Diffusion of cosmic-ray electrons in M 51 observed with LOFAR at 54 MHz

Context. The details of cosmic-ray transport have a strong impact on galaxy evolution. The peak of the cosmic-ray energy distribution is observable in the radio continuum using the electrons as proxy. Aims. We measure the length that the cosmic-ray electrons (CRE) are transported during their lifetime in the nearby galaxy M 51 across one order of magnitude in cosmic-ray energy (approximately 1-10 GeV). To this end we use new ultra-low frequency observations from the LOw Frequency ARay (LOFAR) at 54 MHz and ancillary data between 144 and 8350 MHz. Methods. As the the CRE originate from supernova remnants, the radio maps are smoothed in comparison to the distribution of the star formation. By convolving the map of the star-formation rate (SFR) surface density with a Gaussian kernel, we can linearise the radio-SFR relation. The best-fitting convolution kernel is then our estimate of the CRE transport length. Results. We find that the CRE transport length increases at low frequencies, as expected since the CRE have longer lifetimes. The CRE transport length is $l_{\rm CRE} = \sqrt{4Dt_{\rm syn}}$, where $D$ is the isotropic diffusion coefficient and $t_{\rm syn}$ is the CRE lifetime as given by synchrotron and inverse Compton losses. We find that the data can be well fitted by diffusion, where $D=(2.14\pm 0.13) \times 10^{28}~\rm cm^2\,s^{-1}$. With $D\propto E^{0.001\pm 0.185}$, the diffusion coefficient is independent of the CRE energy $E$ in the range considered. Conclusions. Our results suggest that the transport of GeV-cosmic ray electrons in the star-forming discs of galaxies is governed by energy-independent diffusion.


Introduction
The formation and evolution of galaxies is regulated by the accretion and ejection of matter.The latter occurs in galactic winds, which are the result of star formation and subsequent supernovae.This so-called stellar feedback causes heating of gas, and this hot ionised gas can thermally drive winds (see Veilleux et al. 2020, for a recent review).Recently, it has become evident both from theoretical considerations (Breitschwerdt et al. 1991;Recchia et al. 2016;Yu et al. 2020) and simulations (Salem & Bryan 2014;Pakmor et al. 2016;Jacob et al. 2018) that cosmic rays can also drive winds on galactic scales.Indeed, cosmic-ray-driven winds are thought to be more mass loaded, containing the warm ionised medium with a much higher density (Girichidis et al. 2018).Also, cosmic rays can potentially drive winds in galaxies, where the thermal gas alone is not sufficient to launch a wind (Everett et al. 2008).
Map fits are only available at the CDS via anonymous ftp to cdsarc.cds.unistra.fr(130.79.128.5) or via https:// cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/672/A21 Hence, the influence of cosmic rays could lead to much more ubiquitous galactic winds that are also denser.This would have far-reaching consequences for galaxy evolution and the relation of galaxies with the circumgalactic medium (Tumlinson et al. 2017).
The details of cosmic-ray transport are paramount for properly incorporating them into models of galaxy evolution, with the various contributions from diffusion, advection, and streaming to be included (Hopkins et al. 2020).Most of the understanding of cosmic-ray transport has so far come from our Galaxy, with observations such as those made with the AMS-02 experiment being able to precisely measure the composition and energy spectrum of cosmic rays (Aguilar et al. 2016).While there is no reason to believe that external galaxies should have any different behaviour, their observation provides us with a global view of a galaxy that potentially allows us to study the transport from the disc into the halo (Heesen 2021).This is important as the size of the halo in question, at the boundary of which cosmic rays are assumed to escape in the analysis of Galactic cosmic-ray transport, is a free parameter (Evoli et al. 2020) resulting in a degeneracy between halo size and diffusion coefficient (Chan et al. 2019).Also, the gaseous thin disc and gaseous halo, which have very different magnetic field structures, are essential to study to obtain a meaningful description of cosmicray transport.It has hence been suggested that a vertical gradient of cosmic rays in the halo can lead to streaming of cosmic rays with a velocity of the order of the Alfvén speed along the magnetic field lines (Wiener et al. 2017) which then can lead to a galactic wind (Uhlig et al. 2012).In contrast, the transport of cosmic rays in the disc may be more mediated by magnetic field irregularities that cascade down via turbulence from the injection scale to the gyro (Larmor) radius of 10 11 cm for a 1 GeV CRE in a 10 µG magnetic field (the 'extrinsic turbulence' picture, see Crocker et al. 2021).Then again, cosmic rays can generate their own turbulence and this 'self-confinement' picture is usually invoked in order to explain why cosmic rays are confined for a considerable time to the galaxy (Zweibel 2013).The transport of cosmic-ray electrons may be particularly difficult to model as the usual approximation of a continuous steady state distribution of cosmic ray sources can not be applied.The energy loss time of very highly energetic electrons of about 1 TeV energy is only ≈10 5 yr in the interstellar medium.These particles may diffuse at a given point in a galaxy only from a few nearby sources such as supernova remnants.
One way to study cosmic-ray transport of electrons is to use radio continuum observations of nearby galaxies (see Heesen 2021, for an overview).Radio continuum maps provide us with spatially resolved information via the synchrotron emission from cosmic-ray electrons (CREs) spiralling around magnetic field lines.These electrons, observed in tens of megahertz to tens of gigahertz, have energies of 1-10 GeV, corresponding to the peak of the cosmic-ray energy density.Hence, learning something about the CRE can help us us to study cosmic-ray transport for the dynamically most important part of the energy spectrum.The radio continuum images of galaxies show that the synchrotron emission is widely pervasive and only 10% of the emission stems from supernovae remnants (Lisenfeld & Völk 2000), the most likely source for cosmic-ray acceleration.As the diffuse radio continuum shows the presence of CREs that originate in supernova remnants, we can study cosmic-ray transport by comparing the distribution of massive star formation with that of the radio emission.In particular, one usually compares the distribution of the star-formation rate surface density, Σ SFR , with the radio continuum intensity (e.g., Murphy et al. 2008;Berkhuijsen et al. 2013;Tabatabaei et al. 2013;Heesen et al. 2014Heesen et al. , 2019a;;Vollmer et al. 2020).The radio continuum map is thus the smoothed version of the Σ SFR map, and the smoothing scale is then equated to the CRE transport length.Murphy et al. (2008) compared 1365-MHz emission with 70-µm far-infrared emission, a tracer for star formation, in a sample of 18 nearby galaxies.They found that the smoothing length scale is about 1 kpc, with some variation as a function of the Σ SFR value, with smaller lengths for higher Σ SFR values.At higher Σ SFR values, magnetic field strengths and so synchrotron losses are higher leading to reduced CRE number densities.Hence, this finding is consistent with CRE transport.Vollmer et al. (2020) studied galaxies at two frequencies, 1460 and 4850 MHz, in order to compare the smoothing length scale.They found that most galaxies appear to be diffusion dominated, with a few having dominating cosmic-ray streaming.Heesen et al. (2019a) studied three galaxies, where it was found that they were consistent with diffusion.Mulcahy et al. (2014) used scale lengths at the edge of M 51 in order to measure the diffusion coefficient.A different approach in face-on galaxies is to use the radial radio spectral index distribution and model this with cosmic ray diffusion (Mulcahy et al. 2016;Dörner et al. 2023).This approach is in particular sensitive to the escape of cosmic rays, as the radio continuum spectrum otherwise becomes too steep.This escape can be more directly studied in edge-on galaxies that also allow us to measure the diffusion coefficient in radio haloes that are diffusion dominated (Heesen et al. 2016(Heesen et al. , 2019b;;Schmidt et al. 2019;Stein et al. 2019).
In this work, we expand these kinds of studies to the lowest radio frequencies with new observations of the nearby galaxy M 51 (distance of 8.0 Mpc) using the Low Frequency ARray (LOFAR; van Haarlem et al. 2013).Observations of M 51 were already presented by Heesen et al. (2019a) at 144 MHz using the high-band antenna (HBA) system.We significantly expand the frequency range to lower frequencies using the low-band antenna (LBA) system to obtain a map at 54 MHz.This map is the deepest and best resolved map of this source at this low frequency to date (de Gasperin et al. 2021).By using ancillary data of frequencies up to 8350 MHz, we are now able to span a frequency range in excess of two orders of magnitude, corresponding to the more than an order of magnitude in cosmic-ray energy.We exploit these data in order to determine the so far most accurately measured relation between CRE transport length and lifetime.As we will show, our data are consistent with diffusion where the diffusion coefficient is energy independent and in good agreement with the canonical Galactic value.
This paper is organised as follows.In Sect.2, we describe our radio continuum data including the observations with LOFAR LBA.Section 3 describes the methods we have employed to measure the CRE transport length.We present our results in Sect. 4 and discuss their implications in Sect. 5. We conclude in Sect.6.

LOFAR LBA
We observed with LOFAR LBA between 2017 and 2019 as part of the LOFAR LBA Sky Survey (LoLSS; de Gasperin et al. 2021) in the frequency range of 42-66 MHz.The LBA stations were used in LBA_OUTER mode, meaning that the primary beam has a 4 • full width at half maximum (FWHM) size, closely matched to the HBA primary beam.The (u, v) data include baselines of up to 100 km in length, so a nominal resolution of 15 arcsec can be reached.The final data release of LoLSS will reach this resolution at an rms noise level of 1 mJy beam −1 .This will entail a direction-dependent calibration technique, which is currently being developed and explored (de Gasperin et al. 2020).This paper uses the preliminary LoLSS data release, which has a resolution of 47 arcsec and a median rms noise of 5 mJy beam −1 covering 740 deg 2 around the HETDEX field (de Gasperin et al. 2021).The actual noise of our field is indeed slightly better with 3.3 mJy beam −1 .This means our sensitivity for extended emission on the scale of one synthesized beam is comparable to the final data release.
The data were first calibrated in a standard fashion with the pill pipeline (de Gasperin et al. 2019).As stated before, only direction-independent calibration effects were taken into account.The imaging was performed with wsclean (Offringa et al. 2014) with a Briggs weighting of −0.3 and multiscale cleaning.An outer (u, v) range of 4500λ was used in order to limit direction-dependent ionospheric errors.The 95 direction-dependent calibrated images were combined into a A21, page 2 of 11 single large mosaic.This mosaic formed the basis for our analysis presented here.

Other data
The LOFAR HBA data at 144 MHz were obtained from the LoTSS survey data release 2 (Shimwell et al. 2022), from which we took the 20 arcsec map presented in Heesen et al. (2022).
The 1365 MHz map is from observations with the Westerbork Synthesis Radio Telescope (WSRT) presented by Braun et al. (2007).We also used a 4850 MHz map and a 8350 MHz map from Fletcher et al. (2011), which combined observations with the Very Large Array (VLA) with single-dish data using observations with the 100 m Effelsberg telescope.The largest angular scale that the WSRT can observe at 1365 MHz is 25 arcmin when one considers base lines of longer than 27 m (Heesen et al. 2014).As M 51 has an angular extent of 14 arcmin in the radio continuum, we do not have to correct for missing zero spacings of our interferometric radio data.For the VLA observations a correction was made merging the VLA and Effelsberg maps in the image space (Fletcher et al. 2011).The hybrid SFR maps are taken from Leroy et al. (2008), who use a combina-tion of GALEX far-UV 156 nm and Spitzer mid-IR 24 µm data.We applied a correction for thermal emission using Hα data that were corrected for absorption by dust using a combination of Spitzer 70 and 160 µm maps (Tabatabaei et al. 2007), which will be presented in Tabtabaei et al. (in prep.).The integrated thermal flux density is 55 mJy at 1400 MHz, which is in fair agreement with the 5% thermal fraction reported by Tabatabaei et al. (2017) from radio spectral energy distribution fitting.Maps were transformed to the same coordinate system and convolved with a Gaussian to a resolution to 20 arcsec; the 54 MHz map was left at native resolution.Strong contaminating sources including the nuclei of both M 51a and M 51b were masked, as were a few compact sources within the disc of M 51.These have sizes of 10 milliarcseconds (Rampadarath et al. 2015), and hence they are either supernova remnants within M 51 or active galactic nuclei in the background.In order to ease the comparison with the hybrid star formation rate surface density map, we converted the maps into radio Σ SFR , (Σ SFR ) RC .We used Condon's (Condon 1992) linear relation between radio continuum luminosity and star formation rate.We then converted the radio continuum intensity into (Σ SFR ) RC as described in Heesen et al. (2014Heesen et al. ( , 2019a) assuming a radio spectral index A21, page 3 of 11 of −0.8.This radio spectral index is in good agreement with the integrated radio continuum spectrum of M 51 (Mulcahy et al. 2014).These steps were carried out in the Astronomical Image Processing System (aips), where we made use of the parseltongue python interface (Kettenis et al. 2006).

Morphology of the radio continuum maps
In Fig. 1, we present the radio continuum maps at various frequencies.We can already visually see a clear trend of a decreasing smoothness of the radio continuum emission with increasing frequency (Figs.1a-e) when compared with the hybrid Σ SFR map (Fig. 1f).In particular, in the outskirts the galaxy appears to shrink, and the arm-interarm contrast increases.This is borne out in the radio spectral index maps shown in Fig 2, where in the outskirts of the galaxy the radio spectral index are particularly steep with α < −0.9 (areas in green and blue) both in the 54-144 MHz (Fig. 2a) and 144-1365 (Fig. 2b) radio spectral index maps.At higher frequencies, in the 4850-8350 MHz radio spectral index map (Fig. 2c), the outskirts become invisible due to spectral ageing, but the inter-arm regions have a steep radio spectrum.In contrast, at all frequencies the radio spectrum is consistent with young CREs in the spiral arms, where the radio spectral index is α > −0.6.
The radio spectral index shows areas where the radio continuum spectrum is flatter than expected for an injection spectral index of α ≈ −0.5, in particular in the spiral arms.At low frequencies 144 MHz, this may be caused by thermal absorption where H ii regions become optically thick.Alternatively, the low-energy CRE spectrum may flatten due to the combination of ionisation and bremsstrahlung losses (Basu et al. 2015).
With the new 54 MHz data this can be explored in more detail in the future.At higher frequencies, we also find such a flat radio continuum spectra in the spiral arms, which are prominent in the 4850-8350 MHz radio spectral index map.Since we corrected for thermal emission and the emission would have to be almost 100% thermal, we attribute this to uncertainties in observations of the radio continuum emission.For instance, a small decrease of radio emission at 4850 MHz, which can be caused by the merging of interferometric and single-dish data (VLA and Effelsberg data in the case of Fletcher et al. 2011).Such a deviation is suggested by the radio continuum spectrum of the integrated flux density (see Fig. 2 in Kierdorf et al. 2020).

Radio-star formation rate relation
We first analyse the spatially resolved radio continuum-starformation rate (radio-SFR) relation.We convolve the maps to a spatial resolution of 1.2 kpc corresponding to 30.1 arcsec (FWHM) and bin them to a pixel size of 1.2 × 1.2 kpc 2 (1.8 × 1.8 kpc 2 for 54 MHz).Our analysis was done using the software radio-pixel-plots (rpp) 1 .We take two radio continuum maps at different frequencies and the Σ SFR map and calculate radio and hybdrid Σ SFR values and radio spectral indices.We then fit the radio continuum Σ SFR values as function of the hybrid Σ SFR values with the following function: log 10 [(Σ SFR ) RC ] = a log 10 (Σ SFR ) hyb + b. (1) 1 github.com/sebastian-schulz/radio-pixel-plots For the fitting, we use the orthogonal distance regression (odr) package from the scipy python library, which employs a modified Levenberg-Marquardt algorithm.Its strength is dealing with both x and y errors even in the case of large errors.Even though the measurement uncertainties in our radio and hybrid SFR data are small, the conversion to Σ SFR values has an error of approximately 50% (Leroy et al. 2012).
For the uncertainties, we consider both the map rms noise and a calibration error of 5% (Heesen et al. 2014).After the resolution is converted to 1.2 kpc, we apply a 3σ cutoff.At the same time, a second cutoff based on the radio spectral index is applied.Any pixel with an associated spectral index of α > −0.5 is flagged, plotted as an outlier, and excluded from the fitting data.The motivation for this is to exclude pixels where the radio continuum emission is dominated by thermal emission and thermal absorption (Basu et al. 2015).In practice, the number of points clipped is very small for frequencies below 1.4 GHz, because we expect the radio continuum to be dominated by non-thermal emission (Tabatabaei et al. 2017).
We assume that our errors are in good approximation and described by a normal distribution, and that the ratio of estimated errors for the radio and hybrid Σ SFR is reasonably well known.Such Gaussian statistics are a good approximation for hybrid Σ SFR as demonstrated by Leroy et al. (2012) with a statistical uncertainty of 0.13 dex and a systematic uncertainty of up to 0.2 dex; these statistics can be reasonably extended to radio Σ SFR with a similar level of accuracy (Heesen et al. 2014(Heesen et al. , 2019a)).
In that case, odr works better than the ordinary least-squares regression as it has a smaller bias, lower variance of parameters, and smaller mean-square errors of parameters (Boggs et al. 1989).Best-fitting parameters are presented in Table 1.Berkhuijsen et al. (2013) and Heesen et al. (2014) showed that it is possible to linearise the spatially resolved radio-SFR relation by convolving the hybrid Σ SFR maps with a Gaussian kernel.The choice of kernel is motivated by our assumption that diffusion and not advection (or streaming) is the mechanism for CRE transport in the discs of galaxies.We note, however, that in the haloes of galaxies advection may be the dominating transport mode (see, e.g., Heald et al. 2022, for a recent example).Murphy et al. (2008) and Vollmer et al. (2020) showed that both exponential and Gaussian kernels work equally well in these smoothing experiments, so we cannot simply distinguish the transport mode this way.Hence, we can only later motivate the choice of the Gaussian kernel after investigating the cosmic-ray transport length as a function of CRE lifetime.We also assume the diffusion is cylindric (2D) and isotropic with a preferred diffusion in the disc plane due to the large-scale ordered magnetic field.Hence, we use an elliptical kernel with a standard deviation of σ x along the major axis and a similar σ y along the minor axis, where we define the following:

Smoothing experiment
where i = 20 • is the inclination angle.This is of course a simplification as CR diffusion is anisotropic due to the anisotropy of the eddy scales along the orientation of the 'local background field', which is defined to have a scale that is larger than the eddy scale (see e.g., Sect.2.1 in Zhang et al. 2020).The relevant local field is less regular and stronger than the kpc-scale ordered field found in M 51 and other galaxies.Polarisation observations with spatial resolutions between the outer scale of the turbulence (≈50 pc) and the transport length (≈2 kpc) reveal the ordered A21, page 4 of 11 local fields that could lead to anisotropy.With the best resolution in Fletcher et al. (2011) of ≈150 pc, the field pattern is still ordered with spiral shape, so anisotropic diffusion is indeed probable.
We define the CRE transport length l CRE as half of the FWHM of the Gaussian kernel.Following Heesen et al. (2014), we set the transport length as equal to the diffusion length, and the conversion from FWHM is where σ xy is the standard deviation of the Gaussian kernel.The convolution is done using the convolve_fft method from the astropy library, where the kernel is defined using astropy's Gaussian2DKernel method and takes standard deviations, σ xy, as an input.In order to find the best-fitting kernel size that linearises the radio-SFR relation, we define the following function C : σ xy → a as (i) convolve the hybrid Σ SFR map with a Gaussian kernel of size σ xy ; (ii) convert radio maps to radio Σ SFR maps; (iii) bin maps to a pixel size of 1.2 × 1.2 kpc 2 ; (iv) clip maps using 3σ and α thresholds; and (v) perform the double logarithmic linear fit to measure the slope, a.
With this definition, linearising the radio-SFR relation is equivalent to finding a solution for C(σ xy ) − 1 = 0. rpp uses the iterative root finder scipy.optimize.fsolvefrom scipy, which is based on the fortran library minpack.We use two adjacent frequencies in order to calculate the radio spectral index.For every frequency, except for the lowest and the highest ones, we thus obtain two measurements of the CRE transport length.The difference between these measurements allows us to estimate the error of 0.15 kpc, which is larger than the statistical fitting error of 0.07 kpc.The best-fitting CRE transport lengths are presented in Table 2.
We note that there is some freedom of choosing the shape of the diffusion kernel.Numerical studies on CR transport fit diffusion coefficients using the second moment of their spatial distribution (Qin & Shalchi 2009;Wang & Qin 2019;Xu & Yan 2013;Snodin et al. 2016;Seta et al. 2018).Based on magnetohydrodynamical turbulence simulations, Sampson et al. (2023) find that super-diffusion is ubiquitous in the ISM: super-diffusion would result in kernel shapes that have a narrower core and more extended wings than a Gaussian.The width of the distribution would also be modulated by the density distribution.Our data, which represent just one snapshot in time, do not allow for a A21, page 5 of 11 Table 1.Fitting results of the radio-SFR relation.distinction between diffusive and super-diffusive regimes.However, this is something that could be explored in future work.

Radio-SFR relation
Our findings of the spatially resolved radio-SFR relation are in qualitative agreement with what is theoretically expected.Figures 3 and 4 show that the data points follow a sub-linear radio-SFR relation for all five frequencies.The slope of the relation is flatter for lower frequencies, which is expected; we see older and therefore lower energy CREs at lower frequencies.We divide the points into three groups based on their radio spectral indices: young CREs (−0.50 > α > −0.65), mid-aged CREs (−0.65 ≥ α ≥ −0.85), and old CREs (−0.85 > α).The values for the spectral indices are typical values found in the spiral arms, inter arm regions, and outskirts of galaxies, respectively.Focusing just on the young CREs, the data points are fairly close to the Condon 1:1 relation.This behaviour is explained by the fact that the young CREs have energy spectra close to their injection spectra.They have not yet moved far from their sources and thus have not lost a lot of energy.That implies their spatial distribution should more closely resemble that of the star formation, and they should therefore follow the Condon-relation more closely (Heesen et al. 2019a).

Cosmic-ray electron transport lengths
The CRE transport lengths lie between 1.34 and 5.23 kpc and are thus smaller than the radius of the star-forming disc, which is approximately 10 kpc (Mulcahy et al. 2014).Hence, the galaxy is sufficiently large to explore the transport length to these values.The cosmic-ray transport length decreases with frequency, which confirms that the length increases with lower CRE energy.
For the frequencies where we have two measurements, we see differences in the transport length.These can be attributed to different radio spectral indices and data point numbers due to sensitivity limits.While the differences are larger than the statistical errors, they are still small (approximately 0.3 kpc).
We now calculate the electron lifetime taking into account synchrotron and inverse Compton losses.For that we use the average total (mostly random) equipartition magnetic field strength of B = 12.0 ± 2.6 µG (Heesen et al. 2023).The radiation energy density U rad is the sum of two parts: the CMB energy density, which can be measured directly, and the total energy density U TIR with an additional contribution of stellar light.We used the total infrared luminosity of Dale et al. (2009) and a radius of 16 kpc (equivalent to the star-forming radius of Heesen et al. 2022) to calculate U rad = 13 × 10 −13 erg cm −3 (see Appendix A in Heesen et al. 2018).With U B = B 2 /(8π), we obtain U rad /U B = 0.23 meaning synchrotron losses dominate over inverse Compton losses.
With these results we can write the energy and lifetime of the CREs as functions of parameters we know.Combining numerical factors and using convenient units yields and the following electron lifetime: In Fig. 5, we plot the cosmic-ray transport length as function of the electron lifetime.For cosmic-ray diffusion, we expect following dependence (Syrovatskii 1959): A21, page 6 of 11 where D is the isotropic diffusion coefficient.We find that Eq. ( 6) describes our data sufficiently well (χ 2 ν = 2.0).The resulting diffusion coefficient is (2.14 ± 0.13) × 10 28 cm 2 s −1 .This means that for the relevant energy range considered, 0.53-6.57GeV, the diffusion coefficient is energy independent.This can be quantified further as the CRE energy and the lifetime are related as t syn ∝ E −1 , so by assuming an energy-dependence of D ∝ E µ , we have following relation: We thus find µ = −0.01 ± 0.20, in agreement with an energyindependent diffusion coefficient.We can rule out a linear dependence between transport length and lifetime l CRE ∝ t syn , which would be the case for A21, page 7 of 11 either cosmic-ray streaming or advection.This would result in a much steeper relation than what is observed in Fig. 5. Similarly, energy-dependent diffusion coefficient would result in a flatter relation than what is observed.In Fig. 5, we show the best-fitting relation with D ∝ E 1/3 , effectively meaning l CRE ∝ t 1/3 syn (Eq.( 7)), which is not a good fit to our data (χ 2 ν = 14.4) when compared with the energy-independent case.Our diffusion coefficient is in good agreement with the value measured by Heesen et al. (2019b), which used the same method but only with two frequencies of 144 and 1365 MHz.
What we neglected in our analysis thus far is the influence of CRE escape from the galaxy.Such an escape is required in order to explain the relatively flat radio continuum spectrum.Albeit clearly in agreement with non-thermal synchrotron emission, it is not in agreement with an aged spectrum (Mulcahy et al. 2014).The recent work by Dörner et al. (2023) estimated this escape timescale in the 20-300 Myr range, where a positive gradient with a galactocentric radius is found.If we take an average value at a radius of 7 kpc, we would expect an escape timescale of t esc ≈ 50 Myr (Dörner et al. 2023, their Fig. 4).We then calculated the effective CRE lifetime, 1/t CRE = 1/t syn + 1/t esc , assuming an exponential decay with time, and repeated the analysis.We found that neither diffusion nor streaming can adequately describe the data.Presumably the assumption of a constant escape timescale is too simplistic for our data; at ν < 5 GHz, we expect long escape timescales, whereas at ν >5 GHz we have such short synchrotron loss timescales, so escape does not play much of a role.

Discussion
The measurements presented in this paper deal with CREs, while lots of literature (both observations and theory) deals with cosmic-ray protons.Transport theories (e.g., quasi-linear theory and its non-linear extensions) of charged particles primarily rely on their gyro radius and the magnetic field they propagate through.Consequently, ultra-relativistic cosmic-ray protons and CREs of the same energy share the same transport properties in these theoretical transport models.However, one of the differences between electrons and protons in our GeV energy range is that only electrons are ultra-relativistic, whereas protons are only mildly relativistic at best.It is, however, thus far unclear to what extent this would change the transport.Simulations modelling both protons and electrons with diffusion give good agreement with the local energy spectra near the Earth, measured by Voyager, of both protons and electrons (Werhahn et al. 2021).Similarly, γ-ray observations of nearby galaxies are usually described by diffusion of both electrons and protons (Peretti et al. 2019).Hence, while we have to add the disclaimer that electron and proton transport may be different, observations thus far allow us not to make a distinction, and so in the following we may discuss cosmic-ray transport in general.
Particle experiments near Earth can directly detect the different components of the cosmic rays: electrons, positrons, protons, and heavier nuclei.These are, however, limited to energies above 10 GeV because magnetic fields carried by the solar wind constrain the movements of low-energy cosmic rays.There is also a difference in the probed energy regime; the typical energy for CREs responsible for radio continuum emission is approximately 1-10 GeV, while observations in the Solar System allow us to gain insights about the ion component, which is inaccessible for radio observations.There is, for instance, a relative overabundance of light elements (He, Li, Be, B) by five-to-seven orders of magnitude at ≈1 GeV when compared to the interstellar medium.These can be accounted for if cosmic rays undergo spallation reactions with the interstellar medium (Zweibel 2013), so it is possible to calculate the energy dependence of the diffusion coefficient: where n B /n C is the boron-to-carbon ratio, and the exponent is derived using a Kolomogorov scaling analysis (Becker Tjus & Merten 2020).The physical explanation for the energy dependence is that cosmic rays with higher energies, and hence larger Larmor radii, resonantly interact with fluctuations with larger wavelengths of the Kolmogorov spectrum, leading to larger values of the mean free path and hence to higher diffusion coefficients.
In contrast, we showed that the most likely transport mechanism for CREs in our studied galaxy is energy-independent diffusion.In the following, we discuss possible explanations.
The first theory that we consider suggests that the cosmic rays simply follow the magnetic field lines (Minnie et al. 2009;Reichherzer et al. 2022).Since the turbulence in the interstellar medium is super-Alfvénic, the field lines are also bent and randomised.Then, we can, when viewing on scales larger than the coherence length l c of the turbulence, describe the process of cosmic-ray transport by diffusion due to field line random walk (FLRW) in the turbulence of the ISM.While the particles follow the field lines, they undergo spatial diffusion as the magnetic field lines diffuse in space over time.The resulting decorrelation of the particle trajectories (Jokipii & Parker 1968;Hauff et al. 2010) gives rise to an energy-independent mean free path and thus also an energy-independent diffusion coefficient.Such a picture is applicable if the cosmic-ray gyro radius is much smaller than the field line coherence length: Calculations show that this mechanism is accurate for cosmic rays with energies of up to 100 GeV.For cosmic rays at even higher energies, another explanation is needed for their measured isotropy; however, Reichherzer et al. (2022) showed that the anisotropic diffusion in isotropic Kolmogorov turbulence for such particles may still be described by FLRW.
The second theory is cosmic ray self-confinement.Cosmic rays create their own turbulence via waves that are generated by the cosmic-ray streaming instability (Kulsrud & Pearce 1969;Zweibel 2013).The cosmic rays are then confined by interactions with small-scale fluctuations in the magnetic field for tens of megayears.Due to this confinement and frequent scattering with the magnetic field, their distribution becomes isotropic and their large-scale motions can accurately be described by energyindependent transport.The cosmic rays stream at the Alfvén speed (Skilling 1971), and so the cosmic ray transport is energy independent.However, in the presence of wave damping, cosmic rays may stream faster than at Alfvén speed depending on energy so that the transport is no longer energy independent.Also, the growth rate of the turbulence is energy dependent as well (Farmer & Goldreich 2004).That said, Krumholz et al. (2020) still suggest that the cosmic rays stream along magnetic field lines with a speed close to the Alfvén speed as long as they are decoupled from the neutral gas.A similar result for a fully ionised gas was found by Yan & Lazarian (2008).
The third theory is that of scattering cosmic rays in the turbulent magnetic field, where the source of turbulence is extrinsic.Turbulence is injected at scales of ≈50 pc and cascades down to the resonant scale of the gyro radius.Simple analytic models of such a transport predicts an energy-dependent diffusion coefficient, where the slope depends on the assumed turbulence model.For instance, the standard assumption is D ∝ E 1/3 , as expected for Kolmogorov-type turbulence (Strong et al. 2007).However, as Kempski & Quataert (2022) calculate, the scattering at magnetohydrodynamic fast-mode waves of low-energy cosmic rays in the warm ISM results in energy-independent diffusion.Notably, their calculated diffusion coefficient of around 2 × 10 28 cm 2 s −1 (see Fig. 3 in Kempski & Quataert 2022) is in good agreement with our result.
Unfortunately, the regime of turbulence most relevant to cosmic ray transport of GeV electrons is difficult to access in numerical simulations, so there is no confirmation of either theory from that direction.The reason is that due to the Kolmogorov scaling of the external turbulence, there is little energy in the fluctuations available for resonant scattering of the sub-TeV CREs.This is in contrast to the regimes investigated in test-particle simulations (e.g., Giacalone & Jokipii 1999;Subedi et al. 2017;Reichherzer et al. 2022;Kuhlen et al. 2022), where only parameter spaces above r g /l c ≈ 10 −4 , corresponding to energies 10 TeV, were investigated and for which energy-dependent diffusion is found.
Our approach of measuring CRE transport by comparing the radio continuum map with the SFR surface density is complementary to the radio spectral index analysis by Mulcahy et al. (2016) and Dörner et al. (2023).While our method is sensitive to smoothing of the Σ SFR map by CRE diffusion, the radio spectral index distribution is more governed by the escape of CRE from the galaxy.Our best-fitting diffusion coefficient of A21, page 9 of 11 D ≈ 2 × 10 28 cm 2 s −1 is in good agreement with the results of Dörner et al. (2023) but lower than the value by Mulcahy et al. (2016), who found D ≈ 6 × 10 28 cm 2 s −1 .The difference can be mostly attributed to the fact that Mulcahy et al. (2016) modelled CRE escape only via diffusion, whereas Dörner et al. (2023) used a combination of diffusion and advection.Consequently, the diffusion coefficient has to be higher in order to allow for a fast enough CRE escape (Mulcahy et al. 2016).Our result shows that a lower diffusion coefficient is in agreement both with the smoothing experiment and the radio spectral index analysis, when the effect of a galactic wind is included in the latter.

Conclusions
The transport of cosmic rays in the nearby galaxy M 51 is studied using ultra-low-frequency data at 54 MHz from the LoLSS survey (de Gasperin et al. 2021).We complement these data with radio continuum maps at 144 MHz from LoTSS-DR2 (Shimwell et al. 2019), 1365 MHz from WSRT-SINGS (Braun et al. 2007), and 4850 and 8350 MHz from merged VLA and Effelsberg observations (Fletcher et al. 2011).In order to measure spatially resolved star formation, we use a hybrid Σ SFR map from a combination of Spitzer 24-µm and GALEX 156-nm emission (Leroy et al. 2008).We correct for thermal radio continuum using a combination of Hα and Spitzer infrared observations.These are our main conclusions: -The radio continuum maps appear significantly smoothed out in comparison with the Σ SFR maps.This can be quantified using the spatially resolved radio-SFR relation at a 1.2 kpc resolution.The slopes of the radio-SFR relation are sublinear, meaning an excess of radio continuum emission at low values of Σ SFR and underluminous radio continuum at high values of Σ SFR .As the global (integrated) radio-SFR relation in galaxies is approximately linear (e.g., Smith et al. 2021), this can be understood as the result of cosmic-ray transport.-The slopes of the radio-SFR relation are frequency dependent, where the lowest frequencies have the lowest slopes.This is expected, as the CRE lifetime increases for the lowest frequencies, and so the CRE transport length increases as well, which amplifies the effect of the smoothing; this then lessens the relation between radio continuum emission and star formation.-We smooth the Σ SFR map with a Gaussian kernel in order to mimic cosmic-ray diffusion.Thus, we are able to linearise the spatially resolved radio-SFR relation.The CRE transport length is then half of the FWHM of the smoothing Gaussian function.The CRE transport length decreases with increasing frequency, as expected since the CRE lifetime decreases as well.-When we plot the CRE transport length as function of the CRE lifetime (Fig. 5), we find a square-root dependence exactly as expected for diffusion.When we fit our data as l CRE = 4Dt syn , we can derive the isotropic diffusion coefficient of D = (2.14 ± 0.13) × 10 28 cm 2 s −1 .-The diffusion coefficient with D ∝ E 0.001±0.185 is energy independent across the relevant energy range considered: 0.53-6.57GeV.This result requires some theoretical explanation, such as FLRW, cosmic-ray self-confinement, or cosmic-ray scattering at extrinsic turbulence.One outcome of our work is that the radio spectral index information is useful to identify data points that are dominated by thermal emission or suffer from thermal absorption, or, possibly, a combination of both.In the future, we may be able to correct more accurately for thermal emission and absorption by modelling radio continuum spectra and exploiting the ultralow frequency emission from LoLSS (de Gasperin et al. 2021).Also, spatially resolved optical spectral with integral field unit spectroscopy now allows one to obtain extinction-corrected Hα maps by using the theoretical Hα/Hβ ratio.This is now performed for THINGS galaxies as part of the Metal-several survey (Lara-López et al. 2021).

Fig. 1 .
Fig. 1.Radio continuum maps converted to star formation rate surface density (Σ SFR ).The scaling is logarithmic between 10 −4 M yr −1 kpc −2 and 3 × 10 −1 M yr −1 kpc −2 .In the top row are LOFAR LBA 54 MHz (panel a), LOFAR HBA 144 MHz (panel b), and WSRT 1365 MHz (panel c).In the bottom row are VLA+Effelsberg 4850 MHz (panel d), and 8350 MHz (panel e).In the bottom right (panel f), the hybrid Σ SFR map is shown for comparison.The angular resolution is a 20 arcsec FWHM for all maps, except for LOFAR LBA, where it is 47 arcsec.The radio continuum maps appear to be the smoothed versions of the hybrid Σ SFR map, as expected for CRE diffusion.

Fig. 2 .
Fig. 2. Radio spectral index distribution.The top row shows, from left to right, the non-thermal radio spectral index between 54 and 144 MHz (panel a), between 144 and 1365 MHz (panel b), and between 4850 and 8350 MHz (panel c).Contours are at −1.2, −0.85, and −0.65.The bottom row (panels d-f) shows the corresponding radio spectral index error.The angular resolution is 47 arcsec for the maps between 54 and 144 MHz, and 20 arcsec otherwise.The size of the synthesised beam is shown in the bottom left corner of each panel.

Fig. 3 .
Fig. 3. Cosmic-ray smoothing experiment.Panels show the spatially resolved radio-SFR relation prior to the smoothing (left panels) and following the smoothing of the Σ SFR maps (right panels).The vertical axis shows the radio continuum Σ SFR values converted with a constant factor from the radio continuum intensity.The horizontal axis shows the hybrid Σ SFR values.Each data point represents a pixel of 1.2 × 1.2 kpc 2 .Data points are colour-coded with the non-thermal radio spectral index: data points with α < −0.85 in blue, data points with −0.85 < α < −0.65 in green, and data points with −0.65 < α < −0.50 in red.Data points with α > −0.5 are shown as outliers and not considered for the data analysis; they are shown in grey.The best-fitting relation is shown as a solid line and the Condon relation is shown as a dashed line.Smoothing of the Σ SFR maps allows us to reach a slope of 1 for the radio-SFR relation.The vertical displacement is deviation from the integrated radio-SFR relation.From top to bottom, frequencies are 54 MHz (in combination with 144 MHz, panels a and b), 144 MHz (in combination with 54 MHz, panels c and d), and 1365 MHz (in combination with 144 MHz, panels e and f).

Fig. 4 .
Fig. 4. Figure 3 continued.Top row shows plots at 4850 MHz (in combination with 8350 MHz, panels a and b), bottom row shows plots at 8350 MHz (in combination with 4850 MHz, panels c and d).

Fig. 5 .
Fig.5.Cosmic-ray transport models.We show transport length as function of CRE lifetime.The solid line shows the best-fitting diffusion model with energy-independent diffusion; the dashed line shows the best-fitting streaming model; the dotted line shows the best-fitting diffusion model with an energy-dependent diffusion coefficient with D ∝ E 1/3 .Our data favour energy-independent diffusion.