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

Souvik Manna  National Center for Radio Astrophysics, TIFR,
Pune University Campus, Ganeshkhind, Pune 411007, India  Subhashis Roy
National Center for Radio Astrophysics, TIFR,
Pune University Campus, Ganeshkhind, Pune 411007, India

###### Abstract

We have estimated the magnetic field strengths of a sample of seven galaxies
using their non-thermal synchrotron radio emission at metre wavelengths, and
assuming energy equipartition between magnetic fields and cosmic ray
particles. We tested for deviation of magnetic fields from energy
equipartition with cosmic ray particles, and found that deviations of
$\sim$25% are typical for the sample galaxies. Spatially resolved star
formation rates (SFR) were estimated for the seven galaxies along with five
galaxies studied previously. For the combined sample of twelve galaxies, the
equipartition magnetic fields (B${}_{\textrm{eq}}$) are correlated with the
SFR surface densities ($\Sigma_{\textrm{SFR}}$) at sub-kpc scales with
B${}_{\textrm{eq}}$ $\propto$ $\Sigma_{\textrm{SFR}}^{0.31\pm 0.06}$,
consistent with model predictions. We estimated gas densities
($\rho_{\textrm{gas}}$) for a sub-sample of seven galaxies using archival
observations of the carbon monoxide (CO) rotational transitions and the atomic
hydrogen (Hi) 21 cm line and studied the spatially-resolved correlation
between the magnetic fields and $\rho_{\textrm{gas}}$. Magnetic fields and gas
densities are found to be correlated at sub-kpc scale as B${}_{\textrm{eq}}$
$\propto$ $\rho_{\textrm{gas}}^{0.40\pm 0.09}$. This is broadly consistent
with models, which typically predict B $\propto$ $\rho_{\textrm{gas}}^{0.5}$.

Radio continuum emission — Interstellar medium — Star formation — Magnetic
fields

##  1 Introduction

Magnetic fields are believed to influence several physical processes in a
galaxy at almost every scale (e.g. Elmegreen, 1981; Niklas & Beck, 1997;
Groves et al., 2003; Price & Bate, 2008; Adebahr et al., 2013). Magnetic
fields have been found to consist of two main components: a small-scale
turbulent magnetic field up to a few hundred parsecs (e.g. Batchelor, 1950;
Groves et al., 2003) and a large-scale $``$ordered$"$ or $``$regular$"$
magnetic field component at scales of a few kpcs (e.g. Moss & Shukurov, 1996;
Shukurov et al., 2006; Kulsrud & Zweibel, 2008). Magnetic fields in galaxies
can be measured using their effects on different radiation processes like
Zeeman splitting of emission lines, polarized emission from dust, the
polarization of starlight, Faraday rotation of polarized radio emission, and
intensity of synchrotron emission which we use in this work. Measurement of
the line-of-sight component of the magnetic field via the Zeeman effect in
galaxies other than the Milky Way has been possible for only a few systems
(Kazes et al., 1991; Sarma et al., 2005; Robishaw et al., 2008); a significant
expansion of such studies is very difficult with current-generation
telescopes.

Magnetic fields in galaxies can be measured and studied using synchrotron
emission at radio frequencies, at scales larger than the resolution of the
radio observation. For example, a Very Large Array (VLA) polarization study of
NGC 4736 at 8.46 and 4.86 GHz found that the magnetic field in the galaxy was
ordered in a spiral shape (Chyży & Buta, 2008). An X-shaped structure of the
magnetic field in the galactic halo region was observed by stacking the Karl
G. Jansky VLA polarized emission maps of 16 nearly edge-on spiral galaxies,
obtained as part of the CHANG-ES survey (Krause et al., 2020); such structures
had also been observed in individual spiral galaxies (e.g. Krause et al.,
2006; Krause, 2009; Heesen et al., 2009). However, polarized radio emission
from external individual galaxies is difficult to study at low radio
frequencies due to Faraday depolarization (e.g. Sokoloff et al., 1998).

The average magnetic field strength can also be estimated from the total
intensity of synchrotron radio emission, assuming energy equipartition between
magnetic fields and cosmic ray particles (e.g. Miley, 1980; Beck & Krause,
2005). Equipartition magnetic fields have been studied in several nearby
galaxies, but primarily at frequencies $>$1 GHz (e.g. Chyży et al., 2000;
Soida et al., 2001; Heesen et al., 2009; Fletcher et al., 2011; Adebahr et
al., 2013). Vargas et al. (2018) studied a sample of three nearly edge-on
galaxies from the CHANG-ES survey to separate the thermal Bremsstrahlung from
the non-thermal synchrotron emission at 1.5 and 6 GHz. At these frequencies,
the thermal component is large and hence the correction for the thermal
emission can be as large as $\sim$ 20%, making the derived magnetic field
strengths prone to errors. Conversely, the steep spectral index of synchrotron
emission implies that it will dominate the total emission at frequencies $<1$
GHz, with $\sim$ 95% contribution (Basu et al., 2012b; Roy & Manna, 2021).
Thus, magnetic field strengths derived using observations at $<$1 GHz are very
robust to any correction for thermal emission.

Magnetic fields are believed to play an important role at various stages of
the star-formation process - from the fragmentation of clouds at the few kpc
scales to the final collapse of gas into stars (e.g. Elmegreen, 1981;
Crutcher, 1999; Price & Bate, 2008; Van Loo et al., 2015). To understand the
influence of magnetic fields and star-formation activities on different
physical processes in the ISM at different physical scales, several studies on
radio-infrared correlations have been carried out in the past (e.g. Murphy et
al., 2006a, b, 2008; Tabatabaei et al., 2013). Magnetic fields (B) and star
formation rate surface densities (SFRSD) are expected to be correlated (Niklas
& Beck, 1997). Semi-analytical model also predicts a strong correlation
between B and SFRSDs ($\Sigma_{\textrm{SFR}}$) as B
$\propto\Sigma_{\textrm{SFR}}^{1/3}$ at sub-kpc scales to explain the local
radio-FIR correlation (Schleicher & Beck, 2013, 2016). Observational studies
of the correlation between B and star formation rates (SFR) have been done
primarily in samples of nearby dwarf galaxies. For example, Chyży et al.
(2011) studied 12 local group dwarf galaxies to find that the galaxy-averaged
magnetic field and the SFR follow B $\sim$ SFR0.30±0.04, consistent with the
prediction of B $\propto\Sigma_{\textrm{SFR}}^{1/3}$. However, Jurusik et al.
(2014) found the same power-law index in a sample of Magellanic type dwarf
galaxies to be 0.25$\pm$0.02, somewhat lower than the expectation. Recently, a
study of the dwarf galaxy IC 10 by Basu et al. (2017) provides the only study
of the correlation between spatially-resolved magnetic fields and SFRSDs;
these authors found that the SFRSD is related to the magnetic field as B
$\propto\Sigma_{\textrm{SFR}}^{0.35\pm 0.03}$. Therefore, it is important to
test such predictions by carrying out systematic spatially-resolved studies of
magnetic fields in galaxies and their connection to the star formation rate in
nearby large galaxies.

The energy density of magnetic fields and gas in galaxies are expected to be
in equipartition, which implies B $\propto\sqrt{\rho_{\textrm{gas}}}$ (e.g.
Chandrasekhar & Fermi, 1953; Groves et al., 2003). The observed Radio-FIR
correlation can be explained based on such equipartition between the energy
density of magnetic fields and gas (Niklas & Beck, 1997). Several other
numerical magnetohydrodynamic (MHD) simulations of the ISM have predicted the
coupling constant ($k$) between magnetic fields and gas (B
$\propto\rho_{\textrm{gas}}^{k}$) to be in the range of $\approx$0.4$-$0.6
(Fiedler & Mouschovias, 1993; Kim et al., 2001; Thompson et al., 2006). Niklas
& Beck (1997) studied the correlation between galaxy-integrated equipartition
magnetic fields and gas densities for a sample of 43 galaxies to find a power-
law index of 0.48 $\pm$ 0.05; the observed correlation is consistent with B
$\propto\sqrt{\rho_{\textrm{gas}}}$. Although the correlation between gas
surface densities and SRFSDs has been extensively studied in the nearby
Universe (Kennicutt-Schmidt law; e.g. Kennicutt, 1998a; Onodera et al., 2010;
Roychowdhury et al., 2015), systematic studies of spatially-resolved
correlations between magnetic fields, SFRs and gas densities in nearby
galaxies are yet to be carried out. It is thus important to carry out a
systematic investigation of both the B-$\rho$ and the B-SFR correlations, at
high-spatial resolutions ($\approx$ sub-kpc scales), using direct estimates of
the magnetic fields, gas densities, and star-formation rates, in a sample of
nearby galaxies. In this paper, we present a pilot study of the connection
between spatially resolved magnetic fields, SFRSDs and gas densities in a
sample of nearby galaxies.

We have selected a sample of 46 galaxies (Sample 0; Table 2) from the Spitzer
Local Volume Legacy (LVL) sample of 258 galaxies within 11 Mpc (Dale et al.,
2009). As a pilot project, seven (Sample 1; Table 2) of these 46 galaxies have
been observed with the Giant Metrewave Radio Telescope (GMRT) at 0.33 GHz (Roy
& Manna, 2021). Six of our seven sample galaxies are spirals and the other one
is a dwarf irregular Magellanic-type galaxy.

In this paper, we present spatially resolved equipartition magnetic field
(B${}_{\textrm{eq}}$) maps of the seven galaxies in Sample 1 (Table 2). We
also incorporate the magnetic field maps of five galaxies studied by Basu et
al. (2012a) from previous GMRT observations in our study. We derived SFRSD
maps of all 12 galaxies (Sample 2; Table 2) using extinction-free diagnostics
and used these maps to study the relation between SFRSDs and
B${}_{\textrm{eq}}$ at sub-kpc scales in our pilot study. We used available
archival CO and Hi 21 cm data to measure the gas densities
($\rho_{\textrm{gas}}$) of seven (Sample 3; Table 2) of the combined sample of
12 galaxies and studied the correlation between $\rho_{\textrm{gas}}$ and
B${}_{\textrm{eq}}$ in these galaxies. We also studied the magnetic field-gas
connection through an indirect measurement of their coupling coefficient using
radio$-$FIR correlations of the galaxies in Sample 1.

The paper is organized as follows. The analysis of the data is discussed in
Section 2. In Section 3, we present the results of our analysis, including the
correlation between magnetic fields, SFRSDs and gas densities of the seven
galaxies in Sample 1. In Section 4, we have extended our study to include a
sample of five galaxies of Basu et al. (2012a) . We discuss the results in
Section 5. A summary of this paper is presented in Section 6.

##  2 Data Analysis

As can be seen in Table 1, six of the seven galaxies in Sample 1 are spirals
of varying inclination angles. The seventh galaxy NGC 4449 is a dwarf
irregular galaxy. Basic information about the seven sample galaxies, including
their types, distances, inclination angles, position angles, angular
resolutions, spatial resolutions, and RMS noise obtained on the GMRT and VLA
images are also listed in Table 1. The distances, inclination angles, and
position angles of the galaxies were taken from Dale et al. (2009). Radio
observations and the data reduction procedures are discussed in detail in Roy
& Manna (2021). Briefly, we used GMRT 0.33 GHz observations (covering
0.309$-$0.342 GHz) and archival VLA observations at 1.4 and $\sim$6 GHz to
derive non-thermal emission maps for each galaxy. We used H$\alpha$ and
24$\mu$m observations of the seven galaxies to model free-free emission from
them and subsequently, we subtracted the modelled free-free emission from the
observed radio emission to get the non-thermal radio maps at 0.33, 1.4 and
$\sim$6 GHz (Roy & Manna, 2021). To generate the non-thermal spectral index
maps, we used the non-thermal radio maps at 0.33 and $\sim$6 GHz for NGC 2683,
NGC 3627, NGC 4096, and NGC 4449. For the rest of the galaxies (NGC 4490, NGC
4826, and NGC 5194), we used non-thermal images at 0.33 and 1.4 GHz to
generate the non-thermal spectral index maps (Roy & Manna, 2021). In the
following subsections, we present the analysis of other ancillary data and
relevant measurements.

Table 1: Details of the seven sample galaxies. Note that the images at 0.33 GHz were obtained from observations with the GMRT reported in Roy & Manna (2021) while those at 1.4 GHz were obtained from archival VLA data. The distances to the galaxies were taken from Dale et al. (2009). Galaxies with an asterisk are those for which spatially-resolved CO data are available. Name | Class | Distance | Inclination | Position | uv | Angular | Spatial | RMS | RMS | VLA
---|---|---|---|---|---|---|---|---|---|---
|  | (Mpc) | angle | angle | range | resolution | resolution | (0.33 GHz) | (1.4 GHz) | Project ID
|  |  | (deg) | (deg) |  (k$\lambda$) | $(\textrm{arcsec}^{2})$ | (pc) |  ($\mu$Jy/beam) |  ($\mu$Jy/beam) | (1.4 GHz)
NGC 2683 | Sb | 7.7 | 83 | 43 | 0.19 - 15 |  19 $\times$ 13 | 670 | 200 | 40 | AI23
NGC 3627∗ | SAB | 10. | 65 | 170 | 0.26 - 25 |  16 $\times$ 11 | 760 | 800 | 370 | AS541, AP462
NGC 4096 | SABc | 8.3 | 76 | 20 | 0.14 - 17 |  14 $\times$ 12 | 730 | 100 | 25 | 16A-013
NGC 4449 | Irregular | 4.2 | 0 | 0 | 0.15 - 15 |  26 $\times$ 15 | 360 | 300 | 180 | AB167
NGC 4490 | SBm | 8.0 | 60 | 126 | 0.13 - 14 |  19 $\times$ 18 | 560 | 230 | 100 | AA181
NGC 4826∗ | SAab | 7.5 | 60 | 120 | 0.22 - 20 |  15 $\times$ 14 | 650 | 280 | 70 | AS541
NGC 5194∗ | Sbc | 8.0 | 20 | 10 | 0.15 - 10 |  23 $\times$ 18 | 740 | 310 | 30 | AB505, AN57
Table 2:  List of different samples studied in this paper. Sample Name | Galaxies
---|---
Sample 0 | Full sample containing 46 galaxies from Spitzer LVL survey
Sample 1 | Pilot sample containing 7 galaxies from Sample 0; galaxies listed in Table 1
Sample 2 | Sample 1 + 5 galaxies (NGC 1097, NGC 4736, NGC 5055, NGC 5236 and NGC 6946)
| from Basu et al. (2012b) = 12 galaxies; used to probe the
B${}_{\textrm{eq}}$-SFRSD correlations
Sample 3 | A subset of 7 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
B${}_{\textrm{eq}}$-gas density correlations

###  2.1 Magnetic Field Strengths

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 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

$B_{\textrm{eq}}=[4\pi(K_{\textrm{0}}+1)E_{\textrm{p}}^{1-2\alpha_{nt}}\frac{f(\alpha_{nt})}{C_{4}(i)}\frac{I_{\nu}\nu^{\alpha_{nt}}}{l}]^{\frac{1}{\alpha_{nt}+3}}$
(1)

$K_{\textrm{0}}$, $E_{\textrm{p}}$, $I_{\nu}$, and $\alpha_{nt}$ are the
number density ratio of cosmic ray protons to electrons, the proton rest mass
energy, the intensity of the synchrotron emission at frequency $\nu$, and the
spectral index of synchrotron emission, respectively. $f(\alpha_{nt})$ is a
function of $\alpha_{nt}$ given as
$f(\alpha_{nt})=(2\alpha_{nt}+1)[2(\alpha_{nt}-1)c_{2}(\alpha_{nt})c_{1}^{\alpha_{nt}}]$
(Beck & Krause, 2005). $C_{4}(i)$ 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 $\gamma=(2\alpha_{nt}+1)$. $l$ 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 degree (face-on). For
galaxies with low- and moderate- inclination angles ($<75^{\circ}$), 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 have 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 centre of the galaxy and gradually declining to the edge of the
galaxy. We note that B${}_{\textrm{eq}}$ has only a weak dependence on $l$ as
$B_{\textrm{eq}}(r)=l(r)^{\frac{-1}{\alpha_{nt}+3}}$ and hence is less
sensitive to the exact choice of $l$. Values of $K_{\textrm{0}}$ and
$E_{\textrm{p}}$ were assumed to be 100 and 938.28 MeV, respectively, the same
as used by Beck & Krause (2005). Finally, we used non-thermal radio maps at
0.33 GHz ($I_{\nu}$) and spectral index maps ($\alpha_{nt}$) made using 0.33
and 1.4 or $\sim$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 $\leq$
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 non-
thermal spectral indices being predominantly flat. This could bias the
B${}_{\textrm{eq}}$ for NGC 4449.

####  2.1.1 Uncertainties on Magnetic Field Maps

The procedure we used to estimate the uncertainties on our magnetic field maps
is similar to that of Basu & Roy (2013). We used a Monte Carlo method that
generated $10^{4}$ random flux density values for each pixel in a galaxy map
at 0.33 GHz and either 1.4 GHz or 6 GHz. These flux density values have
Gaussian probability distributions with rms values equal to the measured rms
of each of the 0.33 and 1.4/6 GHz maps. For each of the $10^{4}$ intensity
maps, we computed a magnetic field map using the procedure described in the
beginning of Section 2.1. The rms of these $10^{4}$ magnetic field maps
provided us with the magnetic field uncertainty maps for each of the seven
galaxies in sample 1.

###  2.2 Star Formation Rates

Rest frame H$\alpha$ 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 H$\alpha$ and UV observations are therefore corrected for
the extinction. Dust-corrected SFRs can be estimated by combining far-
ultraviolet (FUV) and H$\alpha$ data with infrared (IR) data to exploit the
complementary strengths at different wavelengths (e.g. Kennicutt & Evans,
2012a; Buat, 1992; Meurer et al., 1995, 1999; Cortese et al., 2008; Leroy et
al., 2012). In addition to the FUV+IR and H$\alpha$+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 star formation rates of our Sample 1
galaxies using FUV+24$\mu$m, H$\alpha$+24$\mu$m, and 1.4 GHz data, which are
discussed, respectively, in the following Sections, 2.2.1, 2.2.2, and 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 IMF (Kroupa, 2001).

####  2.2.1 SFRs using FUV and 24$\mu$m Observations

To estimate SFRSD maps of the seven galaxies in Sample 1 (Table 2) using
FUV+24$\mu$m emission, we used SPITZER 24 $\mu$m IR data (Dale et al., 2009)
and GALEX FUV data (11HUGS survey; Kennicutt et al., 2008). We first convolved
both the 24 $\mu$m 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/sec/pixel and were converted to flux-density units of MJy
$\textrm{Sr}^{-1}$. We also converted the 24$\mu$m images to units of MJy
$\textrm{Sr}^{-1}$ and used the following calibration from Leroy et al. (2012)
to derive SFRSD maps for the sample galaxies:

$\Sigma_{\textrm{SFR}}[\textrm{M}_{\odot}\textrm{yr}^{-1}\textrm{kpc}^{-2}]=0.081\;\textrm{I}_{\textrm{FUV}}[\textrm{MJy}\;\textrm{sr}^{-1}]+0.032\;\textrm{I}_{24\mu\textrm{m}}[\textrm{MJy}\;\textrm{sr}^{-1}]$
(2)

The uncertainties of the coefficients are $\sim$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 emission which are not associated with recent star
formation, etc. (e.g Kennicutt & Evans, 2012b; Leroy et al., 2012).

####  2.2.2 SFRs using H$\alpha$ and 24$\mu$m Observations

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

$\Sigma_{\textrm{SFR}}[\textrm{M}_{\odot}\textrm{yr}^{-1}\textrm{kpc}^{-2}]=634.0\;\textrm{I}_{\textrm{H}\alpha}[\textrm{erg}\;\textrm{s}^{-1}\;\textrm{sr}^{-1}]+0.0025\;\textrm{I}_{24\mu\textrm{m}}[\textrm{MJy}\;\textrm{sr}^{-1}]$
(3)

####  2.2.3 SFRs using 1.4 GHz Observations

Our 1.4 GHz non-thermal 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-$\Sigma_{\textrm{SFR}}$
calibration.

$\frac{\textrm{SFR}_{\textrm{1.4GHz}}}{\textrm{M}_{\odot}\textrm{yr}^{-1}}=6.35\times
10^{-29}\frac{\textrm{L}_{\textrm{1.4GHz}}}{\textrm{erg}\>\textrm{Hz}^{-1}\textrm{s}^{-1}}$
(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${}_{\textrm{FUV}}$) of
the GALEX FUV bands were measured using Table 1 from Bianchi et al. (2017) and
intrinsic fluxes ($\textrm{F}_{\textrm{intrinsic}}$) were estimated from the
following formula:

$\textrm{A}_{\textrm{FUV}}=-2.5\times
log[\frac{\textrm{F}_{\textrm{observed}}}{\textrm{F}_{\textrm{intrinsic}}}]$
(5)

The extinction percentage 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. The extinctions were computed using E(B-V) values along the line of sight to the sample galaxies from Bianchi et al. (2017). Name | Percentage extinction
---|---
NGC 2683 | 22
NGC 3627 | 23
NGC 4096 | 21
NGC 4449 | 15
NGC 4490 | 15
NGC 4826 | 13
NGC 5194 | 27

###  2.3 Gas Densities

Atomic hydrogen (Hi) 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 $\sim$5 and 6 km/s, 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 HI Nearby Galaxy Survey (THINGS; Walter et al., 2008) used VLA
observations to obtain very high spectral ($\leq$ 5.2 km/s) and spatial ($\sim
6^{{}^{\prime\prime}}$) resolution maps of nearby galaxies at 21cm. We used
the publicly available 21cm moment maps from this THINGS survey to estimate
the distribution of Hi in the three galaxies for which CO data are available.
All CO and Hi 21 cm maps were convolved and regridded to a common resolution
and pixel size of the non-thermal radio maps. Gas densities were estimated
(for NGC 3627, NGC 4826 and NGC 5194) following Basu & Roy (2013) assuming CO
to H2 conversion factor of 2 $\times 10^{20}$
$(\textrm{K}\;\textrm{km}\;\textrm{s}^{-1})^{-1}$ (e.g. Bolatto et al., 2013).
A line ratio of 0.8 was assumed to convert $\textrm{CO}_{\textrm{J=2-1}}$ to
$\textrm{CO}_{\textrm{{J=1-0}}}$ (e.g. Leroy et al., 2009). We accounted for
the contribution of helium to the gas density using $\rho_{\textrm{gas}}$=1.36
$\times$ ($\rho_{\textrm{H{\sc i}}}$ \+ $\rho_{\textrm{H}_{2}}$). Line of
sight depths were assumed to be 300 and 400 pc for molecular and atomic gas,
respectively (Basu & Roy, 2013).

##  3 Results

###  3.1 Magnetic Fields in the Galaxies

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 & 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 $\sim$ 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 the axes are
normalized by their maximum values. Here, we have averaged the magnetic field
strengths over an annular elliptical region of width equal to the beam size of
the corresponding map. Position and inclination angle (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
regions) with field strengths up to 50 $\mu$G. Field strengths fall by
$\sim$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.

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

Figure 7 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.

Table 4: Statistical uncertainties on mean magnetic fields for galaxies in Sample 1. Name | Statistical uncertainty
---|---
| on mean magnetic fields
| ($\mu$G)
NGC 2683 | 0.06
NGC 3627 | 0.17
NGC 4096 | 0.04
NGC 4449 | 0.18
NGC 4490 | 0.06
NGC 4826 | 0.11
NGC 5194 | 0.02

Figure 1: The equipartition magnetic field maps of NGC 2683, NGC 3627, NGC
4449 and NGC 4096 (clockwise from top left) (Sample 1). Non-thermal radio
contours at 1.4 GHz are overlaid on magnetic field maps. The magnetic field
strengths are shown in color with non-thermal emission at 1.4 GHz shown as
overlaid contours. Contour levels are presented below each panel in the
figure. The circle in the bottom-left corner of the panels indicates the
angular resolution of the maps. The uncertainties on mean magnetic fields are
0.06$\mu$G, 0.17$\mu$G, 0.04$\mu$G and 0.18$\mu$G for the above galaxies,
respectively.

Figure 2: The equipartition magnetic field maps of NGC 4490 (top left), NGC
4826 (top right) and NGC 5194 (bottom left). The magnetic field strengths are
shown in color with non-thermal emission at 1.4 GHz shown as overlaid
contours. Contour levels are presented below each panel in the figure. The
circle in the bottom-left corner of the panels indicates the angular
resolution of the maps. The uncertainties on mean magnetic fields are
0.06$\mu$G, 0.11$\mu$G and 0.02$\mu$G, respectively. The bottom right panel
presents the radial variation of magnetic field strengths with galactocentric
distance for all seven galaxies in Sample 1.

###  3.2 Star Formation Rates in the Galaxies

We have estimated the global, galaxy-averaged SFRs of Sample 1 galaxies using
1.4 GHz, FUV+24$\mu$m, and H$\alpha$+24$\mu$m emission using calibrations
discussed in Section 2.2. Globally integrated star formation rates 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$\mu$m emission by a factor of 2.2.

As discussed in Section 2.2, we have estimated SFRSD maps of the seven
galaxies (Sample 1) using FUV+24$\mu$m, H$\alpha$+24$\mu$m and 1.4GHz
emission. We show SFRSD maps of the seven galaxies in the Appendix (Figures
8-9), where SFRSDs estimated using 1.4 GHz and FUV+24$\mu$m emission are shown
in contours and colors, respectively. In the Appendix (Figures 10-11), we also
present the SFRSD maps estimated using H$\alpha$+24$\mu$m and 1.4 GHz emission
in colors and contours, respectively. The SFRSD maps of each galaxy in Figures
8$-$11 are shown in the same color scale and contours. To determine the radial
variation of SFRSDs, we have averaged the SFRSD maps of our sample galaxies
over tilted rings centred on the optical centre of each galaxy using their
inclinations and position angles. The width of the tilted rings was taken to
be equal to the beam size of the corresponding image. Figure 3 shows the
radial variation of the average SFRSD, derived using FUV+24$\mu$m and
H$\alpha$+24$\mu$m 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
$1\sigma$ statistical uncertainties, with those derived using FUV+24$\mu$m and
H$\alpha$+24$\mu$m data. Azimuthally averaged SFRSDs of all the seven galaxies
decrease gradually towards the outer region and drop by a factor of 6 to 8 at
the edge.

Table 5: Galaxy-averaged star formation rates of the galaxies in Sample 1, using 1.4 GHz, FUV+24$\mu$m, and H$\alpha$+24$\mu$m data. The uncertainties on the SFR values are $\approx$ 30%. Name | SFR from 1.4 GHz (M⊙yr-1) | SFR from FUV+24$\mu$m (M⊙yr-1) | SFR from H$\alpha$+24$\mu$m (M⊙yr-1)
---|---|---|---
NGC 2683 | 0.28 | 0.25 | 0.33
NGC 3627 | 1.56 | 2.00 | 1.84
NGC 4096 | 0.42 | 0.35 | 0.38
NGC 4449 | 0.37 | 0.38 | 0.32
NGC 4490 | 4.63 | 2.13 | 2.30
NGC 4826 | 0.63 | 0.73 | 0.78
NGC 5194 | 4.16 | 3.88 | 3.65

Figure 3: The variation of SFRSDs (normalized), estimated using FUV+24$\mu$m
(left panel) and H$\alpha$+24$\mu$m (right panel) emission as a function of
galactocentric distance (normalized) for all seven galaxies in Sample 1.

###  3.3 Details of the Individual Galaxies of Sample 1

(i) NGC 2683: In this galaxy, Krause et al. (2020) found very weak linear
polarisation 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 $\approx$31 $\mu$G and the
outer region of the disk has an average value of $\approx$19 $\mu$G (see
Figure 1 and Table 6).

Wiegert et al. (2015) used WISE 22 $\mu$m data to estimate a galaxy-averaged
SFR of $\approx$0.09 M⊙yr-1 for NGC 2683. From our analysis, integrated SFR
was measured to be $\sim$0.24 M⊙yr-1 and $\sim$0.28 M⊙yr-1 using FUV+24$\mu$m
and 1.4 GHz radio emission, respectively. However, we note that Wiegert et al.
(2015) used a distance of 6.27 Mpc for this galaxy, but we have used a
distance of 7.7 Mpc. The SFR is estimated to be 0.16 M⊙yr-1 using FUV+24$\mu$m
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 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$\pm$2 $\mu$G,
assuming a constant non-thermal 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$\pm$1 $\mu$G. They suggested
two distinct magnetic field components of NGC 3627: one for the spiral arms
and another for the inter-arm regions. We have separately studied
equipartition magnetic fields in the arm and interarm regions of the galaxy.
We find that the central region and the edges of the extended bar have
magnetic field strengths of $\approx$ 34 $\mu$G (see Figure 1). The arm region
has a field strength of $\approx$28 $\mu$G (see Table 6). However, the
magnetic field strength in the interarm regions has values $\approx$21 $\mu$G.
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 $\approx$2.0 M⊙yr-1 and $\approx$1.56
M⊙yr-1 from FUV+24$\mu$m 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 $\approx$21 $\mu$G at the centre to $\approx$12 $\mu$G 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 $\approx$ 20
$\mu$G; this is presumably due to its high inclination. The outer part of the
galaxy has an average field strength of $\approx$14 $\mu$G. 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 $\mu$m$-$SFR calibration to measure a
galaxy-averaged SFR of 0.27$\pm$0.02 M⊙yr-1. Our measurement of the galaxy-
averaged SFR is $\approx$0.35 M⊙yr-1 and $\approx$0.43 M⊙yr-1 using
FUV+24$\mu$m 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 $\approx$14 $\mu$G. These authors
also used polarization emission to estimate a regular field of $\approx$8
$\mu$G. 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 have
replaced the pixel values with $\alpha_{\rm nt}<0.55$ with $\alpha_{\rm
nt}=0.55$ while computing the magnetic field for NGC 4449 (see Section 2.1).
The average magnetic field strength is $\approx$17 $\mu$G in this galaxy,
which is comparable to the findings of Chyży et al. (2000).

Our measurements of the galaxy-averaged SFR are $\approx$0.38 M⊙yr-1 and
$\approx$0.37 M⊙yr-1 using FUV+24$\mu$m 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-Wroczyński et al. (2016) observed NGC 4490 at 0.61 GHz
using the GMRT, and at 4.86 $\&$ 8.44 GHz using VLA + Effelsberg. The authors
used these observations to find a mean equipartition magnetic field of
21.9$\pm$2.9 $\mu$G, with typical field strengths in the range of 18 $\mu$G to
40 $\mu$G. We have found a typical equipartition magnetic field strength of
$\approx$40 $\mu$G in the central region, which decreases to $\approx$17
$\mu$G in the outer region (see Figure 2); these values are consistent with
the estimates of Nikiel-Wroczyński et al. (2016). We find a relatively lower
magnetic field strength of $\approx$15 $\mu$G 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 ($\approx$4.63 M⊙yr-1) using 1.4 GHz radio
emission but a factor of $\sim$2 lower SFR (2.13 M⊙yr-1) using the
FUV+24$\mu$m emission (Table 5). Extinction corrections for NGC 4490 are
believed to be higher than those typically assumed and this may lead to an
underestimation of the SFR while using the FUV+24$\mu$m 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 $\approx$38
$\mu$G and $\approx$20 $\mu$G, respectively (see Figure 2 and Table 6). We
find galaxy-averaged SFR of $\approx$0.73 M⊙yr-1 and $\approx$0.63 M⊙yr-1
using FUV+24$\mu$m 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 non-thermal spectral index of 0.1
and 1.1 to find an average equipartition magnetic field strength of 20 $\mu$G
using the revised formula by Beck & Krause (2005). They found a magnetic field
of 20$-$25 $\mu$G in the spiral arms, higher than the 15$-$20 $\mu$G typical
in the interarm regions. Using VLA observations at S-band (2$-$4 GHz)
frequencies, Kierdorf et al. (2020) found the field strength of turbulent and
regular components of the magnetic field in the arm regions of 18$-$24 $\mu$G
and 8$-$16 $\mu$G, respectively. We find an equipartition magnetic field
strength of $\approx$25 $\mu$G in the arm region and $\approx$18 $\mu$G in the
interarm region (see Table 6). The peripheral region has a magnetic field of
$\approx$12 $\mu$G, while the overlapping region between NGC 5194 and NGC 5195
has an average B${}_{\textrm{eq}}$ of $\approx$16 $\mu$G. 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 H$\alpha$+24$\mu$m and H$\alpha$+Pa$\alpha$ emission (Kennicutt et
al., 2007). SFRSDs in different regions were found to be in the range of 0.10
to 0.46 M⊙yr-1kpc-2. Our estimates using the two tracers are consistent with
the estimates of Kennicutt et al. (2007) (See Figures 9 & 11). Furthermore, we
find that the galaxy-integrated SFR derived using FUV+24$\mu$m ($\approx$3.88
M⊙yr-1) and 1.4 GHz data ($\approx$4.16 M⊙yr-1) are consistent with each
other, within 1-sigma statistical uncertainty.

Table 6: Magnetic field strengths in different regions of the galaxies in Sample 1. For the irregular galaxy NGC 4449, we could only measure the galaxy-integrated magnetic field. We have separated the two nearly face-on galaxies (NGC 3627 and NGC 5194) into arm and inter-arm regions. For the rest of the galaxies, we could not separate the arm and inter-arm region due to their higher inclinations. Galaxy | Galaxy-average |  B${}_{\textrm{eq}}$ in |  B${}_{\textrm{eq}}$ in |  B${}_{\textrm{eq}}$ in |  B${}_{\textrm{eq}}$ in
---|---|---|---|---|---
name |  B${}_{\textrm{eq}}$ | central region | disk region | arm region | inter-arm region
|  ($\mu$G) |  ($\mu$G) |  ($\mu$G) |  ($\mu$G) |  ($\mu$G)
NGC 2683 |  24$\pm$6 |  31$\pm$3 |  19$\pm$5 | – | –
NGC 3627 |  25$\pm$4 |  34$\pm$8 | – |  28$\pm$5 |  21$\pm$4
NGC 4096 |  16$\pm$4 |  21$\pm$5 |  14$\pm$3 | – | –
NGC 4449 |  17$\pm$6 | – | – | – | –
NGC 4490 |  23$\pm$10 |  40$\pm$6 |  17$\pm$7 | – | –
NGC 4826 |  23$\pm$9 |  38$\pm$8 |  20$\pm$5 | – | –
NGC 5194 |  16$\pm$6 |  34$\pm$6 | – |  25$\pm$5 |  18$\pm$4

###  3.4 Is the Minimum Energy Condition Valid for the Sample Galaxies?

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
$I_{\textrm{nt}}/I_{\textrm{FIR}}$ will be similar to the distribution of
$\textrm{B}^{1+\alpha_{nt}}$. 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 $I_{\textrm{nt}}/I_{\textrm{FIR}}$ and
$\textrm{B}_{\textrm{eq}}^{1+\alpha_{nt}}$ is expected to follow each other if
$\textrm{B}_{\textrm{eq}}$ is close to B.

To verify the validity of the minimum energy condition in our sample galaxies,
we have followed the procedure as in Hummel (1986) and Basu & Roy (2013). The
CDF of $I_{\textrm{nt}}/I_{\textrm{FIR}}$ and
$B_{\textrm{eq}}^{1+\alpha_{nt}}$ 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 $\alpha_{nt}$, $I_{\textrm{nt}}$ (both at 0.33 and 1.4
GHz), $I_{\textrm{FIR}}$ (70 $\mu$m) and magnetic fields
(B${}_{\textrm{eq}}$), 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 $I_{\textrm{nt}}/I_{\textrm{FIR}}$
and $B_{\textrm{eq}}^{1+\alpha_{nt}}$ at both 0.33 and 1.4 GHz.

We find that the CDFs of $I_{\textrm{nt}}/I_{\textrm{FIR}}$ and
$B_{\textrm{eq}}^{1+\alpha_{nt}}$ 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 5 nearby
large spiral galaxies, but with slight deviations observed in the CDFs of
$I_{\textrm{nt}}/I_{\textrm{FIR}}$ and $B_{\textrm{eq}}^{1+\alpha_{nt}}$ in
the interarm regions of the galaxies.

The observed deviation in the CDFs of $I_{\textrm{nt}}/I_{\textrm{FIR}}$ and
$B_{\textrm{eq}}^{1+\alpha_{nt}}$ 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 $\sigma$. 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 $B_{\textrm{eq}}^{1+\alpha_{nt}}$ using the deviated magnetic field
values. The CDF of $B_{\textrm{eq}}^{1+\alpha_{nt}}$ were then compared to the
observed CDF of $I_{\textrm{nt}}/I_{\textrm{FIR}}$ via a Kolmogorov-Smirnov
(KS) test. This procedure was repeated for a range of $\sigma$ from 0 to 0.2.
We find that the p-values for the KS test comparing the distributions are
maximized when $\sigma$ = 0.1. Indeed, $B_{\textrm{eq}}^{1+\alpha_{nt}}$
derived after deviating the magnetic field using $\sigma$ = 0.1 and
$I_{\textrm{nt}}/I_{\textrm{FIR}}$ are consistent with being derived from the
same distribution, with a KS test p-value of 0.41 and 0.55, when using
$I_{\textrm{nt}}$ at 0.33 and 1.4 GHz, respectively. The bottom panels in
Figure 4 show the CDFs of the two quantities for $\sigma$ = 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 $\sim$ 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.

Figure 4: The top panels show the cumulative distribution function (CDF) of
$\textrm{I}_{\textrm{nt,radio}}/\textrm{I}_{70\mu m}$ (in red) and
$\textrm{B}_{\textrm{eq}}^{1+\alpha_{nt}}$ (in blue), where
$\textrm{I}_{\textrm{nt}}$ is the non-thermal emission at 0.33 GHz (top left)
and 1.4 GHz (top right) (Sample 1). The variables are normalized by their
median values. The bottom panels show the same but now with the magnetic field
perturbed from its measured value using $\sigma$=0.1 (see Section 3.4); the
CDFs of the $\textrm{I}_{\textrm{nt,radio}}/\textrm{I}_{70\mu m}$ and
$\textrm{B}_{\textrm{eq}}^{1+\alpha_{nt}}$ are now consistent with being
derived from the same distribution.

###  3.5 Correlation Between Magnetic Fields and SFRSDs

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
$\approx$360$-$760 pc (Table 1). For the seven sample galaxies, we used the
SFRSD maps estimated using the FUV+24$\mu$m emission. The correlations between
magnetic fields and SFRSDs for the seven galaxies are shown in Figure 5. Each
point represents the logarithms of equipartition magnetic fields and SFRSD
values that are averaged over the beam size of the corresponding maps. da
Silva et al. (2014) found that SFR calibrations could be biased and strongly
affected by stochasticity at small spatial scales where the star formation
rate is low ($\leq$ 10-2.5 M$\odot\textrm{yr}^{-1}$); we have therefore
excluded regions of low star formation rates from the correlation study.

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
$\textrm{B}=\textrm{B}_{0}\ \left(\Sigma_{\textrm{SFR}}\right)^{\eta}$ 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 have 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 ($\eta$) in the range of $\approx 0.27-0.40$ but that the slope is
relatively lower for NGC 4449 with $\eta\approx 0.18$. Averaging over the
slope of all galaxies in Sample 1, we find a mean slope of $0.32\pm 0.06$.

Figure 5: The correlation between magnetic fields and SFRSD for the combined sample of 12 galaxies (Sample 2, Table 2). For the seven galaxies in Sanple 1, the SFRSD estimates shown in the plots were derived using $\textrm{FUV}+24\mu m$ (Section 3.5). The SFRSD estimates for the five galaxies from Basu et al. (2012a) (Sample 2) were derived using $\textrm{H}\alpha+24\mu m$ (Section 4) The red line shows a linear fit to the data points. The black dashed lines show the $\pm 1\sigma$ vertical scatter. Table 7: Best-fit parameters and the scatter of the correlation between magnetic fields and SFRSDs for the seven galaxies in Sample 1. The data were fitted with a power law of the form B=B${}_{0}(\Sigma_{\textrm{SFR}})^{\eta}$. Name | Slope ($\eta$) | Intercept (B0) (log($\mu$G)) | Intercept (B0) ($\mu$G) | Scatter
---|---|---|---|---
NGC 2683 | 0.34 $\pm$ 0.04 | 2.10 $\pm$ 0.07 | 125 $\pm$ 1.2 | 0.05
NGC 3627 | 0.31 $\pm$ 0.03 | 1.71 $\pm$ 0.03 | 51 $\pm$ 1.1 | 0.04
NGC 4096 | 0.33 $\pm$ 0.04 | 1.80 $\pm$ 0.08 | 63 $\pm$ 1.2 | 0.05
NGC 4449 | 0.18 $\pm$ 0.03 | 1.64 $\pm$ 0.04 | 43 $\pm$ 1.1 | 0.03
NGC 4490 | 0.27 $\pm$ 0.02 | 1.90 $\pm$ 0.03 | 79 $\pm$ 1.1 | 0.06
NGC 4826 | 0.38 $\pm$ 0.02 | 1.80 $\pm$ 0.02 | 63 $\pm$ 1.0 | 0.05
NGC 5194 | 0.40 $\pm$ 0.01 | 2.00 $\pm$ 0.02 | 100 $\pm$ 1.0 | 0.07

###  3.6 Correlation Between Magnetic Fields and Gas Densities

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${}_{\textrm{eq}}$ and SFRSDs, we have studied the correlations between
B${}_{\textrm{eq}}$ 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${}_{\textrm{eq}}$ and gas density data points. The
scatters of the three correlations are shown in dashed lines in all the
figures.

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

Figure 6:  The correlations between magnetic fields ($\mu$G) and gas densities
(gm/cm-3) for seven galaxies of Sample 3 (Table 2). The red line shows a
linear fit to the data points. The black dashed lines show the $\pm 1\sigma$
vertical scatter.

##  4 Extending the sample with 5 galaxies from existing GMRT observations

As mentioned earlier, a study of B${}_{\textrm{eq}}$ and radio-FIR
correlations for a sample of five large nearly face-on galaxies was carried
out by Basu et al. (2012a, b); Basu & Roy (2013), using low-radio frequency
observations at 0.33 and 1.4 GHz at sub-kpc linear resolutions. In this paper,
we expand our study of spatially-resolved correlations between magnetic
fields, gas densities, and SFRSDs by including these five galaxies.

We refer readers to Basu et al. (2012a) for a detailed discussion of their
sample, GMRT observations, data reduction procedures, and estimation of non-
thermal spectral indices. It is to be noted that the modelling of the thermal
free-free emission from these galaxies is performed in the same way as was
done for our seven galaxies in Sample 1.

We have estimated the SFRSD maps of these five galaxies using H$\alpha$ data
along with 24 $\mu$m IR data. We obtained H$\alpha$ maps of four of the
galaxies, NGC 1097, NGC 4736, NGC 5055, and NGC 6946 from the ancillary data
at the SINGS website111https://irsa.ipac.caltech.edu/data/SPITZER/SINGS/ and
obtained the H$\alpha$ map of NGC 5236 from 11HUGS (Kennicutt et al., 2008).
We used H$\alpha$ and MIPS 24 $\mu$m data in combination to derive the SFRSD
maps of these galaxies using the calibration from Leroy et al. (2012)
(Equation 3, Section 2.2). To estimate the equipartition magnetic field
strengths of these five galaxies, we have used the non-thermal radio maps at
0.33 and 1.4 GHz from Basu et al. (2012a). The correlations between
equipartition magnetic fields and SFRSDs are shown in Figure 5 where, similar
to the previous correlation studies, each point represents the logarithms of
magnetic fields and SFRSD values that are averaged over the beam size. Similar
to the previous correlations (Section 3.5), we used _orthogonal distance
regression_ in Scipy to fit a power law to the data. We have provided the
best-fit parameters of the power-law fit in Table 8. The scatters of all five
correlations (presented in Table 8) are shown in dashed lines in all the
figures. We find a mean exponent of 0.30$\pm$0.05 for the five galaxies where
the exponent of individual galaxies varies from $\approx$0.25 to
$\approx$0.38.

We have computed maps of cold gas densities of four out of the five galaxies;
NGC 4736, NGC 5055, NGC 5236 and NGC 6946, using the atomic and molecular gas
surface density maps from Basu & Roy (2013). The assumed parameters are taken
to be the same as described in Section 2.3. For the remaining galaxy, NGC
1097, we could not measure gas densities as there are no archival CO data
available for the galaxy. Following the procedures of Section 3.5, we have
also studied the spatially-resolved correlation between equipartition magnetic
fields and gas densities for the four sample galaxies, which are shown in
Figure 6. The best-fit parameters are presented in Table 9. The exponents of
the individual galaxies vary between $\approx$0.25 to $\approx$0.44 where the
mean exponent is found to be 0.35$\pm$0.07.

Table 8: Best-fit parameters and the scatter of the correlation between magnetic fields and SFRSDs for the five galaxies in Basu et al. (2012a) (Sample 2). The data were fitted with a power law of the form B=B${}_{0}(\Sigma_{\textrm{SFR}})^{\eta}$. Name | Slope ($\eta$) | Intercept (B0) (log($\mu$G)) | Intercept (B0) ($\mu$G) | Scatter
---|---|---|---|---
NGC 1097 | 0.27 $\pm$ 0.01 | 1.61 $\pm$ 0.01 | 41 $\pm$ 1.0 | 0.02
NGC 4736 | 0.32 $\pm$ 0.02 | 1.78 $\pm$ 0.05 | 60 $\pm$ 1.1 | 0.04
NGC 5055 | 0.27 $\pm$ 0.04 | 1.26 $\pm$ 0.02 | 18 $\pm$ 1.0 | 0.02
NGC 5236 | 0.38 $\pm$ 0.07 | 1.91 $\pm$ 0.02 | 81 $\pm$ 1.0 | 0.08
NGC 6946 | 0.25 $\pm$ 0.03 | 1.62 $\pm$ 0.05 | 42 $\pm$ 1.1 | 0.05
Table 9: Best-fit parameters and the scatter of the correlation between spatially-resolved magnetic fields and gas densities for the seven galaxies in Sample 3. Galaxies with an asterisk are from the sample of Basu et al. (2012a). Name | Exponent | Scatter
---|---|---
NGC 3627 | 0.40 $\pm$ 0.02 | 0.03
NGC 4826 | 0.49 $\pm$ 0.03 | 0.05
NGC 5194 | 0.53 $\pm$ 0.02 | 0.06
NGC 4736∗ | 0.44 $\pm$ 0.03 | 0.03
NGC 5055∗ | 0.25 $\pm$ 0.02 | 0.02
NGC 5236∗ | 0.40 $\pm$ 0.03 | 0.04
NGC 6946∗ | 0.31 $\pm$ 0.03 | 0.04

##  5 Discussion

Understanding the relationship between the physical condition of the
interstellar medium (ISM) and the star formation process is crucial to
understand galaxy evolution. Gas and magnetic fields are key constituents of
the ISM and therefore it is important to study the interrelations between gas,
magnetic fields, and SFRs. Though the Kennicutt$-$Schmidt relation, i.e. the
relation between gas densities and SFRs, has been extensively studied at high
spatial resolutions in various types of nearby galaxies (e.g. Onodera et al.,
2010; Roychowdhury et al., 2015; Filho et al., 2016; Miettinen et al., 2017),
similar high-resolution observations of how the magnetic fields are related to
SFRs and gas densities are yet to be systematically investigated. Such
observations are critical to understand the validity of several models that
predict strong correlations between the magnetic fields and gas densities
(e.g. Chandrasekhar & Fermi, 1953; Fiedler & Mouschovias, 1993; Cho &
Vishniac, 2000; Groves et al., 2003) as well as magnetic fields and SFRSDs
(e.g. Niklas & Beck, 1997; Schleicher & Beck, 2013, 2016). Here, we have
studied these correlations in a sample of twelve galaxies (Sample 3) at sub-
kpc scales (see Sections 3.5, 3.6 & 4). To our knowledge, this is the first
spatially resolved study of the above correlations in nearby large galaxies.
In this section, we place these findings in the light of predictions made by
various models and in the process attempt to provide physical insights into
the interrelation between magnetic fields, gas densities, and star formation
rates at sub-kpc scales.

###  5.1 Magnetic Fields and SFRSDs

Several Magneto-Hydrodynamical simulations find that galactic magnetic fields
are amplified by gas turbulence in very short timescales (i.e. $\sim$100 Myr)
(e.g. Brandenburg & Subramanian, 2005; Beresnyak, 2012; Schober et al., 2012;
Schleicher & Beck, 2013; Bovino et al., 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 star formation rates 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-kpc scales, magnetic fields and SFRSDs, again at sub-kpc
scales, must be related as B $\propto\Sigma_{\textrm{SFR}}^{1/3}$.

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 of 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${}_{\textrm{eq}}$ and SFRSDs is
0.31$\pm$0.06, i.e B${}_{\textrm{eq}}$ $\propto$
$\Sigma_{\textrm{SFR}}^{0.31\pm 0.06}$(222The uncertainty quoted is the
scatter of the measured value of $\eta$ across the galaxies in Sample 2.),
consistent (at $<$ 1$\sigma$ 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-kpc scales.

We note that the power-law index for the correlation between
B${}_{\textrm{eq}}$ and SFRSDs for NGC 4449 was found to be $0.18\pm 0.03$,
significantly lower than for the remaining galaxies (Table 7) as well as lower
than the model prediction of B $\propto\Sigma_{\textrm{SFR}}^{1/3}$ Schleicher
& Beck (2013) (at $>5\sigma$ significance). For the case of NGC 4449, the
relatively flat spectral index values ($\alpha_{nt}\leq 0.55$) in $\approx
70\%$ of the galaxy meant that the magnetic field values could not be
estimated reliably for a large part of the galaxy (see Section 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-$\Sigma_{\textrm{SFR}}$ correlation depends on several ISM parameters
such as gas density ($\rho_{0}$), the fraction of turbulent kinetic energy
converted into magnetic energy ($f_{sat}$), the injection rate of turbulent
supernova energy (C) and the intercept of Kennicutt-Schmidt (KS) relation
($C_{1}$) (Equation 5).

$\textrm{B}\sim\sqrt{f_{sat}8\pi}\;\rho_{0}^{1/6}\;(\frac{C}{C_{1}})^{1/3}\;\Sigma_{\textrm{SFR}}^{1/3}.$
(6)

Schleicher & Beck (2013) predicted the intercept of the
B-$\Sigma_{\textrm{SFR}}$ correlation to be $\sim$ 26 $\mu$G assuming
$\rho_{0}$ = 10-24 g cm-3 and $f_{sat}$ $\sim$ 5 percent. We have found an
average intercept at 65$\pm$25 $\mu$G of the
B${}_{\textrm{eq}}$-$\Sigma_{\textrm{SFR}}$ correlation of the 12 galaxies in
sample 2 (see Table 7 & 8). Although the mean value is a factor of
$\approx$2.5 higher than the value predicted by Schleicher & Beck (2013), this
value is consistent with the predicted value, within the scatter (at
$\approx$1.6$\sigma$). 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 one or more of $\rho_{0}$, C, and $f_{sat}$. The intercept is
broadly insensitive to the assumed value of $\rho_{0}$ (Equation 6) and
therefore, in order to explain a factor of $\approx 2.5$ higher value of the
intercept, the actual value $\rho_{0}$ has to be higher than the assumed value
of $10^{-24}$ g cm-3 by a factor of $\approx$ 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 $\approx 16$ 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-$\Sigma_{\textrm{SFR}}$ for the dwarf galaxy IC 10 is 51
$\mu$G, similar to our findings of a higher than predicted value of the
intercept.

###  5.2 Magnetic Fields and Gas

Magnetic fields and gas are expected to be correlated as B
$\propto\sqrt{\rho_{\textrm{gas}}}$ (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$\pm$0.09 (see Section 3.6 & 4)333The uncertainty quoted is the
scatter of the measured value of k across the galaxies in Sample 3.. This
value of $k$ is consistent with the numerical simulations that predict $k$
$\approx$ 0.4$-$0.6 and also consistent with the theories that predict B
$\propto\rho_{\textrm{gas}}^{\textrm{0.5}}$. The power-law index of the
correlation between B${}_{\textrm{eq}}$ and gas densities is found to be
0.25$\pm$0.02 and 0.31$\pm$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
($\rho_{\textrm{gas}}$), and between magnetic fields and cosmic ray particles
implies that the non-thermal emission is related to the gas density as
$\textrm{I}_{nt}\propto$ $\rho_{\textrm{gas}}^{k(3+\alpha_{nt})}$ where k is
the power-law index relating magnetic fields and gas densities
($\textrm{B}_{\textrm{eq}}\propto$ $\rho_{\textrm{gas}}^{k}$) (Niklas & Beck,
1997). Further, the Kennicutt-Schmidt law and the radio-FIR correlation imply
that $\textrm{I}_{nt}$ is related to gas densities as (1)
$\textrm{I}_{nt}\propto$ $\rho_{\textrm{gas}}^{m(n+1)}$ for optically thin
dust to UV photons and (2) $\textrm{I}_{nt}\propto$ $\rho_{\textrm{gas}}^{mn}$
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):

$k=\frac{(n+1)m}{3+\alpha_{nt}};\textrm{Optically thin dust}$ (7)

$k=\frac{nm}{3+\alpha_{nt}};\textrm{Optically thick dust}$ (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 Hi 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 ($\alpha_{nt}$) from
Roy & Manna (2021), and (iii) a Kennicutt-Schmidt power-law index of
1.4$\pm$0.15 (Kennicutt, 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.

For the optically thin case, the mean of indirectly-estimated $k$ values of
the sample of seven galaxies are $0.59\pm 0.16$ and $0.53\pm 0.19$ 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$\pm$0.04 and 0.47$\pm$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 ($\approx$1.39 and $\approx$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.

Table 10: Power law index ($k$) of the relation between magnetic fields and gas densities (B $\propto\rho^{k})$ of galaxies in Sample 1, indirectly estimated using the slope of radio-FIR correlation (m) and the slope of the Kennicutt$-$Schmidt law. See Section 5.2.1 for a discussion on these. Name | m | m | $\alpha_{nt}$ | k (Optically thin) | k (Optically thin)
---|---|---|---|---|---
| 0.33 GHz | 1.4 GHz |  | 0.33 GHz | 1.4 GHz
NGC 2683 |  0.54$\pm$0.06 |  0.91$\pm$0.07 |  -0.84$\pm$0.08 |  0.33 $\pm$ 0.04 |  0.57 $\pm$ 0.06
NGC 3627 |  0.55$\pm$0.03 |  0.85$\pm$0.13 |  -1.10$\pm$0.07 |  0.32 $\pm$ 0.03 |  0.50 $\pm$ 0.08
NGC 4096 |  0.74$\pm$0.05 |  0.90$\pm$0.04 |  -0.78$\pm$0.06 |  0.47 $\pm$ 0.04 |  0.57 $\pm$ 0.05
NGC 4449 |  0.77$\pm$0.05 |  0.65$\pm$0.04 |  -0.48$\pm$0.06 |  0.53 $\pm$ 0.05 |  0.45 $\pm$ 0.04
NGC 4490 |  0.68$\pm$0.02 |  0.75$\pm$0.02 |  -0.59$\pm$0.07 |  0.45 $\pm$ 0.03 |  0.50 $\pm$ 0.04
NGC 4826 |  1.39$\pm$0.1 |  1.47$\pm$0.08 |  -0.49$\pm$0.06 |  0.95 $\pm$ 0.09 |  1.00 $\pm$ 0.09
NGC 5194(arm) |  0.50$\pm$0.05 |  0.65$\pm$0.04 |  -0.63$\pm$0.05 |  0.33$\pm$ 0.04 |  0.43 $\pm$ 0.04
NGC 5194(interarm) |  0.73$\pm$0.11 |  1.03$\pm$0.05 |  -0.85$\pm$0.10 |  0.46$\pm$ 0.08 |  0.64 $\pm$ 0.05

##  6 Summary

  1. 1.

We made spatially resolved maps of equipartition magnetic fields in seven
galaxies (Sample 1): NGC 2683, NGC 3627, NGC 4096, NGC 4449, NGC 4490, NGC
4826, and NGC 5194 and find that the magnetic fields are strongest near the
central region and go down by a factor of $\sim$2 at the edge of the magnetic
field maps.

  2. 2.

We have used the tightness of the spatially-resolved radio-FIR correlations to
verify the validity of the equipartition condition between magnetic fields and
cosmic ray particles for the sample galaxies. We find that the magnetic field
values may deviate from the equipartition values by $\sim$25%.

  3. 3.

We have estimated spatially resolved maps of SFRSDs of the galaxies in Sample
1 using FUV+24$\mu$m, H$\alpha$+24$\mu$m, and 1.4 GHz data. Azimuthally
averaged SFRSDs drop by a factor of 6 to 8 at the edge of the galaxies, where
SFRSD values are 5 times the rms of the maps.

  4. 4.

We also included five additional galaxies: NGC 1097, NGC 4736, NGC 5055, NGC
5236, and NGC 6946 from previous GMRT observations of Basu et al. (2012a) and
estimated their equipartition magnetic field, SFRSD and gas density maps.

  5. 5.

We studied the spatial correlation between magnetic fields and star formation
rates at $<$ 1 kpc resolution for the 12 galaxies (Sample 2) and find that
magnetic field strengths and SFRSDs are correlated with an average power-law
index of 0.31$\pm$0.06. This result is in remarkable agreement (at $<$
1$\sigma$ error) with semi-analytical model predictions of
$\textrm{B}\propto\Sigma_{\textrm{SFR}}^{1/3}$ (Schleicher & Beck, 2013,
2016).

  6. 6.

We measure an average intercept of $\approx 65$ $\mu$G from the
B-$\Sigma_{\textrm{SFR}}$ correlations of our galaxies in Sample 2. This is
higher than the predictions of Schleicher & Beck (2013) by a factor of
$\approx 2.5$, and, if confirmed with a larger sample, would imply a
significantly higher efficiency of magnetic field amplification than what is
typically assumed.

  7. 7.

We used spatially resolved gas density maps for seven (Sample 3) of the 12
galaxies, for which archival CO data was available, to find that magnetic
fields are correlated with gas densities as B
$\propto\rho_{\textrm{gas}}^{0.40\pm 0.09}$. This result is consistent with
numerical simulations that predict $k$ $\approx$ 0.4$-$0.6 and broadly
consistent (within $\approx$1 sigma uncertainty) with theories that predict B
$\propto\rho_{\textrm{gas}}^{\textrm{0.5}}$.

  8. 8.

We have indirectly estimated the power-law index ($k$) of the correlation
between the magnetic fields and the gas densities using the slope of the
radio-FIR correlation, the slope of the Kennicutt-Schmidt law, and the non-
thermal spectral index. The mean value of $k$, for optically thin dust, was
found to be 0.52$\pm$0.04 and 0.47$\pm$0.09 at 1.4 and 0.33 GHz respectively
for the six galaxies in Sample 1, with NGC 4826 excluded due to its high value
of $k$. This is consistent with the equipartition between magnetic fields and
gas. The anomalously high values of $k$ (1.0 and 0.95 at 1.4 and 0.33 GHz
respectively) for NGC 4826 are possibly due to the contribution of the central
AGN to the radio emission.

We have started to follow up these pilot study results with a survey of a much
larger sample of galaxies (Sample 0, Table 2). For this, we have already
observed another 24 galaxies using the upgraded GMRT (uGMRT), a Square
Kilometer Array (SKA) pathfinder facility. Sensitivities of the images from
these uGMRT observations are significantly better ($\approx 3$ times) than
those of the observations presented here and the result will be part of a
future publication. In addition, SKA precursors such as the MeerKAT will also
provide very deep images of the diffuse radio-continuum emission around nearby
galaxies. Eventually, the dramatic increase in sensitivity and $\sim$arc-sec
resolution of the SKA has the potential to significantly advance our
understanding of magnetic fields in nearby galaxies. For example, the SKA is
expected to provide sensitive images of polarised synchrotron emission from
nearby galaxies at a few GHz frequencies which would provide information on
the large-scale ordered fields on the plane of the sky (e.g. Johnston-Hollitt
et al., 2015). Further, polarised emission from nearby galaxies at $<\sim$1
GHz, where significant depolarisations take place, could be modelled through
Faraday tomography (e.g. Heald et al., 2015). A combination of the two
approaches could eventually allow us to infer the three-dimensional structure
of the magnetic fields in nearby galaxies. SKA observations will also provide
detailed images of star formation with resolutions of tens of parsecs. These
will help to identify any dependence of SFR and IMF on galaxy type, evolution
and environment within the local volume (Beswick et al., 2015).

We would like to thank Aditya Chowdhury for his help at various stages of this
research. We thank Yogesh Wadadekar, Preeti Kharb, and Dipanjan Mitra for
reading the manuscript and providing useful comments. Aritra Basu provided
their earlier published images and also suggested checking the B vs SFRSD
relation for our sample galaxies. We thank him for the above. We also thank
the anonymous referee whose comments helped significantly improve the
presentation of the paper. We thank the staff of GMRT that allowed these
observations to be made. GMRT is run by National Centre for Radio Astrophysics
of the Tata Institute of fundamental research. We acknowledge the support of
the Department of Atomic Energy, Government of India, under project no.
12-R$\&$D-TFR-5.02-0700.

## References

  * Adebahr et al. (2013) Adebahr, B., Krause, M., Klein, U., et al. 2013, A&A, 555, A23, doi: 10.1051/0004-6361/201220226
  * Bacchini et al. (2020) Bacchini, C., Fraternali, F., Iorio, G., et al. 2020, A&A, 641, A70, doi: 10.1051/0004-6361/202038223
  * Basu et al. (2012a) Basu, A., Mitra, D., Wadadekar, Y., & Ishwara-Chandra, C. H. 2012a, MNRAS, 419, 1136, doi: 10.1111/j.1365-2966.2011.19764.x
  * Basu & Roy (2013) Basu, A., & Roy, S. 2013, MNRAS, 433, 1675, doi: 10.1093/mnras/stt845
  * Basu et al. (2012b) Basu, A., Roy, S., & Mitra, D. 2012b, ApJ, 756, 141, doi: 10.1088/0004-637X/756/2/141
  * Basu et al. (2017) Basu, A., Roychowdhury, S., Heesen, V., et al. 2017, MNRAS, 471, 337, doi: 10.1093/mnras/stx1567
  * Batchelor (1950) Batchelor, G. K. 1950, Proceedings of the Royal Society of London Series A, 201, 405, doi: 10.1098/rspa.1950.0069
  * Beck et al. (1996) Beck, R., Brandenburg, A., Moss, D., Shukurov, A., & Sokoloff, D. 1996, ARA&A, 34, 155, doi: 10.1146/annurev.astro.34.1.155
  * Beck & Krause (2005) Beck, R., & Krause, M. 2005, Astronomische Nachrichten, 326, 414, doi: 10.1002/asna.200510366
  * Bell (2003) Bell, E. F. 2003, ApJ, 586, 794, doi: 10.1086/367829
  * Beresnyak (2012) Beresnyak, A. 2012, Phys. Rev. Lett., 108, 035002, doi: 10.1103/PhysRevLett.108.035002
  * Beswick et al. (2015) Beswick, R., Brinks, E., Perez-Torres, M., et al. 2015, in Advancing Astrophysics with the Square Kilometre Array (AASKA14), 70.  https://arxiv.org/abs/1412.5810
  * Bianchi et al. (2017) Bianchi, L., Shiao, B., & Thilker, D. 2017, ApJS, 230, 24, doi: 10.3847/1538-4365/aa7053
  * Bolatto et al. (2013) Bolatto, A. D., Wolfire, M., & Leroy, A. K. 2013, ARA&A, 51, 207, doi: 10.1146/annurev-astro-082812-140944
  * Bovino et al. (2013) Bovino, S., Schleicher, D. R. G., & Schober, J. 2013, New Journal of Physics, 15, 013055, doi: 10.1088/1367-2630/15/1/013055
  * Brandenburg & Subramanian (2005) Brandenburg, A., & Subramanian, K. 2005, Phys. Rep., 417, 1, doi: 10.1016/j.physrep.2005.06.005
  * Buat (1992) Buat, V. 1992, A&A, 264, 444
  * Chandrasekhar & Fermi (1953) Chandrasekhar, S., & Fermi, E. 1953, ApJ, 118, 113, doi: 10.1086/145731
  * Cho & Vishniac (2000) Cho, J., & Vishniac, E. T. 2000, ApJ, 539, 273, doi: 10.1086/309213
  * Chyży et al. (2000) Chyży, K. T., Beck, R., Kohle, S., Klein, U., & Urbanik, M. 2000, A&A, 355, 128.  https://arxiv.org/abs/astro-ph/0001205
  * Chyży & Buta (2008) Chyży, K. T., & Buta, R. J. 2008, ApJ, 677, L17, doi: 10.1086/587958
  * Chyży et al. (2011) Chyży, K. T., Weżgowiec, M., Beck, R., & Bomans, D. J. 2011, A&A, 529, A94, doi: 10.1051/0004-6361/201015393
  * Clemens et al. (1999) Clemens, M. S., Alexander, P., & Green, D. A. 1999, MNRAS, 307, 481, doi: 10.1046/j.1365-8711.1999.02594.x
  * Cortese et al. (2008) Cortese, L., Boselli, A., Franzetti, P., et al. 2008, MNRAS, 386, 1157, doi: 10.1111/j.1365-2966.2008.13118.x
  * Crutcher (1999) Crutcher, R. M. 1999, in Science with the Atacama Large Millimeter Array (ALMA), 12
  * da Silva et al. (2014) da Silva, R. L., Fumagalli, M., & Krumholz, M. R. 2014, MNRAS, 444, 3275, doi: 10.1093/mnras/stu1688
  * Dale et al. (2009) Dale, D. A., Cohen, S. A., Johnson, L. C., et al. 2009, ApJ, 703, 517, doi: 10.1088/0004-637X/703/1/517
  * Dumas et al. (2011) Dumas, G., Schinnerer, E., Tabatabaei, F. S., et al. 2011, AJ, 141, 41, doi: 10.1088/0004-6256/141/2/41
  * Elmegreen (1981) Elmegreen, B. G. 1981, ApJ, 243, 512, doi: 10.1086/158616
  * Fiedler & Mouschovias (1993) Fiedler, R. A., & Mouschovias, T. C. 1993, ApJ, 415, 680, doi: 10.1086/173193
  * Filho et al. (2016) Filho, M. E., Sánchez Almeida, J., Amorín, R., et al. 2016, ApJ, 820, 109, doi: 10.3847/0004-637X/820/2/109
  * Fletcher et al. (2011) Fletcher, A., Beck, R., Shukurov, A., Berkhuijsen, E. M., & Horellou, C. 2011, MNRAS, 412, 2396, doi: 10.1111/j.1365-2966.2010.18065.x
  * Groves et al. (2003) Groves, B. A., Cho, J., Dopita, M., & Lazarian, A. 2003, PASA, 20, 252, doi: 10.1071/AS03016
  * Heald et al. (2015) Heald, G., Beck, R., de Blok, W. J. G., et al. 2015, in Advancing Astrophysics with the Square Kilometre Array (AASKA14), 106.  https://arxiv.org/abs/1501.00408
  * Heesen et al. (2009) Heesen, V., Beck, R., Krause, M., & Dettmar, R. J. 2009, A&A, 494, 563, doi: 10.1051/0004-6361:200810543
  * Heesen et al. (2014) Heesen, V., Brinks, E., Leroy, A. K., et al. 2014, AJ, 147, 103, doi: 10.1088/0004-6256/147/5/103
  * Hummel (1986) Hummel, E. 1986, A&A, 160, L4
  * Irwin et al. (2012) Irwin, J., Beck, R., Benjamin, R. A., et al. 2012, AJ, 144, 44, doi: 10.1088/0004-6256/144/2/44
  * Johnston-Hollitt et al. (2015) Johnston-Hollitt, M., Govoni, F., Beck, R., et al. 2015, in Advancing Astrophysics with the Square Kilometre Array (AASKA14), 92.  https://arxiv.org/abs/1506.00808
  * Jurusik et al. (2014) Jurusik, W., Drzazga, R. T., Jableka, M., et al. 2014, A&A, 567, A134, doi: 10.1051/0004-6361/201323060
  * Kazes et al. (1991) Kazes, I., Troland, T. H., & Crutcher, R. M. 1991, A&A, 245, L17
  * Kennicutt (1998a) Kennicutt, Robert C., J. 1998a, ApJ, 498, 541, doi: 10.1086/305588
  * Kennicutt (1998b) —. 1998b, ARA&A, 36, 189, doi: 10.1146/annurev.astro.36.1.189
  * Kennicutt et al. (2008) Kennicutt, Robert C., J., Lee, J. C., Funes, J. G., et al. 2008, ApJS, 178, 247, doi: 10.1086/590058
  * Kennicutt et al. (2003) Kennicutt, Robert C., J., Armus, L., Bendo, G., et al. 2003, PASP, 115, 928, doi: 10.1086/376941
  * Kennicutt et al. (2007) Kennicutt, Robert C., J., Calzetti, D., Walter, F., et al. 2007, ApJ, 671, 333, doi: 10.1086/522300
  * Kennicutt & Evans (2012a) Kennicutt, R. C., & Evans, N. J. 2012a, ARA&A, 50, 531, doi: 10.1146/annurev-astro-081811-125610
  * Kennicutt & Evans (2012b) —. 2012b, ARA&A, 50, 531, doi: 10.1146/annurev-astro-081811-125610
  * Kierdorf et al. (2020) Kierdorf, M., Mao, S. A., Beck, R., et al. 2020, A&A, 642, A118, doi: 10.1051/0004-6361/202037847
  * Kim et al. (2001) Kim, J., Balsara, D., & Mac Low, M.-M. 2001, Journal of Korean Astronomical Society, 34, 333, doi: 10.5303/JKAS.2001.34.4.333
  * Krause (2009) Krause, M. 2009, in Revista Mexicana de Astronomia y Astrofisica Conference Series, Vol. 36, Revista Mexicana de Astronomia y Astrofisica Conference Series, 25–29.  https://arxiv.org/abs/0806.2060
  * Krause et al. (2006) Krause, M., Wielebinski, R., & Dumke, M. 2006, A&A, 448, 133, doi: 10.1051/0004-6361:20053789
  * Krause et al. (2020) Krause, M., Irwin, J., Schmidt, P., et al. 2020, A&A, 639, A112, doi: 10.1051/0004-6361/202037780
  * Kroupa (2001) Kroupa, P. 2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x
  * Kulsrud & Zweibel (2008) Kulsrud, R. M., & Zweibel, E. G. 2008, Reports on Progress in Physics, 71, 046901, doi: 10.1088/0034-4885/71/4/046901
  * Leroy et al. (2009) Leroy, A. K., Walter, F., Bigiel, F., et al. 2009, AJ, 137, 4670, doi: 10.1088/0004-6256/137/6/4670
  * Leroy et al. (2012) Leroy, A. K., Bigiel, F., de Blok, W. J. G., et al. 2012, AJ, 144, 3, doi: 10.1088/0004-6256/144/1/3
  * Longair (2011) Longair, M. S. 2011, High Energy Astrophysics
  * Malkan et al. (2017) Malkan, M. A., Jensen, L. D., Rodriguez, D. R., Spinoglio, L., & Rush, B. 2017, ApJ, 846, 102, doi: 10.3847/1538-4357/aa8302
  * Meurer et al. (1999) Meurer, G. R., Heckman, T. M., & Calzetti, D. 1999, ApJ, 521, 64, doi: 10.1086/307523
  * Meurer et al. (1995) Meurer, G. R., Heckman, T. M., Leitherer, C., et al. 1995, AJ, 110, 2665, doi: 10.1086/117721
  * Miettinen et al. (2017) Miettinen, O., Delvecchio, I., Smolčić, V., et al. 2017, A&A, 602, L9, doi: 10.1051/0004-6361/201731157
  * Miley (1980) Miley, G. 1980, ARA&A, 18, 165, doi: 10.1146/annurev.aa.18.090180.001121
  * Moss & Shukurov (1996) Moss, D., & Shukurov, A. 1996, MNRAS, 279, 229, doi: 10.1093/mnras/279.1.229
  * Murphy et al. (2008) Murphy, E. J., Helou, G., Kenney, J. D. P., Armus, L., & Braun, R. 2008, ApJ, 678, 828, doi: 10.1086/587123
  * Murphy et al. (2006a) Murphy, E. J., Braun, R., Helou, G., et al. 2006a, ApJ, 638, 157, doi: 10.1086/498636
  * Murphy et al. (2006b) Murphy, E. J., Helou, G., Braun, R., et al. 2006b, ApJ, 651, L111, doi: 10.1086/509722
  * Murphy et al. (2011) Murphy, E. J., Condon, J. J., Schinnerer, E., et al. 2011, ApJ, 737, 67, doi: 10.1088/0004-637X/737/2/67
  * Nikiel-Wroczyński et al. (2016) Nikiel-Wroczyński, B., Jamrozy, M., Soida, M., Urbanik, M., & Knapik, J. 2016, MNRAS, 459, 683, doi: 10.1093/mnras/stw414
  * Niklas & Beck (1997) Niklas, S., & Beck, R. 1997, A&A, 320, 54
  * Onodera et al. (2010) Onodera, S., Kuno, N., Tosaki, T., et al. 2010, ApJ, 722, L127, doi: 10.1088/2041-8205/722/2/L127
  * Pacholczyk (1970) Pacholczyk, A. G. 1970, Radio astrophysics. Nonthermal processes in galactic and extragalactic sources
  * Price & Bate (2008) Price, D. J., & Bate, M. R. 2008, MNRAS, 385, 1820, doi: 10.1111/j.1365-2966.2008.12976.x
  * Regan et al. (2001) Regan, M. W., Thornley, M. D., Helfer, T. T., et al. 2001, ApJ, 561, 218, doi: 10.1086/323221
  * Robishaw et al. (2008) Robishaw, T., Quataert, E., & Heiles, C. 2008, ApJ, 680, 981, doi: 10.1086/588031
  * Roy & Manna (2021) Roy, S., & Manna, S. 2021, MNRAS, 507, 4734, doi: 10.1093/mnras/stab2441
  * Roychowdhury et al. (2015) Roychowdhury, S., Huang, M.-L., Kauffmann, G., Wang, J., & Chengalur, J. N. 2015, MNRAS, 449, 3700, doi: 10.1093/mnras/stv515
  * Sarazin (1999) Sarazin, C. L. 1999, ApJ, 520, 529, doi: 10.1086/307501
  * Sarma et al. (2005) Sarma, A. P., Momjian, E., Troland, T. H., & Crutcher, R. M. 2005, AJ, 130, 2566, doi: 10.1086/497637
  * Schleicher & Beck (2013) Schleicher, D. R. G., & Beck, R. 2013, A&A, 556, A142, doi: 10.1051/0004-6361/201321707
  * Schleicher & Beck (2016) —. 2016, A&A, 593, A77, doi: 10.1051/0004-6361/201628843
  * Schober et al. (2012) Schober, J., Schleicher, D., Federrath, C., et al. 2012, ApJ, 754, 99, doi: 10.1088/0004-637X/754/2/99
  * Shukurov et al. (2006) Shukurov, A., Sokoloff, D., Subramanian, K., & Brand enburg, A. 2006, A&A, 448, L33, doi: 10.1051/0004-6361:200600011
  * Soida et al. (2001) Soida, M., Urbanik, M., Beck, R., Wielebinski, R., & Balkowski, C. 2001, A&A, 378, 40, doi: 10.1051/0004-6361:20011185
  * Sokoloff et al. (1998) Sokoloff, D. D., Bykov, A. A., Shukurov, A., et al. 1998, MNRAS, 299, 189, doi: 10.1046/j.1365-8711.1998.01782.x
  * Tabatabaei et al. (2013) Tabatabaei, F. S., Berkhuijsen, E. M., Frick, P., Beck, R., & Schinnerer, E. 2013, A&A, 557, A129, doi: 10.1051/0004-6361/201218909
  * Thompson et al. (2006) Thompson, T. A., Quataert, E., Waxman, E., Murray, N., & Martin, C. L. 2006, ApJ, 645, 186, doi: 10.1086/504035
  * Van Loo et al. (2015) Van Loo, S., Tan, J. C., & Falle, S. A. E. G. 2015, ApJ, 800, L11, doi: 10.1088/2041-8205/800/1/L11
  * Vargas et al. (2018) Vargas, C. J., Mora-Partiarroyo, S. C., Schmidt, P., et al. 2018, ApJ, 853, 128, doi: 10.3847/1538-4357/aaa47f
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
  * Walter et al. (2008) Walter, F., Brinks, E., de Blok, W. J. G., et al. 2008, AJ, 136, 2563, doi: 10.1088/0004-6256/136/6/2563
  * Watanabe et al. (2011) Watanabe, Y., Sorai, K., Kuno, N., & Habe, A. 2011, MNRAS, 411, 1409, doi: 10.1111/j.1365-2966.2010.17746.x
  * Wiegert et al. (2015) Wiegert, T., Irwin, J., Miskolczi, A., et al. 2015, AJ, 150, 81, doi: 10.1088/0004-6256/150/3/81
  * Yun et al. (2001) Yun, M. S., Reddy, N. A., & Condon, J. J. 2001, ApJ, 554, 803, doi: 10.1086/323145

##  Appendix A Magnetic Field Uncertainty Maps

We present here (Figure 7) magnetic field uncertainty maps of the galaxies in
Sample 1, generated using the procedure described in Section 2.1.1.

Figure 7: The magnetic field uncertainty maps (in $\mu$G) of NGC 2683 (top
left), NGC 3627 (top centre), NGC 4096 (top right), NGC 4449 (middle left),
NGC 4490 (middle centre), NGC 4826 (middle right) and NGC 5194 (bottom)
(Sample 1), shown in colour scale. Blanked regions (in white colour) in the
centre of each galaxy correspond to regions with spectral index values $\leq$
0.55.

##  Appendix B Star Formation Rate Surface Density maps

We show SFRSD maps of the seven galaxies (Sample 1) in Figures 8 and 9, where
SFRSDs estimated using 1.4 GHz and FUV+24$\mu$m emission are shown in contours
and colors, respectively. In Figures 10 and 11, we have also shown the SFRSD
maps estimated using H$\alpha$+24$\mu$m and 1.4GHz data in colors and
contours, respectively. The SFRSD maps of each galaxy in Figures 8, 9, 10, and
11 have been shown in the same color scale and contours.

Figure 8: SFRSD (M⊙yr-1kpc-2) maps of NGC 2683, NGC 3627, NGC 4449 and NGC
4096(clockwise from top left) (Sample 1). SFRSDs estimated using 1.4 GHz radio
and FUV+24$\mu$m emission are shown in contours and colors, respectively.
Contour levels are listed below each panel of the figure. The circle in the
bottom-left corner of the images indicates the angular resolution of the maps.

Figure 9: SFRSD (M⊙yr-1kpc-2) maps of NGC 4490, NGC 4826 and NGC 5194
(clockwise from top left) (Sample 1). SFRSDs estimated using 1.4 GHz radio and
FUV+24$\mu$m emission are shown in contours and colors, respectively. Contour
levels are listed below each panel of the figure. The circle in the bottom-
left corner of the images indicates the angular resolution of the maps.

Figure 10: SFRSD (M⊙yr-1kpc-2) maps of NGC 2683, NGC 3627, NGC 4449 and NGC
4096 (clockwise from top left) (Sample 1). SFRSDs estimated using 1.4 GHz
radio and H$\alpha$+24$\mu$m emission are shown in contours and colors,
respectively. Contour levels are listed below each panel of the figure. The
circle in the bottom-left corner of the images indicates the angular
resolution of the maps.

Figure 11: SFRSD (M⊙yr-1kpc-2) maps of NGC 4490, NGC 4826 and NGC 5194
(clockwise from top left) (Sample 1). SFRSDs estimated using 1.4 GHz radio and
H$\alpha$+24$\mu$m emission are shown in contours and colors, respectively.
Contour levels are listed below each panel of the figure. The circle in the
bottom-left corner of the images indicates the angular resolution of the maps.