | PGC 65367 | 0.9 | 6.3 | 0.17 $\pm$ 0.01 | 23.77 | -3.75 | -2.71 $\pm$ 0.06 | 7.2 $\pm$ 0.5
| Aquarius Dwarf |  |  |  |  |  |  |

References. — Data as reported in Hunter et al. (2012). Original distance and
metallicity references, respectively. DDO 69: Dolphin et al. (2002), van Zee
et al. (2006). DDO 70: Sakai et al. (2004), Kniazev et al. (2005). DDO 75:
Dolphin et al. (2003), Kniazev et al. (2005). DDO 210: Karachentsev et al.
(2002), Richer, & McCall (1995).

Note. — General information about this galaxy sample, as reported by Hunter, &
Elmegreen (2004, 2006); Hunter et al. (2012). ${\mu_{0}}^{V}$ is the central
$V-$band brightness. RD is the disk scale length. SFRFUV is the star formation
rate determined from $L_{FUV}$, and SFR${}_{D}^{FUV}$ is that divided by
$\pi$R${}_{D}^{2}$. Oxygen abundances for metallicities were determined from H
ii regions.

###  2.2 Observations

Table 2: Summary of Herschel Observations  Galaxy | Instrument | Filters | OBSID | RA (J2000) | DEC (J2000) | Duration | Map Size
---|---|---|---|---|---|---|---
|  | ($\mu$m) |  | (h m s) | (d m s) | (s) | (arcmin)
DDO 69 | SPIRE | 250, 350, 500 | 1342255183 | 09 59 29.30 | +30 44 21.17 | 2019 | 9.2 $\times$ 9.2
| PACS | 100, 160 | 1342255335 | 09 59 26.44 | +30 44 43.37 | 4490 | 6.9 $\times$ 6.9
| PACS | 100, 160 | 1342255336 | 09 59 26.73 | +30 44 47.40 | 4490 | 6.9 $\times$ 6.9
DDO 70 | SPIRE | 250, 350, 500 | 1342255156 | 10 00 01.09 | +05 19 57.42 | 2047 | 9.8 $\times$ 9.8
| PACS | 100, 160 | 1342255962 | 10 00 00.09 | +05 19 56.02 | 4714 | 7.4 $\times$ 7.4
| PACS | 100, 160 | 1342255963 | 10 00 00.11 | +05 19 56.27 | 4714 | 7.4 $\times$ 7.4
DDO 75 | SPIRE | 250, 350, 500 | 1342247243 | 10 10 59.88 | -04 41 31.73 | 2095 | 10.8 $\times$ 10.8
| PACS | 100, 160 | 1342247428 | 10 11 00.79 | -04 41 34.31 | 6172 | 8.1 $\times$ 8.1
| PACS | 100, 160 | 1342247429 | 10 11 00.78 | -04 41 34.26 | 6172 | 8.1 $\times$ 8.1
DDO 210 | SPIRE | 250, 350, 500 | 1342245438 | 20 46 51.48 | -12 51 15.73 | 1201 | 4.8 $\times$ 4.8
| PACS | 100, 160 | 1342245178 | 20 46 51.81 | -12 50 52.50 | 2325 | 3.1 $\times$ 3.1
| PACS | 110, 160 | 1342245179 | 20 46 51.80 | -12 50 51.78 | 2325 | 3.1 $\times$ 3.1

Figure 1: Comparison of emission in the PACS and SPIRE maps with other
observations at several wavelengths. All PACS and SPIRE images are scaled from
–0.3 to 1.0 mJy px-1, and their respective beam sizes are represented by the
black ellipses in the corners. The white outlines represent the extents of the
PACS images, for scale. The green circles denote the apertures used for
photometry, as described in Table 4 and § 3.1.

Figure 2: Comparison of emission in the PACS and SPIRE maps with other
observations at several wavelengths. All PACS and SPIRE images are scaled from
–0.3 to 1.0 mJy px-1. The white outlines represent the extents of the PACS
images, for scale. The green circles denote the apertures used for photometry,
as described in Table 4 and § 3.1.

We used the Herschel PACS (Poglitsch et al., 2010) and SPIRE (Griffin et al.,
2010) instruments to observe our sources from 100–500$\mu$m. We refer the
reader to the current PACS and SPIRE handbooks
111https://www.cosmos.esa.int/documents/12133/996891/PACS+Explanatory+Supplement,
222https://www.cosmos.esa.int/documents/12133/1035800/The+Herschel+Explanatory+Supplement%2C%20Volume+IV+-+THE+SPECTRAL+AND+PHOTOMETRIC+IMAGING+RECEIVER+%28SPIRE%29
for further information on the instrument characteristics as well as
techniques and considerations regarding the processing of their data. The PACS
observations used scan-map mode at medium speed (20″ s-1). The 100$\mu$m and
160$\mu$m bands were observed simultaneously over 2 repetitions to create maps
of the sources.

For SPIRE, the sources were also observed in “large map” mode at the nominal
scan speed of 30″ per second. The 250$\mu$m, 350$\mu$m, and 500$\mu$m bands
were all measured simultaneously over 4 repetitions. Map sizes were set to
2$\times D_{25}$ from LEDA333http://leda.univ-lyon1.fr/ (Makarov et al., 2014)
on a side to ensure background could be determined beyond the measurable
galaxian dust emission.

A summary of the Herschel observations is provided in Table 2. The native full
width at half maximum (FWHM) values for the observations are given in Table 3,
and range from roughly 7″ at 100 $\mu$m to 38″ at 500 $\mu$m. Spitzer mid
infrared images of these galaxies at 24, 70, and 160 $\mu$m were taken from
the Local Volume Survey (Dale et al., 2009). Other ancillary data for much of
the spectrum from UV to radio has been collected for each galaxy as part of
the Little Things survey, and in particular this work utilizes images of
H$\alpha$ (Hunter, & Elmegreen, 2004), $V$–band (Hunter, & Elmegreen, 2006),
FUV (Hunter et al., 2010), and H i (Hunter et al., 2012) emission. See also
Cigan et al., 2016 for more details. We note here that the extents of the
H$\alpha$ images are relatively small, as compared to both the images at other
wavelengths and to the photometry apertures described in § 3.1.

###  2.3 Reduction

Basic data reduction was performed using the Herschel Interactive Processing
Environment v12.1.0 (HIPE; Ott, 2010) with calibration trees 65 and 13.1 for
PACS and SPIRE, respectively. In brief, for the PACS data, the raw
instrumental signals were loaded at this stage, and preliminary processing was
performed with the pipeline: the addition of pointing information, flagging of
saturated pixels, conversion to flux density units (Jy pix-1), flat-field
calibration, electrical crosstalk corrections, and deglitching. The resulting
calibrated scan legs are called “Level 1” data products in the HIPE parlance.
The final steps required to make maps are to remove the bolometer drift noise
and stitch together the many scan legs.

Final flux maps were produced using Scanamorphos (Roussel, 2013) v24.0.
Scanamorphos has been shown to be particularly effective at recovering faint
extended flux in dwarf galaxies, as compared to the MADmap and PhotProject
map-making methods (see Rémy-Ruyer et al., 2013). Scanamorphos addresses low-
frequency bolometer drift “1/$f$” noise by taking advantage of observation
redundancy instead of assuming a noise model. The final maps were constructed
with 1$\farcs$7 pixel sizes at 100$\mu$m and 2$\farcs$85 at 160$\mu$m,
achieving finer sampling than Nyquist. The beam sizes for the different
Herschel bands are listed in Table 3.

For SPIRE, the HIPE Large Map pipeline was used to produce Level 1 data
products, which involves adding pointing data, electrical crosstalk
corrections for the bolometer arrays, deglitching, filter response
calibration, flux conversion (to Jy beam-1), and time response correction. The
final maps were produced with Scanamorphos, with uniform 4$\farcs$50,
6$\farcs$25, and 9″ pixel sizes for all maps in the 250$\mu$m, 350$\mu$m, and
500$\mu$m bands, respectively. See Figures 1 and 2 for a comparison of the
emission in several wavebands for each galaxy. Following the default
procedure, relative gains (to account for beam area differences between
bolometers) were applied in Scanamorphos instead of HIPE.

It should be noted that comparison of results from images produced with
different map-making tools – such as Scanamorphos and those within HIPE – can
have slight systematic differences. Specifically for this work, the PSFs of
SPIRE maps produced by Scanamorphos are slightly modified due to additional
smoothing444https://nhscsci.ipac.caltech.edu//spire/docs/SPIRE_Mapmaking_Report_v6.pdf.
Scanamorphos maps can also have smaller pixel sizes than those made from other
tools, and do not contain blank pixels in their centers.

##  3 PHOTOMETRY

Figure 3: Source and background photometry apertures used to derive the fluxes
for each galaxy. The source apertures, shown in green, are the same size
between the PACS and SPIRE maps for a given galaxy. The background apertures
(annuli for PACS, circles for SPIRE) are shown in blue. All the PACS 100
$\mu$m and SPIRE 250 $\mu$m maps are plotted here with the same flux density
scale, from $-0.3$ to $1.0\ \textrm{mJy px}^{-1}$, also matching that in
Figures 1 and 2.  Table 3:  Herschel and Spitzer Instrument Parameters

Instrument | Band | Beam Size | Calibration
---|---|---|---
|  | (FWHM, arcsec) | Uncertainty
PACS | 100$\mu$m | 7.0 $\times$ 7.4 | 5%
PACS | 160$\mu$m | 10.5 $\times$ 12.3 | 5%
SPIRE | 250$\mu$m | 18.9 $\times$ 18.0 | 4+1.5%
SPIRE | 350$\mu$m | 25.8 $\times$ 24.6 | 4+1.5%
SPIRE | 500$\mu$m | 38.3 $\times$ 35.2 | 4+1.5%
MIPS | 24$\mu$m | 6.5 | 4%
MIPS | 70$\mu$m | 18.7 | 10%
MIPS | 160$\mu$m | 38.8 | 12%

Note. — The two linearly-additive calibration uncertainties listed for SPIRE
include the 4% absolute uncertainty in the model of Neptune and the 1.5%
random uncertainty in Neptune photometry measurements.

References. — Beam sizes come from the PACS and SPIRE Observer’s Manuals.
Instrumental calibration uncertainties are as follows. PACS: The PACS
Observer’s Manual; SPIRE: Bendo et al. (2013); MIPS 24$\mu$m: Engelbracht et
al. (2007); MIPS 70$\mu$m: Gordon et al. (2007); MIPS 160$\mu$m: Stansberry et
al. (2007).

###  3.1 Aperture Selection

We select our source apertures by inspecting the emission in various bands
from the optical to the infrared, to ensure that all of the galaxian emission
is enclosed while still having space outside for background subtraction. The H
i images show that the distribution of the ISM is not always well-predicted by
the optical emission, so the aperture choices should not be driven by optical
emission alone. We use the same source aperture for every PACS and SPIRE band
for a given galaxy, as shown in Figures 1 and 2, so that flux levels are
determined from equivalent areas. Circular apertures are used throughout, for
careful treatment of aperture size effects and background subtraction,
described more fully in the following sections. The aperture centers are the
same as those used for the optical wavelength photometry of each galaxy,
described by Hunter, & Elmegreen (2006). The radii $r$ are based on emission
features from the ancillary data as follows:

DDO 69: $r$=3$\farcm$0\. This encompasses all of the bright emission that
appears in the optical and FUV images. This includes the slightly extended
features in the PACS maps that correspond with H$\alpha$ emission, but
excludes the bright H$\alpha$ spot to the northeast which appears to be a
foreground star. The diffuse PACS 160 $\mu$m emission near the southeast edge
is excluded because it has no H$\alpha$, optical, or FUV counterpart.

DDO 70: $r$=2$\farcm$45\. This includes the bulk of the FUV and optical
emission, and all of the H$\alpha$, while omitting the obvious compact FIR
enhancements that have no optical or FUV counterparts. It is interesting to
note that the apertures selected for the other three galaxies cover the
majority of the H i emission of their hosts, but this aperture for DDO 70 does
not – in fact, the H i extends out to the noisy edge pixels of the PACS maps.
The bright spots in the PACS bands outside of our chosen aperture to the
south, northeast, and northwest do not particularly correspond with any
enhanced neutral hydrogen knots (at the H i map resolution of $\sim$17″),
whereas the emission inside the aperture does, so we treat the external spots
as background sources.

DDO 75: $r$=3$\farcm$15\. This aperture encloses all of the H$\alpha$
emission, and all of the $V$–band and FUV except for a single bright spot 2′
southeast of the edge in both of those images and some low-level emission just
to the northeast in the FUV image. This poses no problem, as the PACS maps
have no corresponding emission in these locations. The bright spot at the
north edge of the chosen aperture has no optical or H i counterpart, and is
very faint at 24 $\mu$m.

DDO 210: $r$=1$\farcm$25\. This aperture avoids almost all background sources
and still covers essentially all of the emission at other wavelengths
including H i. The bright features to the east and southwest of the defined
aperture all appear at 24 $\mu$m, but not in $V$–band or FUV, and they are
clearly outside of the H i extent of the galaxy.

###  3.2 PACS Photometry

The careful calculations of the PACS flux densities and uncertainties
described below build on the treatment described by Rémy-Ruyer et al. (2013)
and Ciesla et al. (2012).

####  3.2.1 Flux Extraction

Since the PACS photometer PSF has a persistent contribution out to around
1000″– larger than the extents of our maps – and since the background regions
in the PACS maps are much closer to the source than in the SPIRE maps, the
procedure for background level and source flux estimation are different from
that of the SPIRE data. In short, the background flux level cannot simply be
computed as the simple median or mean of the pixel flux densities from the
background regions because these contain a small contribution from the source
(a few percent), which would cause the sky level to be overestimated.

Things are simplified if we use a circular source aperture and a concentric
circular annulus for the background region. The extent of the annulus in each
map was chosen to be as large as possible without including large numbers of
background sources, and without reaching the noisy pixels on the outer 1-2′ of
the maps (due to reduced coverage). The background regions used for each map
are shown in Figure 3.

We can measure uncorrected integrated flux densities within the source
aperture and the annulus, labeled in lower case as $f_{\nu,\rm ap}$ and
$f_{\nu,\rm ann}$, respectively. Then, using the enclosed energy fractions
($\Phi$) of the PSF at each radius along with the number of integrated pixels
in each aperture ($N_{\rm ap}$ and $N_{\rm ann}$), we can determine the true
background level with the source contribution removed ($I_{\nu,\rm bg}$
Jy/pixel) and the final source flux $F_{\nu,\rm src}$ with a linear system of
equations.

The raw flux density $f_{\nu,\rm ap}$ recovered from the source aperture is a
fraction ($\Phi_{R0}$) of the total flux actually coming from the source, with
the rest spread out over the 1000″ PSF. Furthermore, the background intensity
$I_{\nu,\rm bg}$ for a given pixel, presumed to be uniform, is present on top
of that. The expression for the uncorrected source aperture flux can then be
written as:

$f_{\nu,\rm ap}=\Phi_{R0}\cdot F_{\nu,\rm src}+N_{\rm ap}\cdot I_{\nu,\rm
bg}.$ (1)

The raw flux summed in the annulus, $f_{\nu,\rm ann}$, would simply be $N_{\rm
ann}\cdot I_{\nu,\rm bg}$ if there were no contribution except for sky
emission. However, the additional component from $F_{\nu,\rm src}$ is present,
and the strength of this component depends on the fraction of PSF power
enclosed between the inner radius $R1$ and outer radius $R2$. That is, the
additional source component in the background annulus is $\Phi_{R2}\cdot
F_{\nu,\rm src}-\Phi_{R1}\cdot F_{\nu,\rm src}=(\Phi_{R2}-\Phi_{R1})F_{\nu,\rm
src}$. Thus the raw integrated flux density in the background annular aperture
is described by:

$f_{\nu,\rm ann}=N_{\rm ann}\cdot I_{\nu,\rm
bg}+(\Phi_{R2}-\Phi_{R1})F_{\nu,\rm src}.$ (2)

Solving these two equations with two unknowns, we obtain the solution for the
corrected $F_{\nu,\rm src}$:

$F_{\nu,\rm src}=\frac{f_{\nu,\rm ann}-\frac{N_{\rm ann}}{N_{\rm
ap}}f_{\nu,\rm ap}}{\Phi_{R2}-\Phi_{R1}-\Phi_{R0}\frac{N_{\rm ann}}{N_{\rm
ap}}}.$ (3)

For a large source aperture radius of $R_{0}$=180″, the enclosed energy
fraction $\Phi_{R0}$ is 0.976, which corresponds to a 2.4% correction for the
source aperture size at 100 $\mu$m. If the background annulus is $R_{2}$=220″
and $R_{1}=R_{0}$, then $\Phi_{R2}$ = 0.987 and the correction for source
leakage into the background annulus is roughly 1.1%. The effect increases for
our smallest source aperture of 75″, where the aperture size correction is
6.4% at 100 $\mu$m, and the leakage into a 115″ background annulus is 4.3%.
Formally, these aperture corrections are for single point source responses,
and while they are good approximations for our maps which are not generally
dominated by bright extended structure, the flux spread beyond the aperture
could be slightly larger than what is accounted for here – in particular for
DDO 75, where there is extended emission near the aperture edge. This may be
tempered somewhat by some background noise potentially being included in the
contribution. We note that the background level could have been overestimated
due to potential contamination from background sources which we do not
estimate, so the source flux can be slightly higher, in principle. These are
relatively small corrections compared to uncertainties from, e.g.,
calibration, which will be discussed in the following section. Color
corrections, typically a few percent or less for these data, were applied
during the fitting process (see § 5.2) instead of at this stage, to better
account for the shape of the FIR emission profile.

####  3.2.2 Uncertainties

The uncertainties $\Delta F_{\nu,\rm src}$ for the PACS map sums are
calculated according to standard error propagation for Equation 3:

$\Delta F_{\nu,\rm src}=\frac{\sqrt{\left(\Delta f_{\nu,\rm
ann}\right)^{2}+\left(\frac{N_{\rm ann}}{N_{\rm ap}}\ \Delta f_{\nu,\rm
ap}\right)^{2}}}{\left|\Phi_{R2}-\Phi_{R1}-\Phi_{R0}\frac{N_{\rm ann}}{N_{\rm
ap}}\right|}\ .$ (4)

We assume there is no uncertainty in the number of pixels or in the $\Phi$
values. $\Delta f_{\nu,\rm ann}$ is a quadratic sum of two terms: the errors
on the intensities of the individual pixels, and the uncertainty on the
integrated background flux. The individual pixel errors, from the “error maps”
produced by Scanamorphos, are summed as $\sqrt{\sum\sigma^{2}_{i,\rm ann}}$.
The uncertainty on the summation of pixel intensities is taken to be the
standard deviation of their values, $\sigma_{\rm sky}$, times $\sqrt{N_{\rm
ann}}$.

$\Delta f_{\nu,\rm ap}$ is similarly composed of the individual pixel errors
$\sigma_{i,\mathrm{ap}}$ and the uncertainty associated with summing
intensities over the source aperture $\Delta_{\rm ap}$. The latter would
ideally be determined in the same manner as was done for the SPIRE data, from
the variance of identical apertures placed at several independent positions in
the image. However, since the source aperture is generally too large to allow
for other full apertures to fit inside the image, we can approximate the
source flux uncertainty by using a smaller integration region to measure the
variance, and scaling the result. We use apertures reduced to $R_{\rm frac}=$
$\nicefrac{{1}}{{2}}$ the source aperture radius, allowing for 6–8 independent
regions. The number of pixels in an aperture goes as $R^{2}$, and uncertainty
in a circular aperture typically goes as $\sqrt{N_{\rm pix}}=\sqrt{R^{2}}=R$
(confirmed for PACS using radii above 100″ by Auld et al., 2013), so we use
$R_{\rm src}/R_{\rm frac}$ as the scaling fraction to obtain the final
$\Delta_{\rm ap}$. Tests on our maps, where full and reduced regions could be
compared, verify that this scaling method yields appropriate results.

The uncertainties on the PACS annulus and source aperture measurements can be
summarized as

$\Delta f_{\nu,\rm ann}=\sqrt{\left(\sqrt{N_{\rm ann}}\sigma_{\rm
sky}\right)^{2}+\sum\sigma^{2}_{i,\rm ann}}$ (5)

and

$\Delta f_{\nu,\rm ap}=\sqrt{\Delta_{\rm ap}^{2}+\sum\sigma^{2}_{i,\rm ap}}$
(6)

for the determination of $\Delta F_{\nu,\rm src}$ with Equation 4. Again, the
dominant contribution to the uncertainty is $\Delta_{\rm ap}$, responsible for
$\sim$90% of the non-systematic error and typically around 12% of the measured
flux. The other components contribute one to a few percent each.

On top of these uncertainties, there is a systematic calibration uncertainty
$\Delta_{\rm cal}$ which is 5% of $F_{\nu,\rm src}$ in each of the PACS bands.
This is added in quadrature to $\Delta F_{\rm src}$, so that

$\Delta F_{\rm tot,PACS}=\sqrt{\left(\Delta{F_{\nu,\rm
src}}\right)^{2}+\Delta^{2}_{\rm cal}}.$ (7)

###  3.3 SPIRE Photometry

####  3.3.1 Flux Extraction

The maps are first converted from units of Jy beam-1 to Jy pix-1, using the
beam sizes listed in Table 3 and the final pixel sizes of 4$\farcs$5 (250
$\mu$m), 6$\farcs$25 (350 $\mu$m), and 9″(500 $\mu$m) as listed in $\S$ 2.3.

Several (8–10) circular apertures surrounding the source region were defined
in each map for the determination of the background flux level and aperture
uncertainty. These background regions, shown in Figure 3, were selected to
fairly represent the sky level, avoiding areas dominated by high emission
(background sources) and low emission levels alike. Our deep observations have
notable contamination of background sources at the longer wavelengths, some of
which undoubtedly fall within our source apertures. To account for the
background galaxy contribution in addition to the sky level, we take the
median of the background aperture fluxes. The total source flux is obtained by
summing the intensities within the source aperture, then subtracting the
background value times the number of pixels in the source aperture.

The monochromatic flux densities produced by the pipeline are assumed to be
for point sources. The so-called “$K_{4}$” term from the SPIRE Observer’s
Manual is a frequency-weighted fraction of the Relative Spectral Response
Function. To correct for extended sources (diameter$>$FWHMbeam), we multiply
the integrated flux by a ratio of the $K_{4}$ factors for extended and point-
like objects. That is, $F_{\rm extended}=F_{\rm integrated}\times
K_{4E}/K_{4P}$. The ratios are 0.9986 at 250 $\mu$m, 1.0015 at 350 $\mu$m, and
0.9993 at 500 $\mu$m. This correction is required if the aperture is greater
than 24″(250 $\mu$m), 34″(350 $\mu$m), or 45″(500 $\mu$m), which applies to
all of our source apertures. Finally, an aperture correction is applied,
dividing by the enclosed energy fraction of the PSF at the source aperture
radius ($\Phi$). As for PACS, color corrections are applied during the fitting
procedure described in § 5.2.

To summarize:

$F_{\nu,\rm tot}=\left(\sum_{i}^{N}I_{\nu,i}-N\cdot I_{\nu,\rm
bg}\right)\frac{K_{4E}}{K_{4P}}\,\frac{1}{\Phi}.$ (8)

####  3.3.2 Uncertainties

The extracted SPIRE flux densities are subject to four types of non-systematic
uncertainty. Noise and flux integration effects related to the source aperture
are estimated by $\Delta_{\rm ap}$. This aperture uncertainty is calculated
empirically as the standard deviation of the sums from each of the background
apertures. Since the background apertures are the same size as that of the
souce, this provides a reliable estimate of the variation that can be expected
from integrating fluxes over that area. The determination of the background
level contributes an uncertainty $\Delta_{\rm bg}$, calculated as the standard
deviation of all background values, multiplied by $N_{\rm ap}/\sqrt{N_{\rm
bg}}$. The effect of individual uncertainties in each pixel’s flux density
from data reduction, summed in quadrature, is $\Delta_{\rm pix}$. Finally, the
4% uncertainty in the beam area is $\Delta_{\rm beam}$. We combine these four
uncertainties in quadrature, so that the total uncertainty in the flux density
determination is

$\Delta_{\rm flux}^{2}=\Delta_{\rm ap}^{2}+\Delta_{\rm bg}^{2}+\Delta_{\rm
pix}^{2}+\Delta_{\rm beam}^{2}.$ (9)

This could in practice be an overestimate of the true uncertainty, as the
components may be slightly correlated.

There is a systematic calibration uncertainty of 5.5% applied to all SPIRE
bands. This is comprised of a 4% absolute uncertainty in the modeled flux
density of Neptune and a 1.5% random component resulting from the repeated
Neptune measurements. As advised in the SPIRE Observer’s Manual, these
calibration errors are added linearly instead of in quadrature. Upper limits
to the flux density ($3\sigma$) are reported for maps where the integrated
flux is less than three times the non-systematic (“internal”) uncertainty.

The dominant source of uncertainty is $\Delta_{\rm ap}$. For example, in the
DDO 70 250 $\mu$m measurement, $\Delta_{\rm ap}$ is 88.3 mJy, comprising about
90% of the non-systematic error, or 15% of the measured flux. $\Delta_{\rm
bg}$ is the smallest component at 5 mJy, less than 1% of the measured flux
density. $\Delta_{\rm pix}$, $\Delta_{\rm beam}$, and the calibration
uncertainty are all similar in scale, around 20 mJy, contributing a few
percent of the flux level each. The final integrated flux densities and
uncertainties for each map are tabulated in Table 4.

###  3.4 Photometry of Smaller Regions

Figure 4: The apertures used to study smaller regions of source emission in
each galaxy are shown in yellow. The original whole-galaxy aperture is shown
in green over the PACS 100 $\mu$m map, for reference. The largest beam size of
the observations – that of SPIRE 500 $\mu$m – is shown in the lower left
corner of each map.

The source of emission in dwarfs can be relatively compact compared to the
full aperture that encompasses all the galaxy FIR emission, leaving large
areas of negligible signal that can reduce the overall signal-to-noise ratio
of the full-aperture integrated fluxes. In order to draw out the maximal
signal and study resolved regions within a particular galaxy, we perform
additional aperture photometry over smaller regions within the full-galaxy
apertures. Flux densities were determined using the same procedures as for the
full-sized apertures, except for the aperture uncertainty in the PACS images:
with the smaller source apertures, equally-sized offset apertures can be used
for the empirical estimate, so no scaling factor is necessary. The regions
used are displayed in Figure 4, and the results of the photometry are listed
along with the full aperture results in Table 4.

We note that the PACS aperture uncertainties are expected to be more
significant for these smaller apertures than for the full galaxy integrations.
Analyzing the large 4°$\times$4° tiles of the Herschel Virgo Cluster Survey,
Auld et al. (2013) found that the aperture uncertainty increases as the usual
$N_{\rm pix}^{1/2}$ at large radii ($\gtrsim 100$″), but that it follows a
steeper $N_{\rm pix}^{3/4}$ trend for smaller radii, with the turnover
occurring at several times the beam FWHM. This is likely due to the large-
scale structure of the PSF causing even relatively distant pixels to be
partially correlated.

Table 4:  Photometry

Galaxy | Region | Aperture Center | Aperture Radius | F100 | F160 | F250 | F350 | F500
---|---|---|---|---|---|---|---|---
|  | RA | DEC | (″) | (pc) | (mJy) | (mJy) | (mJy) | (mJy) | (mJy)
DDO 69 | Whole | 9 59 25.0 | 30 44 42.0 | 180 | 698 | $309\pm 90$ | $336\pm 87$ | $136\pm 29$ | $\leq 101$ | $\leq 116$
DDO 70 | Whole | 10 00 0.9 | 5 19 50.0 | 147 | 926 | $1037\pm 143$ | $862\pm 125$ | $590\pm 99$ | $358\pm 72$ | $211\pm 61$
DDO 75 | Whole | 10 10 59.2 | -4 41 56.0 | 189 | 1191 | $914\pm 141$ | $890\pm 157$ | $365\pm 87$ | $155\pm 88$ | $\leq 199$
DDO 210 | Whole | 20 46 52.0 | -12 50 51.0 | 75 | 327 | $71\pm 21$ | $103\pm 28$ | $87\pm 20$ | $\leq 68$ | $\leq 42$
Subregions
DDO 69 | I | 9 59 32.7 | 30 44 30.3 | 48 | 186 | $126\pm 18$ | $168\pm 24$ | $47\pm 18$ | $\leq 41$ | $\leq 32$
| II | 9 59 18.8 | 30 43 56.9 | 56 | 221 | $154\pm 30$ | $259\pm 40$ | $191\pm 28$ | $94\pm 20$ | $33\pm 15$
DDO 70 | I | 10 00 1.9 | 5 20 31.8 | 45 | 283 | $341\pm 33$ | $301\pm 37$ | $191\pm 23$ | $111\pm 16$ | $45\pm 9$
| II | 9 59 58.7 | 5 19 10.8 | 50 | 315 | $306\pm 35$ | $243\pm 35$ | $176\pm 27$ | $109\pm 18$ | $72\pm 12$
DDO 75 | I | 10 11 6.6 | -4 41 48.1 | 75 | 472 | $577\pm 36$ | $396\pm 100$ | $241\pm 46$ | $86\pm 31$ | $24\pm 23$
DDO 210 | I | 20 46 51.7 | -12 50 36.4 | 35 | 152 | $52\pm 11$ | $59\pm 16$ | $74\pm 17$ | $28\pm 11$ | $9\pm 7$

Note. — Integrated aperture flux densities for each observed band. Upper
limits are given as 3 times the non-systematic uncertainty. All other
uncertainties include the contribution from systematic errors. We note that
the combined subregion flux densities in DDO 69 at 160 and 250 $\mu$m exceed
the full-galaxy values in those bands; this is likely due to the large number
of noisy low and negative flux pixels in the full source aperture. These
values do not contain color corrections, which were determined during the
fitting process and are listed in Table 5.

##  4 Comparison of Observed Bands

###  4.1 Comparison with Spitzer

Figure 5: Comparison of PACS 160 $\mu$m and MIPS 160 $\mu$m measurements. The
horizontal line shows where the fluxes are equal between the instruments. The
full-galaxy and smaller aperture values (stars and diamonds, respectively) are
consistent within the errors. The individual pixels above three times the rms
level in both PACS and MIPS maps (circles) also show some scatter, but fall
near the line of equivalence. The brightest pixels have a slight prevalence
for PACS/MIPS ratios $>$ 1, with ratios in the fainter pixels trending to
slightly below parity.

It is important to compare measurements from different instruments in the same
wavebands, as this gives an impression of the reliability of the data. PACS
and MIPS both have bands at 70 $\mu$m and 160 $\mu$m, however the Little
Things sample only has common data for both at 160 $\mu$m. The MIPS data for
all four galaxies discussed in this photometry study were presented by Dale et
al. (2009) as part of the Spitzer Local Volume Legacy Survey. MIPS fluxes were
integrated using the same apertures as for the PACS measurements to make this
comparison, after the PACS maps were convolved to the MIPS beam size and
reprojected to an identical pixel grid. Figure 5 shows these ratios of
PACS160/MIPS160 surface brightnesses for the Little Things sample, where
ratios are given for the full galaxy integrations, the smaller apertures, and
on a pixel-by-pixel basis.

These objects are all faint, with low absolute fluxes compared to brighter
galaxies. This means that the same absolute offset in fluxes for inherently
faint targets may yield deceptively high or low ratios compared to brighter,
high S/N sources. DDO 75, the brightest in this sample, has a PACS/MIPS ratio
of 0.85$\pm$0.54 – consistent within the error. This mirrors the trend noted
in the spectroscopic study of Little Things dwarf galaxies (Cigan et al.,
2016) where the total infrared luminosity determined from the Galametz et al.
(2013) prescription for PACS 100 $\mu$m was typically about half that
determined from MIPS 24, 70, and 160 $\mu$m. DDO 69 is much fainter overall,
and the PACS flux for the full galaxy is 1.4 times higher than that determined
from the notably noisy MIPS image. DDO 210 is fainter yet, with a small
negative integrated flux from the MIPS 160 $\mu$m map combined with a large
uncertainty: $-0.01\pm 0.14$ Jy. Indeed, no detection is obvious upon manual
inspection of the MIPS image. Thus dividing the PACS value by these results
gives a very large negative ratio and an even larger error. These large
differences occur because the whole-galaxy apertures for these two systems
include many pixels that are essentially just noise. The smaller photometry
regions can vary between instruments by factors of up to 2, though the average
of all their PACS/MIPS intensity ratios is 1.01, in line with unity.

Individual bright pixels (those greater than three times the rms level in each
of their respective maps) show some scatter between the two instruments, but
are consistent with parity overall. The brightest PACS pixels appear to be
slightly brighter in PACS than in MIPS by a factor of $\sim 1.2$ on average,
slowly decreasing to become fainter in PACS by the same factor as they near
the rms level, though the scatter is several times that difference. The pixel
values below the rms level are consistent with random noise, and the noise
pixels in PACS generally correspond to noise pixels in MIPS. The PACS Handbook
provides correction factors for comparing PACS densities with monochromatic
flux densities from MIPS, based on each instrument’s bandpasses, for a variety
of source modified blackbody profiles (discussed more in the following
sections; see Eq. 10). For typical values expected in galaxies of temperatures
in the range of 15–30 K, and $\beta$ between 1–2, these correction factors
range from 0.95–1.05. This is smaller than the scatter seen in the bright
pixels and integrated fluxes in Fig. 5, so the variation cannot be attributed
entirely to differences in bandpass response.

The bandpass overlap between MIPS 70 $\mu$m and PACS 100 $\mu$m is
insufficient for any meaningful comparison, so no attempt was made for this
pair. Although MIPS 70 $\mu$m maps exist for the Little Things sample, they
contain significant noise in stripes that are difficult to cleanly remove from
the images, so we do not consider them for the remainder of this work.

###  4.2 Herschel FIR Colors

Figure 6: FIR color-color diagrams for the DGS and KINGFISH galaxies, with
theoretical modified blackbody curves at fixed T and $\beta$ values, as
presented by Rémy-Ruyer et al. (2013) but using the updated fluxes from Rémy-
Ruyer et al. (2015). Little Things detections are denoted as solid stars
(full-galaxy apertures) and circles (small targeted regions), while the open
symbols are limits.

Figure 7:  Color-Color diagrams at FIR and sub-mm wavelengths, following the
treatment of Rémy-Ruyer et al. (2013). The symbols are the same as in Figure
6.

Color-color diagrams give a qualitative view of the distribution of IR light
in the observed galaxies by relating the relative fluxes in two different
wavebands. This can also be thought of in terms of the SED. For bands near the
peak of a blackbody curve, for example, different colors (relative heights on
the SED) could potentially be a reflection of information such as the
temperature of the dust. The modified blackbody model which is commonly used
to describe the FIR continuum emission in galaxies (discussed in more detail
in § 5.1), depends non-linearly on temperature $T$ and dust emissivity
$\beta$. It scales as $F(\lambda)\propto\lambda^{-\beta}B(\lambda,T)$ where
$B$ is the Planck function, assuming a simple single temperature model of dust
with a specific set of optical properties.

All of the Little Things galaxies were detected in the PACS bands, but only
DDO 70 was detected at 3$\sigma$ significance in all three SPIRE bands.
Reducing the aperture size to the targeted subregions improves the signal to
noise ratio due to the exclusion of many pixels without substantial emission.
In Figures 6 and 7, comparisons of various FIR colors are shown for the Little
Things galaxies, along with the DGS and KINGFISH (Kennicutt et al., 2011)
colors from the fluxes reported by Rémy-Ruyer et al. (2015) (slightly updated
from those in Rémy-Ruyer et al., 2013). Theoretical curves for blackbodies of
varying temperatures and emissivities are also plotted.

The KINGFISH galaxies are generally more tightly clustered than the DGS
galaxies, which spread across a large area of parameter space. Rémy-Ruyer et
al. (2013) attribute this to differences in the dust properties between the
samples, as the KINGFISH galaxies tend to be more metal-rich. The Little
Things sample values fall among the spread seen in the other samples.

Since FIR SEDs of most of these galaxies peak between 100–160 $\mu$m, the
100–160 and 160–250 colors can allow us to study the cold ISM properties
between samples. Flux densities that follow $F_{100}>F_{160}>F_{250}$ would
indicate SEDs that peak shortward of 100 $\mu$m. The KINGFISH galaxies are
clustered in the region depicting cooler dust, while the DGS galaxies show
colors more typical of warmer dust as shown by Rémy-Ruyer et al. (2013). The
Little Things galaxies fall in the mid to low range, indicating cooler dust
than the bulk of the DGS galaxies, with some of the Little Things regions
having very cold dust.

There are several caveats and important factors to note when interpreting
these diagrams. Data do sometimes fall outside of the predicted ranges – the
outliers in the DGS sample were all noted to be faint, low-metallicity
galaxies, and may be due to the fact that a single temperature and $\beta$ is
not always a realistic approximation. Each region should likely have a
distribution of T and $\beta$ (emissivity) values. Generally, larger $\beta$
can be countered by smaller T to produce the same color, and the degeneracy
can be quite pronounced, as seen in the top panel of Figure 6. The SPIRE bands
are more likely to sample the Rayleigh-Jeans limit of a blackbody, as opposed
to the peak of the SED for warm dust or stochastically heated small grain
emission which occurs at shorter wavelengths ($<24$ $\mu$m). Flatter slopes,
or broad SED peaks, can have several causes. One explanation is low $\beta$
values in the dust. However, this could be mimicked by a wide distribution of
dust temperatures in the source.

Many models assume that $\beta$ should be fixed or constrained to be near a
value of 2, as thought to be typical of grains in the Milky Way (Planck
Collaboration et al., 2016). If $\beta$ is assumed to be 2, low 250/500 $\mu$m
ratios can be interpreted as additional emission in the long SPIRE bands (the
so-called “sub-mm excess” e.g., Galliano et al., 2003; Grossi et al., 2010;
Galametz et al., 2011; Draine, & Hensley, 2012), beyond what would be expected
from a single blackbody. This expectation of $\beta=2$ grains in the low-
metallicity environments of dwarfs could simply be flawed, however. Small
$\beta$ values and slightly higher ratios of 500 $\mu$m to shorter wavelength
fluxes can also indicate enhanced emission longward of 500 $\mu$m, as was
found in several DGS dwarfs. Although the 500 $\mu$m detections in DDO 70 are
consistent with small $\beta$ values, the longest-wavelength flux densities
are well-fitted by a single blackbody curve (discussed in § 5), with no hint
of elevated sub-mm emission.

##  5 Determination of FIR Continuum and Dust Properties

###  5.1 SED Model

To derive the properties of dust in our sample of four dwarf galaxies, we
approximate the FIR emission as a modified blackbody function based on three
free parameters: temperature ($T$), dust mass ($M_{\rm dust}$), and the dust
emissivity index ($\beta$). The form of the function is as follows:

$F_{\nu}(\lambda)=\frac{M_{\rm
dust}\,\kappa(\lambda_{0})}{D^{2}}\left(\frac{\lambda}{\lambda_{0}}\right)^{-\beta}B_{\nu}(\lambda,T).$
(10)

Here, $B_{\nu}(\lambda,T)$ is the Planck function, $D$ is the distance to the
source (given in Table 2.1), $\lambda_{0}$ is the reference wavelength of 160
$\mu$m, and $\kappa(\lambda_{0})$ is the dust mass absorption opacity at
$\lambda_{0}$. We adopt a value of $\kappa(\lambda_{0})=1.4$ m2 kg-1 for a
corresponding emissivity of $\beta=2$, from the Zubko et al. (2004) BARE-GR-S
model for a combination of PAHs, graphite, and silicates. This is the same
composition used by Galliano et al. (2011, in their “standard model”) and
Rémy-Ruyer et al. (2013), providing a consistent basis for comparison with
their works. $\beta$ is kept fixed to 2.0 for use with this reference $\kappa$
value, leaving Mdust and T as two remaining free parameters.

###  5.2 Fitting: Parameter and Uncertainty Estimation

Parameter estimation is performed here using Maximum Likelihood Estimation
(MLE), by maximizing the log-likelihood function (or, equivalently, minimizing
its negative). The MLE formalism is invoked here instead of the simpler least
squares regression so that censored data (upper limits) can be considered in
the fits. This becomes particularly useful in the determination of the fit
errors, discussed below, where the fluxes are varied within their
uncertainties and potentially fall above the limit levels. In the case where
all fluxes are detections and errors are gaussian, the MLE result reduces to
the familiar least-squares value.

The log-likelihood function is constructed as follows. Assuming Gaussian-
distributed errors, the probability of $n$ detected bands $i$ having their
measured values $F_{i},F_{i+1},\ldots F_{n}$ is
$\prod\frac{1}{\sqrt{2\pi}\sigma_{i}}\mathrm{exp}\left({-\frac{1}{2}\left(\frac{F_{i}-F_{m,i}(M,\beta,T)}{\sigma_{i}}\right)^{2}}\right)$,
where $F_{m,i}(M,\beta,T)$ is the model Equation 10 evaluated at band $i$ for
parameters $M,\beta,$ and $T$. The appropriate log-likelihood function to
optimize is then

$ln\,\mathcal{L}_{i}(F|M,\beta,T)=-\frac{1}{2}\sum\limits_{i}\left(\frac{F_{i}-F_{m,i}(M,\beta,T)}{\sigma_{i}}\right)^{2}.$
(11)

An additional term can be introduced to account for upper limits. A non-
detection in band $j$ could result from any integrated flux lower than the
limit $F_{lim,j}$. Therefore, the probability of that flux is determined by
integrating the probability distribution over values below the limit:
$P_{j}=\int_{-\infty}^{F_{lim,j}}\mathrm{exp}\left[-\frac{1}{2}\left(\frac{F_{j}-F_{m,j}(M,\beta,T)}{\sigma_{j}}\right)^{2}\right]\,dF_{j}$.
Utilizing the Gauss error function, the log-likelihood for the limits can be
written as

$ln\,\mathcal{L}_{j}(F|M,\beta,T)=\\\
\sum\limits_{j}\,ln\left(\sqrt{2\pi}\sigma_{j}\left[1+\mathrm{erf}\left(\frac{F_{lim,j}-F_{m,j}(M,\beta,T)}{\sqrt{2}\sigma_{j}}\right)\right]\right).$
(12)

A similar derivation of an upper limit term for MLE is described by Sawicki
(2012). The total log-likelihood function can then be expressed by combining
those of the detections and limits:

$ln\,\mathcal{L}_{tot}=ln\,\mathcal{L}_{i}+ln\,\mathcal{L}_{j},$ (13)

with $ln\,\mathcal{L}_{i}$ reducing to the standard $\chi^{2}$ in the case
where all bands are detected.

Uncertainties on the fit parameters are determined by bootstrap resampling –
generating a distribution of additional fits after resampling the flux
densities within their error bars. The uncertainties we quote are the
differences between the MLE best fits and the 16th, and 84th percentiles of
the distributions from 10,000 bootstrap samples. An additional source of
uncertainty that is not formally considered in the fits is the assumed value
of $\kappa$, which can potentially vary substantially depending on the
specific variety of dust considered, and could modify our inferred
$M_{\mathrm{dust}}$ by factors of $\sim$ a few. See, e.g., Clark et al.
(2015), Whitworth et al. (2019), and Cigan et al. (2019) for summary
discussions of the variety of $\kappa$ values used in the FIR.

Color corrections to the flux densities, which are typically on the order of a
few percent or less and dependent on the shape of the SED (specifically,
$\beta$ and temperature), were determined and applied as part of the fitting
process. First, a preliminary fit to the data was performed to determine the
color correction values at each Herschel band. Then a second fit was performed
on the color-corrected data to obtain the true parameters. As the temperature
value can vary slightly from the preliminary to the second fit, the color
correction corresponding to the true fit can also differ slightly, though the
flux density difference generally much smaller than 1%. The 100, 160, 250,
350, and 500 $\mu$m color corrections determined for each aperture are given
in the final column of Table 5.

The SEDs and fits are shown in Figures 8 (full system apertures) and 9 (the
smaller localized apertures). Parameters derived from the model fits are
summarized in Table 5. Statistics including the minimum, median, and maximum
fit values are included for the Little Things derived in this work, alongside
the DGS and KINGFISH samples for comparison. The values of $T$,
$M_{\mathrm{d}}$, and $\beta$ for the DGS and KINGFISH galaxies were taken
from the modified blackbody fits of Rémy-Ruyer et al. (2013).

The number of components in the modified blackbody model can affect the
parameter fits, and allowing for a second component (see, e.g., Dunne, &
Eales, 2001; Clark et al., 2015) can improve the results; a single component
fit will blend aspects of any dust populations contained within the
measurements. For example, if the dust is primarily populated by two
differentiable temperatures, then the overall SED will be broader than for a
single component, which might correct for this by giving a higher temperature
paired with a lower mass. Single-component fitting can also lead to the
appearance of a sub-mm excess, which could be accounted for with a second
(cold) component (e.g., Clark et al., 2015). In the following analysis, a
single-component model is used for DDO 69, DDO 75, and DDO 210, and results in
good fits to the data with small residuals. The FIR emission profiles for DDO
70 however (for the full and reduced apertures) are quite broad, and single
component fits with $\beta$=2.0 result in notable residual emission with the
shape of a blackbody. Therefore, for DDO 70 we employ a second component to
our fits to better characterize the observed emission. Color corrections were
not applied for the DDO 70 two-temperature fits. They are typically small for
the other sources in this sample though, $\sim$3% at 100 $\mu$m and $\sim$1%
at 500 $\mu$m, and have a much smaller potential impact on derived masses than
the assumed dust model and the considerations discussed in the following
section.

Figure 8: Maximum likelihood fits of the flux densities to a modified
blackbody function. The red points are the observed data. Upper limits are
denoted as gray arrows. The blue shading corresponds to the density of
bootstrap resample fits, and reflects the uncertainty in the overall fit. The
MIPS 70 $\mu$m values are shown for reference as grey crosses. The lower sub-
axes indicate the residual differences between the data and the best fit. Both
components are shown in color with the combined profile for DDO 70, and the
single-component fit plus residuals for this target are shown in gray to
illustrate the improvement from including a second component. The observed
data points include color corrections, except for DDO 70.
Figure 9: Modified blackbody fits for the small individual regions within each
galaxy. The red points are the observed data. Upper limits are denoted as gray
arrows. The blue shading corresponds to the density of bootstrap resample
fits, and reflects the uncertainty in the overall fit. The sub-axis below each
fit profile indicates the residual differences between the data and the best
fit. For DDO 70, the individual components are shown in color with the
combined profile, and its single-component fits plus residuals are shown in
gray for reference, showing clear residual structure. The observed data points
include color corrections, except for DDO 70.  Table 5:  Modified Blackbody
Fit Results

Galaxy | Region | Temperature (K) |  $M_{\mathrm{dust}}$ ($M_{\odot}$) |  |  $L_{\mathrm{TIR}}$ ($L_{\odot}$) | TIR/FIR | Color Corrections
---|---|---|---|---|---|---|---
DDO 69 | Whole | $23.7^{+1.5}_{-1.4}$ | ${3.1}^{+0.9}_{-0.7}\,\times{10^{2}}$ |  | ${2.1}^{+0.4}_{-0.3}\,\times{10^{5}}$ | $1.0^{+0.2}_{-0.2}$ | 0.97,1.00,0.99,0.98,0.99
DDO 70 | Whole | $26.5^{+2.1}_{-11.6}$ | ${1.3}^{+0.7}_{-0.7}\,\times{10^{3}}$ |  | ${1.9}^{+1.1}_{-0.3}\,\times{10^{6}}$ | $1.6^{+1.1}_{-1.1}$ |  $\cdots$
|  | $10.0^{+1.8}_{-3.5}$ | ${4.7}^{+2.8}_{-5.6}\,\times{10^{4}}$ |  |  |  |
DDO 75 | Whole | $24.2^{+1.1}_{-1.0}$ | ${2.1}^{+0.5}_{-0.4}\,\times{10^{3}}$ |  | ${1.6}^{+0.2}_{-0.1}\,\times{10^{6}}$ | $1.0^{+0.1}_{-0.1}$ | 0.97,1.00,0.99,0.98,0.99
DDO 210 | Whole | $18.2^{+1.1}_{-1.1}$ | ${4.8}^{+1.8}_{-1.3}\,\times{10^{2}}$ |  | ${6.8}^{+0.8}_{-0.8}\,\times{10^{4}}$ | $1.0^{+0.2}_{-0.2}$ | 0.98,0.96,1.00,0.99,1.00
Fit Results For Individual Regions
DDO 69 | I | $23.0^{+0.9}_{-0.8}$ | ${1.5}^{+0.3}_{-0.3}\,\times{10^{2}}$ |  | ${8.7}^{+0.7}_{-0.6}\,\times{10^{4}}$ | $1.0^{+0.1}_{-0.1}$ | 0.97,0.99,0.99,0.98,0.99
| II | $18.2^{+0.5}_{-0.5}$ | ${8.9}^{+1.3}_{-1.1}\,\times{10^{2}}$ |  | ${1.2}^{+0.1}_{-0.1}\,\times{10^{5}}$ | $1.0^{+0.1}_{-0.1}$ | 0.98,0.96,1.00,0.99,1.00
DDO 70 | I | $23.3^{+0.7}_{-0.7}$ | ${8.5}^{+41.2}_{-1.6}\,\times{10^{2}}$ |  | ${5.5}^{+0.4}_{-5.3}\,\times{10^{5}}$ | $1.9^{+1.8}_{-1.8}$ |  $\cdots$
|  | $11.0^{+0.8}_{-1.5}$ | ${5.5}^{+0.2}_{-1.0}\,\times{10^{3}}$ |  |  |  |
| II | $24.1^{+1.0}_{-0.9}$ | ${6.1}^{+1.5}_{-1.3}\,\times{10^{2}}$ |  | ${5.0}^{+0.3}_{-0.3}\,\times{10^{5}}$ | $1.8^{+0.2}_{-0.2}$ |  $\cdots$
|  | $8.7^{+1.1}_{-1.1}$ | ${2.4}^{+1.8}_{-0.9}\,\times{10^{4}}$ |  |  |  |
DDO 75 | I | $25.0^{+0.8}_{-0.7}$ | ${1.1}^{+0.2}_{-0.2}\,\times{10^{3}}$ |  | ${9.9}^{+0.5}_{-0.4}\,\times{10^{5}}$ | $1.0^{+0.1}_{-0.1}$ | 0.97,1.01,0.99,0.98,0.99
DDO 210 | I | $18.4^{+0.9}_{-0.9}$ | ${3.1}^{+0.9}_{-0.7}\,\times{10^{2}}$ |  | ${4.6}^{+0.4}_{-0.4}\,\times{10^{4}}$ | $1.0^{+0.1}_{-0.1}$ | 0.98,0.96,1.00,0.99,1.00
Statistics for the Little Things, DGS, and KINGFISH Samples
Full Regions | Min$|$Median$|$Max | 10.0 $|$ 23.7 $|$ 26.5 | 3.0e2 $|$ 1.3e3 $|$ 4.7e4 |  | 6.8e4 $|$ 9.0e5 $|$ 1.9e6 | 1.0 $|$ 1.0 $|$ 1.6 |  $\cdots$
Reduced Regions | Min$|$Median$|$Max | 8.7 $|$ 20.7 $|$ 25.0 | 1.5e2 $|$ 8.7e2 $|$ 2.4e4 |  | 4.6e4 $|$ 3.1e5 $|$ 9.9e5 | 1.0 $|$ 1.0 $|$ 1.9 |  $\cdots$
DGS | Min$|$Median$|$Max | 21.0 $|$ 32.0 $|$ 98.0 | 1.0e2 $|$ 1.2e5 $|$ 2.5e7 |  | 1.2e7 $|$ 5.3e8 $|$ 5.3e10 |  $\cdots$ |  $\cdots$
KINGFISH | Min$|$Median$|$Max | 17.0 $|$ 22.5 $|$ 39.0 | 7.0e2 $|$ 1.0e7 $|$ 1.1e8 |  | 3.0e6 $|$ 4.3e9 $|$ 6.9e10 |  $\cdots$ |  $\cdots$

Note. — Fit parameters and uncertainties for the modified blackbody function.
Fitting was performed using the Maximum Likelihood Method with a term in the
probability function to account for upper limits. Uncertainties were estimated
from the 16th and 84th percentiles of fits from 10,000 bootstrap samples
within the flux density errors. $L_{\mathrm{TIR}}$ is determined by
integrating the modified blackbody fit from 3–1000 $\mu$m, and the FIR
luminosity is integrated over 50–650 $\mu$m. The infrared luminosities and
dust masses of the Little Things galaxies are generally dominated by one or
two bright regions. The DGS and KINGFISH values are from the modified
blackbody fits of Rémy-Ruyer et al. (2013). Color corrections were determined
for $\beta=2$ and a temperature from a preliminary fit to the data, as
described in § 5.2; these values are provided in the last column for the 100,
160, 250, 350, and 500 $\mu$m flux densities. The fits for DDO 70 consist of
two components. Color corrections were not applied for this source, though
their effect (typically less than $\sim$3% for the other targets in this
sample) should be much smaller than the uncertainties related to the assumed
dust model as outlined in § 6.1.

##  6 Dust Properties in context with literature samples

###  6.1 Fit Results

DDO 70 was fit with two modified blackbody components due to a single
component with $\beta$=2 being unable to recover the full emission profile
observed. The residuals were large and had the distinctive shape of a second
modified blackbody profile (see Fig. 8), so using an additional component in
the fit model is a natural choice. Fitted masses range from
$\sim$130$M_{\odot}$ in the warm components to a few times 10${}^{4}M_{\odot}$
in the colder components, for combined dust masses of 4.8$\times
10^{4}M_{\odot}$, 6.4$\times 10^{3}M_{\odot}$, and 2.5$\times 10^{4}M_{\odot}$
in the full aperture and regions I & II, respectively. The temperatures ranged
from 23–27K and 9–11K for the warm and cold components, respectively. The
extremely low fitted temperatures in the cold components are not necessarily
physical, however – this could be an artifact of using a simple two-component
model, restricting the assumed dust composition (i.e., $\kappa$, $\beta$),
insufficiently capturing the grain size distribution, or a number of other
considerations. Unfortunately, the full set of dust properties cannot be
uniquely determined from the relatively few data points available. However, a
variety of approaches to fitting the dust masses can explore possible
variations and evaluate the reliability of our general fit results.

Leaving $\beta$ as a free parameter, which can describe the broadness of the
FIR emission at the expense of the reliability of the inferred mass, results
in $\beta$ values near 0 for all apertures in DDO 70. While $\beta=0$ can be
equivalent to pure blackbody emission, a far simpler physical explanation and
more realistic scenario is that the dust has a variety of compositions and
temperatures. To test the effect of varying the dust composition to other
commonly assumed varieties, we performed two-component fits to the DDO 70 full
galaxy aperture fluxes using a number of combinations of full
$\kappa(\lambda)$ curves from the literature, instead of the simpler power law
approximations with slope $\beta$. For example, using the Weingartner & Draine
(2001) LMC model dust with Draine & Lee (1984) graphite results in a combined
dust mass of 5.9$\times 10^{4}M_{\odot}$. A combination of Zubko et al. (2004)
ACAR and Jäger et al. (2003) enstatite dust varieties results in 7.7$\times
10^{4}M_{\odot}$. Generally, all combinations we tested return a warmer
component of $\sim$25–30K and hundreds of $M_{\odot}$, with a cold component
of $\sim$10K and order 103–104 solar masses of dust – very similar to our
adopted model.

We also explored two SED template fitting packages, CIGALE (e.g., Boquien et
al., 2019) and MAGPHYS (da Cunha et al., 2008), to test the variation of
different approaches to fitting dust masses. In both cases, after moderate
tuning of input parameters, reasonable fits to the overall SED of DDO 70 was
achievable, and yielded dust masses of 8–9$\times 10^{4}M_{\odot}$, again
similar to our simple modified blackbody fits. This makes sense, as these
models use similar dust compositions to those considered above. We note,
however, that while the overall SED fits were generally good, the FIR spectrum
was less well-fit, appearing quite similar to the single-component $\beta=2$
fit shown in gray in Fig. 8. Finally, a comparison of our MLE fits with a
hierarchical Bayesian fit (see, e.g. Galliano, 2018, for an extensive
discussion on the topic) yielded nearly identical results, so we do not expect
significant bias from our choice of parameter estimation routine.

Considering all of this, our estimated dust masses are relatively robust to
variations in assumed dust composition and a wide variety of fitting methods
and models. Despite the questionably low temperature of the cold component,
the combined mass is still relatively consistent with estimates from other
methods. The uncertainties reported in Table 5 are appropriate for the
individual fits, though based on the different analysis methods above we
consider wider ranges of uncertainty in later sections – for DDO 70 for
example, on average this ranges from as low as a few hundred solar masses for
the warmer individual components, to $\sim$25% of the fit masses.

The majority of the fits in this new sample, considering both the full-galaxy
and smaller apertures, have temperatures around 20K. The Little Things
galaxies fall toward the low end of the range of dust temperatures seen in the
DGS galaxies, the bulk of which are between $\sim$20 to $\sim$50 K (with two
extreme cases of $\sim$90 K). This may be expected to first order, since many
of the DGS dwarfs are undergoing strong bursts of star formation, and more hot
stars will heat the gas and dust to higher temperatures.

The dust mass fits for the Little Things galaxies are all at the very low end
of what is seen in the DGS sample, whose masses range from 100 to $2.5\times
10^{7}$ $M_{\odot}$ with a median of $1.2\times 10^{5}$ $M_{\odot}$, while
this new sample of dwarfs has dust masses ranging from 305–4.7$\times 10^{4}$
$M_{\odot}$ for the full apertures, and 153–2.4$\times 10^{4}$ $M_{\odot}$ for
the smaller regions.

We see that the infrared maps in our galaxies tends to be dominated by one or
two bright regions (c.f. Figs. 1 and 2). These individual regions can
therefore constitute from 20% to nearly all of the total dust mass (and
integrated luminosities) of their host galaxies. This could have an impact on
studies of distant dwarf galaxies – since observations would effectively only
recover the brightest regions, the bulk properties of the galaxy could be
misinterpreted if it is not resolved. We refer the reader to Polles et al.
(2019) for a study of IC 10 based on infrared spectroscopy at different
spatial scales.

Models of dust emission (e.g., Galliano et al., 2003) predict that smaller
dust molecules, such as PAHs and graphites, are generally warmer than larger
grains, such as the mixed amorphous carbons and silicates. Laboratory
measurements of different dust varieties (e.g., Agladze et al., 1996) have
found that smaller $\beta$ values ($\sim 1$) are more consistent with smaller
carbonaceous grains while larger $\beta$ values ($\sim 2$) are more consistent
with physically larger grains such as silicates. While we should reasonably
expect that the true dust content of these galaxies is a mixture of a various
different grain species, sizes, and temperatures, further observations will be
necessary to disentangle differences in populations. However, the simple
physical model employed here matches the sparsely sampled observed emission
profiles reasonably well, allowing us to characterize the combined bulk dust
properties to first order.

###  6.2 Scaling relations with Metallicity

Figure 10: Blackbody fit parameters plotted against metallicity for the Little
Things, DGS, and KINGFISH samples. The DGS and KINGFISH values are taken from
(Rémy-Ruyer et al., 2013). The Little Things dwarfs observed in this work have
mostly cold and cool dust temperatures, and have among the lowest levels of
detected $L_{\mathrm{TIR}}$ and $M_{\mathrm{d}}$ seen in the DGS sample.

Figure 10 shows the model fit parameters explicitly compared to metallicity
for the Little Things, DGS, and KINGFISH samples. There are two obvious trends
with metallicity: as $Z$ decreases, so does the dust mass and FIR luminosity.
The $M_{\mathrm{d}}$ and $L_{\mathrm{TIR}}$ values in the Little Things sample
follow the metallicity trend tightly, even though the low-$Z$ end of the DGS
sample contains some outlying starbursts with high temperatures.

Rémy-Ruyer et al. (2013) noted that the dust masses were two orders of
magnitude lower in the DGS galaxies than in the KINGFISH galaxies, but that
$M_{\rm dust}$ does not simply scale with galaxy mass. In particular, they
found that as metallicity decreases, the dust mass drops much more rapidly
than stellar mass does. They also determined that it is not an issue with the
“low” fitted grain emissivities $\beta<2$ in many DGS galaxies. Referring to
Eq. 10, a given luminosity that is modeled with a low $\beta$ and a particular
mass can alternately be reproduced by increasing $\beta$ (reducing the
efficiency) and increasing the mass: that is, by assuming there is more dust,
but that it emits less efficiently. However, when the grain emissivities in
the Rémy-Ruyer et al. (2014) sample were fixed to $\beta$=2, the resulting
masses were not increased nearly enough to explain the difference. Ultimately,
Rémy-Ruyer et al. (2013) attribute their difference to the limitations of
using a single modified blackbody – in particular, characterizing only a
single grain state and temperature. In the Rémy-Ruyer et al. (2015) study,
this was shown to be the case when the DGS galaxies were modeled with a more
sophisticated SED model covering the mid infrared to sub-mm, instead of a
simple modified blackbody. Still, all the Little Things dust masses, save for
the DDO 70 cold components, are consistent with those derived for one of the
lowest metallicity (and notably dust-poor) dwarfs, I Zw 18 (Hunt et al., 2014;
Rémy-Ruyer et al., 2015; Schneider et al., 2016).

All three samples show a large spread of temperatures near 12+log(O/H) $\sim$
7.6. The derived single-component temperatures for the KINGFISH galaxies hint
at a slight decrease in $T_{\mathrm{dust}}$ with increasing metallicity but
the DGS and Little Things samples do not show this. With many outliers and
large error bars, there is no real trend when all the samples are considered
in concert.

All of the Little Things TIR luminosities, which have a maximum of $1.9\times
10^{6}$ $L_{\odot}$, are less than the lowest DGS FIR luminosity of $1.2\times
10^{7}$ $L_{\odot}$. Given the small galaxy masses in the Little Things
sample, we would indeed expect them to be less luminous in the FIR than
galaxies such as the KINGFISH spirals with dust reservoirs many times larger.
However, at the very low metallicity end (below $\sim$7.7), the dust masses in
all three samples are roughly similar within the errors, while the
luminosities appear to decrease more steeply in the new Little Things sample.

###  6.3 Dust-to-Gas Ratios

Table 6: Dust-to-Gas Ratios

| Whole Galaxy | Reduced Apertures
---|---|---
| DDO 69 | DDO 70 | DDO 75 | DDO 210 | DDO 69 i | DDO 69 ii | DDO 70 i | DDO 70 ii | DDO 75 i | DDO 210 i
log Mdust [M⊙] | 2.48 | 4.68 | 3.32 | 2.68 | 2.18 | 2.95 | 3.80 | 4.39 | 3.03 | 2.49
log MHI [M⊙] | 6.69 | 7.08 | 7.19 | 6.08 | 5.97 | 5.62 | 6.26 | 6.31 | 6.60 | 5.65
log M${}_{\mathrm{H_{2}}}$ [M⊙] | 8.21 | 8.01 | 8.15 | 7.08 | 7.49 | 7.14 | 7.19 | 7.24 | 7.57 | 6.65
log Mgas [M⊙] | 8.36 | 8.20 | 8.33 | 7.26 | 7.64 | 7.29 | 7.37 | 7.42 | 7.75 | 6.82
Dust-HI Ratio | 6.19$\times 10^{-5}$ | 4.00$\times 10^{-3}$ | 1.35$\times 10^{-4}$ | 3.98$\times 10^{-4}$ | 1.65$\times 10^{-4}$ | 2.13$\times 10^{-3}$ | 3.49$\times 10^{-3}$ | 1.20$\times 10^{-2}$ | 2.66$\times 10^{-4}$ | 7.06$\times 10^{-4}$
Dust-Gas Ratio | 1.33$\times 10^{-6}$ | 3.08$\times 10^{-4}$ | 9.63$\times 10^{-6}$ | 2.64$\times 10^{-5}$ | 3.54$\times 10^{-6}$ | 4.58$\times 10^{-5}$ | 2.69$\times 10^{-4}$ | 9.25$\times 10^{-4}$ | 1.90$\times 10^{-5}$ | 4.68$\times 10^{-5}$

Note. — Typical uncertainties for the dust masses (see § 6.1) and HI are of
order 25% and 10%, respectively, resulting in Dust-HI ratio uncertainties of
$\sim$30%. The uncertainties in M${}_{\mathrm{H_{2}}}$ and Mgas are estimated
as $\sim$2–3 times the inferred masses, and therefore the derived Dust-Gas
ratios are uncertain by similar factors.

Figure 11: Dust to Gas Ratios with metallicity. The uncertainties on the
ratios are factors of $\sim$2–3 (see text). Dust to H i ratios are also shown
for the Little Things sample, with typical uncertainties of order 30%.. We
note that the H2 masses, and therefore $M_{\mathrm{gas}}$, in our galaxies
were determined from [C ii] measurements instead of CO which was used in other
samples. The KINGFISH, Galametz et al. (2011), Galliano et al. (2008), and
Dustpedia (De Vis et al., 2019) samples are overplotted for reference. The DGS
and KINGFISH metallicities are taken from Madden et al. (2013) and Kennicutt
et al. (2011), respectively. For a full list of references for the gas masses
in those samples, we refer the reader to Table A.1 of Rémy-Ruyer et al.
(2014). The dotted and dash-dotted lines are broken power law fits to the DGS
data from Rémy-Ruyer et al. (2014) for $X_{\rm CO}$ taken as the Milky Way
value, and scaled with $Z$, respectively. The Little Things sample roughly
agrees with their noted trend of decreasing DGR with decreasing metallicity.

The dust masses derived in § 5 can be compared with the exquisite Little
Things H i maps (Hunter et al., 2012) to derive ratios of the dust and gas
content in these galaxies. The gas mass in a galaxy for this purpose consists
primarily of H i+H2, with smaller amounts of helium and gas-phase metals. The
H i mass can be calculated from the optically thin limit to the radiative
transfer equation as

$M_{\mathrm{{\rm H{\sc I}}}}[M_{\odot}]=235.6\ (D[\mathrm{Mpc}])^{2}\
S[\mathrm{Jy\ m/s}]$ (14)

using the integrated emission $S$ in the region of interest.

H2 masses are typically estimated from CO measurements in normal galaxies.
However, CO is exceedingly difficult to detect in low-metallicity systems
(Leroy et al., 2009; Cormier et al., 2014, for example). We follow a method of
relating [C ii] to hydrogen in a similar manner to that of Madden et al.
(1997), who determined the amount of H2 for the low-$Z$ dwarf IC 10, and found
it to be a useful method when CO is not a suitable tracer. We also refer the
reader to more recent works using $I_{\mathrm{[C\textsc{II}]}}$ instead of CO
as a proxy for H2, by Velusamy et al. (2010) and Langer et al. (2014) for the
Milky Way, as well as Jameson et al. (2018) for the Magellanic Clouds, as
additional examples.

The H2 masses of these Little Things galaxies are estimated from the [C ii]
measurements of Cigan et al. (2016), who determined that most of the observed
[C ii] in these systems originates from photodissociation regions as opposed
to the ionized ISM. Polles et al. (2019) found the same for IC 10, and Cormier
et al. (2019) have shown that the fraction of [C ii] corresponding to ionized
gas is expected to decrease with metallicity, down to at most a few percent at
the metallicities of these galaxies. For solar-metallicity photodissociation
regions (PDRs), [C ii] only originates from a thin outer layer. As metallicity
decreases, according to the model of Bolatto et al. (1999) for example, the
filling factor of the CO-emitting core shrinks compared to that of the [C
ii]-emitting region. H2 can still be present in the absence of CO – at low
extinction $A_{V}$ (e.g., Tielens & Hollenbach, 1985; Madden et al., 1997) –
and at the extremely low metallicities of the galaxies considered in this
work, [C ii] could trace nearly all of the H2 in PDRs. By modelling the gas
collisional excitation parameters, the total hydrogen content of the neutral
regions can be determined by relating it to the carbon content. The molecular
hydrogen component is then inferred as the excess beyond the atomic component
observed in H i.

In this prescription, the carbon column density is calculated from [C ii]
emission using the optically thin radiative transfer equation (c.f Madden et
al., 1997 Eq. 1, for example):

$I_{\mathrm{[CII]}}=\frac{h\nu
A_{ul}}{4\pi}\left[\frac{g_{u}/g_{l}\,e^{-h\nu/kT}}{1+g_{u}/g_{l}\,e^{-h\nu/kT}+n_{crit}/n}\right]N_{\mathrm{C^{+}}}\quad,$
(15)

where $A_{ul}$ is the Einstein emission coefficient, $g_{u}$ and $g_{l}$ are
the statistical weights of the levels, and $n_{crit}$ is the critical density.
We model the collision parameters $n$ and $T$ using the PDR Toolbox (Pound &
Wolfire, 2008) with measurements of [C ii] and [O i] from Cigan et al. (2016),
along with the integrated TIR continuum. $N_{\mathrm{C^{+}}}$ can be related
to the column density of the colliding hydrogenic nuclei $N_{\mathrm{H}}$,
using the local carbon abundance C/H as the conversion factor
$X_{\mathrm{C^{+}}}=N_{\mathrm{C^{+}}}/N_{\mathrm{H}}$ (e.g., Madden et al.,
1997; Langer et al., 2014). Instead of scaling the Galactic ISM carbon
abundance of 1.42$\times 10^{-4}$ (Sofia et al., 1997, from [C ii]/H i,
assuming equivalence to C/H) by metallicity as in other works, we can
calculate $X_{\mathrm{C^{+}}}$ directly from our [C ii] and H i maps, though
both methods give similar results. Finally, the neutral gas component measured
from H i can be subtracted, leaving the molecular component: $N_{\rm
H_{2}}=(N_{\rm H}-N_{\rm HI})/2$.

The H2 mass is generally several times the H i mass in each galaxy: Stacey et
al. (1991) found that $N_{\rm H_{2}}/N_{\mathrm{H{\sc I}}}\sim 5$ is common in
many luminous galaxies, and Madden et al. (1997) have argued that the same
column density ratio is reasonable for IC 10. We find that this is also a
reasonable approximation for our galaxies, with calculated $N_{\rm
H_{2}}/N_{\mathrm{H{\sc I}}}$ values of $\sim$4 for DDO 70 and DDO 75. DDO 69
has a higher inferred ratio of $\sim$16\. DDO 210 had no [C ii] measurements,
so we adopt $N_{\rm H_{2}}/N_{\mathrm{H{\sc I}}}=5$ for this galaxy.

The total gas mass is calculated as the sum of H i, H2, He, and metals in gas
form: $M_{\rm gas}$ = $M_{\rm H{\sc I}}$ \+ $M_{\rm H_{2}}$ \+ $M_{\rm He}$ \+
$Z_{\rm gal}\times M_{\rm gas}$. If the helium mass is assumed to follow
$M_{\rm He}$ = $Y_{\odot}$ $M_{\rm gas}$, where $Y_{\odot}$ is the Galactic
helium mass fraction of 0.27 (Asplund et al., 2009), then the total gas mass
can be written as:

$M_{\rm gas}=\frac{M_{\rm H{\sc I}}+M_{\rm H_{2}}}{1-Y_{\odot}-Z_{\rm
gal}}\quad.$ (16)

The computed dust-to-gas ratios for the Little Things sample are reported in
Table 6. Jameson et al. (2018) quote a factor of two uncertainty on their H2
masses, dominated by the differences from the various models they considered;
we favor a more conservative factor of $\sim$2–3 estimate for our galaxies
with [C ii] measurements, primarily from the assumptions about how carbon
relates to hydrogen in these systems. The uncertainty for DDO 210 is higher,
as the H2/H i ratio could vary from the assumed value by a factor of a few.

Our DGR values are generally very small compared to galaxies with higher
metallicity, as expected. Though, when plotted (Figure 11) against DGR values
from the DGS and KINGFISH samples (Rémy-Ruyer et al., 2014, 555These DGR
values were calculated with the updated dust masses from Rémy-Ruyer et al.
(2015), from full SED modeling. While the $M_{\mathrm{d}}$ values from Rémy-
Ruyer et al. (2013) were preferable for direct comparison to modified
blackbody fits as in e.g. Fig. 10, the differences in dust masses do not
change the overall trends, so we use these DGR values for consistency with the
published results. with $M_{\mathrm{gas}}$ determined from CO), they broadly
fall in line with the trend fit to the DGS galaxies by Rémy-Ruyer et al.
(2014). In that study they fit their data with a broken power law, once for
CO-based H2 values using the local Milky Way $X_{\mathrm{CO}}$ factor, and a
second time using $X_{\mathrm{CO},Z}$ that is scaled with metallicity. The
breaks occur near 12+log(O/H)=8 as a result of their findings that $M_{\rm
H_{2}}/M_{\rm H{\sc I}}$ ratios in their sample exhibit a break in that
vicinity, as well as the fact that several other studies (such as Draine et
al., 2007; Galliano et al., 2008) have shown that DGR decreases with
metallicity with a slope of –1 down to 12+log(O/H) of 8.0–8.2. It should be
noted that the H2 masses Rémy-Ruyer et al. (2014) infer for metallicities
below $\sim$7.8 come not from direct CO detections in those galaxies, but from
applying the mean $M_{\rm H_{2}}/M_{\rm H{\sc I}}$ ratio for CO detected below
12+log(O/H)$\lesssim 8.1$ to the H i mass in each galaxy. The low-$Z$ DGS
dust/gas ratios did not include a CO-dark H2 component, which would steepen
the non-linear trend further.

We also include in our comparison the Dustpedia archive (Davies et al., 2017),
which gives multiwavelength photometry for 875 local galaxies. De Vis et al.
(2019) first compared the DGR to $Z$ for this sample, which are included in
Fig. 11. The Dustpedia sample fills in more of the parameter space at
moderately low metallicities, and complements the picture of a steeper slope
below 12+log(O/H)$\sim$8 seen in the other samples, namely the Little Things
DGS dwarfs. The dust masses used in these dust to gas ratios for the DGS,
KINGFISH, Dustpedia, Galliano et al. (2008), and Galametz et al. (2011)
samples are all derived from SED models rather than modified blackbody
fitting, and incorporate different dust models, however the overall trends
will be similar despite these differences. The dust masses determined from
modified blackbodies could be lower than those for the SED models by factors
of $\sim$2 to at most 10 (e.g., Rémy-Ruyer et al., 2015).

DDO 69 is a particularly interesting case, in that is has an extremely low DGR
for its metallicity – similar to previous results for I Zw 18 and SBS 0335-052
(as shown by Rémy-Ruyer et al., 2014 and further discussed by Schneider et
al., 2016) and a few sources in the HI-rich sample from De Vis et al. (2017a,
b), suggesting some dwarf galaxies are extremely dust poor at a given $Z$ and
gas mass compared to the bulk of dwarfs studied in the literature. Properly
accounting for a CO-dark H2 component in the lowest-metallicity DGS galaxies
could push them even lower than their extreme positions in Fig. 11. Even
ignoring the H2 content, the Dust/H i ratios for our galaxies still generally
fall below the shallower linear trend line of the higher-metallicity galaxies.

###  6.4 Comparison with dust models

Table 7: Model Summary.

Source | Name | SFH | Dust Source | Other Parameters
---|---|---|---|---
Z14 | Models 1-3 | Bursty | stardust $+$ grain growth | inflows$+$outflows
| Model 4 | Continuous | stardust $+$ grain growth | inflows$+$outflows
| Models 5-7 | Bursty | stardust $+$ grain growth | inflows$+$outflows
DV17b | Model I | MW | stardust |
| Model III | MW | stardust | outflows, gas mass $=10^{9}\,\rm M_{\odot}$
| Model VI | Delayed | stardust ($1/100$) $+$ grain growth | inflows$+$outflows
| Model VII | Bursty | stardust ($1/12$) $+$ grain growth | inflows$+$outflows
This work | Model A | Delayed | stardust $+$ grain growth ($\times 10$) | destruction ($\times 100$)
| Model B | Delayed | stardust $+$ grain growth ($\times 10$) | gas mass ($1/50$)
| Model C | Delayed | stardust ($1/10$) $+$ grain growth ($1/800$) | gas mass ($1/100$), destruction ($\times 10$)

Note. — Comparison of literature models showing the predicted build up of DGR
with metallicity from Z14 - Zhukovska (2014) (all models) and DV17b - De Vis
et al. (2017b). We refer the reader to those works for more details on the
models. Here, we only compare Models I, III and VI and VII from De Vis et al.
(2017b). Models I and III indicate the growth of the DGR with $Z$ for galaxies
with efficient stardust production but no grain growth and a Milky Way like
SFH (Yin et al., 2009), whereas Models VI and VII include grain growth as an
important dust contributor and were shown to be more representative of star
forming (Model VII) and more quiescent dwarf systems (Model VI) when assuming
a bursty and non-bursty SFH respectively. The De Vis et al. (2017b) models
assume an initial gas mass of $10^{9}\,\rm M_{\odot}$ and Models VI and VII
require a reduced fraction of dust formed in stars (shown by the factors in
parentheses). This work takes the De Vis et al. (2017b) model VI but changes
input parameters including initial gas mass, dust destruction, efficiency of
dust production in stars and in grain growth) to explore lower DGRs with
metallicity. This Work Models A-C are therefore the same as DV17b Model VI
unless stated otherwise (in parentheseses).

Figure 12: Dust/Gas Ratio models for various star formation histories from
Zhukovska (2014) (denoted Z M[X] where [X] refers to models 1–7 from their
work) and different dust sources from De Vis et al. (2017b) (denoted DV17 M[X]
where [X] is one of I, III, VI, or VII corresponding to the models of the same
name in their work). For Zhukovska (2014), the specific evolutionary tracks
are determined by the number of bursts, the time and duration, the star
formation timescale, and the gas infall timescale. Models range from periodic
bursty star formation (step patterns, e.g., Z M1) to continuous (smooth
curves, e.g., Z M4). The De Vis et al. (2017b) models compare different dust
sources and include a delayed SFH thought to be appropriate for quiescent
dwarfs. We add new models (MA, MB and MC) here by taking the DV17b Model VI
(appropriate for gas rich, low star-forming dwarfs) and changing the input
parameters to explore lower DGRs than reached in Zhukovska (2014) and De Vis
et al. (2017b). See Table 7 for more details.

Zhukovska (2014) recently modeled the evolution of dust and metallicity in
dwarf irregulars over time, for a variety of star formation histories (SFH).
Their open-box chemical evolution model allows for the slow infall of
primordial gas from outside the galaxy as well as gas outflow from galactic
winds, though they calculate that galactic outflows are negligible. Starting
from zero mass, the galaxy forms by accreting infalling gas until reaching the
final $M_{\mathrm{tot}}$ after the specified infall timescale. Dust is formed
in these models by stars (AGB and type-II supernovae), and they also consider
growth of dust mass by accretion in the ISM onto existing dust grains.

Feldmann (2015) used similar chemical evolution models to argue that dwarf
galaxy samples such as the DGS required strong inflows and outflows to explain
their observed properties, though Rémy-Ruyer et al. (2014) showed that the
models of Asano et al. (2013) can match the DGS values over a wide range of
$Z$ by varying the star formation history. Later, De Vis et al. (2017b) showed
that the Feldmann (2015) and Zhukovska (2014) models fit the more actively
star-forming dwarfs in the DGS but could not explain more quiescent dwarfs
with lower DGRs (e.g., Rémy-Ruyer et al., 2014). Instead, De Vis et al.
(2017b) argued that the gas-rich, less actively star forming dwarfs required
different dust sources – including interstellar grain growth, and reducing AGB
stars and supernova dust production rates – as well as a delayed SFH with less
extreme outflows. We note that Schneider et al. (2016) explained the order of
magnitude variation in DGR measured for two dwarfs at the same metallicity as
being due to more efficient in-situ dust growth in the ISM in one of the
sources where higher cold gas densities were observed. In Figure 12, we
compare the sample of dwarfs observed in this work with all the models from
Zhukovska (2014) and a selection of models from De Vis et al. (2017b) with
details in Table 7.

When compared to the example evolutionary models of Zhukovska (2014) (labeled
Z in Figure 12), the galaxies align with different star formation paths. Thus
the star formation histories of each galaxy in our sample could be quite
different, with a mix of discrete bursts and low-level continuous production.
DDO 210 corresponds most closely to the sample evolutionary tracks with an
early burst of SF. DDO 75 is more closely aligned with the models with
smoother early SF activity and some bursts at later times. DDO 69 falls below
all of the models considered by Zhukovska (2014).

In comparison to the De Vis et al. (2017b) predictions, Models VI and VII
reach similar DGRs at the same $Z$ as galaxies DDO 210 and DDO75, but again
lie well above the source DDO 69. We therefore ran the chemical evolution code
from De Vis et al. (2017b)666available at
https://github.com/zemogle/chemevol/releases/tag/v_de_vis2017 version
v$\\_$de$\\_$vis2017 using their Model VI and changed the input parameters in
order to attempt to reach lower DGRs. Models A–C (Table 7 and Figure 12)
respectively show the outcomes of this analysis: A – the effect of faster
grain growth timescales and faster dust destruction rates; B – decrease in
initial gas mass, and faster grain growth; and C – decrease in initial gas
mass, decrease in stardust contributed by supernovae777In practice reducing
this parameter makes very little difference to the predicted DGR with $Z$
compared to the De Vis et al. (2017b) Model VI since their model had already
reduced the contribution from Type II SN to the dust budget by $100\times$
compared to typical values used, e.g., (Dwek, 1998; Morgan, & Edmunds, 2003;
Rowlands et al., 2014; Zhukovska, 2014)., slower grain growth timescales and
faster destruction rates. In models VI and A–C, the delayed SFH evolves as
$\mathrm{SFH}(t)\propto\tau^{-2}exp(-t/\tau)$ , where $t$ is the age of the
galaxy and the SF timescale $\tau$ is 15 Gyr (see De Vis et al., 2017b, for
more details).

Model A starts with the same DGR at $12+{\rm log(O/H})<6.75$ as Model VI but
diverges significantly once metallicities of $12+{\rm log(O/H})>7.25$ are
reached. Model B predicts lower values of DGRs by $\sim 0.5$ dex at low
metallicities but rapidly increases to match the DGRs reached in Model VI at
metallicities of $12+{\rm log(O/H})\sim 8$. The reduced DGR initially seen in
this model due to the reduction in initial gas mass is counteracted at later
metallicities when the interstellar grain growth “kicks in” (note that this
parameter is 10$\times$ faster than Model VI). Model C lies below the other
models by almost a decade in DGR even at solar metallicities and above, due to
combination of reduced initial gas mass, slower grain growth timescales, and
increased dust destruction rates.

These parameters have different parts to play in the different metallicity
regimes, however. A delayed SFH such as those in Models VI and A–C slows down
the build up of dust and metals compared to the other tracks shown. Next, as
we decrease the supernova dust production, the amount of dust made early on in
the galaxy evolution (at lower metallicities) decreases. Similarly, we see the
same effect when reducing the initial gas mass in Models B and C compared to
Models VI, VII and A. Changing the amount of dust produced in low-intermediate
mass stars only affects the DGR at higher metallicities (given the SFHs used
here) since these stars take $>500$ Myr to reach their dust-production stage
and therefore cannot be responsible for low DGR ratios such as DDO 69.

If one fixes the stardust contribution to the assumed values in Zhukovska
(2014) Models 1-7 or De Vis et al. (2017b) Model VI or VII, then changing the
grain growth timescale in our models has little effect on the DGR values at
low metallicities ($12+{\rm log(O/H})<7.7$) since a critical metallicity is
required to be reached before grain growth “kicks in” and becomes the dominant
dust producer (Zhukovska, 2014; Asano et al., 2013). To reach very low DGRs at
metallicities above this, one requires extremely long grain growth timescales
($\sim$1000-8000$\times$ slower grain growth rates than typical values used in
Zhukovska (2014); Feldmann (2015) and De Vis et al. (2017b)). Increasing the
dust destruction rates again acts to reduce the DGR, though again the increase
needs to be one to two orders of magnitude to move from the parameter space
probed by Models A–C.

We therefore show that the orders of magnitude variation in DGRs at the same
metallicity for chemically young dwarfs might reveal variations in the
relative contributions of dust source in these systems (as also seen in
Schneider et al. 2016). Here we argue that the most dust-poor quiescent dwarfs
can be explained by higher destruction rates and extremely slow grain growth
timescales compared to more massive systems or more actively star-forming
dwarfs.

An interesting future project would be to compare the star formation histories
determined directly from stellar data and color-magnitude diagrams to these
models. If measured star formation histories differ significantly from the
models, they could provide useful information to revise the dust production
models assumed to make these predictions.

##  7 SUMMARY

Four dwarf irregulars from the Little Things sample of dwarf galaxies were
observed with the Herschel PACS and SPIRE photometers to determine the dust
properties of typical dwarf galaxies: those with very low metallicity and
moderate star formation. After carefully measuring the flux densities in five
FIR bands for each, the emission was compared between the different bands, and
compared with previous MIPS 160 $\mu$m data. Dust masses, temperatures, and
emissivities were estimated in these systems from modified blackbody fits to
the FIR emission. This yielded the following results:

  * •

The fitted dust temperatures of the Little Things galaxies show a mix of cool
and warm dust components. Previous work on the DGS dwarfs showed that many of
the lowest-metallicity galaxies had dust temperatures between $\sim$20–50 K.
These new data indicate that very low metallicity dwarfs with moderate star
formation have temperatures typically ranging between 18–26 K.

  * •

All of the these galaxies have extremely low $L_{\rm TIR}$ and dust masses,
with $M_{d}$ ranging from $\sim 10^{4}M_{\odot}$ down to a few hundred
$M_{\odot}$ – consistent with the lower range of $M_{\mathrm{d}}$ values
measured for galaxies in other surveys. The dust masses follow the same trend
of decreasing with lower metallicity as seen in the KINGFISH and DGS samples.

  * •

The integrated $M_{\mathrm{d}}$ and $L_{\rm TIR}$ of the galaxies studied here
are dominated by one or two bright localized regions. Resolving the emission
of other faraway dwarf galaxies is important to avoid misinterpretation of
their bulk properties.

  * •

We show that the the dust temperature does not follow a trend with $Z$ – in
agreement with the DGS results.

  * •

Dust-to-gas ratios in the Little Things sample are smaller than for more
massive and metal-rich galaxies and are mostly consistent with the metallicity
trends noted for other low-$Z$ dwarfs with low star formation rates. The
inferred DGR for DDO 69 is one of the lowest seen to date, at $1.3\times
10^{-6}$. Even ignoring molecular gas, the dust-to-H i ratios of these
galaxies generally fall below the linear DGR trend seen in higher-metallicity
systems.

  * •

Comparison with theoretical models suggests a range of different star
formation histories could explain the differences in DGR–$Z$ for most of the
Little Things dwarfs studied in this work. Changing the assumptions of dust
sources and sinks (destruction rates) can also predict the DGR for these
galaxies without needing bursty SFHs which appear more appropriate for the
more highly star-forming (on average) DGS sample. To reach the lowest DGR
values observed in this sample, a model galaxy with delayed star formation
history, low initial gas mass, extremely efficient dust destruction and
reduced dust contributions from stars and interstellar grain growth is needed,
though they could be consistent with some less extreme models given the
uncertainties. These dwarf galaxies are likely forming very little dust via
stars or grain growth, and have very high destruction rates.

Although this is a relatively small sample of four sources, these new data
have helped to populate the sparsely filled parameter space of extremely
metal-poor, low surface brightness galaxies. Our finding that a small number
of localized regions of the galaxy contribute a significant amount of the
infrared emission and dust mass is relevant for observations of more distant
galaxies, highlighting the importance of resolving the desired features in
these systems.

Funding for this project was provided by NASA JPL RSA grant 1433776. PJC and
HLG acknowledge support from the European Research Council (ERC) in the form
of Consolidator Grant CosmicDust (ERC-2014-CoG-647939, PI H L Gomez). SCM
acknowledges support from the Agence Nationale de la Recherche (ANR) through
the programme SYMPATICO (Program Blanc Projet ANR-11-BS56-0023), and from the
EU FP7 project DustPedia (Grant No. 606847). This work was supported by the
Programme National “Physique et Chimie du Milieu Interstellaire” (PCMI) of
CNRS/INSU with INC/INP cofunded by CEA and CNES. DAH appreciates assistance
for publication provided by the National Science Foundation grant AST-1907492.
The National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc.

## References

  * Agius et al. (2013) Agius, N. K., Sansom, A. E., Popescu, C. C., et al. 2013, MNRAS, 431, 1929
  * Agladze et al. (1996) Agladze, N. I., Sievers, A. J., Jones, S. A., et al. 1996, ApJ, 462, 1026
  * Aniano et al. (2011) Aniano, G., Draine, B. T., Gordon, K. D., et al. 2011, PASP, 123, 1218
  * Asano et al. (2013) Asano, R. S., Takeuchi, T. T., Hirashita, H., et al. 2013, Earth, Planets, and Space, 65, 213
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., et al. 2009, ARA&A, 47, 481
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123
  * Auld et al. (2013) Auld, R., Bianchi, S., Smith, M. W. L., et al. 2013, MNRAS, 428, 1880
  * Beeston et al. (2018) Beeston, R. A., Wright, A. H., Maddox, S., et al. 2018, MNRAS, 479, 1077
  * Bendo et al. (2013) Bendo, G. J., Griffin, M. J., Bock, J. J., et al. 2013, MNRAS, 433, 3062
  * Bolatto et al. (1999) Bolatto, A. D., Jackson, J. M., & Ingalls, J. G. 1999, ApJ, 513, 275
  * Boquien et al. (2019) Boquien, M., Burgarella, D., Roehlly, Y., et al. 2019, A&A, 622, A103. doi:10.1051/0004-6361/201834156
  * Boselli et al. (2010) Boselli, A., Eales, S., Cortese, L., et al. 2010, PASP, 122, 261
  * Ciesla et al. (2012) Ciesla, L., Boselli, A., Smith, M. W. L., et al. 2012, A&A, 543, A161
  * Cigan et al. (2016) Cigan, P., Young, L., Cormier, D., et al. 2016, AJ, 151, 14
  * Cigan et al. (2019) Cigan, P., Matsuura, M., Gomez, H. L., et al. 2019, ApJ, 886, 51
  * Clark et al. (2015) Clark, C. J. R., Dunne, L., Gomez, H. L., et al. 2015, MNRAS, 452, 397
  * Clemens et al. (2013) Clemens, M. S., Negrello, M., De Zotti, G., et al. 2013, MNRAS, 433, 695
  * Cormier et al. (2014) Cormier, D., Madden, S. C., Lebouteiller, V., et al. 2014, A&A, 564, A121
  * Cormier et al. (2019) Cormier, D., Abel, N. P., Hony, S., et al. 2019, A&A, 626, A23
  * Cortese et al. (2012) Cortese, L., Ciesla, L., Boselli, A., et al. 2012, A&A, 540, A52
  * da Cunha et al. (2008) da Cunha, E., Charlot, S., & Elbaz, D. 2008, MNRAS, 388, 1595. doi:10.1111/j.1365-2966.2008.13535.x
  * Dale et al. (2009) Dale, D. A., Cohen, S. A., Johnson, L. C., et al. 2009, ApJ, 703, 517
  * Davies et al. (2017) Davies, J. I., Baes, M., Bianchi, S., et al. 2017, PASP, 129, 044102
  * De Looze et al. (2014) De Looze, I., Cormier, D., Lebouteiller, V., et al. 2014, A&A, 568, A62
  * De Vis et al. (2017a) De Vis, P., Dunne, L., Maddox, S., et al. 2017, MNRAS, 464, 4680
  * De Vis et al. (2017b) De Vis, P., Gomez, H. L., Schofield, S. P., et al. 2017, MNRAS, 471, 1743
  * De Vis et al. (2019) De Vis, P., Jones, A., Viaene, S., et al. 2019, A&A, 623, A5
  * Dolphin et al. (2002) Dolphin, A. E., Saha, A., Claver, J., et al. 2002, AJ, 123, 3154
  * Dolphin et al. (2003) Dolphin, A. E., Saha, A., Skillman, E. D., et al. 2003, AJ, 125, 1261
  * Draine & Lee (1984) Draine, B. T., & Lee, H. M. 1984, ApJ, 285, 89
  * Draine et al. (2007) Draine, B. T., Dale, D. A., Bendo, G., et al. 2007, ApJ, 663, 866
  * Draine, & Hensley (2012) Draine, B. T., & Hensley, B. 2012, ApJ, 757, 103
  * Draine et al. (2014) Draine, B. T., Aniano, G., Krause, O., et al. 2014, ApJ, 780, 172
  * Dunne, & Eales (2001) Dunne, L., & Eales, S. A. 2001, MNRAS, 327, 697
  * Dunne et al. (2011) Dunne, L., Gomez, H. L., da Cunha, E., et al. 2011, MNRAS, 417, 1510
  * Dwek (1998) Dwek, E. 1998, ApJ, 501, 643
  * Eales et al. (2010) Eales, S., Dunne, L., Clements, D., et al. 2010, PASP, 122, 499
  * Edmunds (2001) Edmunds, M. G. 2001, MNRAS, 328, 223
  * Engelbracht et al. (2007) Engelbracht, C. W., Blaylock, M., Su, K. Y. L., et al. 2007, PASP, 119, 994
  * Feldmann (2015) Feldmann, R. 2015, MNRAS, 449, 3274
  * Galametz et al. (2011) Galametz, M., Madden, S. C., Galliano, F., et al. 2011, A&A, 532, A56
  * Galametz et al. (2013) Galametz, M., Kennicutt, R. C., et al., 2013, MNRAS, 431, 1956
  * Galliano et al. (2003) Galliano, F., Madden, S. C., Jones, A. P., et al. 2003, A&A, 407, 159
  * Galliano et al. (2008) Galliano, F., Dwek, E., & Chanial, P. 2008, ApJ, 672, 214
  * Galliano et al. (2011) Galliano, F., Hony, S., Bernard, J.-P., et al. 2011, A&A, 536, A88
  * Galliano (2018) Galliano, F. 2018, MNRAS, 476, 1445
  * Grossi et al. (2010) Grossi, M., Hunt, L. K., Madden, S., et al. 2010, A&A, 518, L52
  * Gordon et al. (2007) Gordon, K. D., Engelbracht, C. W., Fadda, D., et al. 2007, PASP, 119, 1019
  * Gordon et al. (2014) Gordon, K. D., Roman-Duval, J., Bot, C., et al. 2014, ApJ, 797, 85
  * Griffin et al. (2010) Griffin, M. J., Abergel, A., Abreu, A., et al. 2010, A&A, 518, L3
  * Hunt et al. (2014) Hunt, L. K., Testi, L., Casasola, V., et al. 2014, A&A, 561, A49
  * Hunter, & Elmegreen (2004) Hunter, D. A., & Elmegreen, B. G. 2004, AJ, 128, 2170
  * Hunter, & Elmegreen (2006) Hunter, D. A., & Elmegreen, B. G. 2006, ApJS, 162, 49
  * Hunter et al. (2010) Hunter, D. A., Elmegreen, B. G., & Ludka, B. C. 2010, AJ, 139, 447
  * Hunter et al. (2012) Hunter, D. A., Ficut-Vicas, D., Ashley, T., et al. 2012, AJ, 144, 134
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90
  * Jäger et al. (2003) Jäger, C., Dorschner, J., Mutschke, H., et al. 2003, A&A, 408, 193
  * Jameson et al. (2018) Jameson, K. E., Bolatto, A. D., Wolfire, M., et al. 2018, ApJ, 853, 111
  * Jones et al. (2001–) Jones, E., Oliphant, T., Peterson, P., et al. 2001–, SciPy: Open source scientific tools for Python, http://www.scipy.org/
  * Joye & Mandel (2003) Joye, W. A., & Mandel, E. 2003, Astronomical Data Analysis Software and Systems XII, 295, 489
  * Karachentsev et al. (2002) Karachentsev, I. D., Dolphin, A. E., Geisler, D., et al. 2002, A&A, 383, 125
  * Kelly et al. (2012) Kelly, B. C., Shetty, R., Stutz, A. M., et al. 2012, ApJ, 752, 55
  * Kennicutt et al. (2011) Kennicutt, R. C., Calzetti, D., Aniano, G., et al. 2011, PASP, 123, 1347
  * Kniazev et al. (2005) Kniazev, A. Y., Grebel, E. K., Pustilnik, S. A., et al. 2005, AJ, 130, 1558
  * Langer et al. (2014) Langer, W. D., Velusamy, T., Pineda, J. L., et al. 2014, A&A, 561, A122
  * Leroy et al. (2009) Leroy, A. K., Walter, F., Bigiel, F., et al. 2009, AJ, 137, 4670
  * Leroy et al. (2011) Leroy, A. K., Bolatto, A., Gordon, K., et al. 2011, ApJ, 737, 12
  * Madden et al. (1997) Madden, S. C., Poglitsch, A., Geis, N., et al. 1997, ApJ, 483, 200
  * Madden et al. (2013) Madden, S. C., Rémy-Ruyer, A., Galametz, M., et al. 2013, PASP, 125, 600
  * Makarov et al. (2014) Makarov, D., Prugniel, P., Terekhova, N., et al. 2014, A&A, 570, A13
  * Meny et al. (2007) Meny, C., Gromov, V., Boudet, N., et al. 2007, A&A, 468, 171
  * Morgan, & Edmunds (2003) Morgan, H. L., & Edmunds, M. G. 2003, MNRAS, 343, 427
  * Negrello et al. (2013) Negrello, M., Clemens, M., Gonzalez-Nuevo, J., et al. 2013, MNRAS, 429, 1309
  * Newville et al. (2016) Newville, M., Stensitzki, T., Allen, D. B., et al. 2016, Lmfit: Non-Linear Least-Square Minimization and Curve-Fitting for Python, ascl:1606.014
  * Ott (2010) Ott, S. 2010, Astronomical Data Analysis Software and Systems XIX, 139
  * Paradis et al. (2010) Paradis, D., Veneziani, M., Noriega-Crespo, A., et al. 2010, A&A, 520, L8
  * Pérez & Granger (2007) Pérez, F. & Granger, B. E. 2007, Computing in Science & Engineering, 9, 21
  * Pilbratt et al. (2010) Pilbratt, G. L., Riedinger, J. R., Passvogel, T., et al. 2010, A&A, 518, L1
  * Planck Collaboration et al. (2011) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2011, A&A, 536, A1
  * Planck Collaboration et al. (2016) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 586, A132
  * Poglitsch et al. (2010) Poglitsch, A., Waelkens, C., Geis, N., et al. 2010, A&A, 518, L2
  * Polles et al. (2019) Polles, F. L., Madden, S. C., Lebouteiller, V., et al. 2019, A&A, 622, A119
  * Pound & Wolfire (2008) Pound, M. W., & Wolfire, M. G. 2008, Astronomical Data Analysis Software and Systems XVII, 654
  * Rémy-Ruyer et al. (2013) Rémy-Ruyer, A., Madden, S. C., Galliano, F., et al. 2013, A&A, 557, A95
  * Rémy-Ruyer et al. (2014) Rémy-Ruyer, A., Madden, S. C., Galliano, F., et al. 2014, A&A, 563, A31
  * Rémy-Ruyer et al. (2015) Rémy-Ruyer, A., Madden, S. C., Galliano, F., et al. 2015, A&A, 582, A121
  * Richer, & McCall (1995) Richer, M. G., & McCall, M. L. 1995, ApJ, 445, 642
  * Roman-Duval et al. (2014) Roman-Duval, J., Gordon, K. D., Meixner, M., et al. 2014, ApJ, 797, 86
  * Roussel (2013) Roussel, H. 2013, PASP, 125, 1126
  * Rowlands et al. (2014) Rowlands, K., Gomez, H. L., Dunne, L., et al. 2014, MNRAS, 441, 1040
  * Sakai et al. (2004) Sakai, S., Ferrarese, L., Kennicutt, R. C., et al. 2004, ApJ, 608, 42
  * Sawicki (2012) Sawicki, M. 2012, PASP, 124, 1208
  * Schneider et al. (2016) Schneider, R., Hunt, L., & Valiante, R. 2016, MNRAS, 457, 1842
  * Skibba et al. (2011) Skibba, R. A., Engelbracht, C. W., Dale, D., et al. 2011, ApJ, 738, 89
  * Smith et al. (2012) Smith, M. W. L., Eales, S. A., Gomez, H. L., et al. 2012, ApJ, 756, 40
  * Smith et al. (2012) Smith, M. W. L., Gomez, H. L., Eales, S. A., et al. 2012, ApJ, 748, 123
  * Sofia et al. (1997) Sofia, U. J., Cardelli, J. A., Guerin, K. P., et al. 1997, ApJ, 482, L105
  * Stacey et al. (1991) Stacey, G. J., Geis, N., Genzel, R., et al. 1991, ApJ, 373, 423
  * Stansberry et al. (2007) Stansberry, J. A., Gordon, K. D., Bhattacharya, B., et al. 2007, PASP, 119, 1038
  * Tielens & Hollenbach (1985) Tielens, A. G. G. M., & Hollenbach, D. 1985, ApJ, 291, 722
  * van der Walt et al. (2011) Van der Walt, S., Colbert, S., & Varoquaux, G. 2011, Computing in Science & Engineering, 13, 22
  * van Zee et al. (2006) van Zee, L., Skillman, E. D., & Haynes, M. P. 2006, ApJ, 637, 269
  * Velusamy et al. (2010) Velusamy, T., Langer, W. D., Pineda, J. L., et al. 2010, A&A, 521, L18
  * Weingartner & Draine (2001) Weingartner, J. C., & Draine, B. T. 2001, ApJ, 548, 296
  * Whitworth et al. (2019) Whitworth, A. P., Marsh, K. A., Cigan, P. J., et al. 2019, MNRAS, 489, 5436
  * Yin et al. (2009) Yin, J., Hou, J. L., Prantzos, N., et al. 2009, A&A, 505, 497
  * Zhukovska (2014) Zhukovska, S. 2014, A&A, 562, A76
  * Zubko et al. (2004) Zubko, V., Dwek, E., & Arendt, R. G. 2004, ApJS, 152, 211