A Radio Continuum Study of Dwarf Galaxies: 6 cm Imaging of LITTLE THINGS

The LITTLE THINGS sample consists of 40 gas-rich dwarf galaxies within 11 Mpc (see Hunter et al. 2012 for sample details) and is listed in Table 1. The sample spans 4 dex in both SFR and gas mass, and a factor of 50 in metallicity.

Table 1.  The Galaxy Sample

		D	MV	RHb	RDb		log10 ΣSFR(Hα)	log10 ΣSFR(FUV)
Galaxy	Other Namesa	(Mpc)	(mag)	(arcmin)	(kpc)	E(B − V)c	(M⊙yr−1 kpc−2)d	(M⊙yr−1 kpc−2)d	$12+{\mathrm{log}}_{10}{\rm{O}}/{\rm{H}}$ e
Im Galaxies
CVnIdwA	UGCA 292	3.6	−12.4	0.87	0.57 ± 0.12	0.01	−2.58 ± 0.01	−2.48 ± 0.01	7.3 ± 0.06
DDO 43	PGC 21073, UGC 3860	7.8	−15.1	0.89	0.41 ± 0.03	0.05	−1.78 ± 0.01	−1.55 ± 0.01	8.3 ± 0.09
DDO 46	PGC 21585, UGC 3966	6.1	−14.7	...	1.14 ± 0.06	0.05	−2.89 ± 0.01	−2.46 ± 0.01	8.1 ± 0.1
DDO 47	PGC 21600, UGC 3974	5.2	−15.5	2.24	1.37 ± 0.06	0.02	−2.70 ± 0.01	−2.40 ± 0.01	7.8 ± 0.2
DDO 50	PGC 23324, UGC 4305, Holmberg II, VIIZw 223	3.4	−16.6	3.97	1.10 ± 0.05	0.02	−1.67 ± 0.01	−1.55 ± 0.01	7.7 ± 0.14
DDO 52	PGC 23769, UGC 4426	10.3	−15.4	1.08	1.30 ± 0.13	0.03	−3.20 ± 0.01	−2.43 ± 0.01	(7.7)
DDO 53	PGC 24050, UGC 4459, VIIZw 238	3.6	−13.8	1.37	0.72 ± 0.06	0.03	−2.42 ± 0.01	−2.41 ± 0.01	7.6 ± 0.11
DDO 63	PGC 27605, Holmberg I, UGC 5139, Mailyan 044	3.9	−14.8	2.17	0.68 ± 0.01	0.01	−2.32 ± 0.01	−1.95 ± 0.01	7.6 ± 0.11
DDO 69	PGC 28868, UGC 5364, Leo A	0.8	−11.7	2.40	0.19 ± 0.01	0.00	−2.83 ± 0.01	−2.22 ± 0.01	7.4 ± 0.10
DDO 70	PGC 28913, UGC 5373, Sextans B	1.3	−14.1	3.71	0.48 ± 0.01	0.01	−2.85 ± 0.01	−2.16 ± 0.01	7.5 ± 0.06
DDO 75	PGC 29653, UGCA 205, Sextans A	1.3	−13.9	3.09	0.22 ± 0.01	0.02	−1.28 ± 0.01	−1.07 ± 0.01	7.5 ± 0.06
DDO 87	PGC 32405, UGC 5918, VIIZw 347	7.7	−15.0	1.15	1.31 ± 0.12	0.00	−1.36 ± 0.01	−1.00 ± 0.01	7.8 ± 0.04
DDO 101	PGC 37449, UGC 6900	6.4	−15.0	1.05	0.94 ± 0.03	0.01	−2.85 ± 0.01	−2.81 ± 0.01	8.7 ± 0.03
DDO 126	PGC 40791, UGC 7559	4.9	−14.9	1.76	0.87 ± 0.03	0.00	−2.37 ± 0.01	−2.10 ± 0.01	(7.8)
DDO 133	PGC 41636, UGC 7698	3.5	−14.8	2.33	1.24 ± 0.09	0.00	−2.88 ± 0.01	−2.62 ± 0.01	8.2 ± 0.09
DDO 154	PGC 43869, UGC 8024, NGC 4789A	3.7	−14.2	1.55	0.59 ± 0.03	0.01	−2.50 ± 0.01	−1.93 ± 0.01	7.5 ± 0.09
DDO 155	PGC 44491, UGC 8091, GR 8, LSBC D646-07	2.2	−12.5	0.95	0.15 ± 0.01	0.01	−1.44 ± 0.01	...	7.7 ± 0.06
DDO 165	PGC 45372, UGC 8201, IIZw 499, Mailyan 82	4.6	−15.6	2.14	2.26 ± 0.08	0.01	−3.67 ± 0.01	...	7.6 ± 0.08
DDO 167	PGC 45939, UGC 8308	4.2	−13.0	0.75	0.33 ± 0.05	0.00	−2.36 ± 0.01	−1.83 ± 0.01	7.7 ± 0.2
DDO 168	PGC 46039, UGC 8320	4.3	−15.7	2.32	0.82 ± 0.01	0.00	−2.27 ± 0.01	−2.04 ± 0.01	8.3 ± 0.07
DDO 187	PGC 50961, UGC 9128	2.2	−12.7	1.06	0.18 ± 0.01	0.00	−2.52 ± 0.01	−1.98 ± 0.01	7.7 ± 0.09
DDO 210	PGC 65367, Aquarius Dwarf	0.9	−10.9	1.31	0.17 ± 0.01	0.03	...	−2.71 ± 0.06	(7.2)
DDO 216	PGC 71538, UGC 12613, Peg DIG, Pegasus Dwarf	1.1	−13.7	4.00	0.54 ± 0.01	0.02	−4.10 ± 0.07	−3.21 ± 0.01	7.9 ± 0.15
F564-V3	LSBC D564-08	8.7	−14.0	...	0.53 ± 0.03	0.02	...	−2.79 ± 0.02	(7.6)
IC 10	PGC 1305, UGC 192	0.7	−16.3	...	0.40 ± 0.01	0.75	−1.11 ± 0.01	...	8.2 ± 0.12
IC 1613	PGC 3844, UGC 668, DDO 8	0.7	−14.6	9.10	0.58 ± 0.02	0.00	−2.56 ± 0.01	−1.99 ± 0.01	7.6 ± 0.05
LGS 3	PGC 3792, Pisces dwarf	0.7	−9.7	0.96	0.23 ± 0.02	0.04	...	−3.88 ± 0.06	(7.0)
M81 dwA	PGC 23521	3.5	−11.7	...	0.26 ± 0.01	0.02	...	−2.26 ± 0.01	(7.3)
NGC 1569	PGC 15345, UGC 3056, Arp 210, VIIZw 16	3.4	−18.2	...	0.38 ± 0.02	0.51	0.19 ± 0.01	−0.01 ± 0.01	8.2 ± 0.05
NGC 2366	PGC 21102, UGC 3851, DDO 42	3.4	−16.8	4.72	1.36 ± 0.04	0.04	−1.67 ± 0.01	−1.66 ± 0.01	7.9 ± 0.01
NGC 3738	PGC 35856, UGC 6565, Arp 234	4.9	−17.1	2.40	0.78 ± 0.01	0.00	−1.66 ± 0.01	−1.53 ± 0.01	8.4 ± 0.01
NGC 4163	PGC 38881, NGC 4167, UGC 7199	2.9	−14.4	1.47	0.27 ± 0.03	0.00	−2.28 ± 0.13	−1.74 ± 0.01	7.9 ± 0.2
NGC 4214	PGC 39225, UGC 7278	3.0	−17.6	4.67	0.75 ± 0.01	0.00	−1.03 ± 0.01	−1.08 ± 0.01	8.2 ± 0.06
Sag DIG	PGC 63287, Lowal's Object	1.1	−12.5	...	0.23 ± 0.03	0.14	−2.97 ± 0.04	−2.11 ± 0.01	7.3 ± 0.1
UGC 8508	PGC 47495, I Zw 60	2.6	−13.6	1.28	0.27 ± 0.01	0.00	−2.03 ± 0.01	...	7.9 ± 0.2
WLM	PGC 143, UGCA 444, DDO 221, Wolf-Lundmark-Melott	1.0	−14.4	5.81	0.57 ± 0.03	0.02	−2.77 ± 0.01	−2.05 ± 0.01	7.8 ± 0.06
BCD Galaxies
Haro 29	PGC 40665, UGCA 281, Mrk 209, I Zw 36	5.8	−14.6	0.84	0.29 ± 0.01	0.00	−0.77 ± 0.01	−1.07 ± 0.01	7.9 ± 0.07
Haro 36	PGC 43124, UGC 7950	9.3	−15.9	...	0.69 ± 0.01	0.00	−1.86 ± 0.01	−1.55 ± 0.01	8.4 ± 0.08
Mrk 178	PGC 35684, UGC 6541	3.9	−14.1	1.01	0.33 ± 0.01	0.00	−1.60 ± 0.01	−1.66 ± 0.01	7.7 ± 0.02
VIIZw 403	PGC 35286, UGC 6456	4.4	−14.3	1.11	0.52 ± 0.02	0.02	−1.71 ± 0.01	−1.67 ± 0.01	7.7 ± 0.01

Notes. See Hunter et al. 2012 for references to galaxy distances and oxygen abundances.

^aSelected alternate identifications obtained from NED. ^bR_H is the Holmberg radius, the radius of the galaxy at a B-band isophote, corrected for reddening of 26.7 mag arcsec⁻². R_D is the disk scale length measured from V-band images (table from Hunter & Elmegreen 2006). ^cForeground reddening from Burstein & Heiles (1984). ^dΣ_SFR(Hα) is the star formation rate density (SFRD) measured from Hα, calculated over the area $\pi {R}_{D}^{2}$, where R_D is the disk scale length (Hunter & Elmegreen 2004). Σ_SFR(FUV) is the SFRD determined from GALEX FUV fluxes. ^eValues in parentheses were determined from the empirical relationship between oxygen abundance and M_B and are particularly uncertain.

Download table as:  ASCIITypeset images: 1 2

Observations of the LITTLE THINGS sample were obtained (project ID: 12A-234) with the VLA at C-band (6 cm: 4–8 GHz) and in its C-configuration in nine observing runs between March and May of 2012. All observing runs included one of four NRAO primary calibrators to calibrate the flux scale and a calibrator within 10° of each dwarf galaxy to correct the complex gain on timescales of around 10 minutes. For the details of the various calibrators used, see Table 2. One of the primary goals of these observations is to resolve the faint low surface brightness emission associated with dwarf galaxies. The C-configuration provided the best compromise between resolution and surface brightness sensitivity. We note that IC 1613 is 0.7 Mpc away and so has a large angular size. We utilized archival observations taken in D-configuration (project ID: AH1006) to minimize the effect of missing large-scale emission for this galaxy. At the C-band, we expect a roughly equal mix of RC_Th and RC_Nth emission and sensitivity to spatial scales up to ∼4'. Given that most galaxies have angular sizes smaller than this, we do not expect significant loss of large-scale flux.

We calibrated the data using the Common Astronomy Software Applications (casa⁹ ; McMullin et al. 2007) package following standard procedures that we present in the following subsections.

2.2.1. Flagging

2.2.2. Calibration

2.2.3. Imaging

The LITTLE THINGS project has acquired a large collection of multiwavelength and spatially resolved data on each of the 40 dwarf galaxies (see Hunter et al. 2012; Zhang et al. 2012 for details). We make use of the following ancillary data in this study:

• 1.  
  Hα line emission: the FWHM of the filter used for the Hα observations was 30 Å, centered on 6562.8 Å (Hunter & Elmegreen 2004), while the FWHM of the point-spread function (PSF) was ∼2''. The maps were continuum subtracted and the fluxes corrected for [N ii] contribution. Hunter et al. (2012) used Burstein & Heiles (1982) values to correct the Hα and FUV maps for foreground reddening. Internal extinction in dwarf galaxies can generally be ignored because they have low metallicity and consequently a low dust-to-gas ratio with respect to spirals (Ficut-Vicas 2016). However, internal extinction may be important in some of the more actively star-forming dwarfs. We discuss this further in Section 4.1.2.
• 2.  
  Far-ultraviolet broadband emission: the FUV data were taken with GALEX in the 1350–1750 Å band (effective wavelength of 1516 Å) with a resolution of 4'' at the FWHM. The data were calibrated with the GR4/5 pipeline except for DDO 165 and NGC 4214, which were processed with the GR6 pipeline (Zhang et al. 2012). The resulting images were sky subtracted and geometrically transformed to match the optical V-band orientation. UGC 8508 was not observed due to bright foreground stars, and IC 10 was not observed due to its low Galactic latitude, placing it in a region of high extinction. For surface brightness measurements, and hence for extended emission, the estimated uncertainty for the GALEX FUV maps is 0.15 mag (Gil de Paz et al. 2007).
• 3.  
  Infrared (IR) broadband emission: the IR data were taken with the Spitzer Space Telescope using the Multiband Imaging Photometer for Spitzer (MIPS). The two bands used were mid-infrared (MIR), with an effective wavelength of 24 μm with a resolution of 6'' at FWHM, and FIR, with an effective wavelength of 70 μm and a resolution of 175 at FWHM. The Spitzer 24 and 70 μm maps were taken from either the Local Volume Legacy (LVL) survey (see Dale et al. 2009, for details) or the Spitzer Infrared Nearby Galaxies Survey (SINGS). A pixel-dependent background subtraction was performed, and images were convolved with a custom kernel to make a near-Gaussian PSF. For the Spitzer 24 μm maps, the photometric uncertainty is 2% for both unresolved sources and extended emission (Engelbracht et al. 2007).

Contamination by background sources in the RC is an issue since their emission is often brighter than, or similar to, the emission originating from the dwarf galaxy (Padovani 2011). Low-resolution observations reported in the literature are predominantly from single-dish observations and will have suffered from contamination to varying degrees. Our resolved maps make it possible to remove the effects of contamination by identifying emission unrelated to our galaxies.

We inspected each of our RC images and classified features in a manner similar to Chomiuk & Wilcots (2009). Flux was attributed as originating from one of the following:

• 1.  
  the dwarf galaxy (exactly coincident with an SF tracer), or
• 2.  
  a background galaxy, or
• 3.  
  ambiguous emission of unknown origin (i.e., unable to discern between (a) background origin or (b) nonthermal emission from unresolved supernova remnants (SNRs) or diffuse nonthermal emission).

We applied a two-step process to classify the RC emission in our images into these three categories. First, we cross-matched our RC sources with the literature. Following this, we applied a procedure designed to isolate RC emission features from background galaxies based on their proximity to Hα emission. We describe these two steps in more detail below.

3.1.1. Cross-matching with Line-of-sight Optical Counterparts

3.1.2. Isolating Obvious Background Galaxies

3.1.3. Ambiguous Sources

After cross-matching with NED and isolating ambiguous sources by comparing to Hα, there remained sources that we could not attribute as coming from a background galaxy, but at the same time were not close enough to an SF site to be confidently classified as originating from the target galaxy; we refer to these sources of RC emission as ambiguous. To illustrate our definition of ambiguous RC emission, we present four of our observations that contained such a source in Figure 2. A strong unresolved source can be seen in DDO 46 and DDO 63, while DDO 69 and IC 1613 demonstrate galaxies with significantly extended sources.

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Most of our observations contained at least one ambiguous source; none of these had a nonthermal luminosity that exceeded a reference threshold—that of a known bright SNR (1 × 10¹⁹ W Hz⁻¹ or 3.3 mJy at 5 Mpc at 6 GHz). This reference luminosity was based on observations of SNR N4449–12, which resides in the dwarf galaxy NGC 4449 at a distance of 4.2 Mpc. In 2002, this SNR had a luminosity of S_6cm = 4.84 mJy with a spectral index of α = −0.7 between 20 cm and 6 cm (Chomiuk & Wilcots 2009). For comparison, this is 10 times the luminosity of Cassiopeia A. Since the luminosity terminally declines for the majority of the SNR's lifetime, we treat the observed luminosity of SNR N4449–12 in 2002 as an approximate empirical upper limit to the luminosity of an SNR. We justify our use of SNR N4449–12 as it was the most luminous from a sample of 43 SNRs from four irregular galaxies (35 of which are in galaxies that overlap with our sample, namely: NGC 1569, NGC 2366, and NGC 4214).

Using the method above, we are able to classify all of the observed RC emission in our images. As an example, we show DDO 133 in Figure 3 along with the classification attributed to each source of RC emission.

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Owing to the way that interferometers function, large angular structures in the sky can be completely missed if their corresponding visibilities are not recorded by the interferometer. The largest angular scale (θ_LAS) that the VLA is sensitive to in C-configuration (shortest baseline of 35 m) at 6 cm is ∼4 arcminutes. This assumes an observation of 12 hr that is uniformly weighted and untapered. Observations of a shorter duration will have a slightly lower θ_LAS value, and for weighting schemes closer to natural weighting, the θ_LAS will be larger. In our observations, angular scales of ∼4 arcminutes and above may not be adequately sampled, leading to a lower than expected flux density; there are only seven galaxies with an angular size greater than 4' (see column 4 of Table 3). Under the assumption that RC emission coincides with optical emission, it is only these galaxies that are vulnerable to having large angular structures absent in the (u, v) data. Even so, the SF in dwarf galaxies is intermittent on scales of one to a few Gyr, whereas CRe age over much shorter timescales of tens of Myr; therefore, in the majority of our sample, no significant emission is expected from, for example, a CRe halo. We note that NGC 1569 was found to have an extended radio halo extending beyond the optical emission when observed between 0.6 and 1.4 GHz (Israel & de Bruyn 1988). This is attributed to the post-starburst nature of the galaxy, which is not reflected in the majority of targets in our sample. We do not see any evidence of such a halo in our 6 GHz image. This may be due to spatial filtering or spectral aging, which has shifted the halo emission below our detection threshold.

With background and ambiguous sources removed (see Section 3.1), emission from our RC and ancillary images was integrated within each of the dwarf galaxy's optical disks (hereafter the disk mask; see Table 3 for the disk parameters). We also extract the integrated properties including the ambiguous sources; these can be found in Appendix C in Table 7. The semimajor axis of the disk was based on optical isophotes: using either the Holmberg radius (defined as the isophote where the B-band surface brightness drops to a magnitude of 26.66; Hunter & Elmegreen 2006) or three times the V-band disk scale length (Hunter & Elmegreen 2006) if the B-band radius was not defined. All emission outside this radius was masked.

The majority of our dwarf galaxy sample only exhibits significant RC emission in isolated regions, which is attributed to both the episodic nature of SF in dwarf galaxies (e.g., Stinson et al. 2007) and the surface brightness sensitivity of our RC observations, which limits our RC maps to to detecting SFRDs greater than ∼5 × 10⁻³ M_⊙ yr⁻¹ kpc⁻². When integrated over the disk, the signal from most galaxies is dominated by the contribution of noise from the individual beams within the integration area. The uncertainty, δN, is given by ${\sigma }_{\mathrm{rms}}\sqrt{N}$, where σ_rms is the rms noise level and N is the number of individual beams. This motivates the use of masks to isolate genuine emission from background noise (i.e., reduce the integration area, which is proportional to N) in order to improve the RC S/N.

3.4.1. Radio Continuum-based Mask

To test the robustness of our source identification and extraction approach, we determine the radio continuum source counts from our images. We compare these to Huynh et al. (2012), who performed 5.5 GHz observations with the Australia Telescope Compact Array of a 900 arcmin² region with a restoring beam of 49 × 20 and an rms noise of 12 μJy beam⁻¹. After correcting for incompleteness and resolution bias, they present normalized source counts in 10 flux density bins ranging between 50 and 5000 μJy (see their Table 2).

Our images were generated using a restoring beam of approximately 3'' and attained an rms noise of ∼6 μJy beam⁻¹. Therefore, the sensitivity per beam in our data is comparable to that from Huynh et al. (2012). We scale the Huynh et al. (2012) bins to 6.2 GHz, the effective frequency for most of our images, assuming a spectral index of −0.7 ± 0.2. This assumption is supported by various studies that show that the average spectral index of star-forming galaxies is narrowly concentrated around ∼−0.7 with a small dispersion of ≲0.2 (Condon 1992; Niklas et al. 1997; Lisenfeld & Volk 2000). For each bin, we cycled through our images, counting all sources with flux densities in the range ΔS. We count sources only from within a 4' circular aperture centered on the image pointing reference to avoid regions where the primary beam response leads to higher noise levels. Sources are assigned to three different groups following our source classification approach described in Section 3.1. The first group includes all sources in the field including the galaxy emission, the second counts only sources we are confident are background sources, and the final group consists of both background and ambiguous sources. Sources were not counted if, in the given bin, the low end of the bin was less than five times the rms noise from the image (this only affected the two lowest bins because of a few high rms images). No attempt was made to count resolved sources originating from the same source (e.g., radio lobes, multiple SF regions from a dwarf, etc.).

To estimate the completeness of our source catalog, we follow a similar approach to Huynh et al. (2012) and perform a Monte Carlo simulation. We inject a synthetic Gaussian source with a randomly generated position and brightness from 30 to 3000 μJy into our image and then apply the fellwalker source detection algorithm following the same approach as described in Section 3.4 to see if the source is recovered. We do this 8000 times and find that sources with flux densities of 5σ_rms (∼50 μJy) have a detection rate of 50%, where σ_rms is the rms noise in the image. The detection rate rises steeply to 90% at 120 μJy. We also correct for the resolution bias following the same approach as Huynh et al. (2012). This correction accounts for sources with weak extended emission and large total integrated flux densities that have peaks that fall below the detection threshold. Given our slightly higher sensitivity and resolution ,we find lower resolution correction factors than Huynh et al. (2012), with values of 1.24 in our lowest bin and 1.03 in our highest bin.

We present the results of our source counts in Table 5. For each bin, we present the raw source counts (N) and the counts corrected for completeness and resolution bias (N_c). We determine the RC source count rate (dN_c/dS), which corresponds to the number of sources found per steradian normalized to the midpoint of the flux bin. Finally, we normalize our corrected source counts by dividing by the expected number of sources (N_exp) derived from a non-evolving Euclidean model using the relation $N(\gt {S}_{6\mathrm{cm}})\,=60\ast {S}_{6\,\mathrm{cm}}^{-1.5}$. The Poissonian errors are presented for the normalized and corrected counts with the resolution and completeness correction uncertainties (10% and 2%–5%, respectively) added in quadrature.

In Figure 4, we present a comparison of our source counts using all sources (black squares), only background sources (blue triangles), and both background and ambiguous sources (green pentagons). We compare our results to the corrected and normalized source counts of Huynh et al. (2012; red circles). This plot clearly shows that our counts are consistent with those of Huynh et al. (2012) until ∼10³ μJy. Beyond this flux, we see that including galaxy emission in our source counts leads to higher counts than those found in Huynh et al. (2012), particularly at flux densities above 8.6 mJy. Ideally, we would like to use the source counts to test the reliability of our source identification approach. In particular, we would like to test whether sources we define as ambiguous are background sources or associated with the galaxy emission. If we assume that our source identification approach has reliably identified the galaxy emission and background sources and that the bulk of our ambiguous sources are associated with one of these groups, then we should see a signature of this in our source counts. If the ambiguous sources belong to the background sources group, we would expect that including them in the source counts while excluding the galaxy emission would lead to source counts that are similar to those of Huynh et al. (2012). Conversely, if the ambiguous sources are background sources and we do not count them while also excluding the galaxy emission, we would expect to see lower source counts than expected. In Figure 4, we do see some tentative evidence that suggests the ambiguous sources are background sources with the background-only source counts (blue triangles) tending to be lower than the source counts including both the background and ambiguous sources (green pentagons). However, due to the small number of sources in each bin and the associated errors, we are prevented from stating that, statistically, the ambiguous sources belong to the population of background sources.

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4.1.1. Comparison with Literature Flux Densities

4.1.2. Composition of the Radio Continuum: Thermal and Nonthermal Contributions

The total RC emission comprises two contributions: RC_Th and RC_Nth. Since the Hα and RC_Th both have their origins in hot (∼10⁴ K) plasma associated with H ii regions, a tight spatial and temporal correlation between the two is expected (e.g., Deeg et al. 1997; Murphy et al. 2011). The Hα–RC_Th relation taken from Deeg et al. (1997) assumes the form

$$\begin{eqnarray}\displaystyle \frac{{\mathrm{RC}}_{\mathrm{Th}}}{{\rm{W}}\,{{\rm{m}}}^{-2}} & = & 1.14\times {10}^{-25}{\left(\displaystyle \frac{\nu }{\mathrm{GHz}}\right)}^{-0.1}\\ & & \times {\left(\displaystyle \frac{{T}_{{\rm{e}}}}{{10}^{4}{\rm{K}}}\right)}^{0.34}\,\displaystyle \frac{{F}_{{\rm{H}}\alpha }}{\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}},\end{eqnarray} \tag{ 1 }$$

where ν is the observed frequency in GHz, T_e is the electron temperature, which is assumed to be 10⁴ K, and F_Hα is the Hα luminosity. On a spatially resolved basis, the RC_Th flux density can be subtracted from the total RC, yielding the RC_Nth flux density distribution.

We do not correct our Hα estimates for internal extinction, following the same approach as Heesen et al. (2014). As our later analysis utilizes the SFR derived by combining the 24 μm and FUV emission, we wish to avoid using the 24 μm to correct for internal extinction so as not to introduce a spurious correlation. Dwarf galaxies are expected to have low internal extinction due to their low metallicity, and therefore, this is thought to generally not have a significant impact on our results. To verify this assertion, we estimate the internal extinction in our Hα maps following the method of Kennicutt et al. (2009):

$$\begin{eqnarray}&&{I}_{{\rm{H}}\alpha ,\mathrm{corr}}={I}_{{\rm{H}}\alpha ,\mathrm{obs}}+0.02{\nu }_{24\mu {\rm{m}}}{I}_{24\mu {\rm{m}}},\end{eqnarray} \tag{ 2 }$$

where I_Hα,obs is the observed Hα intensity, which has been corrected for foreground reddening, I_Hα,corr is the Hα intensity corrected for internal extinction, and I_24μm is the 24 μm intensity. Our most intensely star-forming galaxies are IC 10 and NGC 1569. We calculate the average internal extinction toward these galaxies and find values of 38% and 35%, respectively. We have explored the extinction toward NGC 1569 in Westcott et al. (2017) using a Bayesian approach to separate the RC emission. We were able to estimate an average internal extinction of ∼20%, slightly lower than our estimate above. Galaxies with lower SFR, <0.1 M_⊙ yr⁻¹, that make up the bulk of our sample have much lower internal extinctions of <10% as derived from the 24 μm intensity. For example, VIIZw 403 and DDO 50 both have an internal extinction of ∼8%. In light of these results, we caution that in our subsequent analysis the RC_Th flux estimates in galaxies with higher SFRs may be underestimated.

The uncertainty in our estimate of the RC_Th emission is dominated by the foreground Galactic extinction correction and T_e. The uncertainty in the Galactic extinction correction for our sample is ±0.015 mag for values of E(B − V) ≤ 0.015 and ∼10% for E(B − V) > 0.015 (Burstein & Heiles 1982). We assume a single value for the foreground extinction across each galaxy. The foreground extinction may vary considerably across each galaxy, particularly for those galaxies in close proximity to the Milky Way, such as IC 10, where the foreground extinction has been shown to vary across the face of the galaxy from −60% to +25% of our assumed value (Basu et al. 2017).

The value of T_e is assumed to be the standard value of 10⁴ K but the electron temperature in H ii regions has been shown to vary considerably. For example, a sample of 61 Galactic H ii regions where found to have values of T_e ranging from 0.25 × 10⁴ K to 1.16 × 10⁴ K (Hindson et al. 2016). In a study by Nicholls et al. (2014) the mean electron temperature of 17 H ii regions in 14 dwarf irregular galaxies was T_e = 1.4 × 10⁴ K. Variations in the electron temperature from our assumed value could give rise to up to ∼20% error in the estimated thermal emission.

After the removal of known background galaxies and ambiguous sources, we apply our RC and disk masks to isolate the RC_Th (scaled Hα) emission. When integrating over the RC mask, we find that the average thermal fraction for our sample is ∼50% ± 10% (upper limit). When integrating over the entire disk, we find a higher thermal fraction of 70% ± 10%. For comparison, we scale thermal fractions reported for dwarf galaxies in the literature to 6.2 GHz assuming a spectral index of −0.1 and −0.7 for thermal and nonthermal components, respectively. The scaled thermal fractions in dwarf galaxies have been quoted as 51% for a sample of stacked faint dwarfs (Roychowdhury & Chengalur 2012), 53% in IC 10 (Heesen et al. 2011), 41% in NGC 1569 (Lisenfeld et al. 2004), and 41% in NGC 4449 (Niklas et al. 1997). Our estimate of the thermal fraction integrated over the RC mask is consistent with these literature values. The thermal fraction integrated over the disk mask is significantly greater. We ignore internal extinction in our estimate of the RC_Th, which may lead to slightly lower values of the thermal fraction in the high-SFR galaxies such as NGC 1569 and IC 10. It is also possible that on the scale of the disk, we are missing some flux associated with large-scale RC emission, which would lead to higher thermal fractions in the most extended galaxies. A more robust measure of the RC_Th emission may be obtained using a Bayesian approach (Tabatabaei et al. 2017; Westcott et al. 2017); however, this requires a large number of observations across the radio SED.

To estimate the RC_Nth emission, we subtract the RC_Th emission from the total RC. We caution that the RC_Nth emission may in some cases turn out to be rather an upper limit because of the previous points.

We estimate the SFR following the approach of Leroy et al. (2012). This corrects the FUV-inferred SFR for internal extinction, which is only relevant for our more actively star-forming dwarfs. The FUV has been proven to be a reliable SF indicator at low SFR in comparison to Hα-inferred SFRs (e.g., Lee et al. 2009; Ficut-Vicas 2016), and the timescale of RC_Nth emission is closer to the FUV-inferred SF timescales than to, e.g., the Hα-inferred SF timescales. Galactic foreground extinction is taken into account separately (see Hunter et al. 2012 for details). To correct for internal extinction, Bigiel et al. (2008) and Leroy et al. (2012) use Spitzer 24 μm dust emission to empirically correct GALEX FUV fluxes for the fraction of dust-obscured SF on the assumption that a proportion of energy absorbed by internal dust is re-radiated at 24 μm (this is based on the original idea by Calzetti et al. 2007, who use Hα instead of FUV). We use

$$\begin{eqnarray}\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{SFR}}}{{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}} & = & 0.081\displaystyle \frac{{I}_{\mathrm{FUV}}}{\mathrm{MJy}\,{\mathrm{sr}}^{-1}}\\ & & +0.0032\displaystyle \frac{{I}_{24\mu {\rm{m}}}}{\mathrm{MJy}\,{\mathrm{sr}}^{-1}},\end{eqnarray} \tag{ 3 }$$

where the FUV and 24 μm intensity are in units of MJy sr⁻¹, and Σ_SFR represents the star formation rate density (SFRD). We show a map of the SFRD for DDO 50 in Figure 1. For those galaxies where Spitzer 24 μm data were not available (see Tables 3 and 4, column 7), we used the FUV-inferred SFR without any correction. Due to the low dust content of the majority of our sample, the FUV dominates the SFR estimates. The error associated with our SFR estimates is ∼20%. When compared to other methods of deriving the SFR, this approach was found the have a scatter of ∼50% down to a Σ_SFR of 10⁻⁴ M_⊙ yr⁻¹ (Leroy et al. 2012).

In Figure 5, we present the RC–SFR relation of our sample when considering the optical disk mask (top) and RC-based mask (bottom). We are able to determine the RC and SFR for 11 and 19 galaxies in the disk- and RC-based masks, respectively. The left panels of Figure 5 show the relation when we include the ambiguous RC sources, whereas the right panels show the relation with ambiguous sources removed. If we do not remove the ambiguous sources, we find a significant flattening and increase in the scatter of the fit to the data particularly in the case of the RC-based mask results. The most likely cause for this is that these ambiguous sources are background radio galaxies. We therefore continue our analysis focusing only on the results where ambiguous sources are removed. In doing so, we may remove at most 10% of genuine RC emission in the form of SNRs as according to Chomiuk & Wilcots (2009), RC emission from SNRs contribute <10% of the total RC in dwarf galaxies

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The data points of our sample of galaxies in Figures 5–11 are color-coded based on the galaxy's metallicity. This was done to investigate if there are any trends with metallicity. We find that in general the lowest metallicity objects congregate toward the low-radio continuum, low-SFR end of the plot whereas the high end is populated by the higher metallicity galaxies. This is a direct consequence of the metallicity–luminosity relation (Skillman et al. 1989) and the fact that more luminous, hence more massive, galaxies tend to have a higher SFR.

We compare our data points with the RC–SFR relation presented by Murphy et al. (2011). They derive an expression for the RC_Th and RC_Nth emission and combine these to determine the total RC in a galaxy. The thermal component is derived from the ionizing photon rate, which is directly proportional to the thermal spectral luminosity assuming an optically thin plasma, giving

$$\begin{eqnarray}\left(\displaystyle \frac{{\mathrm{SFR}}_{\nu }^{\mathrm{Th}}}{{M}_{\odot }{\mathrm{yr}}^{-1}}\right) & = & 4.6\times {10}^{-21}{\left(\displaystyle \frac{{T}_{{\rm{e}}}}{{10}^{4}{\rm{K}}}\right)}^{-0.45}\\ & & .{\left(\displaystyle \frac{\nu }{\mathrm{GHz}}\right)}^{0.1}\left(\displaystyle \frac{{L}_{\nu }^{\mathrm{Th}}}{{\rm{W}}\,{\mathrm{Hz}}^{-1}}\right),\end{eqnarray} \tag{ 4 }$$

where T_e is the electron temperature and ${L}_{\nu }^{\mathrm{Th}}$ is the thermal radio luminosity. This equation assumes solar metallicity, continuous SF, and a Kroupa IMF. Using a Kroupa IMF results in SFR estimates that are ∼2.5 times larger than those found by Condon (1992). We assume an electron temperature of 10⁴ K. As mentioned previously, this value may vary considerably. A value of T_e = 1.4 × 10⁴ K (Nicholls et al. 2014) would lead to a 14% decrease in the SFR. The expected RC_Nth is derived using

$$\begin{eqnarray}\left(\displaystyle \frac{{\mathrm{SFR}}_{\nu }^{\mathrm{Nth}}}{{M}_{\odot }{\mathrm{yr}}^{-1}}\right) & = & 6.64\times {10}^{-22}{\left(\displaystyle \frac{\nu }{\mathrm{GHz}}\right)}^{{\alpha }_{\mathrm{Nth}}}\\ & & .\left(\displaystyle \frac{{L}_{\nu }^{\mathrm{Nth}}}{{\rm{W}}\,{\mathrm{Hz}}^{-1}}\right).\end{eqnarray} \tag{ 5 }$$

This relationship is derived using the starburst99 population synthesis code (Leitherer et al. 1999) and the empirical relationship between the supernova rate and nonthermal spectral luminosity of the Milky Way. We assume a value for the nonthermal spectral index of α_Nth = −0.7 ± 0.2. Finally, the total RC is the combination of RC_Th and RC_Nth, leading to

$$\begin{eqnarray}\left(\displaystyle \frac{{\mathrm{SFR}}_{\nu }}{{M}_{\odot }{\mathrm{yr}}^{-1}}\right) & = & {10}^{-20}\left[2.18{\left(\displaystyle \frac{{T}_{{\rm{e}}}}{{10}^{4}{\rm{K}}}\right)}^{-0.45}\right.\\ & & \times \left.{\left(\displaystyle \frac{\nu }{\mathrm{GHz}}\right)}^{0.1}+15.1{\left(\displaystyle \frac{\nu }{\mathrm{GHz}}\right)}^{{\alpha }_{\mathrm{Nth}}}\right]\left(\displaystyle \frac{{L}_{\nu }}{{\rm{W}}\,{\mathrm{Hz}}^{-1}}\right),\end{eqnarray} \tag{ 6 }$$

where we use a frequency ν of 6.2 GHz. These expected relations are plotted as the gray shaded area in Figure 5. The width of the band reflects the overall uncertainty based on a typical error in the spectral index of 0.2 and a canonical factor of 2 uncertainty in the SFR.

We performed a bivariate linear regression to quantify the relation between the RC luminosity and SFR, assuming the data follow a power-law function of the form $y={{Ax}}^{n}$ or $\mathrm{log}(y)=n\mathrm{log}(x)+c$, where $c=\mathrm{log}(A)$. We used the odr¹² module from scipy, which accepts four arrays of data points ($\mathrm{log}x$ and $\mathrm{log}y$, and the 1σ errors in log-space: $\tfrac{\delta x}{x}$ and $\tfrac{\delta y}{y}$) and the model function, and works to minimize the squares of the orthogonal distance between the data points and the model, ultimately returning best-fit values and their standard deviations.

We find that the disk mask RC–SFR relation (Figure 5 top-right panel) results are consistent with a linear relationship with $n=0.93\pm 0.14$, but the RC luminosity is lower than expected based on the observed SFR by approximately a factor of ∼5. We note that IC 1613 falls below the relation we find. If we exclude this galaxy, we find that the average offset is a factor of ∼3. We find that the RC mask-integrated RC–SFR relation in Figure 5 (bottom-right panel), where the RC mask is applied to both the RC and the SFR map, is marginally shallower than the Murphy et al. (2011) relation with a gradient of n = 0.86 ± 0.04 with a scatter of 0.17 dex. If we perform the fit excluding NGC 1569 (Figure 5, dashed black line), we find a value of n = 0.91 ± 0.04. In Figure 6, we compare the results of our disk-integrated and mask-integrated RC–SFR relation to the study of 18 spiral galaxies at 20 cm by Heesen et al. (2014). We extend their parameter space by 2 dex, down to SFRs of 10⁻⁴ M_⊙ yr⁻¹. At 20 cm, the Heesen et al. (2014) study found a slope of the RC–SFR relation of n = 1.24 ± 0.04, which is significantly steeper than our results.

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The relationship between the RC_Th and RC_Nth emission and SFR integrated over the disk and RC mask is shown in Figure 7. When integrating the two components over the optical disk mask, we find slopes of n = 1.20 ± 0.09 and n = 0.82 ± 0.06 for RC_Th and RC_Nth, respectively. We find a slope of n = 0.82 ± 0.05 and n = 0.79 ± 0.06 for RC_Th and RC_Nth emission integrated over the RC mask, respectively. If we exclude NGC 1569 from the fit, we find marginally steeper slopes. Our results for RC_Th agree with those of Murphy et al. (2011) when integrating over the RC mask but appear to diverge at the low-SFR levels (<10⁻² M_⊙ yr⁻¹) in the disk mask. This may be due to the stochastic nature of SF particularly in the faintest galaxies. It is important to note that the RC_Th plot is essentially an Hα–FUV plot. The RC_Th values are based on the Hα emission and are thus not independently determined; in turn, the SFR relies heavily on the FUV. The scatter, especially for the least active dwarf galaxies (SFR ≲ 10⁻² M_⊙ yr⁻¹), is likely due to the Hα emission underestimating the SFR in comparison to that from the FUV by a factor of up to 10 (Lee et al. 2009). These authors argue that as only the highest mass stars (M ≳ 18 M_⊙) produce a significant number of photons to ionize the surrounding H i, having a deficit of these stars significantly reduces the amount of Hα emission, while the FUV emission is not affected as much since a larger fraction of the stellar population contributes to the FUV. On the other hand, Koda et al. (2012) find O stars in stellar clusters as small as 100–1000 M_⊙, coming to the conclusion that the stellar IMF is not necessarily truncated; it could be stochastically populated at the high-mass end, accounting for the observed variations in Figure 7. We discuss this further in Section 4.6.

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The RC_Nth results are shallower than expected based on the predictions of Murphy et al. (2011) in both masking cases. Not only is the slope shallower, we also see that when using the disk mask the RC_Nth emission falls below the expected SFR by a factor of 2–4. This agrees with Bell (2003), who found that the radio emission of low-luminosity galaxies must be suppressed by at least a factor of two to account for the RC–FIR relation at low luminosity. Our results also agree with the findings of Price & Duric (1992), who found that the power-law dependence of the synchrotron luminosities and SFR has a slope of n = 0.8. Using the same method applied here but observing at 20 cm, Heesen et al. (2014) found a slope of the RC_Nth component for spiral galaxies to be n = 1.16 ± 0.08, significantly steeper than our results (Figure 6). We note that the RC_Nth may be underestimated particularly for large-scale galaxies that have high SFRs such as NGC 1569. This would lead to the RC_Nth–SFR relation being steeper than what we see in Figure 7. However, when we remove NGC 1569 from our fitting, our results remain consistent with those with NGC 1569 included.

The RC surface-brightness–SFRD relation is presented in Figure 8, where the SFRD is derived over the extent of the galaxy. We find a tight, linear RC–SFRD relation with a slope of n = 0.99 ± 0.11 and n = 0.88 ± 0.02 for the disk- and RC-based masks, respectively. Within the errors, these slopes are the same as those found for the relations plotted in Figure 5. Unlike the luminosity plots in Figure 5 though, this is independent of the distance and so errors introduced by distance uncertainties forcing a linear relation due to flux-to-luminosity scaling are avoided. Figure 8 could thus be used as a baseline for future studies of normal star-forming galaxies—especially those studies that do not have reliable distance measurements (e.g., only photometric redshifts of optical counterparts).

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The FIR is often used as a proxy for SFR in studies of unresolved galaxies. It is therefore instructive to examine the relationship between the FIR and SFR integrated within the disk and RC masks in Figure 9 of our sample. We compare our estimate to the monochromatic 70 μm calibration of Calzetti et al. (2010) using

$$\begin{eqnarray}&&\left(\displaystyle \frac{{\mathrm{SFR}}_{70\mu {\rm{m}}}}{{M}_{\odot }{\mathrm{yr}}^{-1}}\right)=\left(\displaystyle \frac{{L}_{70\mu {\rm{m}}}}{{\rm{W}}\,{\mathrm{Hz}}^{-1}}\right)\cdot 2.52\times {10}^{-24},\end{eqnarray} \tag{ 7 }$$

where L_70μm is the 70 μm luminosity in W Hz⁻¹. We find that our best-fit line is the same within the errors for both the disk- and RC-based masks (n = 1.07 ± 0.09 and 1.05 ± 0.06, respectively) and runs parallel to the Calzetti et al. (2010) relationship. However, for any given SFR, we find that our measurement of the integrated 70 μm emissions is underestimated compared to the expected 70 μm luminosity by a factor of ∼10. Given the fact that dwarf galaxies have low metallicity, this is not surprising. The metallicity of all our galaxies falls below a value of 12 + log(O/H) ∼ 8.1, below which Calzetti et al. (2010) found the FIR to be an unreliable tracer of the SFR. At these low metallicities, the galaxies become basically optically thin, and FUV photons can escape before being reprocessed by dust and re-emitted in the FIR. This was also suggested as the cause of the ratio between the total IR and FUV being <1 in low-luminosity galaxies in a study by Bell (2003).

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The RC–FIR relation based on 1809 galaxies culled from the NRAO VLA Sky Survey (NVSS; Condon et al. 1998) catalog and the 1.2 Jy IRAS Redshift Survey catalog (Strauss et al. 1992) was investigated by Yun et al. (2001). They related the integrated 1.4 GHz RC of an unresolved galaxy to the IRAS 60 μm luminosity and found

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{6\mathrm{GHz}}}{{\rm{W}}\,{\mathrm{Hz}}^{-1}}=(2.24\pm 0.67)\times {10}^{-3}{\left(\displaystyle \frac{{L}_{70\mu {\rm{m}}}}{{\rm{W}}{\mathrm{Hz}}^{-1}}\right)}^{0.99},\end{eqnarray} \tag{ 8 }$$

where we converted the IRAS 60 μm luminosity to a "luminosity density" (i.e., from W to W Hz⁻¹) by noting that the response from the IRAS 60 μm filter is equivalent to a perfectly transmitting filter with a bandwidth of 2.6 × 10¹² Hz. The IRAS 60 μm "luminosity density" was further converted to the equivalent Spitzer 70 μm luminosity by scaling it up by a factor of 1.27. This value is based on a graybody model for dust emission with β = 1.82 and T_dust = 35 K; this assumes that the Yun et al. (2001) galaxies are in a quiescent mode of star formation, and that there is no significant emission from warm dust. The Yun et al. (2001) VLA 1.4 GHz RC data were reduced by a factor of 2.83 to derive predicted equivalent VLA 6 GHz flux densities assuming a constant spectral index of −0.7 ± 0.2 between 20 and 6 cm for the galaxies in their sample.

In Figure 10, we show the RC–FIR relation for our dwarf galaxies and compare this to the results of Yun et al. (2001). The RC–FIR relation traditionally samples the parameter space above FIR luminosities of ∼10²² W Hz⁻¹; we extend this to lower luminosities by 3 dex. The uncertainty presented in Figure 10 takes into account an uncertainty in the spectral index of 0.2 and 15 K in dust temperature. We show the RC–FIR relation for our dwarf sample where emission is integrated over the entire disk (Figure 10, left) and from regions of significant RC emission only (i.e., the RC-based mask; Figure 10, right). In Figure 11, we show the RC–FIR relation for just our dwarf galaxy sample integrated over the disk mask (left) and RC mask (right). The top panels of this figure show the luminosity and the bottom panels the flux density to illustrate any dependence on distance. We find that when integrated over the disk, our results for the luminosity match those found by Yun et al. (2001) with a slope of 1.02 ± 0.10. The flux density derived slope is slightly shallower at 0.87 ± 0.07. However, when we integrate the RC and 70 μm emission using our RC mask, we find that our results diverge from the Yun et al. (2001) relation in both the luminosity and flux density plots with a flatter slope of 0.81 ± 0.03 and 0.80 ± 0.03 for the luminosity and flux density, respectively. We discuss the possible reasons behind this in Section 4.6.

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An alternative way of exploring the RC–FIR relation described by Yun et al. (2001) is through the q parameter. This is the natural logarithm of the ratio of the IRAS FIR (a weighted combination of 60 and 100 μm flux) to the VLA 1.4 GHz flux densities of the Yun et al. (2001) sample and is described by

$$\begin{eqnarray}&&{q}_{\mathrm{FIR}:1.4}=\mathrm{log}\left(\displaystyle \frac{\mathrm{FIR}\,[\mathrm{Jy}]}{\mathrm{RC}\,[\mathrm{Jy}]}\right).\end{eqnarray} \tag{ 9 }$$

The average q_FIR:1.4 parameter was found to be 2.34 ± 0.01. Since our RC and FIR measurements were made in different bands from those used by Yun et al. (2001), we convert their q_FIR:1.4 to q_70:6 = 2.68 ± 0.12, where the subscripts 70 and 6 refer to the 70 μm FIR and 6 GHz RC, respectively. As before, the uncertainty is calculated by assuming a 0.2 uncertainty in the spectral index between 20 and 6 cm, and a 15 K uncertainty in dust temperature.

We plot q_70:6 values as a function of 70 μm FIR luminosity in Figure 12. The LITTLE THINGS dwarfs are consistent with the Yun et al. (2001) sample when integrated over the disk (Figure 12; left). This reveals that the RC–FIR "conspiracy" continues in our dwarf galaxy sample. However, we see in the right panel of Figure 12 that when we integrate the emission over the RC mask there is a considerable excess of RC emission compared to 70 μm for sources that have a low radio luminosity, as was already evident in Figures 10 and 11 (right-hand panels).

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The relationship between the RC, FIR, and SFR can be summarized by three equations. The RC–SFR and FUV–SFR relations can be expressed as

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\mathrm{RC}}}{{\rm{W}}\,{\mathrm{Hz}}^{-1}}={10}^{A}{\left(\displaystyle \frac{\mathrm{SFR}}{{M}_{\odot }{\mathrm{yr}}^{-1}}\right)}^{n}\end{eqnarray} \tag{ 10 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\mathrm{FIR}}}{{\rm{W}}\,{\mathrm{Hz}}^{-1}}={10}^{A}{\left(\displaystyle \frac{\mathrm{SFR}}{{M}_{\odot }{\mathrm{yr}}^{-1}}\right)}^{n},\end{eqnarray} \tag{ 11 }$$

while the RC–FIR relation can be described by

$$\begin{eqnarray}&&\displaystyle \frac{\mathrm{RC}}{{\rm{W}}\,{\mathrm{Hz}}^{-1}}={10}^{A}{\left(\displaystyle \frac{70\mu {\rm{m}}\mathrm{FIR}}{{10}^{20}{\rm{W}}{\mathrm{Hz}}^{-1}}\right)}^{n}.\end{eqnarray} \tag{ 12 }$$

We summarize the results for the various fits for these relations over the disk and RC masks in Table 6.

We use the Pearson (P) and Spearman (S) coefficients to describe the strength and direction of the correlation in our relationships, where a value of −1.0 indicates a strong anti-correlation, 0.0 indicates the relationship is random or non-existent, and +1.0 indicates a strong, positive correlation. The Pearson coefficient assumes a purely linear relationship and so will approach 0.0 if there is an inconsistent relationship while the Spearman coefficient evaluates the monotonic relationship between two variables and so will remain high. Thus, a strong relationship that deviates from linearity will have a lower Pearson than Spearman score. Both the Pearson and Spearman coefficients indicate a strong correlation between all components of the RC and SFR, with values ranging from 0.77 to 1.00 depending on the relation and type of mask used.

The fitted parameters for A and n for each relation in Table 6 vary significantly based on the type of mask used except in the case of the FIR–SFR relation. The FIR–SFR relation has lower Pearson and Spearman scores compared to the RC-based relations, suggesting a weaker relationship and deviation from linearity and showing a larger scatter. Since the RC-based mask is able to probe significantly lower SFR galaxies, this suggests that the physical processes responsible for the FIR–SFR relation operate in the same way regardless of the level of SFR in our sample. The varying parameters that fit the various RC–SFR relations in the disk and RC masks indicate that there may be some change in the physical processes operating within regions on the scale of the entire disk and resolved scales traced by the RC-based masks.

One of the most striking results of this study is the divergence we see in the RC–FIR relation at low luminosities when integrating over the RC mask (Figure 11, right). This appears to be caused by the FIR emission being underestimated in the RC mask relative to the expected Calzetti et al. (2010) SFR (Figure 9), whereas the RC emission continues to follow the SFR down to low values (Figure 5). Our disk-masked results on the other hand underestimate both the RC and FIR luminosity compared to the predicted SFR, leading to the linear slope that continues the RC–FIR "conspiracy" seen in the left panel of Figure 11. In the case of RC emission excess, we find evidence that it is the RC_Nth component that is responsible for the suppression of the RC on the scale of the disk (Figure 7).

We propose two possible scenarios that may be responsible for the relations we observe. The first is that dwarf galaxies do not act as calorimeters, and CRe and dust-heating photons are able to escape the galaxy before they are able to generate the total RC_Nth and FIR emission associated with their host SF. The second possibility is that we are witnessing the effect of stochastic SF resulting in a partially sampled and/or truncated IMF. This may lead to the RC_Nth and FIR underestimating the SFR compared to studies of larger galaxies. It is possible that some combination of these scenarios is responsible for our results.

4.6.1. Calorimeter Breakdown

4.6.2. Stochastic Star Formation

Our results suggest that it is the suppression of the RC_Nth emission that is responsible for the RC–FIR relation remaining consistent at low SFRs. To explore the source of the RC_Nth emission, we investigate the synchrotron emissivity in an optically thin region,

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{NTh}}\propto {n}_{\mathrm{CR}}{B}_{\perp }^{\tfrac{\gamma +1}{2}},\end{eqnarray} \tag{ 13 }$$

where n_CR is the number density of CRe present in the dwarf's galactic magnetic field, B_⊥ is the strength of the transverse magnetic field, and γ is the power-law slope of the CRe injection spectrum.

The RC_Nth emission depends both on the energy density contained within the magnetic field and that of the population of CRe. The combined energy density associated with the magnetic field and CRe is usually assumed to be at a minimum (see Section 16.5 of Longair 1981). In galaxies, this is a reasonable assumption. If the energy densities are not equal, they will tend to balance: for example, if the energy density is dominated by the CRe, then they will rise out of the galactic disk in Parker lobes due to their buoyancy, expand, and escape, thus reducing their energy density until it is balanced with that in the magnetic field.

It is, however, conceivable that dwarf galaxies in particular deviate from equipartition. This would lead to a reduction in synchrotron emission (see Figure 7) in two different ways:

• (1)  
  a low number density of CRe (n_CR) present in the dwarf's galactic magnetic field. Dwarf galaxies in particular are prone to galactic outflows since they have low masses and correspondingly shallow gravitational potentials, and winds can advect plasma and resident CRe away from the galaxy;
• (2)  
  the magnetic field strength (B) being lower than the equipartition value at which the energy density of the magnetic field is equal to that of the cosmic rays (electrons and protons combined). In the standard paradigm of a mean field α–Ω dynamo, the key ingredients are turbulence and shear. Dwarf galaxies may be sites of weak, large-scale, ordered magnetic fields, so magnetic field amplification may be less efficient. Studies by Chyży et al. (2011) and Roychowdhury & Chengalur (2012) found global magnetic field strengths on the order of <5 μG toward dwarf galaxies. However, the turbulent field in and around the SF regions may not necessarily be weaker than that found in spirals (e.g., Tabatabaei et al. 2013) as 30 μG is observed across some 100 pc regions.

In the following, the magnetic field strength in our sample of dwarf galaxies is estimated under the assumption of equipartition; this is the only practical way of estimating the field strengths given our current data set. We apply the equipartition formula for the total magnetic field following Equation (3) from Beck & Krause (2005). We made the standard assumptions of a spectral index of −0.7 ± 0.2, proton-to-electron number density ratio ${\boldsymbol{K}}$ of 100 ± 50 (Beck & Krause 2005; Murphy et al. 2011), and that the dwarf galaxy has a scale height of 400 ± 200 pc independent of distance from the galaxy center (Banerjee et al. 2011; Elmegreen & Hunter 2015). We note that these assumptions are prone to significant uncertainty. The value of ${\boldsymbol{K}}$ depends on the acceleration process, propagation, and energy losses of the protons and electrons. As CRe propagate away from their sites of acceleration, they rapidly lose energy, leading to values of ${\boldsymbol{K}}\gt 100$. If this is the case in our dwarf galaxies and if CRe are escaping the galaxy, altogether this will lead to an underestimate of the equipartition magnetic field. The scale height of dwarf galaxies is also prone to large variations due to their low mass and the potential for outflows. Typical scale heights have values ranging from 200–400 pc in the inner regions and 600–1000 pc in the outer regions.

The average transverse magnetic field strength of our sample is 5.2 ± 2.6 μG and 7.5 ± 3.3 μG when integrated over the RC and disk masks, respectively (see Tables 3 and 4 for galaxy specific values). These values are greater than the ∼2 μG found in the ∼50 faint dwarf galaxies from the NVSS catalog (Roychowdhury & Chengalur 2012), and within the errors they are consistent with the 4.2 μG found toward the 12 local group dwarf galaxies reported in Chyży et al. (2011). The transverse field strength we measure in both masks is lower than the 9.7 μG found in the larger spiral galaxies in the WSRT SINGS sample (Heesen et al. 2014). The disk-integrated magnetic field strength is greater than the mask-integrated value because only the brightest galaxies can be integrated over the entire optical disk.

Our data allow the magnetic field strength to be measured on a resolved basis. In a few dwarf galaxies (e.g., NGC 1569, NGC 2366, NGC 4214), we find numerous regions where the magnetic field strength was measured to be as high as 30–50 μG in localized 100 pc regions (approximate area of the synthesized beam). In fact, the brightest RC_Nth flux density in our sample comes from a ∼100 pc region in NGC 1569—the flux density from this unresolved region implied a magnetic field strength of ∼50 μG. Heesen et al. (2015) analyzed in detail the RC_Nth spectrum of the nonthermal super bubble in IC 10, deriving a magnetic field strength of 44 μG. These are all strong magnetic fields akin to those found in the SF regions of larger spirals (e.g., the turbulent magnetic fields in NGC 6946's SF regions; Tabatabaei et al. 2013). With such high magnetic field strengths, CRe could lose all their energy before diffusing into the ISM (rendering the region a local calorimeter). In IC 10, for example, Heesen et al. (2015) find that at 6.3 GHz, the CRe lifetime in the nonthermal super bubble is only 0.9 Myr, comparable with the age of 1 Myr derived from the observed curvature of the spectrum. For less intense SF regions, such as DDO 168, DDO 47, and DDO 53, we find peak local magnetic field strengths of 10–15 μG where CRe may have sufficient time to escape into the ISM. Once there:

• (1)  
  the CRe, now losing energy through synchrotron radiation at a much slower rate, diffuse or are advected into the intergalactic medium (IGM) before they have time to radiate all their energy—this is the "non-calorimetric" situation that leads to the RC–FIR "conspiracy" (e.g., Bell 2003; Dale et al. 2009; Lacki et al. 2010), or
• (2)  
  the CRe continue to diffuse to 1 kpc but, because they continue to radiate and lose energy, the frequency of synchrotron emission shifts gradually to lower frequencies to the extent that emission falls outside of the 6 cm window.

Exploring these possibilities and their impact on the RC_Nth–SFR relation requires further information regarding the spectral index of the RC emission so that we can explore the CRe transport and aging timescales.