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A&A
Volume 669, January 2023
Article Number A8
Number of page(s) 38
Section Extragalactic astronomy
DOI https://doi.org/10.1051/0004-6361/202243328
Published online 20 December 2022

© The Authors 2022

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe-to-Open model. Subscribe to A&A to support open access publication.

1. Introduction

Magnetic fields are a key ingredient for galaxy formation and evolution. They regulate the transport of highly energetic particles, the cosmic rays (Becker Tjus & Merten 2020), and are essential in any theory of star formation (Krumholz & Federrath 2019) as they can decelerate the formation of massive stars (Tabatabaei et al. 2018). Magnetic fields need to be taken into account for modelling stellar feedback, as cosmic-ray-driven winds are now thought to be behind galactic winds, which are very important for galaxy evolution (Veilleux et al. 2020); as such, steady state models have been developed (e.g., Breitschwerdt et al. 1991; Everett et al. 2008; Recchia et al. 2016; Yu et al. 2020) and simulations include them now as well (Salem & Bryan 2014; Pakmor et al. 2016; Girichidis et al. 2018; Jacob et al. 2018). Radio continuum observations of nearby galaxies show that high-energy cosmic rays are scattered in the tangled magnetic field near star-forming regions, causing winds and outflows (Tabatabaei et al. 2017). Hence, the question of origin and what regulates magnetic fields in galaxies is of wide-ranging importance for understanding galaxies as they appear in today’s Universe (Vogelsberger et al. 2020). In particular, it is now acknowledged that magnetic fields in galaxies have a high field strength, of the order of 10 μG, and so the magnetic field is dynamically important in comparison with the other phases of the interstellar medium (ISM; Beck 2007).

Observations have been able to reveal some of the structure and role that magnetic fields play in galaxies (see Beck 2015a, for an overview). There is a variety of processes able to generate the necessary seed fields, which can be distinguished between primordial and astrophysical sources. The amplification of magnetic fields in galaxies is thought to be related to the turbulence in the ISM, which is then referred to as the small-scale dynamo as the amplification happens on a scale smaller than the turbulent injection scale (Beck et al. 2012; Rieder & Teyssier 2016; Pakmor et al. 2017). The magnetic field strength is expected to saturate following a phase of exponential amplification (e.g., Bhat et al. 2019). This phase is then followed by the ordering of fields due to differential rotation. In order to generate so-called regular fields, the mean-field galactic dynamo is used; it is the leading theory that operates at scales larger than the turbulence injection scale (e.g., Brandenburg & Subramanian 2005; Chamandy 2016). Observations indicate approximate energy equipartition between the magnetic energy density and the kinetic energy density of neutral gas in the ISM (Beck 2007; Tabatabaei et al. 2008; Basu & Roy 2013; Beck 2015b).

The magnetic field is closely related to star formation, and Schleicher & Beck (2013) propose a theoretical explanation for the observed relation between the magnetic field strength (B) and the star-formation rate surface density (ΣSFR). This B–star-formation rate (SFR) relation is B∝ Σ SFR 1/3 $ B\propto \Sigma_{\rm SFR} ^{1/3} $. They assumed that the kinetic energy density of the gas is regulated by the star formation, and so this relation is the result of a saturating small-scale dynamo. However, as Steinwandel et al. (2020) point out, a similar relation is also found when the magnetic field is amplified by adiabatic compression only, with Kennicutt–Schmidt slopes of 1.4 for the atomic gas and 1.0 for the molecular gas. In galaxy samples, the magnetic field strength has so far only been analysed for global measurements (Niklas 1995; Chyży et al. 2011; Tabatabaei et al. 2017), although there are a few case studies for spatially resolved observations of individual galaxies (e.g., Tabatabaei et al. 2008; Chyży 2008; Tabatabaei et al. 2013) and several more with reviews of the observations presented in Fletcher (2010) and Beck et al. (2019). A previous study using LOw Frequency ARay (LOFAR) data for M 51 was presented by Mulcahy et al. (2014), who find magnetic fields with a strength up of to 30 μG in the central region, 10–20 μG in the spiral arms, and 10–15 μG in the inter-arm regions. What has been missing, however, is a survey of galaxies where the magnetic field strength has been calculated on a spatially resolved basis with a consistent approach. With this work, we attempt to fill this gap. We use low-frequency radio continuum observations obtained with LOFAR (van Haarlem et al. 2013), which has the advantage of low thermal emission, and take advantage of an interferometric telescope that gives us good angular and spatial resolution. Our work analyses the first statistically meaningful sample of galaxies to estimate the magnetic field strength and study both the B–ΣSFR and B–ΣH I + H2 relations at 0.3–1.2 kpc spatial scales.

Our galaxy sample consists of a mix of galaxies from the SIRTF Nearby Galaxies Survey (SINGS; Kennicutt 2003), the Continuum Halos in Nearby Galaxies: An EVLA Survey (CHANG-ES; Irwin et al. 2012), and Key Insights on Nearby Galaxies: A Far-Infrared Survey with Herschel (KINGFISH; Kennicutt 2011). We calculate magnetic field strength maps from LOFAR 144 MHz intensity maps and spectral index maps from Heesen et al. (2022). The galaxies were chosen to have rich ancillary data, and we include observations from the H I Nearby Galaxy Survey (THINGS; Walter et al. 2008) and the HERA CO-Line Extragalactic Survey (HERACLES; Leroy et al. 2009). Both are surveys that trace the atomic (H I) and molecular (H2) hydrogen in the ISM, respectively, and we use their results for the comparison of the magnetic field to the surface and energy densities of our sample. We also use maps of the SFR from Leroy et al. (2008), who calculated the SFR surface density in 23 nearby galaxies using far-ultraviolet data from the Galaxy Evolution Explorer (GALEX; Gil de Paz et al. 2007) SINGS 24 μm maps.

This paper is organised as follows. In Sect. 2 we present an overview of both the radio continuum and ancillary data. Section 3 summarises the methodology used, including the calculation of equipartition magnetic field strengths, a consideration of thermal emission, and a method to account for cosmic-ray transport. We present our results in Sect. 4 and the discussion in Sect. 5. We conclude in Sect. 6. The appendix contains more information about the correct approach for calculating equipartition magnetic field strengths (Appendix A), the atlas of magnetic fields in galaxies (Appendix B), the magnetic field–gas relations in individual sample galaxies (Appendix C), and radial magnetic field profiles for moderately inclined galaxies (Appendix D).

2. Data

2.1. LOFAR Two-metre Sky Survey

LOFAR is a radio interferometer that consists of multiple stations with combined thousands of radio antennas spread across Europe with its core in the Netherlands. The LOFAR Two-metre Sky Survey (LoTSS; Shimwell et al. 2017) is a survey observing the northern sky at a frequency of 120–168 MHz. It uses the LOFAR high band antennas, which are sensitive in a range of 120–240 MHz. Currently, only stations in the Netherlands are included in the measurements, which results in a resolution limit of ∼6 arcsec, but in the future, with very long baseline interferometry (LOFAR-VLBI), the resolution could be improved up to 0.3 arcsec for small sources. The first LoTSS data release, which is described by Shimwell et al. (2019), was released in 2019 and included around of 20% of the northern sky. In this paper we use data of the second LoTSS data release (LoTSS-DR2; Shimwell et al. 2022). The data consist of maps of 39 nearby galaxies at a frequency of 144 MHz, which were presented by Heesen et al. (2022), and we use the data to calculate spatially resolved magnetic field strength maps.

We used the LoTSS-DR2 144 MHz maps at 6 arcsec angular resolution as presented by Heesen et al. (2022). We also use their radio spectral index maps, or if not available, the integrated radio spectral index. Since the equipartition formula is only valid for a spectral index in the range of −1.1 ⪅ α ⪅ −0.6, values outside this range were clipped to the boundaries (see Appendix A for a discussion). In some galaxies the radio spectral index error is larger than the allowed spectral index range, so we clipped the error at a value of 1.0. The radio continuum intensity maps were masked below 3σ6″, where σ6″ is the rms noise of the 6 arcsec radio continuum map. We note that the radio spectral index maps have a lower angular resolution of 20 arcsec (for a few galaxies, slightly worse), which is not matched to the intensity maps.

2.2. Ancillary data

For a subset of nine galaxies (NGC 925, 2841, 2976, 3184, 3198, 4736, 5055, 5194, and 7331), we use a range of ancillary data for further analysis. We use H I maps from the THINGS survey (Walter et al. 2008) including moment 0 (H I surface mass density) and moment 2 (velocity dispersion). We also use SFR surface density maps created from Spitzer 24-μm and GALEX 156 nm emission (Leroy et al. 2008). To calculate H2 masses, we use CO (J = 2 → 1) maps from the HERACLES survey (Leroy et al. 2009). Details of the ancillary data are given in Table 1. Since the angular resolution of the THINGS maps is approximately 6 arcsec, the full resolution of the LoTSS data can be exploited. The ΣSFR and CO maps have a resolution of ≈13 arcsec, so that we need to lower the resolution of the LoTSS maps accordingly.

Table 1.

Magnetic field strengths and ancillary data in the LoTSS-DR2 sample.

The 21 cm line emission is then converted into the surface mass density of the atomic gas ΣH I where we used a correction factor of 1.36 to account for helium. The CO(2-1) emission is converted to the mass surface density of the molecular gas ΣH2 again correcting with a factor of 1.36 to account for helium (Leroy et al. 2009). For the CO data we assumed a 20% calibration error. We calculate the kinetic energy density of the combined atomic and molecular gas as u H  I+H 2 =1/2ρ v t 2 $ u_{\rm H\,I + \mathrm{H}_2}=1/2\,\rho\,{{\it v}}_{\rm t}^2 $. The velocity dispersion vt is the average of the CO and H I velocity dispersion in each individual galaxy by Mogotsi et al. (2016, see Table 1 for values). NGC 5194 and 7331 were not included in their sample so we use their mean velocity dispersions of 11.7 and 7.3 km s−1 for CO and H I, respectively. We assume an error of 25% for the velocity dispersion. In order to derive the gas density ρ, we assume a path length of 400 ± 40 pc for the H I gas and 100 ± 10 pc for the H2 gas, which we then correct for with the inclination angle using a factor of 1/cos(i). The scale height of the H I disc in THINGS galaxies was measured by Bagetakos et al. (2011) and a scale height of ≈200 pc is a reasonable approximation across the star-forming disc. This is further motivated by Imamura & Sofue (1997), who found that in the Milky Way the transition from H2 near the mid-plane to H I in the halo happens at a height of ≈50 pc, where the H I gas extends a few hundred parsec away from the mid-plane.

All our galaxies with ancillary data have angular sizes smaller than 15 arcmin, so that missing spacings for the interferometric H I observations from THINGS are not an issue (Walter et al. 2008). The CO data from the HERACLES survey were obtained with the IRAM 30 m telescope, so that missing spacings do not have to be considered.

3. Methodology

3.1. Equipartition magnetic field strength

In order to estimate the total magnetic field strength from the intensity of radio continuum emission, one has to assume a relation between the energy densities of the cosmic rays (uCR) and the magnetic field (uB). In this work, we assume energy equipartition, which ensures that the total energy density uB + uCR is close to the minimum value possible for a given radio continuum luminosity. The equipartition assumption may fail, in particular on scales smaller than the cosmic ray diffusion length (Seta & Beck 2019). Hence, we attempted a correction for diffusion by using either integrated relations or by smoothing with the diffusion length. We used the revised equipartition formula from Beck & Krause (2005):1

B eq , ⊥ = ( 4 π ( 2 α − 1 ) ( K 0 + 1 ) I ν E p 1 + 2 α ( ν 2 c 1 ) − α ( 2 α + 1 ) c 2 ( α ) l eff ( i ) · c 4 ( i ) ) 1 3 − α , $$ \begin{aligned} B_{\rm eq,\perp } = \left( \frac{4\pi (2\alpha - 1) (K_0 + 1) I_\nu E_{\rm p}^{1+2\alpha } \left(\frac{\nu }{2c_1}\right)^{-\alpha } }{(2\alpha + 1) c_2(\alpha ) l_{\mathrm{eff} }(i) \cdot c_4(i) } \right)^{\frac{1}{3 - \alpha }}, \end{aligned} $$(1)

with Beq, ⊥ the total magnetic field strength in the sky plane, the constant c1, the parameter c2, which depends on spectral index, the proton rest mass Ep, and the effective path length leff(i) through the source. The parameter c4 depends on inclination and spectral index. We assume isotropic magnetic fields, so

c 4 = ( 2 3 ) ( 1 − α ) / 2 . $$ \begin{aligned} c_4 = \left(\frac{2}{3}\right)^{(1 - \alpha )/2}. \end{aligned} $$(2)

Equation (1) is valid under the assumption of a constant ratio of protons to electrons, K0. We assumed a constant K0, which should be valid at particle energies of a few GeV. If cosmic-ray electrons (CREs) suffer from energy losses, K0 is a function of particle energy at lower as well as at higher energies (see Appendix A). Also, magnetic field fluctuations lead to an overestimate of the total magnetic field strength (Beck et al. 2003).

For the path length we assumed leff = 1.4 kpc/cos i for the mildly inclined galaxies (i < 78°), which is motivated by the results of Krause et al. (2018), who found a radio scale height of 1.4 ± 0.7 kpc at 1.5 GHz (see Sect. 4.1 for details). Also, i is the inclination angle of the galaxy disc relative to the sky plane (i = 0° is face-on). For the edge-on galaxies, that is, the galaxies from the CHANG-ES survey, leff is assumed to be the star-forming radius, as defined by the extent of the 24 μm emission (Wiegert et al. 2015). Since leff and K0 are only estimates, we adopted a 50% uncertainty on both values to calculate the error of the magnetic field. For the calculation of galaxy-wide averages of the magnetic field strength ⟨Beq⟩, we used the integrated flux density S6″ from the 6 arcsec data and calculated the mean intensity using the 6 arcsec major axis as radius.

3.2. Thermal emission

The low-frequency radio continuum emission is dominated by non-thermal synchrotron emission, so we did not correct for thermal emission when calculating equipartition magnetic field strengths. In this section we justify this assumption, by exploring the influence of thermal emission for a subset of five galaxies (see Table 2). The thermal and non-thermal components of the radio continuum emission were separated using the thermal radio tracer approach in which the H α line emission is used as a template for the free–free emission after correcting for extinction (Tabatabaei et al. 2007, 2013, 2018). Hence, extinction maps are first constructed for galaxies using the Herschel PACS 70 μm and 160 μm data taken as part of the KINGFISH project (Kennicutt 2011). Following Tabatabaei et al. (2013), the brightness temperature of the free–free radio continuum emission, Tb, is given by

T b = T e ( 1 − e − A I H α ) K , $$ \begin{aligned} {T_{\rm b}=T_{\rm e}(1-e^{-A\,I_{\mathrm{H}\,\alpha }})}\,\mathrm K, \end{aligned} $$(3)

Table 2.

Influence of the correction for thermal emission on the magnetic field strength.

where the factor A is

A = 3.763 ( ν GHz ) − 2.1 ( T e 10 4 K ) − 0.3 10 290 K T e . $$ \begin{aligned} A=3.763\,\left(\frac{{\nu }}{\mathrm{GHz}}\right)^{-2.1}\, \left(\frac{T_{\rm e}}{10^4\,\mathrm{K}}\right) ^{-0.3}\, 10^{\frac{290\,\mathrm{K}}{T_{\rm e}}}. \end{aligned} $$(4)

Here, IHα is the de-reddened H α intensity in units of erg cm−2 s−1 sr−1 and Te the electron temperature. The brightness temperature is derived for Te = 104 K. A 30% variation in Te would change the thermal fraction by about 23%. The thermal free-free emission, obtained after the Kelvin-to-Jy/beam conversion of Tb, is subtracted from the observed radio continuum emission resulting in a map of the synchrotron emission for each galaxy. We then also re-calculated non-thermal radio spectral index maps between 144 and 1365 MHz using the maps from the Westerbork Synthesis Radio Telescope–SINGS survey (Braun et al. 2007). With the non-thermal radio continuum map at 144 MHz and the non-thermal radio spectral index map, we re-calculated the non-thermal magnetic field strength Bnt.

In Fig. 1 the difference between the magnetic field strengths ΔB = Beq − Bnt is shown for the specific case of NGC 5194 (M 51). The difference in the estimate of the magnetic field is below the 1σ difference and Beq is smaller than Bnt for most parts of the galaxy. Thus, the non-thermally corrected estimate on the magnetic field strength gives a lower bound of the real strength, but lies within the uncertainties. The mean of the difference (see Table 2) is for all galaxies smaller than σB, and hence it is within the magnetic field uncertainties. Notably also the maximum magnetic field strength in the map is hardly changed by the thermal correction. We hence can justify to neglect thermal correction for the remainder of the sample.

thumbnail Fig. 1.

Difference in magnetic field strength, ΔB = Beq − Bnt, due to the thermal correction in NGC 5194/5.

3.3. Correlations

We correlated the maps of the magnetic field strength with our ancillary data. Maps that needed to be correlated were transformed to the same coordinate system with the imrebin function of the Common Astronomy Software Applications (CASA) package (McMullin et al. 2007). The pixel size is increased to be equal or slightly larger than the full width at half maximum of the lowest resolution map with CASA’s imrebin function. Maps are masked below 3σ, although we also tested the influence of using either 4σ or 5σ as a cut-off level and found no significant difference with regard to the best-fitting correlations. The retained pixels are displayed in a log-log diagram (Fig. 2) according to the radio spectral index: data points with α < −0.85, a spectral index steeper than the injection index corresponding to aged cosmic rays; data points with −0.85 ≤ α < −0.65, a spectral index near the injection index corresponding to young cosmic rays; and data points with −0.65 ≤ α, a spectral index flatter than the injection spectrum, possibly due to, for example, thermal emission, synchrotron self-absorption, or thermal absorption. Since the number of data points for the B–ΣH I correlation is very large, we chose to colour-code the point density instead.

thumbnail Fig. 2.

Spatially resolved relations between the magnetic field strength and the SFR surface density (B–ΣSFR) (a), the atomic gas mass surface density (B–ΣH I) (b), molecular gas mass surface density (B–ΣH2) (c), and the combined atomic and molecular gas mass surface density (B–ΣH I + H2) (d). Data points are coloured according to the radio spectral index and, in panel (b), according to the point density. Best-fitting relations are shown as sold lines and the theoretical expectations as dashed lines.

We fit a power-law with the following parametrisation:

y = a ( x c ) b , $$ \begin{aligned} { y} = a\left(\frac{x}{c}\right)^b, \end{aligned} $$(5)

where we consider errors in both axes using orthogonal distance regression. On the fitted graphs, the 95% (2σ) confidence interval is highlighted in grey. The theoretical prediction is indicated by a red dashed line and their origin is explained in the corresponding section. Because the spatially resolved relations are dominated by the gas rich ‘grand spiral’ galaxies NGC 5055, 5194/5, and 7331 that each span two orders of magnitude in gas mass surface densities, one may speculate that these galaxies entirely dominate the relations. Hence, we also investigated relations using only the remaining relatively gas poor galaxies NGC 925, 2841, 2976, 3184, 3198, and 4736 shown in Fig. 3. As we explain in more detail below, these galaxies show relations that are broadly in line with the full sample.

thumbnail Fig. 3.

As Fig. 2, but only for galaxies that are relatively gas poor. These are all galaxies except the grand spiral galaxies NGC 5055, 5194/5, and 7331. As can be seen, these galaxies approximately conform with the relations that are found for the complete sample.

We also present global correlations, where we plot the mean quantities for individual galaxies. For the mean magnetic field strength, we used the mean intensity, which was then converted as described in Sect. 3.1. For the ancillary data, we then used the integrated values from Table 1, which are then divided by the surface area to obtain an average value for the galaxy. For the error, we used the standard deviation of the maps. The resulting global relations are presented in Fig. 4.

thumbnail Fig. 4.

Global relations between the mean magnetic field strength and the mean SFR surface density (a), the mean atomic gas mass surface density (b), mean molecular gas mass surface density (c), and the mean combined atomic and molecular gas mass surface density (d). Best-fitting relations are shown as sold lines and the theoretical expectations as dashed lines.

3.4. Accounting for the influence of cosmic-ray transport

At the low frequencies of LOFAR, the effect of cosmic-ray transport can be significant, resulting in the smoothing the radio continuum maps. This is because the CRE lifetime increases at low frequencies and so the cosmic rays travel further from their sites of acceleration. When one studies the spatially resolved radio–SFR relation, the relation becomes sub-linear unlike the integrated radio–SFR relation, which is approximately linear when one accounts for the influence of cosmic-ray transport (Smith et al. 2021; Heesen et al. 2022). This influence of the cosmic-ray transport on spatially resolved observations (Heesen et al. 2014; Heesen & Buie 2019) is supported further by Mulcahy et al. (2016) who analysed the radial variation of the radio spectral index in NGC 5194. To measure the cosmic-ray transport length, we convolved the ΣSFR maps in order to linearise the radio–SFR relation (Berkhuijsen et al. 2013). The process is explained in more detail in Heesen & Buie (2019) and we present in Table 3 the resulting cosmic-ray transport lengths for seven galaxies.

Table 3.

Diffusion lengths used to account for cosmic-ray transport.

To account for the cosmic-ray transport effect, we convolved the ΣSFR and ΣH I + H2 maps with a Gaussian kernel, converting the cosmic-ray transport length into the standard deviation, σxy, as follows:

l CR = 1 2 ( 2 2 ln 2 ) σ xy . $$ \begin{aligned} l_\mathrm{CR} = \frac{1}{2} \left( 2\sqrt{2\ln 2} \right) \sigma _{x{ y}}. \end{aligned} $$(6)

We used ASTROPY’s convolve_fft for the convolution and analysed the correlations as described in Sect. 3.3. The resulting relations where we accounted for the influence of cosmic-ray transport are presented in Fig. 5.

thumbnail Fig. 5.

Spatially resolved relations with cosmic-ray transport accounted for between the magnetic field strength and the SFR surface density (a) and with the combined atomic and molecular gas mass surface density (b). The ΣSFR and the ΣH I + H2 maps have been smoothed with a Gaussian kernel in order to account for cosmic-ray transport. Data points are coloured according to the radio spectral index and, in panel (b), according to the point density. Best-fitting relations are shown as sold lines and the theoretical expectations as dashed lines.

4. Results

4.1. Magnetic field strength and scale length

In Table 1 we list the mean and maximum equipartition magnetic field strength in our galaxy sample along with the assumed inclination angles and path lengths that are needed as input parameters. The corresponding equipartition magnetic field strength maps are presented in Appendix B. The mean magnetic field strength varies between 3.6 and 12.5 μG and the maximum magnetic field strength between 7.5 and 41.1 μG. Taken as a mean across the sample, the mean magnetic field strength is 7.9 ± 2.0 μG and the maximum magnetic field strength is 19.8 ± 8.6 μG.

In Appendix D we present the 1D radial profiles of the magnetic field strengths in our sample galaxies that have moderate inclination angles (i ⪅ 75 degree). We determined the magnetic field radial scale-lengths in our sample galaxies with moderate inclination angles by fitting the radial profiles using the form

B eq ( r ) = B 0 e − r / r B , $$ \begin{aligned} B_{\rm eq}(r) = B_0\,\mathrm{e}^{-r/r_B}, \end{aligned} $$(7)

where B0 provides an estimate of the magnetic field strength at the centre of a galaxy, and rB is the radial scale-length. In Table 4, we present the results of the fitted profiles. We found scale-lengths between 3.1 and 88.6 kpc disregarding cases where the error is larger than 50%. In some cases the magnetic field strength is nearly constant with radius (rB > 100 kpc), namely, NGC 925, 2976, 3077, 3938, 4625, and 5457. These galaxies are all of a very late type (Sc, Sd, Sm, or P). The median magnetic field scale-length in our sample is rB = 10.4 ± 26.5 kpc or, expressed as a ratio of R25, the median scale-length is rB = (1.0 ± 1.8)R25.

Table 4.

Radial scale length of the magnetic field.

Table 5.

Correlations with the magnetic field strength.

4.2. Magnetic field–star formation relation

In Fig. 2a the combined plot for the B–SFR relation is presented. There is a strong correlation with a Pearson correlation coefficient of r = 0.69. The slope of the relation is 0.182 ± 0.004, which is significantly flatter than the observed relation of b ≈ 1/3 for the global magnetic field strength (Niklas & Beck 1997; Heesen et al. 2014; Tabatabaei et al. 2017). A similar slope is found when we consider only gas poor galaxies as shown in Fig. 3a. In Fig. 4a, we present the corresponding global relation of the mean magnetic field strength as a function of the mean SFR surface density. The slope is now higher at b = 0.22 ± 0.10, although is still well below integrated B–SFR measurements. Integrated measurements can be regarded as upper limits for the slopes as higher mass galaxies have steeper radio continuum spectra and thus higher calorimetric fractions of the CREs (Smith et al. 2021). On the other hand, spatially resolved relations are lower limits due to the effects of cosmic-ray transport, which is particularly important at low frequencies. The effects of CRE transport can be seen in Fig. 2a, where the data points with flatter spectral indices fall predominantly below the dashed line shown for comparison, B ∝ Σ SFR 1 / 3 $ B\propto \Sigma_{\mathrm{SFR}}^{1/3} $, which is, as well as the integrated relation, in line with a simplifying theory assuming energy equipartition and a constant velocity dispersion (Schleicher & Beck 2013). The B–ΣSFR relation was already investigated previously in individual galaxies. Tabatabaei et al. (2013) found a slope of 0.14 ± 0.01, whereas Chyży (2008) found a slope of 0.18 ± 0.01. The B–ΣSFR relation is closely related to the radio continuum–star-formation rate (RC–SFR) relation because for equipartition, Iν ∝ B3.6 assuming a non-thermal spectral index of −0.6. As the slope of the RC–SFR relation changes between arm and inter-arm region with higher slopes in the arm region (Basu & Roy 2013), the B–ΣSFR relation follows accordingly. The large scatter of the B–ΣSFR relation is in part a consequence of treating the arm and inter-arm region the same, while a differentiation either using morphology (Dumas et al. 2011; Basu & Roy 2013) or radio spectral index (Heesen & Buie 2019) may lead to a reduced scatter. But since in this work we would like to concentrate on the relation between magnetic field strength and the gas surface density, we did not investigate these influences any further.

Because supernovae as the probable sources of CREs are rare in outer discs and CRE diffusion from inside is limited to a few kiloparsecs, it is well possible that energy equipartition no longer holds in the outer discs. Then the equipartition estimates, derived from the synchrotron intensities that are limited by the low density of CREs, yield underestimates for the field strength. The reason is that any under-equipartition CRE density has to be balanced by a higher field strength to explain the observed synchrotron intensity, so that the field strength has to increase. This can be tested by Faraday rotation measures (RMs) of background sources located behind outer discs. RM data of intervening galaxies in front of quasi-stellar objects indicate indeed that the extents of magnetic fields are much larger than those of the star-forming disc (see e.g., Kim et al. 2016). The field strengths at low ΣSFR in Figs. 2a and 5a are above the expected values. The above correction for a possible non-equipartition in the outer discs would even increase this discrepancy, not remove it.

To circumvent the problematic study of magnetic fields in outer discs in emission, one may hence turn in the future to the usage of RM studies. Those with LOFAR give access to weak magnetic fields as are expected for nearby star-forming galaxies (O’Sullivan et al. 2020).

4.3. Magnetic field–gas relation

The spatially resolved magnetic field–gas relation for the atomic gas is shown in Figs. 2b and 3b. The scatter of 0.14 dex is the largest in our investigated magnetic field correlations. Consequently, the correlation coefficient for the full sample r = 0.36 is fairly small, suggesting that the correlation is not significant. This result does not change for the global B–ΣH I relation as presented in Fig. 4b, where the correlation is even weaker. We note that the data points seem to cluster around several areas, fitting all these with just a single line and average them for the global relations, might underestimate the complexity of the distribution. Hence, neither the fitted nor the theoretical line are accurate descriptions of the data. The H I surface density rarely exceeds 10 M⊙ pc−2 where H I ‘saturates’ and the transition from H I to H2 happens (Leroy et al. 2008).

For the molecular gas, the spatially resolved B–ΣH2 relation is shown in Figs. 2c and 3c. The scatter is fairly small with 0.08 dex and correlation is significant with r = 0.84 (both refer to the full sample). Clearly, the magnetic field is much better correlated with the distribution of the molecular gas than with the atomic gas. We also note that the slope is identical to the B–ΣSFR relation within the uncertainties, but the scatter is slightly smaller. A similar slope is expected when one considers ΣSFR ∝ ΣH 2 (Bigiel et al. 2008). A similar result was also found in the global relation presented in Fig. 4c, where the scatter is reduced further to 0.05 dex and the slope increases to b = 0.25 ± 0.10. Again, this slope is in agreement with the global B–ΣSFR relation.

In Fig. 2d we show the magnetic field–gas relation for the combined atomic and molecular gas surface densities. We find that the B–ΣH I + H2 relation has the steepest slope with b = 0.309 ± 0.006 in conjunction with a small scatter of 0.08 dex indicating a tight correlation (r = 0.80) for the full sample with similar results for the gas poor sample (Fig. 3d). For the global relation shown in Fig. 4d, the slope is increased further to b = 0.39 ± 0.19. We note that for the spatially resolved correlation only pixels were taken into account, where CO emission was detected. The higher slope of the B–ΣH I + H2 in comparison to the B–ΣH2 relation may be related to the fact that the lowest mass surface densities are removed as there is always H I gas even if there is only little H2 gas.

It seems that the relation between B and ΣH I has different exponents in regions of different density, that is, a flat relation at low densities but a steeper one at high densities. This is consistent with results from Zeeman splitting in our Milky Way (see Fig. 1 in Crutcher et al. 2010). A similar but weaker effect is also seen in the relation between B and ΣH2 that also flattens at low densities. The probable reason is that B is not only related to gas density, but also to the turbulent velocity dispersion, which may not be not constant but varies with gas density (Pakmor et al. 2017). Figure 2 may tell us that the variation of turbulent velocity across and between galaxies is larger in H I gas than in H2 gas, possibly related to the different dynamics of molecular clouds (Sun et al. 2018) and of atomic gas (Tamburro et al. 2009).

4.4. Influence of cosmic-ray transport

In Fig. 5 we present the spatially resolved B–ΣSFR and the B–ΣH I + H2 relations, where we have accounted for cosmic-ray transport by convolving the ΣSFR and ΣH I + H2 maps with a Gaussian kernel that matched the cosmic-ray transport length in each galaxy. In this section we want to concentrate only on these two most relevant relations with the magnetic field strength. The slope of the B–ΣSFR relation increases to b = 0.284 ± 0.006, as expected, and the correlation coefficient increases, too, when compared with the relation of the un-convolved maps. For the B–ΣH I + H2 relation, the slope increases to b = 0.393 ± 0.009, and the correlation coefficient rises slightly. Hence, we conclude that cosmic-ray transport does affect our results of the magnetic field–gas relation, but not by quite as much as the B–ΣSFR relation. This is because the molecular gas is only detected where the ΣSFR-values are high, whereas cosmic-ray transport affects mostly galaxy outskirts and inter-arm regions. This can be seen by the various slopes of the RC–SFR relation in arm and inter-arm regions, where the arm regions are almost linear (e.g., Dumas et al. 2011; Heesen & Buie 2019) in good agreement with the global relation. In contrast, the ΣSFR maps cover the full extent of the galaxies including the outskirts.

4.5. Kinetic energy density

In Fig. 6 we show a comparison between the magnetic and the total kinetic energy density of the gas. The best-fitting relation is

u B = ( 2.53 ± 0.62 ) × 10 − 12 erg cm − 3 ( u H I + H 2 10 − 12 erg cm − 3 ) 0.461 ± 0.009 , $$ \begin{aligned} u_{B} = (2.53 \pm 0.62) \times 10^{-12}\ \mathrm{erg\, cm^{-3}} \left( \frac{u_{\rm H\,I + H_2}}{10^{-12}\,\mathrm {erg\, cm}^{-3}} \right)^{0.461 \pm 0.009}, \end{aligned} $$(8)

thumbnail Fig. 6.

Relation between the magnetic energy density and the total kinetic energy density. The best-fitting relation is shown as a solid line and the energy equipartition as a dashed line. Data points are coloured according to their radio spectral index, highlighting that these relations are universal for all galaxies.

with r = 0.78 and a scatter of 0.20 dex. The slope is with 0.50 smaller than what would be expected if energy equipartition with the gas were to hold everywhere. While on average the magnetic field is in equipartition, at the low energy densities we find an excess of magnetic energy density, whereas at the high kinetic energy densities, the magnetic field is fairly weak. As stated in Sect. 4.4, we can rule out cosmic-ray transport as the sole reason. It is possible that the gas velocity dispersion changes across the galaxy, possibly being correlated with ΣSFR (Tamburro et al. 2009). A positive correlation with ΣSFR as observed in H I (Tamburro et al. 2009) as well as in CO meaning H2 (Colombo et al. 2014) would mean that we underestimate the kinetic energy density in areas of high ΣSFR, hence compounding the domination of kinetic over magnetic energy density.

5. Discussion

5.1. What regulates magnetic fields in galaxies?

If the magnetic field is frozen into the gas, we would expect a correlation with the gas density as

B ∝ ρ κ , $$ \begin{aligned} B \propto \rho ^\kappa , \end{aligned} $$(9)

where κ depends on physical properties of the medium and on geometry. For isotropic compression in dense media with supersonic turbulence, κ = 2/3, as observed from Zeeman splitting data (Crutcher et al. 2010), whereas for linear compression such as in a shock κ = 1 for the field component parallel to the shock front and κ = 0 for the component perpendicular to the shock front. We note that magnetic field–gas relation in multi-phase simulations is very complex and cannot be easily described by a single power-law relationship consistent with these simple gas compressions. In particular, the relation depends on the Alfvénic Mach number, where κ = 2/3 is found in high Mach number regions of the cold ISM dominated by molecular hydrogen (Seta & Federrath 2022). However, as such regions would likely occupy a small fraction of the ISM and H I is mostly sub-to trans-sonic, flux freezing does not hold.

On the other hand, if the magnetic energy density is in equipartition with the kinetic energy density of the gas, we expect

B 2 8 π = f 2 ρ v t 2 , $$ \begin{aligned} \frac{B^2}{8\pi } = \frac{f}{2}\,\rho \, { v}_{\rm t}^2 , \end{aligned} $$(10)

where vt is the turbulent velocity. A constant vt leads to κ = 1/2, so that B ∝ ρ0.5. At the moderate densities of the warm ISM, vt is trans- to subsonic (Burkhart et al. 2012) and similar to the sound velocity vs. If gas pressure were constant, vt ∝ vs ρ0.5, so that B would not vary with density. Realistically, pressure is not constant in the ISM (de Avillez & Breitschwerdt 2005).

The fraction f of magnetic energy density to kinetic energy density is mostly found to be close to unity (Beck 2007; Gent et al. 2013; Beck 2015b). A 100% efficient small-scale dynamo would, in this picture, correspond to f = 1. In some regions f is even larger than unity, which cannot be solely explained by the small-scale dynamo and another source of energy sustaining the dynamo action is needed (Beck 2015a). For a constant velocity dispersion, we obtain B ∝ ρ0.5. If one now assumes a Kennicutt-Schmidt relation between the gas and the SFR surface density, Σ SFR ∝ Σ gas N $ \Sigma_{\rm SFR}\propto \Sigma_{\rm gas}^N $, we can infer, assuming N = 1.5 (Schleicher & Beck 2013),

B ∝ Σ SFR 1 / 3 . $$ \begin{aligned} B \propto \Sigma _{\rm SFR}^{1/3}. \end{aligned} $$(11)

For molecular dominated gas, we have N = 1 (Leroy et al. 2008), which would result in a steeper B–ΣSFR relation of B ∝ Σ SFR 1 / 2 $ B \propto \Sigma_{\mathrm{SFR}}^{1/2} $. This was supported by the numerical simulations by Steinwandel et al. (2020) that yielded slopes between 0.3 and 0.45. Equations (9)–(11) are hence the basic theoretical relations we can test with our observations.

We find the best correlations with the smallest scatter and the highest correlation coefficient in the B–ΣH2 correlation, when considering only the spatially resolved correlations. However, the dependence of the magnetic field on gas density is only weak with B∝ Σ H 2 0.183 $ B\propto \Sigma_{\rm H_2}^{0.183} $. The relation between magnetic field and total gas density, which has the same small scatter but a much larger slope of B∝ Σ H  I+H 2 0.309 $ B\propto \Sigma_{\rm H\,I+H_2}^{0.309} $, appears more significant. This slope increases further when we either consider global relations or the spatially resolved relation B∝ Σ H  I+H 2 0.393±0.009 $ B\propto \Sigma_{\rm H\,I+H_2}^{0.393\pm 0.009} $, where the cosmic-ray diffusion is accounted for. This slope is already fairly close to the slope as expected for the saturated dynamo. The remaining difference may stem from a decreasing velocity dispersion in regions with larger gas densities. What appears to be rather certain is that pure compression cannot account for the observed B–ρ behaviour as the κ is too low.

The B–ΣSFR correlation of B ∝ Σ SFR 0.182 $ B\propto \Sigma_{\mathrm{SFR}}^{0.182} $ agrees with expectation for a saturated small-scale dynamo, where the slope of the relation is slightly too low but increases to 0.284 when cosmic-ray transport is accounted for. The magnetic energy density is in approximate equipartition with the kinetic energy density further supporting the saturation of the small-scale dynamo. However, locally we find deviations from equipartition, in particular in areas of high kinetic energy densities where the magnetic field is sub-dominant. In contrast, the magnetic field dominates in areas of low kinetic energy densities. This is in agreement with radial profiles of the energy densities, which show that the magnetic field dominates in the galaxy outskirts (Beck 2007, 2015b). Large-scale gravitational instabilities as a source of turbulence are thought to be efficient only in high-density gas (Brucy et al. 2020). A more probable source of turbulence in the outer discs of galaxies is the magneto-rotational instability (Gressel et al. 2013). Other possibilities are a hypothetical change of efficiency of the large-scale dynamo that operates more effectively in the galaxy outskirts and the influence of the thermal gas pressure on the magnetic field (Basu & Roy 2013).

5.2. Limitations of the equipartition approach

When we apply the equipartition estimate for the magnetic field strength (Sect. 3.1), we ‘trim’ the radio spectral index to the range of −1.1 ≤ α ≤ −0.6. This is in our opinion the best option. The equipartition formula yields unphysically large B values for α > −0.6 (very flat) and also for α < −1.1 (very steep). The first case indicates thermal absorption or ionisation losses, the second strong synchrotron losses. In both cases the proton-to-electron ratio K0 is larger than the canonical value of K0 = 100, without knowing the correct value. We performed a few experiments that showed that trimming the spectral index gives more reliable B estimates than trying to correct the value of K0. These experiments are presented in Appendix A. Weak and mild synchrotron losses, such as those that we have at low radio frequencies, do not affect the equipartition estimate of the total field.

Another limitation is the assumption of a constant magnetic field strength along the line of sight, which is obviously a simplification and a more complex model would be necessary. The mean magnetic field is systematically lower in the edge-on galaxies as we are seeing the radio halo, where the field strength is weaker. In face-on galaxies, we assume a path length of 1.4 kpc (Sect. 3.1), which can only be a crude approximation. Observations of edge-on galaxies show that reality is more complicated. According to Krause et al. (2018), the radio emission in many edge-on galaxies has two components, a (thin) disc and a halo. The radio disc is a result of the strong magnetic fields in the star-forming regions, while the radio halo mainly reflects the distribution of the CREs leaving the disc. The intensity and vertical extent of the radio disc varies locally, following the tracers of star formation. The radio halo is more homogeneous. Krause et al. (2018) found average values for the halo scale heights of 1.1 ± 0.3 kpc at 6000 MHz and 1.4 ± 0.7 kpc at 1500 MHz. Thermal emission was not subtracted, so that the scale heights of pure synchrotron emission are expected to be somewhat larger. The scale heights of the (thin) disc have large uncertainties due to the limited resolution.

In the case of mildly inclined galaxies, a superposition of emission from both components along the line of sight is observed, so that it is hard to estimate their relative contributions. The ratio of disc-to-halo emission has been so far only poorly studied. Stein et al. (2020) found ratios of order unity with the ratio rising with the ΣSFR-values, so that some galaxies have relatively high ratios of disc-to-halo emission. In summary, our assumed path length for the face-on galaxies is a good approximation for galaxies with strong star formation in the disc, whereas for weakly star-forming galaxies an average between disc and halo path lengths would be more appropriate.

Finally, there is the question whether equipartition is applicable at all in galaxies. Global state-of-the-art magnetohydrodynamic models of star-forming galaxies including cosmic rays were investigated recently in a series of papers: Werhahn et al. (2021b,a) and Pfrommer et al. (2022). These simulations show that equipartition (measured on scales of 10 kpc) is reached in Milky-Way-mass galaxies but fails for small galaxies (see Fig. 1 in Pfrommer et al. 2022). From similar models, Buck et al. (2020) show radially averaged profiles of magnetic and cosmic-ray energy densities (their Fig. 4, panels 2 and 4) that match quite well, although there is a slight suppression of the cosmic-ray energy density compared to the magnetic energy density in galaxy outskirts. If this is indeed the case in our galaxies, this would lead to an overestimation of the magnetic field strength from our equipartition estimates. This could then explain our high magnetic energy densities compared with the kinetic energy densities in these areas. While Seta & Beck (2019) show that equipartition does not hold on very small scales, these scales are not relevant for our work as they are unresolved. They suggest that equipartition could be tested directly in the local ISM, from solar system-corrected cosmic ray data and Voyager magnetic fields data from outside the solar system. Nevertheless, equipartition does hold to the best of our knowledge on the scale relevant here. Lastly, γ-ray observations also confirm the validity of energy equipartition for surface SFRs of ΣSFR ⪅ 100 M⊙ yr−1 kpc−2 (Yoast-Hull et al. 2016).

6. Conclusions

We used the LOFAR data from LoTSS-DR2 at 144 MHz (Shimwell et al. 2022) to calculate magnetic field strengths in a sample of 39 nearby galaxies (Heesen et al. 2022) assuming energy equipartition between cosmic rays and the magnetic field. We used radio spectral index maps to assess the CRE spectrum. In agreement with former studies, we find that our sample of galaxies shows magnetic fields with strengths of up to 40 μG, but with mean values of around 10 μG. For a subset of nine galaxies, we studied the relation between magnetic fields, star formation, and gas density. These are our main conclusions:

  • The magnetic field strength has the tightest and steepest correlation with the total gas surface density. We find B∝ Σ H  I+H 2 0.309±0.006 $ B\propto \Sigma_{\rm H\,I+H_2}^{0.309\pm 0.006} $ with a scatter of 0.08 dex.

  • Similarly, we find tight correlations between the magnetic field strength and both the SFR and molecular gas surface densities, albeit with smaller slopes of B ∝ Σ SFR 0.182 ± 0.004 $ B\propto \Sigma_{\mathrm{SFR}}^{0.182\pm 0.004} $ and B∝ Σ H 2 0.183±0.003 $ B\propto \Sigma_{\rm H_2}^{0.183\pm 0.003} $, respectively.

  • We find no significant correlation between the magnetic field strength and the atomic mass surface density, ΣH I. The scatter around the best-fitting relation is the largest, with 0.14 dex.

  • When we account for cosmic-ray transport, smoothing the ΣSFR and ΣH I + H2 maps with a Gaussian kernel, we find steeper slopes of B ∝ Σ SFR 0.284 ± 0.006 $ B\propto \Sigma_{\mathrm{SFR}}^{0.284\pm 0.006} $ and B∝ Σ H  I+H 2 0.393±0.009 $ B\propto \Sigma_{\rm H\,I+H_2}^{0.393\pm 0.009} $. Both relations can be explained by a saturated small-scale dynamo, where the magnetic energy density is in equipartition with the kinetic energy density.

  • Comparing the energy densities, we find that while on average they do agree, locally the magnetic field is sub-dominant in regions of high kinetic energy density, whereas the opposite is the case for regions with a low kinetic energy density.

  • Possible explanations are that the efficiency of the small-scale dynamo is lower in the inner parts of the disc, the magnetic field strength is boosted by the magneto-rotational instability in the outer parts of the disc, or we need to take other ISM components, such as the thermal gas, into account.

The most surprising aspect of our work is that the magnetic field depends most on the total gas mass surface density and less so on the SFR surface density. This suggests a B–ρ coupling, where the slope is in agreement with equipartition between magnetic and kinetic energy density. Our angular resolution corresponds to 0.3–1.2 kpc at the distances of our galaxies. A related correlation between CO and the radio continuum was found in a sample of galaxies at sub-kiloparsec scales (Murgia et al. 2005; Paladino et al. 2006, 2008); the same correlation between CO and radio continuum luminosity was confirmed again in NGC 5194/5 at the 40 pc cloud scale (Schinnerer et al. 2013). A B–ρ relation also has wider implications for the integrated radio continuum–far-infrared relation of galaxies (Niklas & Beck 1997) and the RC–SFR relation (Heesen et al. 2014). New insights may be gained in the future from expanding the LoTSS-DR2 nearby galaxy sample to the full LoTSS sky coverage and new CO observations at high angular resolution, as provided with the PHANGS–ALMA survey (Leroy et al. 2021).

1

We use the radio spectral index convention Iν ∝ να.

2

https://alasky.u-strasbg.fr/hips-image-services/hips2fits

Acknowledgments

We thank the anonymous referee for an insightful review, which helped to improve the quality of the manuscript. We also thank Amit Seta for a fruitful discussion about this work. This paper is based (in part) on data obtained with the International LOFAR Telescope (ILT). LOFAR (van Haarlem et al. 2013) is the Low Frequency Array designed and constructed by ASTRON. It has observing, data processing, and data storage facilities in several countries, that are owned by various parties (each with their own funding sources), and that are collectively operated by the ILT foundation under a joint scientific policy. The ILT resources have benefitted from the following recent major funding sources: CNRS-INSU, Observatoire de Paris and Université d’Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science and Technology Facilities Council, UK; Ministry of Science and Higher Education, Poland. This work made use of the SCIPY project https://scipy.org. M.B. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2121 ‘Quantum Universe’ – 390833306. The Jülich LOFAR Long Term Archive and the German LOFAR network are both coordinated and operated by the Jülich Supercomputing Centre (JSC), and computing resources on the supercomputer JUWELS at JSC were provided by the Gauss Centre for Supercomputing e.V. (grant CHTB00) through the John von Neumann Institute for Computing (NIC). MS acknowledges funding from the German Science Foundation DFG, within the Collaborative Research Center SFB1491 “Cosmic Interacting Matters – From Source to Signal”.

References

  1. Ahumada, R., Prieto, C. A., Almeida, A., et al. 2020, ApJS, 249, 3 [Google Scholar]
  2. Bagetakos, I., Brinks, E., Walter, F., et al. 2011, AJ, 141, 23 [NASA ADS] [CrossRef] [Google Scholar]
  3. Basu, A., & Roy, S. 2013, MNRAS, 433, 1675 [NASA ADS] [CrossRef] [Google Scholar]
  4. Beck, R. 2007, A&A, 470, 539 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  5. Beck, R. 2015a, A&ARv, 24, 4 [Google Scholar]
  6. Beck, R. 2015b, A&A, 578, A93 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  7. Beck, R., & Krause, M. 2005, Astron. Nachr., 326, 414 [Google Scholar]
  8. Beck, R., Shukurov, A., Sokoloff, D., & Wielebinski, R. 2003, A&A, 411, 99 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  9. Beck, A. M., Lesch, H., Dolag, K., et al. 2012, MNRAS, 422, 2152 [NASA ADS] [CrossRef] [Google Scholar]
  10. Beck, R., Chamandy, L., Elson, E., & Blackman, E. G. 2019, Galaxies, 8, 4 [NASA ADS] [CrossRef] [Google Scholar]
  11. Becker Tjus, J., & Merten, L. 2020, Phys. Rep., 872, 1 [NASA ADS] [CrossRef] [Google Scholar]
  12. Bell, A. R. 1978, MNRAS, 182, 443 [Google Scholar]
  13. Berkhuijsen, E. M., Beck, R., & Tabatabaei, F. S. 2013, MNRAS, 435, 1598 [Google Scholar]
  14. Bhat, P., Subramanian, K., & Brandenburg, A. 2019, ArXiv e-prints [arXiv:1905.08278] [Google Scholar]
  15. Bigiel, F., Leroy, A., Walter, F., et al. 2008, AJ, 136, 2846 [NASA ADS] [CrossRef] [Google Scholar]
  16. Brandenburg, A., & Subramanian, K. 2005, Phys. Rep., 417, 1 [NASA ADS] [CrossRef] [Google Scholar]
  17. Braun, R., Oosterloo, T. A., Morganti, R., Klein, U., & Beck, R. 2007, A&A, 461, 455 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  18. Breitschwerdt, D., McKenzie, J. F., & Voelk, H. J. 1991, A&A, 245, 79 [NASA ADS] [Google Scholar]
  19. Brucy, N., Hennebelle, P., Bournaud, F., & Colling, C. 2020, ApJ, 896, L34 [NASA ADS] [CrossRef] [Google Scholar]
  20. Buck, T., Pfrommer, C., Pakmor, R., Grand, R. J. J., & Springel, V. 2020, MNRAS, 497, 1712 [NASA ADS] [CrossRef] [Google Scholar]
  21. Burkhart, B., Lazarian, A., & Gaensler, B. M. 2012, ApJ, 749, 145 [NASA ADS] [CrossRef] [Google Scholar]
  22. Chamandy, L. 2016, MNRAS, 462, 4402 [NASA ADS] [CrossRef] [Google Scholar]
  23. Chyży, K. T. 2008, A&A, 482, 755 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  24. Chyży, K. T., Weżgowiec, M., Beck, R., & Bomans, D. J. 2011, A&A, 529, A94 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  25. Colombo, D., Hughes, A., Schinnerer, E., et al. 2014, ApJ, 784, 3 [NASA ADS] [CrossRef] [Google Scholar]
  26. Crutcher, R. M., Wandelt, B., Heiles, C., Falgarone, E., & Troland, T. H. 2010, ApJ, 725, 466 [NASA ADS] [CrossRef] [Google Scholar]
  27. Cummings, A. C., Stone, E. C., Heikkila, B. C., et al. 2016, ApJ, 831, 18 [CrossRef] [Google Scholar]
  28. de Avillez, M. A., & Breitschwerdt, D. 2005, A&A, 436, 585 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  29. Dumas, G., Schinnerer, E., Tabatabaei, F. S., et al. 2011, AJ, 141, 41 [NASA ADS] [CrossRef] [Google Scholar]
  30. Everett, J. E., Zweibel, E. G., Benjamin, R. A., et al. 2008, ApJ, 674, 258 [CrossRef] [Google Scholar]
  31. Fletcher, A. 2010, in The Dynamic Interstellar Medium: A Celebration of the Canadian Galactic Plane Survey, eds. R. Kothes, T. L. Landecker, & A. G. Willis, ASP Conf. Ser., 438, 197 [NASA ADS] [Google Scholar]
  32. Gent, F. A., Shukurov, A., Sarson, G. R., Fletcher, A., & Mantere, M. J. 2013, MNRAS, 430, L40 [NASA ADS] [CrossRef] [Google Scholar]
  33. Gil de Paz, A., Boissier, S., Madore, B. F., et al. 2007, ApJS, 173, 185 [Google Scholar]
  34. Girichidis, P., Naab, T., Hanasz, M., & Walch, S. 2018, MNRAS, 479, 3042 [NASA ADS] [CrossRef] [Google Scholar]
  35. Gressel, O., Elstner, D., & Ziegler, U. 2013, A&A, 560, A93 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  36. Heesen, V., Brinks, E., Leroy, A. K., et al. 2014, AJ, 147, 103 [NASA ADS] [CrossRef] [Google Scholar]
  37. Heesen, V., Buie, II. E., Huff, C. J., et al. 2019, A&A, 622, A8 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  38. Heesen, V., Staffehl, M., Basu, A., et al. 2022, A&A, 664, A83 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  39. Imamura, K., & Sofue, Y. 1997, A&A, 319, 1 [NASA ADS] [Google Scholar]
  40. Irwin, J., Beck, R., Benjamin, R. A., et al. 2012, AJ, 144, 43 [Google Scholar]
  41. Jacob, S., Pakmor, R., Simpson, C. M., Springel, V., & Pfrommer, C. 2018, MNRAS, 475, 570 [NASA ADS] [CrossRef] [Google Scholar]
  42. Kennicutt, R. C., Jr., Armus, L., Bendo, G., et al. 2003, PASP, 115, 928 [NASA ADS] [CrossRef] [Google Scholar]
  43. Kennicutt, R. C., Jr., Calzetti, D., Aniano, G., et al. 2011, PASP, 123, 1347 [NASA ADS] [CrossRef] [Google Scholar]
  44. Kim, K. S., Lilly, S. J., Miniati, F., et al. 2016, ApJ, 829, 133 [NASA ADS] [CrossRef] [Google Scholar]
  45. Krause, M., Irwin, J., Wiegert, T., et al. 2018, A&A, 611, A72 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  46. Krumholz, M. R., & Federrath, C. 2019, Front. Astron. Space Sci., 6, 7 [NASA ADS] [CrossRef] [Google Scholar]
  47. Lasker, B. M., Doggett, J., McLean, B., et al. 1996, in Astronomical Data Analysis Software and Systems V, eds. G. H. Jacoby, & J. Barnes, ASP Conf. Ser., 101, 88 [NASA ADS] [Google Scholar]
  48. Leroy, A. K., Walter, F., Brinks, E., et al. 2008, AJ, 136, 2782 [Google Scholar]
  49. Leroy, A. K., Walter, F., Bigiel, F., et al. 2009, AJ, 137, 4670 [Google Scholar]
  50. Leroy, A. K., Schinnerer, E., Hughes, A., et al. 2021, ApJS, 257, 43 [NASA ADS] [CrossRef] [Google Scholar]
  51. McMullin, J. P., Waters, B., Schiebel, D., Young, W., & Golap, K. 2007, in Astronomical Data Analysis Software and Systems XVI, eds. R. A. Shaw, F. Hill, & D. J. Bell, ASP Conf. Ser., 376 [Google Scholar]
  52. Mogotsi, K. M., de Blok, W. J. G., Caldú-Primo, A., et al. 2016, AJ, 151, 15 [NASA ADS] [CrossRef] [Google Scholar]
  53. Mulcahy, D. D., Horneffer, A., Beck, R., et al. 2014, A&A, 568, A74 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  54. Mulcahy, D. D., Fletcher, A., Beck, R., Mitra, D., & Scaife, A. M. M. 2016, A&A, 592, A123 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  55. Murgia, M., Helfer, T. T., Ekers, R., et al. 2005, A&A, 437, 389 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  56. Niklas, S. 1995, PhD Thesis, Univ. Bonn [Google Scholar]
  57. Niklas, S., & Beck, R. 1997, A&A, 320, 54 [NASA ADS] [Google Scholar]
  58. O’Sullivan, S. P., Brüggen, M., Vazza, F., et al. 2020, MNRAS, 495, 2607 [Google Scholar]
  59. Pakmor, R., Gómez, F. A., Grand, R. J. J., et al. 2017, MNRAS, 469, 3185 [NASA ADS] [CrossRef] [Google Scholar]
  60. Pakmor, R., Pfrommer, C., Simpson, C. M., & Springel, V. 2016, ApJ, 824, L30 [NASA ADS] [CrossRef] [Google Scholar]
  61. Paladino, R., Murgia, M., Helfer, T. T., et al. 2006, A&A, 456, 847 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  62. Paladino, R., Murgia, M., Tarchi, A., Moscadelli, L., & Comito, C. 2008, A&A, 485, 679 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  63. Pfrommer, C., Werhahn, M., Pakmor, R., Girichidis, P., & Simpson, C. M. 2022, MNRAS, 515, 4229 [NASA ADS] [CrossRef] [Google Scholar]
  64. Recchia, S., Blasi, P., & Morlino, G. 2016, MNRAS, 462, 4227 [NASA ADS] [CrossRef] [Google Scholar]
  65. Rieder, M., & Teyssier, R. 2016, MNRAS, 457, 1722 [NASA ADS] [CrossRef] [Google Scholar]
  66. Salem, M., & Bryan, G. L. 2014, MNRAS, 437, 3312 [CrossRef] [Google Scholar]
  67. Schinnerer, E., Meidt, S. E., Pety, J., et al. 2013, ApJ, 779, 42 [Google Scholar]
  68. Schleicher, D. R. G., & Beck, R. 2013, A&A, 556, A142 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  69. Seta, A., & Beck, R. 2019, Galaxies, 7, 45 [NASA ADS] [CrossRef] [Google Scholar]
  70. Seta, A., & Federrath, C. 2022, MNRAS, 514, 957 [NASA ADS] [CrossRef] [Google Scholar]
  71. Shimwell, T. W., Röttgering, H. J. A., Best, P. N., et al. 2017, A&A, 598, A104 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  72. Shimwell, T. W., Tasse, C., Hardcastle, M. J., et al. 2019, A&A, 622, A1 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  73. Shimwell, T. W., Hardcastle, M. J., Tasse, C., et al. 2022, A&A, 659, A1 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  74. Smith, D. J. B., Haskell, P., Gürkan, G., et al. 2021, A&A, 648, A6 [EDP Sciences] [Google Scholar]
  75. Stein, Y., Dettmar, R. J., Beck, R., et al. 2020, A&A, 639, A111 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  76. Steinwandel, U. P., Dolag, K., Lesch, H., et al. 2020, MNRAS, 494, 4393 [NASA ADS] [CrossRef] [Google Scholar]
  77. Sun, J., Leroy, A. K., Schruba, A., et al. 2018, ApJ, 860, 172 [NASA ADS] [CrossRef] [Google Scholar]
  78. Tabatabaei, F. S., Beck, R., Krügel, E., et al. 2007, A&A, 475, 133 [CrossRef] [EDP Sciences] [Google Scholar]
  79. Tabatabaei, F. S., Krause, M., Fletcher, A., & Beck, R. 2008, A&A, 490, 1005 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  80. Tabatabaei, F. S., Schinnerer, E., Murphy, E. J., et al. 2013, A&A, 552, A19 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  81. Tabatabaei, F. S., Schinnerer, E., Krause, M., et al. 2017, ApJ, 836, 185 [Google Scholar]
  82. Tabatabaei, F. S., Minguez, P., Prieto, M. A., & Fernández-Ontiveros, J. A. 2018, Nat. Astron., 2, 83 [Google Scholar]
  83. Tamburro, D., Rix, H. W., Leroy, A. K., et al. 2009, AJ, 137, 4424 [Google Scholar]
  84. van Haarlem, M. P., Wise, M. W., Gunst, A. W., et al. 2013, A&A, 556, A2 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  85. Veilleux, S., Maiolino, R., Bolatto, A. D., & Aalto, S. 2020, A&ARv, 28, 2 [NASA ADS] [CrossRef] [Google Scholar]
  86. Vogelsberger, M., Marinacci, F., Torrey, P., & Puchwein, E. 2020, Nat. Rev. Phys., 2, 42 [Google Scholar]
  87. Walter, F., Brinks, E., de Blok, W. J. G., et al. 2008, AJ, 136, 2563 [Google Scholar]
  88. Werhahn, M., Pfrommer, C., & Girichidis, P. 2021a, MNRAS, 508, 4072 [NASA ADS] [CrossRef] [Google Scholar]
  89. Werhahn, M., Pfrommer, C., Girichidis, P., Puchwein, E., & Pakmor, R. 2021b, MNRAS, 505, 3273 [NASA ADS] [CrossRef] [Google Scholar]
  90. Wiegert, T., Irwin, J., Miskolczi, A., et al. 2015, AJ, 150, 81 [Google Scholar]
  91. Yoast-Hull, T. M., Gallagher, J. S., & Zweibel, E. G. 2016, MNRAS, 457, L29 [NASA ADS] [CrossRef] [Google Scholar]
  92. Yu, B. P. B., Owen, E. R., Wu, K., & Ferreras, I. 2020, MNRAS, 492, 3179 [NASA ADS] [CrossRef] [Google Scholar]

Appendix A: Equipartition method

The equipartition estimate of total magnetic field strengths by Beck & Krause (2005, BK05;) is the standard method to estimate magnetic field strengths in star-forming galaxies. Energy equipartition between total cosmic rays and total magnetic fields is thought to be valid on scales larger than the CRE propagation length (Seta & Beck 2019). A ratio K0 of number densities of cosmic-ray protons (CRps) to the total number of cosmic rays needs to be assumed to extrapolate the CRE spectrum (traced by radio synchrotron emission) to the spectrum of CRps that dominate the total cosmic-ray energy. K0 = 100 is the standard value for the GeV range that is relevant for gigahertz emission. K0 ≈ 100 is expected from the theory of diffusive shock acceleration (Bell 1978) and has also been measured in the local ISM (e.g. Cummings et al. 2016).

Concerns may arise whether the BK05 method is applicable to outer discs and haloes of spiral galaxies, where CRE have lost a significant fraction of their energy. As energy losses of CRps are much weaker, the p/e ratio K0 of number densities becomes larger than the standard value of 100 and varies with frequency. However, considering this complication is not needed because the BK05 method is quite robust against the effects of energy losses of CRE, as shown by the five following examples.

First, as the standard case, we assume a star-forming galaxy with inclination i = 30°, path length l = 1.0 kpc/cos i, 15″ beam, spectral index α = −0.7, flux density Sν = 1.0 mJy beam−1 at 1.4 GHz, corresponding to 0.286 mJy beam−1 at 8.35 GHz, degree of polarisation p = 10 %, and K0 = 100. Application of the BFELD tool (written by M. Krause) yields a total field of Beq = 13.4 μG that is regarded to be the ‘correct’ one.

The second is weak synchrotron loss at the higher frequency: α = −0.9 and flux density Sν = 1.0 mJy beam−1 at 1.4 GHz (no loss) gives 0.2 mJy beam−1 at 8.35 GHz (loss by a factor of about 1.4 compared to the standard case) and Beq = 13.5 μG, very close to the correct one.

The third is strong synchrotron loss at the higher frequency: α = −1.1 and flux density Sν = 1.0 mJy beam−1 at 1.4 GHz (no loss) gives 0.14 mJy beam−1 at 8.35 GHz (loss by a factor of about 2 compared to the standard case). When naively assigning the same low spectral index to CRps, we get Beq = 14.3 μG, still close to the correct value. The reason is that the integral over the steep CRp energy spectrum, extrapolated from the steep CRE energy spectrum using a constant K0, remains almost constant, even though the real CRp spectrum is flatter.

The fourth is strong spectral steepening at the higher frequency due to an energy cutoff, for example by time-dependent injection: α = −1.5 and flux density Sν = 1.0 mJy beam−1 at 1.4 GHz (no loss) gives 0.07 mJy beam−1 at 8.35 GHz (loss by a factor of about 4 compared to the standard case). When assigning the same low spectral index to CRps, we get Beq = 16.4 μG, a significant overestimate compared to the correct value.

The last is ionisation loss or free-free absorption: α = −0.55 and flux density Sν = 1.0 mJy beam−1 at 1.4 GHz (no loss) gives 3.55 mJy beam−1 at 0.1 GHz (loss by a factor of about 1.8 compared to the standard case). When assigning the same high spectral index to CRps, we get Beq = 16.7 μG, again an overestimate.

Figure A.1 shows the magnetic field strength as a function of a wide range of spectral indices. The overestimates are large for very steep and very flat spectra.

thumbnail Fig. A.1.

Equipartition estimates of the total magnetic field strength as a function of synchrotron spectral index, assuming a fixed flux density at 1.4 GHz and a constant value of the proton-to-electron ratio of K0 = 100.

We finish this section with our conclusions and some recommendations on how to calculate equipartition magnetic field strengths. First, synchrotron loss of CRE hardly affects the equipartition estimate of the total field. The error is are smaller than the uncertainties of the other input parameters, K0 and l. The reason is that the BK05 method extrapolates and integrates the spectrum of CRps over all energies (with a break at the proton rest mass), so that no particles are ‘lost’.

Second, for very steep CRE spectra the low-energy part of the extrapolated CRp spectrum is overestimated, and so is the field strength. The standard BK05 formula gives approximately correct values only for synchrotron spectra with spectral indices of α > −1.1, while steeper spectra lead to significant overestimates of the field strength. Hence, regions with α < −1.1 should be clipped.

Third, the BK05 method should only be applied for synchrotron spectra with α < −0.6. Flatter spectra, when assumed to apply also to CRps, lead to massive overestimates of the field strength because the high-energy part of the CRp spectrum dominates. The energy integral over the CRp spectrum even diverges for α ≥ −0.5. Hence, regions with α > −0.6 should be clipped.

Finally, energy losses of CRE increase the p/e ratio K0, for example to values of 140 (case b), 200 (case c), and 400 (case d). However, using such high values for the equipartition estimate would lead to even larger overestimates of the field strengths. Hence, K0 = 100 should be used in all cases.

Appendix B: Atlas of magnetic fields in galaxies

In this Appendix, we present the atlas of magnetic fields. For each galaxy, we show the equipartition magnetic field strength as a map made from LOFAR 144 MHz data with 6 arcsec angular resolution. The first panel shows a grey-scale representation, where the extent of the 6 arcsec emission to the 3σ contour line is shown as a blue ellipse (data taken from Heesen et al. 2022). The second panel shows an overlay of the magnetic field strength as contour levels onto an optical image. These rgb images are from the Sloan Digital Sky Survey (SDSS) data release 16 (Ahumada et al. 2020) or from the Digitized Sky Survey (DSS2; Lasker et al. 1996) for NGC 891 and 925, which we got from the hips2fits2 service. In the case of NGC 891 and 925, DSS2 was used because the galaxy is outside the SDSS footprint. These maps are presented in Figs. B.1–B.39.

thumbnail Fig. B.1.

NGC 855. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.2.

NGC 891. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.3.

NGC 925. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.4.

NGC 2683. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.5.

NGC 2798. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.6.

NGC 2820. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.7.

NGC 2841. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.8.

NGC 2976. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.9.

NGC 3003. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.10.

NGC 3077. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.11.

NGC 3079. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.12.

NGC 3184. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.13.

NGC 3198. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.14.

NGC 3265. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.15.

NGC 3432. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.16.

NGC 3448. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.17.

NGC 3556. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.18.

NGC 3877. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.19.

NGC 3938. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.20.

NGC 4013. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.21.

NGC 4096. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.22.

NGC 4125. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.23.

NGC 4157. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.24.

NGC 4217. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.25.

NGC 4244. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.26.

NGC 4449. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.27.

NGC 4625. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.28.

NGC 4631. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.29.

NGC 4725. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.30.

NGC 4736. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.31.

NGC 5033. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.32.

NGC 5055. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.33.

NGC 5194. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.34.

NGC 5195. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.35.

NGC 5297. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.36.

NGC 5457. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.37.

NGC 5474. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.38.

NGC 5907. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

thumbnail Fig. B.39.

NGC 7331. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).

Appendix C: Magnetic field–gas relation in individual galaxies

In this appendix we present the magnetic field–gas relation in the sub-sample of nine galaxies, where we have analysed the data. We show the individual relations, separated as B–ΣH I relation for the atomic gas only, as B–ΣH2 for the molecular gas only and as B–ΣH I + H2 for the combined atomic and molecular gas. We present for each galaxy an overlay of the equipartition field strength as contours on the gas mass surface densities and the corresponding relations. The contour levels of the magnetic field strength can read of from the colour bars of the maps. The data points in the individual relations are coloured according to the radio spectral index. Best-fitting relations are shown as solid lines and 1σ confidence levels as grey-shaded areas. The theoretical lines for simplified assumptions such as constant velocity dispersion are shown as red dashed lines. The relations are presented in Figs. C.1–C.9.

thumbnail Fig. C.1.

NGC 925. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).

thumbnail Fig. C.2.

NGC 2841. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).

thumbnail Fig. C.3.

NGC 2976. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).

thumbnail Fig. C.4.

NGC 3184. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).

thumbnail Fig. C.5.

NGC 3198. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).

thumbnail Fig. C.6.

NGC 4736. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).

thumbnail Fig. C.7.

NGC 5055. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).

thumbnail Fig. C.8.

NGC 5194. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).

thumbnail Fig. C.9.

NGC 7331. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).

Appendix D: Radial profiles of the magnetic field strengths

To determine radial profiles of the magnetic field strength, we estimated the mean magnetic field within ellipsoidal annuli, where each annulus has a width of 6 arcsec. Errors were estimated by adding in quadrature the mean magnetic field error and the error of the mean in each annulus. For the latter we divided the standard deviation of the magnetic field strength by the square-root of the number of beams. We disregarded pixels, where the magnetic field strength could not be determined. The position angle of the ellipses were taken from the literature, and the eccentricity of the ellipses were determined using the 3σ contour lines from the 6 arcsec intensity maps (see Heesen et al. 2022, for details). These ellipses are hence bound by the ellipses presented already in the atlas of magnetic fields (Appendix B).

thumbnail Fig. D.1.

Radial profiles of the equipartition field strengths in our galaxy sample. Solid lines show best-fitting exponential relations. Vertical lines show the extent of the radio disc (r = a6″) and the extent of the optical stellar disc (r = r25).

thumbnail Fig. D.2.

Radial profiles of the equipartition field strengths in our galaxy sample. Solid lines show best-fitting exponential relations. Vertical lines show the extent of the radio disc (r = a6″) and the extent of the optical stellar disc (r = r25).

All Tables

Table 1.

Magnetic field strengths and ancillary data in the LoTSS-DR2 sample.

In the text
Table 2.

Influence of the correction for thermal emission on the magnetic field strength.

In the text
Table 3.

Diffusion lengths used to account for cosmic-ray transport.

In the text
Table 4.

Radial scale length of the magnetic field.

In the text
Table 5.

Correlations with the magnetic field strength.

In the text

All Figures

thumbnail Fig. 1.

Difference in magnetic field strength, ΔB = Beq − Bnt, due to the thermal correction in NGC 5194/5.
In the text
thumbnail Fig. 2.

Spatially resolved relations between the magnetic field strength and the SFR surface density (B–ΣSFR) (a), the atomic gas mass surface density (B–ΣH I) (b), molecular gas mass surface density (B–ΣH2) (c), and the combined atomic and molecular gas mass surface density (B–ΣH I + H2) (d). Data points are coloured according to the radio spectral index and, in panel (b), according to the point density. Best-fitting relations are shown as sold lines and the theoretical expectations as dashed lines.
In the text
thumbnail Fig. 3.

As Fig. 2, but only for galaxies that are relatively gas poor. These are all galaxies except the grand spiral galaxies NGC 5055, 5194/5, and 7331. As can be seen, these galaxies approximately conform with the relations that are found for the complete sample.
In the text
thumbnail Fig. 4.

Global relations between the mean magnetic field strength and the mean SFR surface density (a), the mean atomic gas mass surface density (b), mean molecular gas mass surface density (c), and the mean combined atomic and molecular gas mass surface density (d). Best-fitting relations are shown as sold lines and the theoretical expectations as dashed lines.
In the text
thumbnail Fig. 5.

Spatially resolved relations with cosmic-ray transport accounted for between the magnetic field strength and the SFR surface density (a) and with the combined atomic and molecular gas mass surface density (b). The ΣSFR and the ΣH I + H2 maps have been smoothed with a Gaussian kernel in order to account for cosmic-ray transport. Data points are coloured according to the radio spectral index and, in panel (b), according to the point density. Best-fitting relations are shown as sold lines and the theoretical expectations as dashed lines.
In the text
thumbnail Fig. 6.

Relation between the magnetic energy density and the total kinetic energy density. The best-fitting relation is shown as a solid line and the energy equipartition as a dashed line. Data points are coloured according to their radio spectral index, highlighting that these relations are universal for all galaxies.
In the text
thumbnail Fig. A.1.

Equipartition estimates of the total magnetic field strength as a function of synchrotron spectral index, assuming a fixed flux density at 1.4 GHz and a constant value of the proton-to-electron ratio of K0 = 100.
In the text
thumbnail Fig. B.1.

NGC 855. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.2.

NGC 891. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.3.

NGC 925. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.4.

NGC 2683. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.5.

NGC 2798. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.6.

NGC 2820. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.7.

NGC 2841. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.8.

NGC 2976. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.9.

NGC 3003. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.10.

NGC 3077. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.11.

NGC 3079. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.12.

NGC 3184. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.13.

NGC 3198. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.14.

NGC 3265. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.15.

NGC 3432. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.16.

NGC 3448. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.17.

NGC 3556. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.18.

NGC 3877. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.19.

NGC 3938. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.20.

NGC 4013. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.21.

NGC 4096. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.22.

NGC 4125. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.23.

NGC 4157. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.24.

NGC 4217. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.25.

NGC 4244. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.26.

NGC 4449. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.27.

NGC 4625. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.28.

NGC 4631. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.29.

NGC 4725. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.30.

NGC 4736. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.31.

NGC 5033. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.32.

NGC 5055. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.33.

NGC 5194. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.34.

NGC 5195. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.35.

NGC 5297. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.36.

NGC 5457. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.37.

NGC 5474. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.38.

NGC 5907. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. B.39.

NGC 7331. Equipartition magnetic field strength as grey-scale map (a) and as contour map (b).
In the text
thumbnail Fig. C.1.

NGC 925. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).
In the text
thumbnail Fig. C.2.

NGC 2841. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).
In the text
thumbnail Fig. C.3.

NGC 2976. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).
In the text
thumbnail Fig. C.4.

NGC 3184. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).
In the text
thumbnail Fig. C.5.

NGC 3198. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).
In the text
thumbnail Fig. C.6.

NGC 4736. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).
In the text
thumbnail Fig. C.7.

NGC 5055. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).
In the text
thumbnail Fig. C.8.

NGC 5194. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).
In the text
thumbnail Fig. C.9.

NGC 7331. Overlay of the equipartition magnetic field strength as contours on the H I mass surface density (a), H2 mass surface density (c) and combined H I and H2 mass surface density (e). Spatially resolved B–ΣH I relation (b), B–ΣH2 relation (d) and B–ΣH I + H2 relation (f).
In the text
thumbnail Fig. D.1.

Radial profiles of the equipartition field strengths in our galaxy sample. Solid lines show best-fitting exponential relations. Vertical lines show the extent of the radio disc (r = a6″) and the extent of the optical stellar disc (r = r25).
In the text
thumbnail Fig. D.2.

Radial profiles of the equipartition field strengths in our galaxy sample. Solid lines show best-fitting exponential relations. Vertical lines show the extent of the radio disc (r = a6″) and the extent of the optical stellar disc (r = r25).
In the text

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