Energetic Particles in the Central Starburst, Disc, and Halo of NGC253

Detailed modelling of the spectro-spatial distributions of energetic electrons and protons in galactic discs and haloes of starburst galaxies (SBGs) is needed in order to follow their interactions with the magnetized interstellar medium and radiation fields, determine their radiative yields, and for estimating their residual spectral densities in intergalactic environments. We have developed a semi-analytical approach for calculating the particle spectro-spatial distributions in the disc and halo based on a diffusion model for particle propagation from acceleration sites in the central SB and disc regions, including all their relevant interaction modes. Important overall normalization of our models is based on previous modelling of the Galactic disc (with the GALPROP code), scaled to the higher star-formations rate in NGC253, and on spatially resolved radio measurements of the central SB and disc. These provide the essential input for determining the particle distributions and their predicted radiative yields in the outer disc and inner halo for a range of values of the key parameters that affect diffusion rate and energy losses. Results of our work clearly indicate that quantitative description of non-thermal emission in SBGs has to be based on modelling of the particle distributions in the entire disc, not just the central SB region.


INTRODUCTION
Emission by energetic particles ('cosmic rays') in some of the nearby star-forming galaxies has been measured extensively over a sufficiently wide spectral range that provides a reasonably good basis for detailed studies of the particle spectro-spatial distributions (SSDs) in interstellar space.Such studies are particularly well motivated in the case of several galaxies from which non-thermal (NT) emission has been measured not only in radio and X-ray, but also at γ-ray energies, mostly by the Fermi satellite (e.g., Andromeda, M82, and NGC253; Abdo et al. 2010a;Abdo et al. 2010b).The latter two galaxies are of special interest due to their relatively high star-formation rate (SFR) that directly translates to more intense NT emission by a higher flux of energetic particles than in 'normal' spirals.Indeed, these galaxies are the closest examples of starburst galaxies (SBGs), in which the enhanced SFR and SN activity are manifested by intense optical and IR emission.
Interest in studies of NT emission in SBGs began long ago with quantitative modelling the radio and hard X-ray emission by energetic electrons in the disc and halo of NGC253 (e.g., Goldshmidt & Rephaeli 1995); this was expanded to include high energy emission by electrons and protons (Paglione et al. . 1996;Völk, Aharonian & Breitschwerdt 1996;Torres 2004;Domingo-Santamaría & Torres 2005), followed by detailed predictions of the high energy γ-ray spectra of both M82 and NGC253 by Persic, Rephaeli, & Arieli 2008, de Cea del Pozo et al. 2009, Rephaeli, Arieli, & Persic 2010, Lacki et al. 2011.Indeed, the levels of γ-ray emission predicted in some of these works were proven to be quite realistic when these galaxies were detected in the GeV-TeV range by Fermi, VERITAS, and H.E.S.S telescopes (Abdo et al. 2010a;Acciari et al. 2009;Acero et al. 2009).
Main objectives in most of the studies of energetic electrons and protons in SBGs were the spectra and levels of (low and high) energy γ-ray emission in the galactic disc.Given that in SBGs a significant fraction of the total SF activity takes place in a relatively small and dense central ('nuclear') region, in some previous calculations quantitative predictions of NT emission were based on the assumption that γ-ray emission is dominated by energetic protons in the central SB (CSB) region.Due to the relative smallness of this compact region, all relevant quantities characterizing the particles and media -gas density, magnetic and radiation fields -were assumed to be uniform.The main objective of such an approximate treatment was estimation of the high energy γ-ray emission, based on the presumption that the high SFR and gas density in the CSB region imply that most of the total galactic GeV-TeV emission is produced by neutral (π 0 ) decays as result of energetic proton interactions with ambient gas protons in this central region.Clearly, for a more realistic and useful description of the particles and their radiative yields, a detailed spectra-spatial description is required in order to account for the different distributions of particle acceleration sites and effective propagation mode in the central CSB and disc regions, as well as an assessment of the relative strengths of the radio, hard X-ray , and γ-ray emission from these regions.
This approximation was shown to be quantitatively inadequate when emission from the full galactic disc was modelled based on the GALPROP (e.g., Moskalenko et al. 2003) code.A modified version of this code was developed to predict the spectral energy distribution of M82 (Persic et al. 2008) and NGC 253 (Rephaeli, Arieli, & Persic 2010).Normalizing the particle spectra by the SFR and the measured radio spectra in the central and full disc regions, it was found that a significant fraction of the total galactic γ-ray emission in M82 and NGC 253 comes from outside the CSB region.It is thus clear that a more realistic description is required of the particle and (magnetic and radiation) fields distributions across the full galactic disc; whereas an approximate model could be sufficient in order to estimate the overall level of γ-ray emission, mostly for assessing the likelihood of its detection (originally, by Fermi /LAT), detailed analyses of radio and γ-ray measurements necessitate modelling of the full particle SSD in the disc.
The intense SF and SN activity, the efficient acceleration of electrons and protons, and their coupling to IS media, radiation and magnetic fields, motivated a more comprehensive evolutionary approach to the study of NT phenomena in SBGs, one that is based on magnetohydrodynamical (MHD) simulations.This necessarily more detailed approach that, in principle, enables a more complete description of relevant hydrodynamical, thermal, and NT processes, is clearly more ambitious and challenging than the above-mentioned empirical (largely numerical) approach that focuses only on energetic particles and their interactions with magnetic and radiation fields and coupling to (non-evolving) IS media.Simulation-based studies of energetic particles and their radiative yields were carried out, among others, by Chan et al. (2019), Hopkins et al. (2020), and Werhahn et al. , (2021, 2023).
MHD simulations could possibly afford a more realistic description of the complexity of physical phenomena in IS media, and -in particular -the various plasma processes that couple energetic electrons and protons to the magnetized gas and determine their modes of propagation (e.g., Ruszkowski et al. 2017, Thomas et al. 2023).While these studies can provide a more detailed, relatively fine-grained description of the particle SSDs as compared with the empirical (necessarily coarse-grained) approach, inherent uncertainties in the description of the evolutionary history of the galaxy on the one hand, and the limited quality of observational spectro-spatial data (even from nearby galaxies), imply that both approaches are not only viable, but also can provide useful (and possibly complementary) insight.
We continued our empirically based study of energetic electron and proton SSDs and their radiative yields in the disc and halo of a SFG by developing a new semi-analytical approach based on a diffusion model for particle propagation from acceleration sites and interactions in both disc and halo.Important overall normalization of the particle SSDs was set by the observationally estimated SFR, similarly to what was done in global modelling of the Galaxy with the GALPROP code (Strong et al. 2010).Fitting predicted radio spectra to the measured spectral and spatial profiles of radio measurements fully determined the SSDs in the outer disc and provided a reasonably reliable basis for predicting the particle distributions and radiative yields in the halo for a range of values of key parameters affecting energy losses and diffusion.In the first analysis using this semi-analytical approach we (Rephaeli and Sadeh 2019) selected NGC 4631 and NGC 4666, two of 35 nearby edge-on SFG included in the EVLA radio continuum survey CHANG-ES (Wiegert et al. 2015), which yielded the first co-added median map of the halo of a spiral galaxy at 1.5 GHz, with emission extending to more than 20 kpc above the disc.Doing so enabled not only the determination of key NT quantities across the full disc, but also provided a more adequate basis for extending the quantitative description of the particle SSD to the galactic halo.
For a realistic description of energetic particles in a spiral galaxy with a high SFR, and in which a significant fraction of the SF activity occurs in a compact CSB region, improved modelling of the particle SSDs and radiative yields is needed to account for the much higher particle fluxes, magnetic and radiation fields.The elevated values of all these quantities in the CSB are reflected in a much more intense emission in essentially all spectral bands than would otherwise be predicted from the central part of a one-zone disc.This need is even stronger in the case of the two nearby well probed SBGs M82 and NGC 253.Clearly, the nearly optimal capability of spectro-spatial radio mapping of these galaxies, and in light of future spatially resolved measurements in high X-and-γ-ray measurements, it is quite essential to model NT emission in the CSB, disc, and halo regions of these iconic SBGs.
Here we apply our modified multizone code to the study of NT emission in NGC 253.In contrast to M82, NT emission in NGC 253 is thought to be largely a result of stellar-driven activity, with only a minor contribution by a massive blackhole (MBH) in the galactic centre.As such, it is a better suited example for the study of NT phenomena in a SBG.We describe proton and electron SSDs in the CSB, disc, and halo regions of NGC 253 based on our diffusion-based model for their propagation as they traverse the magnetized gas and lose energy by all relevant processes.In Section 2, we present the basic model and results of its implementation.Our quantitative treatment provides a reliable basis for predicting the levels of radio, X-&-γ-ray emission in all three regions, as detailed in Section 3, and further discussed in Section 4.

METHODOLOGY
It is well established that particle acceleration in SFGs is largely a stellar-related phenomenon driven by SN shocks and pulsar wind nebulae.Traversing the multiphase IS media the particles lose energy by interactions with the magnetized IS gas and radiation fields, eventually reaching the dilute halo where their propagation is much faster and energy losses much weaker than in the disc.As stated above, we treat separately the CSB and disc-halo regions.Full treatment of the radio and hard X-ray emission produced by the interaction of primary and secondary electrons with the ambient magnetic fields, and of the γ-emission produced by the decay of π 0 mesons created in hadronic collisions between energetic protons and galactic gas, is presented in full details elsewhere (Rephaeli and Sadeh, 2019).
We focus on the main NT properties of the nearby SBG NGC 253, namely the particle source distribution, the magnitude and morphology of the magnetic field, gas density, and diffusion coefficient in particular, as well as the structure of the visual and far-infrared (FIR) radiation fields in the CSB and disc.It should be noted that in the common diffusion-based approach of energetic particle propagation in IS space, the diffusion coefficient is an effective quantity in the description of what realistically is a more complex propagation process that may also include other modes of transport, such as convection and streaming (e.g., Zweibel 2013), possibly as part of a galactic wind.Our approach here is necessarily simplified due to the complexity of IS media and the lack of higher sensitivity and spatial resolution measurements.Accordingly, the effective diffusion process along the direction perpendicular to the disc is taken to be constant in the small CSB region, spatially variable in the disc and halo.In addition to spatial dependence, diffusion could also depend appreciably on the particle energy.Therefore, we have expanded our previous treatment -which was based on energy-independent diffusion -to include also the case when the diffusion coefficient varies with energy in the form deduced by Werhahn et al. (2021) based on simulations with the AREPO MHD code of a sample of model galaxies.The predicted γ-ray spectra from energetic protons and electrons in these simulations were normalized to γ-ray observations of three nearby galaxies that included NGC 253 (in addition to M82 and NGC 2146).

Central SB region
We model the CSB region as a sphere with radius 300 pc (Kapińska et al. 2017) and uniform magnetic field, gas density, and diffusion coefficient.Particle sources are assumed to follow a Gaussian distribution with variance σR =100 pc.Specifically, where ri is the radial position of source i. Integrating the Green function solution of the diffusion equation (e.g.Atoyan, Aharonian, & Völker 1995) over all sources we obtain the spatio-temporal spectral distribution of primary electrons and protons: fe,p(t, γ, ro) = ∆N (γt)P (γt) where rs, ro, and r d are, respectively, the CSB radius, observation point, and diffusion radius; γt and γ are the pre-and post-cooling energies of the diffusing particle, ∆N is the particle spectral injection rate, and P(γ) is the energy loss-rate.The latter expression is convolved with the synchrotron and Compton (Blumenthal & Gould 1970), and π 0 decay (Mannheim & Schlickeiser 1994) emissivities, then time-integrated to obtain the full spatio-temporal spectral radiation emissivity at distance ro, observation time to, and radio frequency ν, or γ-photon energy Eγ .To calculate the radio, hard X-ray, and γ radiative yields the magnetic field, B, gas density, n, and diffusion coefficient, D, need to be selected.We consider three cases differing by their diffusion coefficient and the power index of the particle injection spectrum, α, as specified in table (1).Following Strong et al. (2010), the fiducial particle injection spectrum is assumed to be a power-law with index α = 1.8 below break energy of 4 GeV, and 9 GeV for electrons and protons, respectively, and α = 2.25 above the break.
Whereas the diffusion coefficient is assumed to be spatially uniform in this (small) region, we consider both cases of energydependent and energy-independent diffusion (hereafter, EDD and EID).Given that EDD is physically motivated (e.g., Evoli et al. 2020), the EID case can be regarded as a limiting case of very weak energy dependence.We adopt the scaling D = D0(E/E0) δ , with E0=3 GeV, and δ=0.3 (Werhahn et al. 2021); the latter value was found to provide a better fit to results of the authors' (AREPO) simulations than the somewhat higher value (0.5) that is assumed sometimes (e.g., Evoli et al. 2020).We note that in comparison with EID, diffusion is slower (faster) below (above) E0, namely γe ∼5900 and γp ∼4 for electrons and protons, respectively.This will be useful in elucidating the differences between the calculated radiative yields in the EDD and EID models. 1 × 10 28 1 × 10 28 3 × 10 28 α 1.8 1.5 1.6 Table 1.CSB parameters sets, cases C1-C3: gas density (n), magnetic field (B), diffusion coefficient (D) -all taken to be spatially uniform across the CSB region, and power index of the particle injection spectrum.
The optical and FIR radiation fields are taken to be uniform greybodies at temperatures 5000 • K and 30 • K, with emissivities ∼2.6 ×10 −14 and 0.017, respectively.

Disc-halo region
It is estimated that ∼ 70% of the energetic particle sources in NGC 253 are in the CSB region where the estimated SFR is ∼ 5 M⊙yr −1 (Wik et al. 2014).The total source distribution in the full disc is represented as a sum of two Gaussians with a relative normalization corresponding to this fraction of sources in the central region.
where σa = 12 kpc (somewhat smaller than ∼13.8 kpc, the disc radius inferred by Lucero et al. 2015) and σc = 1 kpc is a measure of the scale height of the galactic disc, and the constants k1 and k2 are determined by the relative fraction of sources in the CSB, (5) where erf is the error function.

Key physical quantities
To account for the spatial variation in the relevant quantities, we describe the variability by (integrable) functions that approximate the complex disc morphology.We model the spatial distributions of key quantities based on the observationally determined radio morphology of the disc and inner halo of (an edge on) spiral galaxy.As noted in the Introduction, this was achieved by Wiegert et al. (2015) from co-added 1.5 GHz maps of 30 nearby SFGs observed as part of the CHANG-ES EVLA survey.The median map of the emission profile well beyond the galactic disc provides a quantitative basis for modelling the spatial distributions of the relevant quantities that affect the propagation of energetic electrons and protons in the disc and halo -gas density, magnetic field, and diffusion coefficient.The disc-halo region is described by a sequence of concentric ellipsoidal shells, each with a uniform gas density, magnetic field, and diffusion coefficient.The ellipsoids are taken to be highly eccentric towards the centre, gradually becoming spherical at ∼ 20 kpc, represented in terms of the function where e1 = 1 − (c/a) 2 ≈ 0.96, a = 12 kpc, c = 3.4 kpc (Heesen et al. 2009), and ai is a 500-element vector in the plane of the disc extending from the galactic centre to a radial distance of 350 kpc.The vectors ai and ei fully determine the sequence of 500 concentric ellipsoids, the surfaces of which are expressed by (x 2 + y 2 )/a 2 i + z 2 /c 2 i = 1, where ci = ai 1 − e 2 i .Next, the gas density, magnetic field, and diffusion coefficient have to be specified in each shell.Since for each point on a D1 D2 D3 n 0 (cm −3 ) 1.5 (1.5) 1.5 (1.5) 1.5 (1.5) B 0 (µG) 30 ( 20 given shell B, n, and D, are constants by construction, it is convenient to associate with every shell their values at the shell intersection with the x-axis, i.e.where (x,y,z)=(ai, 0, 0).We describe the spatial distribution of these quantities by β-King-like profiles, with the physically motivated scaling B∝ n 2/3 , and with 3 vectors of 500 elements specified each at the shell-x-axis intersection: and the diffusion coefficient, in both EDD and EID cases, is assumed to increase as where n0, B0, and D0 are their central values and rc, β, βD are treated as free shape parameters to be determined by comparison with measurements.These are supplemented by the disc and total gas mass, which is computed by integrating the gas density profile over the respective volumes, using n0, rc and β.
We construct a 3D Cartesian grid with 50×50×100 elements in the (x, y, z) axes, each of which extends out to 350 kpc.As with the CSB, the spatio-temporal distributions of primary electrons and protons is computed at each grid point as These are convolved with the synchrotron, Compton, and γ emissivities, and time-integrated to yield the full spatio-temporal spectral emissivity at distance ro, observation time to, and radio frequency ν, or γ-photon energy Eγ .(B, n, and D, at each grid point are inferred by interpolating between their counterparts on values in the 2 adjacent ellipsoidal shells, as these do not necessarily coincide with the Cartesian grid.)For the computation of the radio-and γ-emissivities produced in the disc-halo region, we have considered 3 cases whose parameter sets are listed in table (2).As with the CSB region, the optical and FIR radiation fields are taken to be uniform greybodies at temperatures 5000 • K and 30 • K, with greybody emissivities ∼9.4×10 −14 and 3×10 −5 , respectively.Outside the disc the radiation fields fall radially as r −2 .

RESULTS
Spectral radio, hard X-ray, and γ-emissivity grids were computed for both the CSB and disc-halo regions.The radio emissivities were calculated at 0.61, 1.41, 1.45, 1.47, 1.66, 4.52, 4.85, 4.89, 6.70, 7.00, and 8.09 GHz; the hard X-ray spectrum at 7-200 keV, and the γ-spectrum was calculated at each decade in the 1 MeV-10 TeV range.These were integrated over the respective volumes and converted to fluxes taking 3.5 Mpc for the distance to NGC253, resulting in spectral radio, hard X-ray, and γ-fluxes.
Radio emissivity grid of the disc-halo was computed at 227 MHz and integrated along the los at increasing heights above the galactic plane, then convolved with a Gaussian beam of σB=1.7 arcmin, to produce spectral brightness profiles.Radio fluxes originating in the CSB and the disc-halo are plotted in the left-hand panels of Fig. 1, together with measured data compiled by Kapińska et al. (2017).Our results are normalized to the level of the measured flux at 4.55 GHz, for comparison of the computed and measured spectra.The total computed to measured normalization factors for EDD vary from ∼ 0.5 to 2.1 (cases 1 and 3, respectively), whereas those calculated for EID are between ∼0.7 (case 1) and 2 (case 3).The corresponding normalization factors for the CSB alone are ∼1.4-2.6 and 0.5-1 for EDD and EID, respectively.
The calculated radio flux spectra in the EDD case (upper left panel) are marginally consistent with the measurements, while clearly steeper than in the EID case (lower left panel) which provides a better overall fit to the data.As noted in Section 2, the scaling of the diffusion coefficient with energy suggests that particles with energies lower than E0=3 GeV, namely γ ∼5900 for electrons, diffuse slower than when D=D0.With a central field value of ∼20-30µG, the synchrotron spectrum peaks around ν ∼3-4 GHz; consequently, the lower frequency part of the spectrum originates mainly from lower energy electrons that spend longer times in the higher magnetic field (and denser) regions due to slower diffusion.While these electrons enhance the radio emission at lower frequencies, higher energy electrons diffuse more rapidly out of these regions and therefore contribute less to the higher frequency part of the spectrum.These considerations can explain the somewhat steeper EDD spectra.The right-hand panels of the figure describe the radio profiles of the disc-halo for cases D1-D3, together with the measured profile inferred from Fig. 2 of Kapińska et al. (2017).Here too the calculated brightness profiles are normalized to the measured profile.Examination of the figure indicates that disc-halo case D1 best matches the measurements up to a height of z ∼ 3.6 kpc above the disc midplane for EDD, whereas case D3 best fits the data for EID.Generally, and up to a certain altitude above the disc plane, EDD profiles are somewhat flatter than the EID profiles since at 227 MHz most of the energetically relevant electrons diffuse slower than predicted in the EID case.Spectral 7-200 keV luminosities of the CSB and the (entire) galaxy are shown in Fig. 2. The integrated CSB 7-20 keV fluxes predicted in case C1 are ∼1.1×10−14 erg s −1 cm −2 for the EID model, and 1.4×10 −14 erg s −1 cm −2 for the EDD, both well below the 90% confidence upper limit of ∼1.8×10 −13 erg s −1 cm −2 deduced by Wik et al. (2014) from NuStar observations.This limit was obtained from an estimate of the measured residual emission after accounting for detected point sources in a central region roughly comparable in size to that of the CSB.The corresponding integrated CSB fluxes for cases C2 and C3 are also well below this limit.As shown in Fig. 2, most of the predicted hard X-ray emission originates outside the CSB; the 7-20 keV fluxes from the full disc are ∼2.1×10−13 erg s −1 cm −2 (EID), and 2.3×10 −13 erg s −1 cm −2 (EDD).All the quantitative limits set by Wik et al. (2014) are for varying central regions of up to ∼ 1 kpc in radius.Our results imply that the levels of Compton-produced 7-20 keV emission in both the CSB and the full disc are quite comparable in the EID and EDD models, with slightly steeper spectra in the latter case as a result of the somewhat steeper electron spectrum (due to D ∝ E 0.3 ).
The spectral γ-luminosity of the CSB and disc-halo in the 1 MeV-10 TeV range are shown in Fig. 3 for the EDD (upper) and EID (lower) models.We emphasize that with the particle distributions normalized by the radio measurements, and the selected (case-dependent) gas densities, the predicted γ-ray spectra are fully determined.The figures include the spectral flux measured by H.E.S.S. and Fermi /LAT in the 60 MeV-5.75TeV range (Abdalla et al. 2018).Cases T1-T3 are the sum of the CSB case C2 and the respective disc-halo emission.Los-integrated spectral γ emissivities at different heights above the disc plane are shown in the right-hand panels of the figure.As is clear from the figures, the emission from the CSB constitutes a significant fraction of the total disc emission; specifically, the central region contributes 80-90 per cent of the integrated 10 MeV-10 TeV in both EID and EDD models.Still, for viable range of values of key disc parameters, γ-ray emission from the rest of the disc has to be accounted for, particularly at low energies (below ∼ 10 GeV).
As mentioned in the previous section, the AREPO MHD simulations of Werhahn et al. (2021) specifically included a galaxy whose basic properties were chosen to resemble those of NGC253.Model spectra of the hadronic γ-ray emission produced by a steady-state distribution of energetic protons were normalized to the Fermi /LAT-H.E.S.S. observations.As such, their spectra are roughly similar to our calculated spectra, which fit the (same) data reasonably well.In this (specific) regard the main difference between our phenomenological approach to their simulated-based study stems from the fact that we normalize the particle spectral distributions by comparison of their (combined) synchrotron emission with the good-quality radio spectra of Kapińska et al. (2017).
We note in passing that (as has already been noted in several previous studies) the leptonic contributions to γ-ray emission by (NT) bremsstrahlung and Compton scattering are small, particularly so at energies below 100 GeV (e.g. , Rephaeli, Arieli, andPersic 2010, Werhahn et al. 2021).
Finally, whereas there is little variation in the EDD and EID spectra in the respective C1-C3 and T1-T3 cases, at high energies the EDD spectra are somewhat steeper than the corresponding EID spectra.Specifically, the slope of the high-energy part of the EDD spectra of both the CSB and total flux is ∼ 1.49 − 1.52, as compared with 1.25 for the EID models, which, as expected, retraces the power index of the proton progenitors.Clearly, the faster EDD of high-energy protons out of the central, denser regions accounts for the steepening of the spectrum at the higher γ-energies.Note that in their study of Galactic γ-emission from π 0 -decay Aharonian & Atoyan (1996) infer a scaling relation between the power-law index of the γ-ray spectral index and the indices of the proton injection spectrum and that of the diffusion coefficient, α ′ = α + (3/2)δ, valid at high (γ-ray) energies.This relation would imply α ′ = 1.7 for α = 1.25 and δ = 0.3, higher by 40 per cent than our deduced value.The difference can be explained by the fact that particle sources in our treatment are distributed throughout the disc, whereas Aharonian & Atoyan (1996) assumed a point-source model.Clearly, the spectrum is flatter when at any point in the disc the proton population includes freshly injected protons in addition to previously injected (thus, energy-degraded) protons.not contribute appreciably to the measured CSB emission, particularly so to the level of the measured radio flux which is used to determine the overall normalization factor.Given the various modelling and inherent observational uncertainties stemming from the limited spatial resolution of the CSB radio emission, the practical consideration is whether the MBH contributes at a level which is more than (roughly) 20% to the measured flux.As noted in the Introduction, the selection of NGC253 for this study is partly based on the expectation that in this SBG (unlike M82) the relative contribution of the central source is small both in radio (as can be deduced from the work of Kapińska et al. 2017) and also in γ-ray (Gutiérrez et al. 2020).Nevertheless, the main impact of an appreciable contribution of a central source on our quantitative results would amount to somewhat reduced values of the overall normalization factors specified in the previous section.
As we have previously noted (in the Introduction), the diffusion-based modelling of energetic particle propagation in IS space is meant to be an approximate description of what realistically is a more complex mode of transport that involves also advection, possibly as part of (a large scale) galactic wind.Clearly, inclusion of the latter propagation mode introduces additional free parameters; this is unwarranted given the current level of precision of the observational data base.By including spatial and energy dependence of the diffusion coefficient we can (at least partly) gauge the impact of advection with flow speed that varies with (vertical) distance from the disc through a corresponding increase in the diffusion coefficient.For example, in the simulations of Thomas et al. (2023), the wind speed reaches high values (> 300 km s −1 ) only at large distances from the disc, whose possible impact is partly accounted for by the high values of the diffusion coefficient in our models.
NT X-ray emission is mainly by Compton scattering of energetic electrons off the intense optical and FIR radiation fields in the disc.Our estimates of the predicted emission in the 7-20 keV band are below the bounds from NuStar measurements set by Wik et al. (2014;table 3) on diffuse (point-source subtracted) emission in the central disc region.However, the range of models considered here, the predicted full-disc emission is considerably higher than that of the CSB.
For the range of the EID models considered in our work, the spectral (energy) index of the γ-ray emission at energies higher than 100 MeV (i.e., beyond the π 0 -decay peak) is 1.0 − −1.25 for emission from the CSB and also for emission from the rest of the disc.These values are roughly in the range deduced by Abdalla et al. (2018) from Fermi /HESS measurements, ∼ 1.4 ± 0.3.The predicted spatial profile of the disc γ-ray emission is quite steep; at ∼ 1 kpc the flux drops by a factor of 2-3 (for all three cases) with respect to its value at the boundary of the CSB.As we have already noted (in the previous section), most of the disc emission in the (combined Fermi /LAT-H.E.S.S.) 10 MeV-10 TeV flux, a fraction of ∼ 0.8-0.9,originates in the CSB.However, CSB emission clearly does not dominate the total galactic emission, neither in radio nor in hard X-ray.Therefore, modelling the galactic emission as originating largely from the small CSB region (as assumed in some previous works) seems quite unrealistic.Also, for the range of realistic values of the key parameters assumed here, NT emission from the galactic halo is practically negligible.While this is expected, our predicted γ-ray spatial profile can be useful for estimating the size of the inner halo region that has to be observed when detection of the full galactic emission is sought.
The result that a significant part of the γ-ray emission in NGC253 originates outside the CSB region has theoretical and observational implications.It clearly indicates that fitting the measured spectrum by modelling only the CSB can lead to significant bias in the basic derived quantities, especially those characterizing the particle spectra and gas densities in this region.Nuclear SB regions of nearby SBGs have been studied in several previous works (cited in the Introduction) and more recently by Peretti et al. (2019), whose main objective was to determine the conditions for energetic particle confinement in efficiency and the contribution of SFGs to the proton energy flux in IG space.The highest estimated calorimetric efficiencies (case 1, in both EID and EDD models) are obtained for the lower value of the central diffusion coefficient (in both the CSB and disk), as well as with lower βD, for which the growth of the diffusion profile is milder.The resulting slower proton diffusion (and consequently the more frequent interactions with ambient protons) in these denser regions enhances (the pion-decay) γ-ray yield, and thus also the respective calorimetric efficiencies.This dependence of the calorimetric efficiency on the diffusion coefficient has already been deduced in previous works (e.g., Werhahn et al. 2021).
Reliable estimates of the calorimetric efficiency of protons in SFGs are needed in the study of NT phenomena in galaxy clusters.In the central regions of rich clusters the gas density and magnetic fields are significantly enhanced with respect to their levels in IG space outside clusters, with typical values of O(10 −3 ) cm −3 and O(1) µG, higher than corresponding values in the outer halo of a SFG that is not a cluster member.The higher concentration of SFGs in a cluster and stronger coupling of protons to the magnetized intracluster gas result in the creation of secondary electrons and the emission of significant radio and NT X-&-γ emission in the central cluster region (Rephaeli and Sadeh 2016).

Figure 2 .
Figure2.Hard X-ray spectral flux calculated in the 7-200 keV range.Shown are the CSB (dotted) and total (solid) spectra for case 1, with energy-dependent and independent diffusion.

Table 2 .
Disc and halo parameter sets, cases D1-D3: central gas density (n 0 ), magnetic field (B 0 ), diffusion coefficient (D 0 ), and core radius (rc).β and β D of the King β type profiles are, respectively, the shape parameters of the gas density and diffusion coefficient.Also listed are the disc-halo (Mg d ), and the total (Mg t ) gas masses, respectively.Figures in parentheses denote the respective parameters for EID coefficient.