Cosmic-ray electron transport in the galaxy M 51

Context. Indirect observations of the cosmic-ray electron (CRE) distribution via synchrotron emission is crucial for deepening the understanding of the CRE transport in the interstellar medium, and in investigating the role of galactic outﬂows. Aims. In this paper, we quantify the contribution of diffusion-and advection-dominated transport of CREs in the galaxy M51 considering relevant energy loss processes. Methods. We used recent measurement from M 51 that allow for the derivation of the diffusion coefﬁcient, the star formation rate, and the magnetic ﬁeld strength. With this input, we solved the 3D transport equation numerically including the spatial dependence as provided by the measurements, using the open-source transport framework CRPropa (v3.1). We included 3D transport (diffusion and advection), and the relevant loss processes. Results. We ﬁnd that the data can be described well with the parameters from recent measurements. For the best ﬁt, it is required that the wind velocity, following from the observed star formation rate, must be decreased by a factor of 5. We ﬁnd a model in which the inner galaxy is dominated by advective escape and the outer galaxy is composed by both diffusion and advection. Conclusions. Three-dimensional modelling of cosmic-ray transport in the face-on galaxy M51 allows for conclusions about the strength of the outﬂow of such galaxies by quantifying the need for a wind in the description of the cosmic-ray signatures. This opens up the possibility of investigating galactic winds in face-on galaxies in general.


Introduction
In this age of multi-wavelength and multi-messenger astronomy, nearby galaxies that allow for spatially resolved substructures can be observed at different wavelengths in such detail that there is differential information about the population of basically all of the ingredients needed to describe spatially resolved cosmic-ray transport and interaction of cosmic rays.In particular, information on the 3D magnetic eld structures, a differential view on the star formation rate and secondary properties such as the spectral index of cosmic rays, the cosmic-ray diffusion coefcient, as well as the advection velocity of the plasma can be provided for both edge-on (Heesen et al. 2018;Miskolczi et al. 2019;Heald et al. 2022) and face-on (Murphy et al. 2008;Tabatabaei et al. 2013;Mulcahy et al. 2016) galaxies.This wealth of data brings 1D transport models beyond their limits.
One-dimensional models try to either describe a galaxy in edge-on or face-on geometry.Clearly, while this is a useful sim-plication, it neglects the true 3D structure of a galaxy.For the edge-on case, one hence neglects the local concentration of star formation, such as in spiral arms where advection is more important than diffusion.Also, the conservation of magnetic ux means that at least extensions to quasi 1D models are required in order to lead to a decreasing magnetic eld strength, such as the 'ux tube' model (Heald et al. 2022).On the other hand, in face-on galaxies the radio continuum emission is smeared out in comparison with the star formation, so that this map can be convolved to either minimize the difference between the two maps (Murphy et al. 2008) or linearize the cross-correlation between them (Berkhuijsen et al. 2013).This subsequently provides an estimate of the cosmic-ray diffusion length.A shortcoming of this method is that it neglects the inuence of the radio halo along the line of sight.Also, as Mulcahy et al. (2016) have shown, cosmic-ray electrons (CREs) escape from the galaxy and this cannot be neglected, but it is indeed required in order to explain the radio continuum spectrum.The model by Mulcahy et al. (2016) was a rst attempt to move beyond purely descriptive work in face-on galaxies, solving the diffusion-loss equation to model the radio spectral index.Obviously, the most promising way would be to go beyond this simplication and describe a galaxy with both cosmicray diffusion and advection in a 3D model.In this paper, we use the publicly available Monte-Carlo code CRPropa (Batista et al. 2016;Merten et al. 2017;Alves Batista et al. 2022) to describe the 3D transport in M 51.While originally written to describe the extragalactic transport of hadronic cosmic rays via the solution of the equation of motion, CRPropa has been extended to a second propagation method, that is solving the transport equation via the approach of stochastic differential equations (SDEs).The conversion of a Fokker-Planck equation into an SDE is useful here, as the particle densities are derived from the pseudo-particle trajectories.This way, the equation of motion approach and the transport equation can be treated in one framework since both work with individual particle trajectories A&A 669, A111 (2023) (Merten et al. 2017).This approach also allows for both continuous and catastrophic losses for the production of full particle showers in the interactions that can be followed up on, etc.
Modelling the 3D transport of CREs in M 51 mainly depends on the following three assumptions: (1) the diffusion coefcient and its energy scaling, (2) the escape scale height for CREs, and (3) the advection speed prole.We subsequently implement these properties in the transport code as described in the following Sect. 2 and t the parameters to the observed properties.The results are discussed in Sect. 3 and conclusions are made in Sect. 4.

Transport model
The transport of CREs can be described by the diffusionadvection equation (e.g.Becker Tjus & Merten 2020) assuming isotropic diffusion, where n is the particle density distribution, D is the diffusion coefcient, ∂E/∂t is the energy loss term described in Sect.2.1, S is the source term, and u is the advection speed derived from the Star Formation Rate surface Density (SFRD) as described in Sect.2.2.Here, no leakage term is included, as in our 3D simulations all particles that reach the boundary of the galaxy (see Sect. 2.5) leave the simulation volume.

Energy loss
Energy loss terms ∂E/∂t for synchrotron emission and inverse Compton scattering are taken into account in the CRE transport equation.In our simulation, we applied the energy loss as a continuous process following the parametrization given by Mulcahy et al. ( 2016) where E is the CRE energy in GeV, U rad = 1 eV is the energy density of the interstellar radiation eld, and B(r) is the root mean square of the magnetic eld.The magnetic eld model was implemented to decrease exponentially with the height z above the scale height h b : Here, r gc denotes the galactocentric radius, and z is the height above the galactic plane.The radial prole of the magnetic eld strength was measured by Heesen et al. (2022) and shown in Fig. 1.The exponential cutoff scale h b is listed in Table 1.

Source distribution
The acceleration of CREs is believed to happen in star forming regions, possibly at the shock front of supernova remnants; interested readers can refer to Becker Tjus & Merten (2020), for example, for a review.Diffusive reacceleration in these regions is possible as well (Tolksdorf et al. 2019).Other possible sources are pulsar wind nebula (López-Coto et al. 2022).While the different acceleration scenarios can inuence the spectral energy distribution of the sources, the origin for all of them are the star Table 1.Parameters for the magnetic eld scale height h b and the height of the disc h d . Model Notes. (a) Variable scale heights as a function of the galactocentric radius as presented in Mulcahy et al. (2016).
formation regions.Therefore, we assume that the radial source position of the electrons follows the observed SFRD (see Fig. 1).
The SFRD was estimated using a hybrid star formation map from a combination of GALEX 156-nm far-ultraviolet and Spitzer 24-µm mid-infrared data (Leroy et al. 2008).

Cosmic-ray advection
Even the strength of the galactic wind is assumed to be proportional to the SFRD.This is motivated both by a similarity analysis of planar blast waves (Vijayan et al. 2020) and radio continuum observations of radio haloes in edge-on galaxies (Heesen 2021).The galactic wind speed as measured from ionized gas depends on the SFRD.Hence, we used this as our parametrization.
We took a galactic wind in z-direction u(r) = sgn(z) v(r gc ) e z , where is the best-t wind velocity following the radial-dependent SFRD found in Heesen et al. (2018) and sgn denotes the sign function.In this model, the wind velocity does not depend on the z position.In galactic wind models, the wind speed is zero in the galactic disc and then with height.Depending on the assumptions of geometry and energy and mass injection, this acceleration can be either gradual over a few kiloparsecs (Everett et al. 2008) or rapid over a few hundred parsecs (Yu et al. 2020).So far, no consensus has been reached as to the vertical acceleration prole either, as the properties of the CRE distribution and magnetic eld strength are difcult to disentangle (Heesen 2021).Hence, we make the simplifying assumption of a constant wind speed.
A111, page 2 of 8 Notes.Here, l min and l max denote the range of the propagation steps and T max is the maximal simulation time until a stationary solution was reached.

CRE diffusion
Deections of CREs in the Galactic magnetic eld introduce diffusive transport behaviour, which is characterized by the diffusion tensor D that enters the transport equation.It is known that in galaxies, spatial diffusion can be anisotropic-or isotropicdependent on the environment (Sampson et al. 2022).In the absence of detailed knowledge of the 3D magnetic eld structure for M 51, we assume scalar isotropic diffusion.We note that a more realistic, 3D modelling of the diffusion tensor requires knowledge of the relation between parallel and perpendicular diffusion coefcient components (Reichherzer et al. 2022).Assuming that the magnetic eld lines are mainly orientated in the galactic plane, the escape would be dominated by perpendicular diffusion.In this case, by choosing an isotropic diffusion, the transport in the plane would be underestimated.However, parallel escape is suppressed by the geometry of the large, at galactic disc anyways (see Sect. 2.5) and we probably suppress this component a bit more by isotropic diffusion.
The diffusion coefcient dependency on the parameters of the CREs and the environment relies on the dominant scattering mechanism.Recent observational data, for example, spectra of positrons and their parent protons in the Milky Way (Cowsik & Huth 2022) or analytical considerations employing advectiondominated escape models (Recchia et al. 2016), suggest energyindependent diffusion coefcients of charged particles up to several GeV.This relates to gyroradii of the order of 10 −7 pc in approximately µG magnetic elds shown in Fig. 1 1 .However, even for diffusion-dominated escape, various explanations exist for energy-independent diffusion (see e.g.Kempski & Quataert 2022;Cowsik & Huth 2022 and references therein).Possible explanations include resonant scattering of CREs by self-excited uctuations when these waves are excited through the streaming instability in the absence of damping (Kempski & Quataert 2022).Energy-independence can also be achieved for particle scattering in pre-existing magnetohydrodynamic turbulence (Cowsik & Huth 2022) or through the inuence of the Parker instability, causing the leakage of cosmic rays out of the galaxy (Parker 1966(Parker , 1969)).Also, the eld-line-random walk that may contribute to perpendicular diffusion at these energies exhibits energy-independent diffusion (Minnie et al. 2009;Reichherzer et al. 2020).Regardless of which of the effects or combination described above holds, we assume energy-independent diffusion.In lack of a theoretically motivated diffusion coefcient, we compared the observation-based diffusion coefcient D ≈ 2 × 10 28 cm 2 s −1 (Heesen et al. 2019) with the best-t model from Mulcahy et al. (2016).We compared our result to energydependent diffusion, which led to a signicantly worse t for the data (see Appendix B).Such a behaviour is also suggested by the 1D diffusion models of Mulcahy et al. (2016) and the convolution experiments of Heesen et al. (2019).

Geometry of M51
To model the geometry of M51, we took a cylindrical form of the galaxy with a maximal radius of R max = 15 kpc and a height h d , allowing a z-position −h d ≤ z ≤ h d .The parameter for h d is not fully known.Therefore, we present the results for three different models as follows: Model A considers a large-scale height for the galactic height and for the magnetic eld of h d = h b = 7 kpc.This value is not realistic, but it was chosen to see the impact of the parameter.Model B is based on the observed synchrotron emission scale height of 1.5 kpc (Krause et al. 2018).Therefore, the height of the disc is considered to be h d = 3 kpc and, for the magnetic eld height, we used h b = 6 kpc.Model C follows the variable scale height presented in Mulcahy et al. (2016).Here, a scale height reads as 3.2 kpc, r gc ≤ 6 kpc 3.2 kpc + 5.6 6 (r gc − 6 kpc), else 8.8 kpc, r gc ≥ 12 kpc . (5) All model parameters are summarized in Table 1.

Simulation setup
To solve the transport equation, we used the method of stochastic differential equations (SDEs) implemented in CRPropa 3.1 (Batista et al. 2016;Merten et al. 2017).We simulated 10 5 pseudo-particles in the energy range of 0.1 GeV to 50 GeV.We assumed a source with an injection dN/dE The CRE density distribution was considered for 1000 time steps up to 500 Myr to calculate the stationary solution of the transport equation (Eq.( 1)) following Merten et al. (2017).All particles reaching the boundary have been lost to the intergalactic medium.Also, particles propagating longer than the maximum simulation time T max were taken out of the simulation.Here, T max = 2.5 Gyr is just an assumption.In Mulcahy et al. (2016), it is shown that the CRE distribution reaches a steady state after 500 Myr.Therefore, our choice of a 5 times higher simulation time is more conservative.The details of the used modules and given parameters for the simulation are given in Table 2.We analysed the CRE spectrum in a slightly smaller energy range than simulated to minimize numerical artefacts.The range of the power law ts is 0.5 ≤ E/GeV ≤ 6.
A111, page 3 of 8 Fig. 2. CRE spectral index as a function of the galactocentric radius.The left panel shows the simulation results using the measured diffusion coefcient from Heesen et al. (2019) and the right panel uses the best-t value from Mulcahy et al. (2016).The model parameters are shown in Table 1.Green points indicate simulations without advection and blue point show those with advection.The data are taken from Heesen et al. (2019).

Results
Taking the model as described before, the resulting CRE spectral index is presented in Fig. 2, where the model without advection (green points) and including advection as described in Sect.2.2 (blue points) is compared for two diffusion coefcients.In the case of the lower diffusion and neglecting advection, the spectra for all models are too steep due to the high retention time and corresponding high energy loss.Only model B undershoots the observed data in a range slightly, but it does not show the correct radial behaviour.In the case of higher diffusion, model C (green circle) ts the data in the inner galaxy (r gc < 6 kpc) well.Only A111, page 4 of 8 Fig. 4. Determining the dominant timescale.Upper panel: timescales for the escape via diffusion (green line) and advection (blue line) to the z-direction.Additionally, the energy loss timescale is given for three different energies (red lines; the line style denotes energy).Lower panel: ratio between the advection and diffusion timescale.
in the outer part of the galaxy can a difference between the data and the model be seen.This is due to different data for the magnetic eld strength, star formation rate, and spectral synchrotron index in Mulcahy et al. (2016).Another reason for the difference could be the difference between the 1D diffusion model used by Mulcahy et al. (2016) and the 3D approach in this work.
In the case where advection is taken into account (Fig. 2, blue points), the observed CRE spectral index is near the injection spectrum ∝ E −2 .This is due to high advection speed and a quick loss of all particles.The case of the high variation for the low diffusion models in the outer galaxy can be explained by the low number of observed pseudo-particles in this domain.In this part of the galaxy, the SFRD is so low that there is nearly no production of high energy CREs.But due to the high advection speed, the particles leave the simulation volume before they can diffuse in the outer galaxy.
Following these observation, a galactic wind signicantly weaker than indicated by the SFRD is necessary to match the observed data.Taking this into account, we introduced a scaling factor f adv in Eq. ( 4) and optimized this value to t the data best.Details on the optimization are given in Appendix A. The nal CRE indices using the optimal value for the advection normalization are shown in Fig. 3.It can be seen that the lower diffusion coefcient shows a signicantly better t to the data.The best t provides model C. In this case, the optimal normalization factor is f adv = 0.2.The models with a constant scale height (model A and model B) do not t the radial gradient.In the inner galaxy (r gc .7 kpc) the CRE spectra are too steep and in the outer galaxy too at.
Taking the geometry of model C and the lower diffusion coefcient as the best-t model, analysing the timescales shows the dominant processes.Here, the diffusion timescale is dened as τ di = h(r gc ) 2 /D 0 and the advection timescale as τ adv = h(r gc )/v(r gc ).In Fig. 4 it is shown that the escape inner galaxy (r gc .7 kpc) is dominated by advection.The escape in the outer galaxy is composed by both diffusion and advection.In the relevant energy range (E > 2 GeV), the energy loss time is much shorter than the escape timescale.This leads to a steepening of the CRE spectrum.The rise of the energy loss timescales (red lines) at the edge of the galaxy is due to the vanishing magnetic eld strength in the outer part (see Fig. 1).This region is excluded in our analysis.

Conclusions
Our best-t model to the radial gradient of the observed CRE spectra has the following settings: 1.The diffusion coefcient is independent of the energy with D 0 = 2 × 10 28 cm 2 s −1 .This result is in agreement with the measurement from Heesen et al. (2019).2. The scale height for the escape of CREs depends on the galactocentric radius.We used h d = 3.2 kpc for the inner galaxy (r gc ≤ 6 kpc) and increased it linearly up to h d = 8.8 kpc at r gc = 12 kpc.3. The advection speed following the SFRD was reduced by a factor of 5 compared to the measurements in Heesen et al. (2018).The discrepancy can possibly be explained by the fact the radio continuum observations use global SFRD values with Σ SFR = SFR/(πr 2 ?), where r ? is the radial extent of the star-forming disc.If the wind is launched from the central area of the galaxy, the SFRD would be correspondingly higher if one were to use an effective radius r e ≈ r ?/2; this would reduce the advection speed normalization in Eq. ( 4) by a factor of 2. While these advection speeds may still be slightly too high, the wind velocities of the ionized gas as measured by Heckman & Borthakur (2016) are in fair agreement with our new results.We conclude that the escape of CREs is governed by different mechanisms in the inner and outer part of M 51: the inner galaxy (r gc ≤ 7 kpc) appears as an advection-dominated region; whereas, in the outer galaxy, both diffusion and advection have to be taken into account.This is basically consistent with the picture of a wind being present.In contrast to previous results, however, our best-t model results in a wind that is a factor of 5 smaller than derived indirectly from the star-formation rate.Finally, we can show here that with a 3D transport model, it is possible to constrain the propagation environment of CREs, concerning diffusion and advection.More specically, the 3D modelling represents an additional way of indirectly deducing the strength of a wind velocity of the face-on galaxy M 51, opening the possibility to systematically investigate galactic winds for face-on galaxies in general.

Fig. 1 .
Fig.1.Radial dependence of the root mean square of the magnetic eld strength (blue) and the Star Formation Rate surface Density (SFRD, orange).Data are taken fromHeesen et al. (2019Heesen et al. ( , 2022)).The grey dotted line indicates the restriction of our model.

Fig. 3 .
Fig. 3. Radial variation of the CRE spectral index using the optimized wind speed.The model parameters are shown in Table1.The left panel shows the observed diffusion coefcient fromHeesen et al. (2019) and the right panel shows the best-t value for the diffusion coefcient fromMulcahy et al. (2016).
This article is published in open access under the Subscribe-to-Open model.Subscribe to A&A to support open access publication.
A111, page 1 of 8Open 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.

Table 2 .
Parameters and modules for the simulation in CRPropa3.1.