Magnetic Fields, Star Formation Rates, and Gas Densities at Sub-kiloparsec Scales in a Pilot Sample of Nearby Galaxies

The average magnetic field strengths can be estimated from the observed synchrotron flux densities, assuming energy equipartition between cosmic-ray particles and magnetic fields (classical equipartition formula, e.g., Pacholczyk 1970; Miley 1980; Longair 2011). The equipartition condition is achieved when the total energy in magnetic fields and cosmic-ray particles is minimum.

The classical equipartition formalism has shortcomings that lead to an overestimation of the magnetic field strength (B) at regions of steep spectral indices and an underestimation of B at flat spectral index regions. To overcome these shortcomings of the classical equipartition formula, Beck & Krause (2005) proposed a revised formula to estimate the average magnetic field strength. The formula is expressed as

$$\begin{eqnarray}&&{B}_{\mathrm{eq}}={\left[4\pi ({K}_{0}+1){E}_{{\rm{p}}}^{1-2{\alpha }_{\mathrm{nt}}}\displaystyle \frac{f({\alpha }_{\mathrm{nt}})}{{C}_{4}(i)}\displaystyle \frac{{I}_{\nu }{\nu }^{{\alpha }_{\mathrm{nt}}}}{l}\right]}^{\tfrac{1}{{\alpha }_{\mathrm{nt}}+3}}.\end{eqnarray} \tag{ 1 }$$

K₀, E_p, I_ν , and α_nt are the number density ratio of cosmic-ray protons to electrons, the proton rest mass energy, the intensity of the synchrotron emission at frequency ν, and the spectral index of synchrotron emission, respectively. f(α_nt) is a function of α_nt given as $f({\alpha }_{\mathrm{nt}})=(2{\alpha }_{\mathrm{nt}}+1)[2({\alpha }_{\mathrm{nt}}-1){c}_{2}({\alpha }_{\mathrm{nt}}){c}_{1}^{{\alpha }_{\mathrm{nt}}}]$ (Beck & Krause 2005). C₄(i) is a constant that depends on the inclination angle (i) of the galaxy and is expressed as ${C}_{4}(i)={[\cos (i)]}^{(\gamma +1)/2}$, where γ = (2α_nt + 1). l is the path length of the synchrotron emission. The path length was assumed to be 1 kpc for a galaxy with an inclination angle of 0° (face-on). For galaxies with low- and moderate-inclination angles (<75°), the assumed path length was corrected for the inclinations of the galaxies as l/cos(i). For the two nearly edge-on galaxies in Sample 1, NGC 2683 and NGC 4096, we assumed an oblate spheroidal shape of the synchrotron emission, such that the diameter on the plane of the galaxy is equal to its major axis. The path lengths (l) were then appropriately calculated, with the path length being maximum (equal to the galaxy's major axis) at the optical center of the galaxy and gradually declining to the edge of the galaxy. We note that B_eq has only a weak dependence on l as ${B}_{\mathrm{eq}}{(r)=l(r)}^{\tfrac{-1}{{\alpha }_{\mathrm{nt}}+3}}$ and hence is less sensitive to the exact choice of l. Values of K₀ and E_p were assumed to be 100 and 938.28 MeV, respectively, the same as those used by Beck & Krause (2005). Finally, we used nonthermal radio maps at 0.33 GHz (I_ν ) and spectral index maps (α_nt) made using 0.33 and 1.4 or ∼6 GHz radio observations (Roy & Manna 2021) to produce magnetic field maps of the sample galaxies using Equation (1).

The revised equipartition formula diverges for spectral index values ≤0.5 because such flat spectra indicate energy loss of electrons through ionizations or Coulomb interactions (Sarazin 1999). The central bulge and arm regions have a mostly flatter spectrum due to the association of star-forming regions and the estimates of equipartition magnetic fields in such regions might be affected by systematic uncertainties. This issue affects the derived magnetic field strengths for 8%, 12%, 3%, 70%, 17%, 7%, and 6% of the projected total surface area of NGC 2683, NGC 3627, NGC 4096, NGC 4449, NGC 4490, NGC 4826, and NGC 5194, respectively. We note that a large fraction of the derived magnetic field values are affected for NGC 4449 due to its nonthermal spectral indices being predominantly flat. This could bias the B_eq for NGC 4449.

2.1.1. Uncertainties on Magnetic Field Maps

Rest-frame Hα and ultraviolet (UV) observations are the best tracers of recent SFRs as the radiation from these predominantly originate in newly formed massive stars. However, the observations are affected by extinction caused by interstellar dust in both the host galaxy as well as the Milky Way. SFRs estimated from Hα and UV observations are therefore corrected for the extinction. Dust-corrected SFRs can be estimated by combining far-ultraviolet (FUV) and Hα data with infrared (IR) data to exploit the complementary strengths at different wavelengths (e.g., Buat 1992; Meurer et al. 1995, 1999; Cortese et al. 2008; Kennicutt Jr. & Evans 2012; Leroy et al. 2012). In addition to the FUV+IR and Hα+IR tracers, the low-frequency radio emission from galaxies, which is predominantly optically thin synchrotron emission, can be used to estimate their dust-unobscured SFRs via the radio–FIR correlation (e.g., Yun et al. 2001).

We estimated the spatially resolved SFRs of our Sample 1 galaxies using FUV+24 μm, Hα+24 μm, and 1.4 GHz data, which are discussed, respectively, in Sections, 2.2.1–2.2.3. We used data of these different frequencies as tracers in order to (1) get a fair comparison between different SFR diagnostics and (2) for studying star formation history at different timescales. All SFRs in this paper assume a Kroupa initial mass function (IMF; Kroupa 2001).

2.2.1. SFRs Using FUV and 24 μm Observations

To estimate SFRSD maps of the seven galaxies in Sample 1 (Table 2) using FUV+24 μm emission, we used SPITZER 24 μm IR data (Dale et al. 2009) and GALEX FUV data (11HUGS survey; Kennicutt Jr. et al. 2008). We first convolved both the 24 μm and the FUV maps of all galaxies to the same resolutions as our magnetic field maps. The FUV data were corrected for extinction due to dust in the Milky Way (see Section 2.2.4). The FUV images were in units of counts s⁻¹ pix⁻¹ and were converted to flux-density units of MJy Sr⁻¹. We also converted the 24 μm images to units of MJy Sr⁻¹ and used the following calibration from Leroy et al. (2012) to derive SFRSD maps for the sample galaxies:

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Sigma }}}_{\mathrm{SFR}}[{M}_{\odot }\,{\mathrm{yr}}^{-1}{\mathrm{kpc}}^{-2}]=0.081\,{I}_{\mathrm{FUV}}[\mathrm{MJy}\,{\mathrm{sr}}^{-1}]\\ \,+\,0.032\,{I}_{24\mu {\rm{m}}}[\mathrm{MJy}\,{\mathrm{sr}}^{-1}]\end{array}\end{eqnarray} \tag{ 2 }$$

Table 2. Samples Studied in This Paper.

Sample Name	Galaxies
Sample 0	Full sample containing 46 galaxies from the Spitzer LVL survey
Sample 1	Pilot sample containing seven galaxies from Sample 0; galaxies listed in Table 1
Sample 2	Sample 1 + five galaxies (NGC 1097, NGC 4736, NGC 5055, NGC 5236, and NGC 6946)
	from Basu et al. (2012b) = 12 galaxies; used to probe the Beq–SFRSD correlations
Sample 3	A subset of seven galaxies (NGC 3627, NGC 4826, NGC 5194, NGC 4736, NGC 5055, NGC 5236, and
	NGC 6946) from Sample 2, which have archival CO data; used to study the Beq–gas density correlations

Download table as:  ASCIITypeset image

The uncertainties of the coefficients are ∼10%–30%. Note that the uncertainty in SFR estimates arises from issues such as the error in sampling the stellar IMF of different star-forming regions, determining the contribution of different emissions which are not associated with recent star formation, etc. (e.g Kennicutt Jr. & Evans 2012; Leroy et al. 2012).

2.2.2. SFRs Using Hα and 24 μm Observations

To estimate SFRSD maps using Hα+24 μm as a tracer, we used 24 μm emission along with Hα emission from 11HUGS (Kennicutt Jr et al. 2008), for all but NGC 5194, for which we used data from the SINGS survey (Kennicutt Jr. et al. 2003). All the maps were convolved and regridded to the resolution and pixel size of the magnetic field maps. For the Hα maps from 11HUGS and SINGS, the flux-density units were converted to erg s⁻¹ cm⁻². We used the following calibration from Leroy et al. (2012) to estimate SFRSDs of the galaxies in Sample 1:

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Sigma }}}_{\mathrm{SFR}}[{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}]=634.0\,{I}_{{\rm{H}}\alpha }[\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{sr}}^{-1}]\\ \,+\,0.0025\,{I}_{24\mu {\rm{m}}}[\mathrm{MJy}\,{\mathrm{sr}}^{-1}]\end{array}\end{eqnarray} \tag{ 3 }$$

2.2.3. SFRs Using 1.4 GHz Observations

Our 1.4 GHz nonthermal maps of the galaxies (Sample 1) (Roy & Manna 2021) and an SFR calibration from Murphy et al. (2011) were used to derive SFRSD maps (Equation (4)). The calibration is based on the observed radio–FIR correlation in a sample of nearby star-forming galaxies (Bell 2003) and has a scatter of 0.26 dex. We used this galaxy-integrated calibration (Equation (4)) to derive the formula for spatially resolved radio–Σ_SFR calibration:

$$\begin{eqnarray}&&\displaystyle \frac{{\mathrm{SFR}}_{1.4\mathrm{GHz}}}{{M}_{\odot }\,{\mathrm{yr}}^{-1}}=6.35\times {10}^{-29}\displaystyle \frac{{L}_{1.4\mathrm{GHz}}}{\,}\mathrm{erg}\ {\mathrm{Hz}}^{-1}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 4 }$$

The spatially resolved calibration is consistent with the calibration of Heesen et al. (2014). We used the above relation to estimate the SFRSD maps of the sample galaxies from the measured 1.4 GHz surface brightness.

2.2.4. Galactic Extinction Correction for FUV Emission

We corrected for the extinction of FUV emission due to dust in the Milky Way using the E(B − V) values along the line of sight to the sample galaxies from Bianchi et al. (2017). The extinction coefficients (A_FUV) of the GALEX FUV bands were measured using Table 1 from Bianchi et al. (2017) and intrinsic fluxes (F_intrinsic) were estimated from the following formula:

$$\begin{eqnarray}&&{A}_{\mathrm{FUV}}=-2.5\times \mathrm{log}\left[\displaystyle \frac{{F}_{\mathrm{observed}}}{{F}_{\mathrm{intrinsic}}}\right].\end{eqnarray} \tag{ 5 }$$

The percentage of extinction of the FUV emission is listed in Table 3.

Table 3. FUV Extinction Values of the Sample 1 Galaxies Due to the Milky Way Foreground Dust

Name	Percentage of Extinction
NGC 2683	22
NGC 3627	23
NGC 4096	21
NGC 4449	15
NGC 4490	15
NGC 4826	13
NGC 5194	27

Note. The extinctions were computed using E(B − V) values along the line of sight to the sample galaxies from Bianchi et al. (2017).

Download table as:  ASCIITypeset image

Atomic hydrogen (H i) and molecular hydrogen (H₂) predominantly contribute to the total gas mass of galaxies. H₂ is best traced using rotational transitions in CO (e.g., Bolatto et al. 2013). Spatially resolved observations of CO transitions exist for only three of our seven galaxies in Sample 1 (Table 2). We have used CO J = 2–1 line data of NGC 3627 and NGC 5194 from the HERA CO-Line Extragalactic Survey (HERACLES; Leroy et al. 2009) and CO J = 1−0 data of NGC 4826 from the BIMA Survey of Nearby Galaxies (BIMA SONG; Regan et al. 2001). The HERACLES and BIMA survey have a spatial resolution of 13'' and 6'', respectively. The velocity resolution of the HERACLES and BIMA spectral cubes are ∼5 and 6 km s⁻¹, respectively. We restrict our study of the connection between gas densities and magnetic fields to only these three galaxies for which spatially resolved CO data are available.

The H i Nearby Galaxy Survey (THINGS; Walter et al. 2008) used VLA observations to obtain very high spectral (≤5.2 km s⁻¹) and spatial (∼6'') resolution maps of nearby galaxies at 21 cm. We used the publicly available 21 cm moment maps from this THINGS survey to estimate the distribution of H i in the three galaxies for which CO data are available. All CO and H i 21 cm maps were convolved and regridded to a common resolution and pixel size of the nonthermal radio maps. Gas densities were estimated (for NGC 3627, NGC 4826, and NGC 5194) following Basu & Roy (2013) assuming a CO to H₂ conversion factor of 2 ×10²⁰ ${({\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1})}^{-1}$ (e.g., Bolatto et al. 2013). A line ratio of 0.8 was assumed to convert CO_J=2−1 to CO_J=1−0 (e.g., Leroy et al. 2009). We accounted for the contribution of helium to the gas density using ρ_gas = 1.36 ×(ρ_HI + ${\rho }_{{{\rm{H}}}_{2}}$). Line-of-sight depths were assumed to be 300 and 400 pc for molecular and atomic gas, respectively (Basu & Roy 2013).

We have estimated spatially resolved revised equipartition magnetic field maps for seven galaxies in Sample 1, using the procedures of Section 2.1; these maps are shown in Figures 1 and 2. Flux-density contours of 1.4 GHz observations are overlaid on magnetic field maps. The resolution of these maps corresponds to spatial scales of ∼0.4−0.8 kpc (see Table 1). The bottom right panel of Figure 2 shows the radial variation of the magnetic field with Galactocentric radius of all the seven galaxies where both axes are normalized by their maximum values. Here, we averaged the magnetic field strengths over an annular elliptical region of width equal to the beam size of the corresponding map. The position and inclination angles (Table 1) of each galaxy were used while selecting the elliptical regions. We find magnetic fields to be stronger at the central region and at the star formation sites (arm region) with field strengths up to 50 μG. Field strengths fall by ∼50% at the edges of the magnetic field maps. The Milky Way also shows such a trend in the variation of magnetic field strengths (Beck et al. 1996). We note that our analysis was limited to distances where the signal-to-noise ratio in spectral index maps is > 5; the magnetic field strengths at these distances are thus likely to be reliable.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We note that, compared to the magnetic field strengths obtained using the classical equipartition expression, these values are higher by ∼1.3−1.5 for a nonthermal spectral index of −0.6, and they match for a spectral index of −0.75 (Beck & Krause 2005).

Figure 7 (Appendix A) shows the uncertainties in the magnetic field values for Sample 1 derived using the Monte Carlo method described in Section 2.1.1. Statistical uncertainties on mean magnetic fields for these seven galaxies are provided in Table 4.

We have estimated the global, galaxy-averaged SFRs of Sample 1 galaxies using 1.4 GHz, FUV+24 μm, and Hα+24 μm emission using the calibrations discussed in Section 2.2. Globally integrated SFRs of the sample galaxies are given in Table 5. No systematic offset was found in the SFR values estimated using these tracers. The differences in the SFR values for our galaxies are much less than the calibration uncertainty except for NGC 4490. For NGC 4490, SFR calculated from 1.4 GHz emission is higher than the same from FUV+24 μm emission by a factor of 2.2.

As discussed in Section 2.2, we estimated the SFRSD maps of the seven galaxies (Sample 1) using FUV+24 μm, Hα+24 μm, and 1.4 GHz emission. We show SFRSD maps of the seven galaxies in the Appendix B (Figures 8 and 9), where SFRSDs estimated using 1.4 GHz and FUV+24 μm emission are shown in contours and colors, respectively. In Appendix B (Figures 10 and 11), we also present the SFRSD maps estimated using Hα+24 μm and 1.4 GHz emission in colors and contours, respectively. The SFRSD maps of each galaxy in Figures 8−11 are shown in the same color scale and contours. To determine the radial variation of SFRSDs, we averaged the SFRSD maps of our sample galaxies over tilted rings centered on the optical center of each galaxy using their inclinations and position angles. The width of the tilted rings was taken to be equal to the beam size of the corresponding image. Figure 3 shows the radial variation of the average SFRSD, derived using FUV+24 and Hα+24 μm emission, with Galactocentric distance where both the axes are normalized to their maximum values. We also derived the radial variation of SFRSDs for the galaxies using 1.4 GHz emission and it is consistent within 1σ statistical uncertainties, with those derived using FUV+24 and Hα+24 μm data. Azimuthally averaged SFRSDs of all seven galaxies decrease gradually toward the outer region and drop by a factor of 6–8 at the edge.

Zoom In Zoom Out Reset image size

(i) NGC 2683: In this galaxy, Krause et al. (2020) found very weak linear polarization using C-band and L-band VLA observations. Based on the optical image, we could separate the central region from the disk. The average magnetic field in the central region is found to be ≈31 μG and the outer region of the disk has an average value of ≈19 μG (see Figure 1 and Table 6).

Wiegert et al. (2015) used Wide-field Infrared Survey Explorer 22 μm data to estimate a galaxy-averaged SFR of ≈0.09 M_⊙ yr⁻¹ for NGC 2683. From our analysis, the integrated SFR was measured to be ∼0.24 and ∼0.28 M_⊙ yr⁻¹ using FUV+24 μm and 1.4 GHz radio emission, respectively. However, we note that Wiegert et al. (2015) used a distance of 6.27 Mpc for this galaxy, but we have used a distance of 7.7 Mpc. The SFR is estimated to be 0.16 M_⊙ yr⁻¹ using FUV+24 μm emission, assuming the same distance as used by Wiegert et al. (2015). Taking the calibration uncertainties and the assumed distance into account, our estimated SFR is hence consistent with that of Wiegert et al. (2015). We note that the contours on the background sources (Figure 8) are not real SFRSDs, as these are likely to be background active galactic nuclei (AGNs).

(ii) NGC 3627: NGC 3627 was observed at 8.46 GHz and 4.85 GHz using the VLA in its D-configuration (Soida et al. 2001). These authors estimated the magnetic field strengths using the classical equipartition formula (Longair 2011) and found an average equipartition magnetic field strength of 11 ± 2 μG, assuming a constant nonthermal spectral index of 0.9 and a disk thickness of 2 kpc. Soida et al. (2001) also studied the polarized emission at these frequencies to find a regular magnetic field of 4 ± 1 μG. They suggested two distinct magnetic field components of NGC 3627: one for the spiral arms and another for the inter-arm region. We have separately studied equipartition magnetic fields in the arm and inter-arm regions of the galaxy. We find that the central region and the edges of the extended bar have magnetic field strengths of ≈34 μG (see Figure 1). The arm region has a field strength of ≈28 μG (see Table 6). However, the magnetic field strength in the inter-arm region has values ≈21 μG. We note that our estimates of the equipartition magnetic field strengths in the galaxy are higher than those found by Soida et al. (2001); this difference likely arises from the fact that Soida et al. (2001) estimated the magnetic field strengths using the classical equipartition formula, which is known to significantly underestimate the magnetic field in the star-forming regions.

We measured a galaxy-averaged SFR of ≈2.0 and ≈1.56 M_⊙ yr⁻¹ from FUV+24 μm and 1.4 GHz emission, respectively. Our measurements of spatially resolved SFRs in different regions are consistent, within calibration uncertainties, with the SFR estimates of Watanabe et al. (2011).

(iii) NGC 4096: Our estimate of the equipartition magnetic field in NGC 4096 varies from ≈21 μG at the center to ≈12 μG at the edge (Table 6). The magnetic field strength in both the central region and northern periphery is quite similar, with typical field strengths of ≈20 μG; this is presumably due to its high inclination. The outer part of the galaxy has an average field strength of ≈14 μG. NGC 4096 was observed (Irwin et al. 2012; Wiegert et al. 2015) with its B field and further studied by Krause et al. (2020) who found very little polarized emission from the galaxy.

Wiegert et al. (2015) used the 22 μm−SFR calibration to measure a galaxy-averaged SFR of 0.27 ± 0.02 M_⊙ yr⁻¹. Our measurement of the galaxy-averaged SFR is ≈0.35 M_⊙ yr⁻¹ and ≈0.43 M_⊙ yr⁻¹ using FUV+24 μm and 1.4 GHz emission, respectively. Considering the calibration uncertainties, our estimates are consistent with that of Wiegert et al. (2015).

(iv) NGC 4449: This is an optically bright irregular starburst galaxy. Chyży et al. (2000) used VLA 4.86 and 8.46 GHz observations to find a galaxy-averaged equipartition magnetic field of ≈14 μG. These authors also used polarization emission to estimate a regular field of ≈8 μG. The equipartition magnetic field map of NGC 4449 from our study is shown in Figure 1. As noted in Section 2.1, about 70 % of the total projected area of this galaxy has spectral index values of less than 0.55. We replaced the pixel values with α_nt < 0.55 with α_nt = 0.55 while computing the magnetic field for NGC 4449 (see Section 2.1). The average magnetic field strength is ≈17 μG in this galaxy, which is comparable to the findings of Chyży et al. (2000).

Our measurements of the galaxy-averaged SFR are ≈0.38 and ≈0.37 M_⊙ yr⁻¹ using FUV+24 μm and 1.4 GHz emission, respectively, which are consistent with the SFR of 0.47 M_⊙ yr⁻¹ estimated by Chyży et al. (2011).

(v) NGC 4490: Nikiel-Wroczynski et al. (2016) observed NGC 4490 at 0.61 GHz using the GMRT, and at 4.86 and 8.44 GHz using VLA + Effelsberg. The authors used these observations to find a mean equipartition magnetic field of 21.9 ± 2.9 μG, with typical field strengths in the range of 18–40 μG. We found a typical equipartition magnetic field strength of ≈40 μG in the central region, which decreases to ≈17 μG in the outer region (see Figure 2); these values are consistent with the estimates of Nikiel-Wroczynski et al. (2016). We find a relatively lower magnetic field strength of ≈15 μG in both the interacting region and the companion galaxy NGC 4485. Therefore, a gradual decrease in the average magnetic field strength occurs from the center to the outer region.

Clemens et al. (1999) used radio observations to find a galaxy-averaged SFR of 4.7 M_⊙ yr⁻¹. We found a similar SFR (≈4.63 M_⊙ yr⁻¹) using 1.4 GHz radio emission but a factor of ∼2 lower SFR (2.13 M_⊙ yr⁻¹) using the FUV+24 μm emission (Table 5). Extinction corrections for NGC 4490 are believed to be higher than those typically assumed and this may lead to an underestimation of the SFR while using the FUV+24 μm diagnostics (Clemens et al. 1999).

(vi) NGC 4826: No spatially resolved maps of magnetic fields and SFRSDs are available in the literature. We measure the central and outer regions of the galaxy to have an average equipartition magnetic field strength of ≈38 and ≈20 μG, respectively (see Figure 2 and Table 6). We find galaxy-averaged SFR of ≈0.73 and ≈0.63 M_⊙ yr⁻¹ using FUV+24 μm and 1.4 GHz data, respectively.

(vii) NGC 5194: Fletcher et al. (2011) used VLA C-band observations of the galaxy and assumed a constant thermal and nonthermal spectral index of 0.1 and 1.1 to find an average equipartition magnetic field strength of 20 μG using the revised formula by Beck & Krause (2005). They found a magnetic field of 20−25 μG in the spiral arms, higher than the 15−20 μG typical in the inter-arm region. Using VLA observations at S-band (2−4 GHz) frequencies, Kierdorf et al. (2020) found the field strength of turbulent and regular components of the magnetic field in the arm region of 18−24 and 8−16 μG, respectively. We find an equipartition magnetic field strength of ≈25 μG in the arm region and ≈18 μG in the inter-arm region (see Table 6). The peripheral region has a magnetic field of ≈12 μG, while the overlapping region between NGC 5194 and NGC 5195 has an average B_eq of ≈16 μG. Considering our use of Equation (1) (Beck & Krause 2005), measurements are roughly consistent with the earlier study of Fletcher et al. (2011) and Kierdorf et al. (2020).

Spatially resolved SFRs were measured in several star-forming regions of NGC 5194 using Hα+24 μm and Hα+Paα emission (Kennicutt Jr. et al. 2007). SFRSDs in different regions were found to be in the range of 0.10–0.46 M_⊙ yr⁻¹ kpc⁻². Our estimates using the two tracers are consistent with the estimates of Kennicutt Jr. et al. (2007) (see Figures 9 and 11). Furthermore, we find that the galaxy-integrated SFR derived using FUV+24 μm (≈3.88 M_⊙ yr⁻¹) and 1.4 GHz data (≈4.16 M_⊙ yr⁻¹) are consistent with each other, within 1σ statistical uncertainty.

We have estimated magnetic fields for the galaxies in Sample 1 assuming the minimum energy condition or equipartition condition, i.e., by assuming that the energy density in the magnetic field is approximately equal to the energy density in cosmic-ray particles. Therefore, it is important to verify the validity of this assumption in our sample galaxies. The tightness of the spatially resolved radio−FIR correlation can be used to estimate the deviation of the energy densities from the minimum energy condition (Hummel 1986; Basu & Roy 2013). According to the simplified model of Hummel (1986), when the minimum energy condition is satisfied, the distribution of I_nt/I_FIR will be similar to the distribution of ${{B}}^{1+{\alpha }_{\mathrm{nt}}}$. The model assumes the following to be constant across galaxies: (a) the ratio of the number densities of relativistic electrons and dust-heating stars, (b) the volume ratio of radio and FIR emitting regions, and (c) the ratio of efficiency factors for both the radio and FIR emission. In this model, the cumulative distribution function (CDF) of the quantity I_nt/I_FIR and ${B}_{\mathrm{eq}}^{1+{\alpha }_{\mathrm{nt}}}$ is expected to follow each other if B_eq is close to B.

To verify the validity of the minimum energy condition in our sample galaxies, we followed the procedure as in Hummel (1986) and Basu & Roy (2013). The CDFs of I_nt/I_FIR and ${B}_{\mathrm{eq}}^{1+{\alpha }_{\mathrm{nt}}}$ were estimated using our radio maps of the sample galaxies at both 0.33 and 1.4 GHz. We used an ensemble of spatially resolved values of α_nt, I_nt (both at 0.33 and 1.4 GHz), I_FIR (70 μm), and magnetic fields (B_eq), which are averaged over the beam size from all the galaxies in Sample 1 (Table 2) to generate these distributions. The CDFs of all quantities were normalized by their median values. The top panels in Figure 4 show the median-normalized CDFs of I_nt/I_FIR and ${B}_{\mathrm{eq}}^{1+{\alpha }_{\mathrm{nt}}}$ at both 0.33 and 1.4 GHz.

Zoom In Zoom Out Reset image size

We find that the CDFs of I_nt/I_FIR and ${B}_{\mathrm{eq}}^{1+{\alpha }_{\mathrm{nt}}}$ at both 0.33 and 1.4 GHz broadly follow each other but with slight deviations at high and low ends (see top panels in Figure 4). This implies that the minimum energy condition is broadly valid and is consistent with earlier findings. For example, Hummel (1986) found the distribution of the two quantities is similar in a sample of Sbc galaxies while Basu & Roy (2013) reached similar conclusions in a study of five nearby large spiral galaxies, but with slight deviations observed in the CDFs of I_nt/I_FIR and ${B}_{\mathrm{eq}}^{1+{\alpha }_{\mathrm{nt}}}$ in the inter-arm regions of the galaxies.

The observed deviation in the CDFs of I_nt/I_FIR and ${B}_{\mathrm{eq}}^{1+{\alpha }_{\mathrm{nt}}}$ for our sample galaxies imply a corresponding deviation from the minimum energy condition. In order to quantify this deviation, we performed a Monte Carlo simulation originally proposed by Hummel (1986). In this simulation, random numbers (X) were drawn from a Gaussian distribution with standard deviation σ. Thereafter, we multiplied 10^X with the observed equipartition magnetic fields to introduce deviations from the minimum energy condition. We thus constructed the CDF of ${B}_{\mathrm{eq}}^{1+{\alpha }_{\mathrm{nt}}}$ using the deviated magnetic field values. The CDF of ${B}_{\mathrm{eq}}^{1+{\alpha }_{\mathrm{nt}}}$ was then compared to the observed CDF of I_nt/I_FIR via a Kolmogorov–Smirnov (K-S) test. This procedure was repeated for a range of σ from 0–0.2. We find that the p-values for the K-S test comparing the distributions are maximized when σ = 0.1. Indeed, ${B}_{\mathrm{eq}}^{1+{\alpha }_{\mathrm{nt}}}$ derived after deviating the magnetic field using σ = 0.1 and I_nt/I_FIR are consistent with being derived from the same distribution, with a K-S test p-value of 0.41 and 0.55, when using I_nt at 0.33 and 1.4 GHz, respectively. The bottom panels in Figure 4 show the CDFs of the two quantities for σ = 0.1 at 0.33 and 1.4 GHz; it is clear that the CDFs follow each other. This implies the actual magnetic field values may deviate from the equipartition values by ∼25% in our galaxies in Sample 1. We note that any violation of the assumptions made by Hummel (1986) may also lead to the observed deviation in the CDFs.

We have studied the correlation between the spatially resolved equipartition magnetic field and SFRSDs for the galaxies in Sample 1 (Table 2) at scales of ≈360−760 pc (Table 1). For the seven sample galaxies, we used the SFRSD maps estimated using the FUV+24 μm emission. The correlations between magnetic fields and SFRSDs for the seven galaxies are shown in Figure 5. Each point represents the logarithms of equipartition magnetic fields and SFRSD values that are averaged over the beam size of the corresponding maps. da Silva et al. (2014) found that SFR calibrations could be biased and strongly affected by stochasticity at small spatial scales where the SFR is low (≤10^−2.5 M_⊙ yr⁻¹); we therefore excluded regions of low SFRs from the correlation study.

Zoom In Zoom Out Reset image size

We find that the equipartition magnetic field and the SFSRD are correlated in all seven sample galaxies. We use orthogonal distance regression in SciPy (Virtanen et al. 2020) to fit a power law of the form $B={B}_{0}\ {\left({{\rm{\Sigma }}}_{\mathrm{SFR}}\right)}^{\eta }$ to the magnetic field−SFRSD data points; the spatially resolved uncertainty maps of equipartition magnetic fields and rms noise on the SFRSD maps were used to estimate the uncertainties on each data point during the fitting procedure. The best-fit parameters of the power law are given in Table 7. We also estimated the scatter (rms of the data points along the y-axis) of the correlations which are presented in Table 7 and are shown in dashed lines in the corresponding plots (Figure 5). We find that six of the seven galaxies have slopes (η) in the range of ≈0.27–0.40 but that the slope is relatively lower for NGC 4449 with η ≈ 0.18. Averaging over the slope of all galaxies in Sample 1, we find a mean slope of 0.32 ± 0.06.

We have studied the correlation between spatially resolved equipartition magnetic fields and gas densities for three of the galaxies in Sample 1, NGC 3627, NGC 4826, and NGC 5194, for which spatially resolved CO observations were available (see Section 2.3). Similar to the study of correlations between B_eq and SFRSDs, we studied the correlations between B_eq and gas density values, both averaged over the beam size of the corresponding maps. The correlations between magnetic fields and gas densities of NGC 3627, NGC 4826, and NGC 5194 are shown in Figure 6. We have again used orthogonal distance regression in SciPy (Virtanen et al. 2020) to fit a power law to the B_eq and gas density data points. The scatters of the three correlations are shown in dashed lines in all the figures.

Zoom In Zoom Out Reset image size

The measured best-fit power-law indices are 0.40 ± 0.02, 0.49 ± 0.03, and 0.53 ± 0.02 (Table 9) for NGC 3627, NGC 4826, and NGC 5194, respectively. The mean of the power-law indices is 0.47 ± 0.05.

Several magnetohydrodynamical simulations have found that galactic magnetic fields are amplified by gas turbulence at very short timescales (i.e., ∼100 Myr) (e.g., Brandenburg & Subramanian 2005; Beresnyak 2012; Schober et al. 2012; Bovino et al. 2013; Schleicher & Beck 2013). The primary driver of gas turbulence in the ISM of galaxies is supernova explosion (Bacchini et al. 2020), the rate of which is in turn directly coupled to the SFR in the galaxy. Therefore, it is expected that the SFRs and the magnetic fields in a galaxy will be correlated. Indeed, using semi-analytical models, Schleicher & Beck (2013, 2016) found that in order to explain the radio–FIR correlation at sub-kiloparsec scales, magnetic fields, and SFRSDs, again at sub-kiloparsec scales, must be related as B ∝ ${{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}$.

Studies in the literature on the correlation between magnetic fields and SFRSDs have focused on dwarf galaxies and those studies were carried out using galaxy-integrated magnetic fields and SFRSDs. As mentioned in Section 1, to our knowledge, there is only one published work on the spatially resolved study of the correlation between magnetic fields and SFRSDs (Basu et al. 2017).

For the 12 galaxies in Sample 2 (Table 2), we find that the mean value of the power-law index of the correlation between B_eq and SFRSDs is 0.31 ± 0.06, i.e B_eq ∝ ${{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.31\pm 0.06}$( ² ), consistent (at < 1σ error) with the model of Schleicher & Beck (2013, 2016). Thus, it appears that the semi-analytical models that are based on the amplification of magnetic fields due to supernova-driven gas turbulence work remarkably well for the pilot sample in predicting the correlation between magnetic fields and SFRSDs down to sub-kiloparsec scales.

We note that the power-law index for the correlation between B_eq and SFRSDs for NGC 4449 was found to be 0.18 ± 0.03, significantly lower than for the remaining galaxies (Table 7) as well as lower than the model prediction of B ∝ ${{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}$ Schleicher & Beck (2013) (at >5σ significance). For the case of NGC 4449, the relatively flat spectral index values (α_nt ≤ 0.55) in ≈70% of the galaxy meant that the magnetic field values could not be estimated reliably for a large part of the galaxy (see Sections 2.1 and 3.5). This could lead to biases in the correlation, and therefore, the low value of the power-law index for NGC 4449 should be taken with caution.

5.1.1. Intercept of the Correlation

According to the model proposed by Schleicher & Beck (2013), the intercept of the B–Σ_SFR correlation depends on several ISM parameters such as gas density (ρ₀), the fraction of turbulent kinetic energy converted into magnetic energy (f_sat), the injection rate of turbulent supernova energy (C) and the intercept of the Kennicutt–Schmidt relation (C₁) (Equation (5)).

$$\begin{eqnarray}&&{B}\sim \sqrt{{f}_{\mathrm{sat}}8\pi }\,{\rho }_{0}^{1/6}\,{\left(\displaystyle \frac{C}{{C}_{1}}\right)}^{1/3}\,{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}.\end{eqnarray} \tag{ 6 }$$

Schleicher & Beck (2013) predicted the intercept of the B–Σ_SFR correlation to be ∼26 μG assuming ρ₀ = 10⁻²⁴ g cm⁻³ and f_sat ∼5%. We have found an average intercept at 65 ±25 μG of the B_eq–Σ_SFR correlation of the 12 galaxies in Sample 2 (see Tables 7 and 8). Although the mean value is a factor of ≈2.5 higher than the value predicted by Schleicher & Beck (2013), this value is consistent with the predicted value, within the scatter (at ≈1.6σ). Future follow-up studies, such as using our full survey (Sample 0 which consists of 46 galaxies), are required to draw statistically robust conclusions about the value of the intercept.

If the value of f_sat is indeed higher, this would imply a higher than assumed value of 1 or more of ρ₀, C, and f_sat. The intercept is broadly insensitive to the assumed value of ρ₀ (Equation (6)), and therefore, in order to explain a factor of ≈2.5 higher value of the intercept, the actual value ρ₀ has to be higher than the assumed value of 10⁻²⁴ g cm⁻³ by a factor of ≈240; such high gas densities are unphysical and are not observed in typical regions of a galaxy. The other possibility that the assumed value of the injection rate of turbulent supernova energy (C) is higher by a factor of ≈16 is also contrary to expectation; Basu et al. (2017) found that under reasonable conditions the value of C can be higher by at most a factor of 1.4. Therefore, f_sat must be higher than 0.05 to explain a significantly higher value of the intercept. An understanding of how galaxies can achieve such efficient amplification of magnetic fields with f_sat much greater than 5% requires detailed MHD simulations. We note that Basu et al. (2017) found that the value of the intercept for B–Σ_SFR for the dwarf galaxy IC 10 is 51 μG, similar to our findings of a higher than predicted value of the intercept.

Magnetic fields and gas are expected to be correlated as B ∝ $\sqrt{{\rho }_{\mathrm{gas}}}$ (e.g., Chandrasekhar & Fermi 1953; Groves et al. 2003). We find that equipartition magnetic fields are correlated with gas densities for the seven galaxies (Sample 3) with an average power-law index, k = 0.40 ± 0.09 (see Sections 3.5 and 4) ³ . This value of k is consistent with the numerical simulations that predict k ≈0.4−0.6 and also consistent with the theories that predict B ∝ ${\rho }_{\mathrm{gas}}^{0.5}$. The power-law index of the correlation between B_eq and gas densities is found to be 0.25 ± 0.02 and 0.31 ± 0.03 for NGC 5055 and NGC 6946 respectively, significantly lower than the model predictions and as compared to the other galaxies in Sample 3. A lower value of k could mean that either the efficiency of the amplification of the magnetic field is less or that the magnetic field strengths derived assuming the minimum energy condition are underestimated (Dumas et al. 2011). Strong synchrotron or inverse Compton losses of cosmic-ray electrons could suppress the radio synchrotron emission, which would then cause the equipartition magnetic fields to be underestimated.

5.2.1. Magnetic Fields, Gas Densities, and the Radio–FIR Correlations

Energy equipartition between the magnetic field (B) and the gas density (ρ_gas), and between magnetic fields and cosmic-ray particles implies that the nonthermal emission is related to the gas density as I_nt ∝ ${\rho }_{\mathrm{gas}}^{k(3+{\alpha }_{\mathrm{nt}})}$ where k is the power-law index relating magnetic fields and gas densities (B_eq ∝ ${\rho }_{\mathrm{gas}}^{k}$) (Niklas & Beck 1997). Further, Kennicutt–Schmidt law and the radio–FIR correlation imply that I_nt is related to gas densities as (1) I_nt ∝ ${\rho }_{\mathrm{gas}}^{m(n+1)}$ for optically thin dust to UV photons and (2) I_nt ∝ ${\rho }_{\mathrm{gas}}^{{mn}}$ for optically thick dust to UV photons, where m is the power-law index of the radio–FIR correlation and n is the power-law index of the Kennicutt–Schmidt law. Therefore, we can obtain the following relation between the power-law index of all four correlations (Dumas et al. 2011):

$$\begin{eqnarray}&&k=\frac{(n+1)m}{3+{\alpha }_{\mathrm{nt}}};\,\mathrm{optically\; thin\; dust},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&k=\frac{{nm}}{3+{\alpha }_{\mathrm{nt}}};\,\mathrm{optically\; thick\; dust}.\end{eqnarray} \tag{ 8 }$$

We can use the above equations to indirectly estimate the power-law index, k, of the correlation between magnetic fields and gas densities. For the three galaxies, NGC 3627, NGC 4826, and NGC 5194 (Roy & Manna 2021), we have estimated gas densities using CO and H i observations. Now we can compare the direct measurement of k with an indirect estimate of k using Equations (7) and (8); this will provide additional information on the validity of both the minimum energy conditions that were assumed between magnetic fields and the gas densities as well as the magnetic fields and cosmic-ray particles. For the galaxies from Basu et al. (2012a), this study was already presented and discussed in Basu et al. (2012b).

We have estimated k for all the seven sample galaxies from Roy & Manna (2021) (Sample 1), using the assumption of optically thin dust to UV photons, using (i) the slope of radio–FIR correlation (m) as derived in Roy & Manna (2021), (ii) the measured galaxy-averaged spectral index (α_nt) from Roy & Manna (2021), and (iii) a Kennicutt–Schmidt power-law index of 1.4 ± 0.15 (Kennicutt Jr. 1998b).

Table 10 provides the relevant values as well as estimated values of k derived using the measured value of m using radio emission at both 0.33 and 1.4 GHz. For two of the galaxies, NGC 3627 & NGC 5194, the value of k estimated using Equation (7) is comparable to the direct measurement of k. This broadly validates the assumption of energy equipartition between magnetic fields and cosmic-ray particles in these two galaxies.

Table 10. Power-law Index (k) of the Relation between Magnetic Fields and Gas Densities (B ∝ ρ^k ) of Galaxies in Sample 1, Indirectly Estimated Using the Slope of Radio–FIR Correlation (m) and the Slope of the Kennicutt−Schmidt Law

Name	m	m	αnt	k (Optically Thin)	k (Optically Thin)
	0.33 GHz	1.4 GHz		0.33 GHz	1.4 GHz
NGC 2683	0.54 ± 0.06	0.91 ± 0.07	−0.84 ± 0.08	0.33 ± 0.04	0.57 ± 0.06
NGC 3627	0.55 ± 0.03	0.85 ± 0.13	−1.10 ± 0.07	0.32 ± 0.03	0.50 ± 0.08
NGC 4096	0.74 ± 0.05	0.90 ± 0.04	−0.78 ± 0.06	0.47 ± 0.04	0.57 ± 0.05
NGC 4449	0.77 ± 0.05	0.65 ± 0.04	−0.48 ± 0.06	0.53 ± 0.05	0.45 ± 0.04
NGC 4490	0.68 ± 0.02	0.75 ± 0.02	−0.59 ± 0.07	0.45 ± 0.03	0.50 ± 0.04
NGC 4826	1.39 ± 0.1	1.47 ± 0.08	−0.49 ± 0.06	0.95 ± 0.09	1.00 ± 0.09
NGC 5194 (arm)	0.50 ± 0.05	0.65 ± 0.04	−0.63 ± 0.05	0.33 ± 0.04	0.43 ± 0.04
NGC 5194 (inter-arm)	0.73 ± 0.11	1.03 ± 0.05	−0.85 ± 0.10	0.46 ± 0.08	0.64 ± 0.05

Note. See Section 5.2.1 for a discussion of these.

Download table as:  ASCIITypeset image

For the optically thin case, the mean of indirectly estimated k values of the sample of seven galaxies are 0.59 ± 0.16 and 0.53 ± 0.19 at 1.4 and 0.33 GHz, respectively. However, this includes the galaxy NGC 4826, which shows an anomalously high value of k = 1.0 and 0.95 derived at 1.4 and 0.33 GHz, respectively. Excluding this galaxy from the mean calculation, we find that k = 0.52 ± 0.04 and 0.47 ± 0.09 at 1.4 and 0.33 GHz, respectively. Remarkably, for all the galaxies except NGC 4826, the k value at 1.4 GHz, for the optically thin case, is consistent with 0.5 within error bars. Thus, the indirectly estimated values of k are consistent with equipartition between magnetic fields and gas energy densities (Chandrasekhar & Fermi 1953; Fiedler & Mouschovias 1993; Cho & Vishniac 2000; Groves et al. 2003). This is similar to the findings of Niklas & Beck (1997) for their sample of 43 galaxies and Basu et al. (2012b) for their sample of four galaxies.

The value of k derived for NGC 4826, for the optically thin case, is a consequence of the anomalously high value of the power-law index of the radio–FIR correlation (≈1.39 and ≈1.47 for 0.33 and 1.4 GHz, respectively, Table 10), which is different from the other six galaxies in the sample. NGC 4826 has been classified as a Seyfert 2 galaxy in the past (Malkan et al. 2017), and therefore, the emission from the core contributes to the observed power-law index of the radio–FIR correlation (Roy & Manna 2021). It is likely that the significant contribution of the AGN to the radio emission makes the estimate of k for NGC 4826 unreliable.