Issue 
A&A
Volume 651, July 2021



Article Number  A116  
Number of page(s)  11  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/202140799  
Published online  28 July 2021 
Spectral index of synchrotron emission: insights from the diffuse and magnetised interstellar medium
^{1}
INAF–Osservatorio Astrofisico di Arcetri,
Largo E. Fermi 5,
50125
Firenze,
Italy
email: marco.padovani@inaf.it
^{2}
Ruđer Bošković Institute,
Bijenička cesta 54,
10000
Zagreb, Croatia
^{3}
Observatoire de Paris, LERMA, Sorbonne Université, CNRS, Université PSL,
75005
Paris, France
Received:
12
March
2021
Accepted:
20
June
2021
Context. The interpretation of Galactic synchrotron observations is complicated by the degeneracy between the strength of the magnetic field perpendicular to the line of sight (LOS), B_{⊥}, and the cosmicray electron (CRe) spectrum. Depending on the observing frequency, an energyindependent spectral energy slope s for the CRe spectrum is usually assumed: s = −2 at frequencies below ≃400 MHz and s = −3 at higher frequencies.
Aims. Motivated by the high angular and spectral resolution of current facilities such as the LOw Frequency ARray (LOFAR) and future telescopes such as the Square Kilometre Array (SKA), we aim to understand the consequences of taking into account the energydependent CRe spectral energy slope on the analysis of the spatial variations of the brightness temperature spectral index, β, and on the estimate of the average value of B_{⊥} along the LOS.
Methods. We illustrate analytically and numerically the impact that different realisations of the CRe spectrum have on the interpretation of the spatial variation of β. We use two snapshots from 3D magnetohydrodynamic simulations as input for the magnetic field, with median magnetic field strength of ≃4 and ≃20 μG, to study the variation of β over a wide range of frequencies (≃0.1−10 GHz).
Results. We find that the common assumption of an energyindependent s is only valid in special cases. We show that for typical magnetic field strengths of the diffuse ISM (≃2−20 μG), at frequencies of 0.1−10 GHz, the electrons that are mainly responsible for the synchrotron emission have energies in the range ≃100 MeV−50 GeV. This is the energy range where the spectral slope, s, of CRe varies to the greatest extent. We also show that the polarisation fraction can be much smaller than the maximum value of ≃70% because the orientation of B_{⊥} varies across the beam of the telescope and along the LOS. Finally, we present a lookup plot that can be used to estimate the average value of B_{⊥} along the LOS from a set of values of β measured at different frequencies, for a given CRe spectrum.
Conclusions. In order to interpret the spatial variations of β observed from centimetre to metre wavelengths across the Galaxy, the energydependent slope of the Galactic CRe spectrum in the energy range ≃100 MeV−50 GeV must be taken into account.
Key words: cosmic rays / ISM: magnetic fields / ISM: clouds / ISM: structure / radio continuum: ISM / radiation mechanisms: nonthermal
© ESO 2021
1 Introduction
Studies of diffuse synchrotron emission and its polarisation play a key role in constraining properties of the magnetic fields and of the interstellar medium (ISM) in the Milky Way, especially the cosmicray electron (CRe) energy spectrum. They are also relevant for measurements of the cosmic microwave background radiation at high radio frequencies (e.g. Planck Collaboration IV 2020, and references therein) and the cosmological 21 cm radiation from Cosmic Dawn and Epoch of Reionisation (EoR) at low radio frequencies (e.g. Bowman et al. 2018; Gehlot et al. 2019; Mertens et al. 2020; Trott et al. 2020). Galactic synchrotron emission is one of the main foreground contaminants in these cosmological experiments and its emission dominates the radio sky at frequencies below 10 GHz. It is therefore of great importance to obtain a detailed understanding of the spectral and spatial variations of Galactic synchrotron emission in order to successfully mitigate their effects on cosmological observations (for more details see a review by Chapman & Jelić 2019).
The spectrum of Galactic synchrotron emission is usually expressed in terms of the brightness temperature T_{ν} (defined in Sect. 2.1) which is characterised by a frequencydependent spectral index β(ν) = d logT_{ν}∕d logν. Spatial variations ofβ in a given region of the sky reflect spatial variations of the CRe and magnetic field properties in the ISM along the line of sight (LOS) across that region. Full sky maps of Galactic synchrotron emission clearly show spatial variations of β already at the angular resolution of ≃5° (Guzmán et al. 2011). Facilities providing at least three times this angular resolution such as the LOw Frequency ARray, LOFAR (van Haarlem et al. 2013), and in the near future the Square Kilometre Array, SKA (Dewdney et al. 2009), will be able to investigate even finer variations ofβ.
In terms of frequency, the observed synchrotron spectrum is flatter at low radio frequencies than at high radio frequencies (Roger et al. 1999; Guzmán et al. 2011). Typical values of β estimated from observations at mid and high Galactic latitudes are − 2.59 < β < −2.54 ± 0.01 between 50 and 100 MHz (Mozdzen et al. 2019) and − 2.62 ± 0.02 < β < −2.60 between 90 and 190 MHz (Mozdzen et al. 2017), as measured recently by the Experiment to Detect the Global EoR Signature (EDGES). In contrast, in the frequency range 1.4−7.5 GHz, β is − 2.81 ± 0.16 (Platania et al. 1998). This difference in the spectral index at low and high radio frequencies is related to aging of the CRe energy spectrum (hereafter simply ‘CRe spectrum’), j_{e}(E).
As CRe propagate through the ISM, they lose energy by a number of energyloss mechanisms that involve interactions with matter, magnetic fields, and radiation (Longair 2011). These processes deplete the population of relativistic electrons and change their original energy (injection) spectrum. The Galactic CRe spectrum presents many challenges but significant advances have been made in recent years. A number of relevant results have been achieved at high energies (above ≃ 10 GeV) thanks to detections made by the Fermi Large Area Telescope (FermiLAT; Ackermann et al. 2010), the balloonborne Pamela experiment (Adriani et al. 2011), and the Alpha Magnetic Spectrometer (AMS02) on board the International Space Station (Aguilar et al. 2014). Only recently, the two Voyager probes crossed the heliopause, overcoming the problem of solar modulation, and constraining j_{e} (E) down to E ≃ 3 MeV. (Cummings et al. 2016; Stone et al. 2019). Nevertheless, the origin and propagation of CRe are only partly understood because of the degeneracy of a number of parameters and uncertainty about the role of reacceleration, convection, and on the diffusion coefficient (see, e.g. Strong et al. 2007 and Grenier et al. 2015 for comprehensive reviews on this topic).
The uncertainties on the Galactic CRe spectrum limit the interpretation of Galactic synchrotron emission. Depending on the frequency of observation, it is usually assumed that the spectrum of the electrons contributing to the emission can be characterised by a single energy slope. In this paper we show that this assumption results in an oversimplification and needs to be replaced by a more accurate modelling when interpreting observations from centimetre to metre wavelengths. This is especially the case for the recent LOFAR (van Haarlem et al. 2013) polarimetric observations (e.g. Jelić et al. 2015; Van Eck et al. 2017), where observed polarised structures were possibly associated with synchrotron radiation from neutral clouds. As suggested by Van Eck et al. (2017) and further supported by Bracco et al. (2020), the observed polarised synchrotron emission might be originating from lowcolumndensity clouds of interstellar gas along the sight line composed of a mixture of warm and cold neutral hydrogen media, referred to as WNM and CNM, respectively.
In light of these recent results, we partly focus on the effects of the energydependent CRe spectral energy slope at a few hundred MHz. However, we also highlight the impact of an energydependent energy slope at higher frequencies. This paper is organised as follows. In Sect. 2 we introduce the theory of synchrotron emission and illustrate the effects on the brightness temperature spectral index depending on the parameterisation of the CRe spectrum; in Sect. 3 we apply the above results first to a cloud modelled as a uniform slab, and then to the diffuse, multiphase ISM with the help of 3D magnetohydrodynamic (MHD) simulations, the results of which we then use in Sect. 4 to compute brightness temperature maps, the spectral index, and the polarisation fraction. In Sect. 5 we discuss the effect of the angular resolution of the observations and present a procedure for predicting the average strength of the magnetic field perpendicular to the LOS, B_{⊥}, once the CRe spectrum is set. In Sect. 6 we summarise our main findings.
2 Fundamentals of synchrotron emission
Above ≃10 GeV, the CRe spectrum (i.e. the number of electrons per unit energy, time, area, and solid angle), is approximately a power law in energy, j_{e} (E) ∝ E^{s}. The spectral energy slope (hereafter simply ‘spectral slope’), s(E) = d logj_{e}∕d logE, has been measured in the solar neighbourhood by several probes: FermiLAT established a spectral slope s = − 3.08 ± 0.05 in the energy range 7 GeV–1 TeV (Ackermann et al. 2010), the Pamela experiment found s = − 3.18 ± 0.05 above the energy region influenced by the solar wind (>30 GeV; Adriani et al. 2011), and AMS02 determined s = −3.28 ± 0.03 in the energy range 19.0–31.8 GeV and s = −3.15 ± 0.04 in the range 83.4–290 GeV (Aguilar et al. 2014). At low energies, the spectral slope measured by the two Voyager probes in the energy range ≃3−40 MeV is − 1.30 ± 0.05 (Cummings et al. 2016; Stone et al. 2019). Thus, at energies below ≃10 GeV, the spectral slope is energydependent. As we see below, this has significant consequences for the spectrum of the synchrotron emission observed at frequencies of hundreds of MHz up to tens of GHz.
Generally, models and simulations developed for the interpretation of Galactic synchrotron emission assume that CRe contributing to the emission above and below 408 MHz have s = − 3 and s = − 2, respectively (see, e.g. Sun et al. 2008; Waelkens et al. 2009; Reissl et al. 2019; Wang et al. 2020). This simplification is usually made to avoid timeconsuming calculations, and is based on observations supporting a flatter spectrum below 408 MHz (see, e.g. Reich & Reich 1988a,b; Roger et al. 1999). In Sect. 2.2 we show that this assumption turns out to be inaccurate at both low and high frequencies, leading to misinterpretation of the synchrotron observations, in particular of the spatial variations of the spectral index β. To support this claim, we consider the two realisations of the local CRe spectrum by Orlando (2018) and Padovani et al. (2018) shown in Fig. 1. The former is based on multifrequency observations, from radio to γ rays, and Voyager 1 measurements through propagation models, and is representative of intermediate Galactic latitudes (10° < b < 20°) that include most of the local radio synchrotron emission within a radius of ~1 kpc around the Sun; the latter is given by an analytical fourparameter fitting formula that perfectly reproduces the powerlaw behaviour measured at low and high energies (see also Ivlev et al. 2015). As shown by the inset in Fig. 1, the spectra by Orlando (2018) and that of Padovani et al. (2018) differ by less than ~ 25% over the range of energies of interest here, below E ≃ 50 GeV. As we see in Sect. 3.1, synchrotron observations can also constrain these two parameterisations.
CRe at a given energy emit over a broad range of frequencies and, conversely, the synchrotron emission observed at a given frequency comes from a broad range of CRe energies. Before focusing on the frequency range of LOFAR observations (115–189 MHz; see Sect. 4), we consider frequency ranges characteristic of allsky Galactic radio surveys both at low (45–408 MHz; Guzmán et al. 2011) and high frequencies (1–10 GHz; Platania et al. 1998). For B_{⊥} ≃ 2 to 20 μG, as expected in the diffuse ISM (e.g. Heiles & Troland 2005; Beck 2015; Ferrière 2020), CRe that account for nearly all of the observed synchrotron emission have energies ranging from ≃ 100 MeV to 50 GeV (see Sect. 2.2)^{1}. In this energy range, the spectral slope s shows large variation with energy: between − 1.8 and − 3.3 and between − 1.5 and − 3.2, for the spectra modelled by Orlando (2018) and Padovani et al. (2018), respectively (see Fig. 2).
Fig. 1 CRe spectrum according to Orlando (2018; green line) and Padovani et al. (2018; violet line). The inset shows the ratio of the spectra (O = Orlando, P = Padovani; solid black line). Data: Voyager 1 (orange squares; Cummings et al. 2016); Voyager 2 (grey squares; Stone et al. 2019); FermiLAT (magenta triangles; Ackermann et al. 2010); Pamela (redcircles; Adriani et al. 2011); AMS02 (blue squares; Aguilar et al. 2014). 
Fig. 2 Spectral slope vs. energy of the CRe spectra by Orlando (2018; green line) and Padovani et al. (2018; violet line) in the energy range relevant for our study (100 MeV−50 GeV). The inset shows the difference Δs between the spectral slope of the two spectra. 
2.1 Basic equations
Here we summarise the basic equations for the calculation of the synchrotron brightness temperature, T_{ν} (see, e.g. Ginzburg & Syrovatskii 1965, for details). At any given position r in a cloud, the specific emissivity^{2} can be split into two components linearly polarised along and across the component of the magnetic field perpendicular to the LOS, B_{⊥} (r), (1)
are thepower per unit frequency emitted by an electron of energy E at frequency ν for the two polarisations. Here, B_{⊥} = B_{⊥}, v_{e} is the electron velocity, x = ν∕ν_{c}, and ν_{c} is the critical frequency given by (3)
The functions F(x) and G(x) are defined by (4)
where K_{5∕3} and K_{2∕3} are the modified Bessel functions of order 5/3 and 2/3, respectively. The corresponding Stokes Q_{ν} and U_{ν} specific emissivities are (6)
where φ(r) is the local polarisation angle counted positively clockwise. The orientation of B_{⊥} rotated by ± 90° gives the local polarisation angle (modulo 180°). The emissivities are integrated along the LOS to obtain the specific intensity (brightness) for each polarisation, I_{ν,∥} and I_{ν,⊥}, and the Stokes parameters Q_{ν} and U_{ν}. We also compute the polarisation fraction (8)
where I_{ν} = I_{ν,∥} + I_{ν,⊥}. Finally, the flux density, S_{ν}, obtained from the convolution of the specific intensity integrated along the LOS with the telescope beam is converted into brightness temperature by the relation (9)
where , θ_{b} is the full width of the beam at half its maximum intensity, and k_{B} is the Boltzmann constant. Here, S_{ν} is in Jy beam^{−1} and the other quantities are in cgs units. If the CRe spectrum is a single power law, j_{e} (E) ∝ E^{−s}, then S_{ν} ∝ ν^{α}, with α = (s + 1)∕2, and T_{ν} ∝ ν^{β}, with β = (s − 3)∕2 (Ginzburg & Syrovatskii 1965).
Fig. 3 Left panel, upper plot: integrand of the specific emissivity, dε_{ν}∕dE, as a functionof the energy E, normalised to its maximum value and summed over the two polarisations (Eqs. (1)), computed at 100 MHz, 1 GHz, and 10 GHz (green, orange, and violet line, respectively), for B_{⊥} = 2 μG. The coloured area below each curve corresponds to 95% of the total emissivity. Right plot: Same as the upper plot, as a function of the CRe spectral slope s. Main plot: Spectral slope as a function of the energy for the CRe spectrum by Orlando (2018; black solid line). Coloured lines show the range of energies (and the corresponding spectral slopes) contributing to 95% of the specific emissivity. Right panel: same as left panel, but for B_{⊥} = 20 μG. 
2.2 Contributions to synchrotron emission from 100 MHz to 10 GHz
In Sect. 2 we mention that the assumption of an energyindependent slope s can introduce severe biases in the reliability of models and numerical simulations and ultimately affect the interpretation of synchrotron observations, such as the spatial variation of β. To prove this, we consider three observing frequencies (100 MHz, 1 GHz, and 10 GHz) representative of allsky Galactic radio surveys (Platania et al. 1998; Guzmán et al. 2011) and two extreme values for B_{⊥} (2 and 20 μG) consistent with the magnetic field strength expected in the diffuse medium (Heiles & Troland 2005; Beck 2015; Ferrière 2020). To identify the range of energies contributing to the specific emissivity for a given value of ν and B_{⊥}, we examine the integrands dε_{ν}∕dE summed over the two polarisations (Eqs. (1)). As shown in the upper panels of Fig. 3, d ε_{ν} ∕dE has a welldefined maximum. For each frequency, we compute the energy range around the energy of the maximum, where the specific emissivity is equal to 95% of its total. This fiducial 95% level is meant to show that most of the synchrotron emission at a given frequency originates from a definite, albeit broad, energy range of CRe. The resulting energy ranges (and the corresponding ranges of spectral slope s derived for the CRe spectrum by Orlando 2018) are listed in Table 1. Looking at the ranges of s at each frequency, it is clear that only in the extreme case of high frequencies (ν ≳ 10 GHz) and very small B_{⊥} (≃2 μG) is the hypothesis of an energyindependent spectral slope accurate (in this case s ≃ −3.3). We note that the energy and the spectral slope intervals are the same for a given ν∕B_{⊥} ratio. This is because the total power per unit frequency, , emitted at a frequency ν by an electron of energy E in a field B_{⊥}, peaks at an energy proportional to (see, e.g. Longair 2011). Figure 3 shows that, for the typical frequency range of synchrotron observation in the diffuse ISM and for the expected values of B_{⊥}, CRe with energies between 100 MeV and 50 GeV are responsible for almost all the nonthermal emission. In this energy range, the spectral slope has the largest variation, and so the assumption of an energyindependent s is not correct.
For the sake of simplicity, we have only considered the case of the CRe spectrum by Orlando (2018). Using that of Padovani et al. (2018) instead, the energy intervals we obtain are similar, as the two spectra differ on average by 25% (see Sect. 2 and Fig. 1). However, the spectral slope intervals change (see Fig. 2). This means that the use of a particular CRe spectrum has a major influence on the interpretation of the observations. In the following section we show how observational estimates of the brightness temperature spectral index help to constrain the accuracy of a CRe spectrum.
Ranges of energy E and spectral slope s for two valuesof B_{⊥} and for a set of frequencies.
3 Modelling synchrotron emission
In this section we first consider a cloud modelled as a simple uniform slab to show how specific realisations of the CRe spectrum can be ruled out by comparison with observational estimates of β (Sect. 3.1). We then describe the numerical simulations that we use in the rest of the paper to study the spatial variations of β and of the polarisation fraction (Sect. 3.2).
Fig. 4 Upper panel: brightness temperature T_{ν} as a functionof B_{⊥} computed atν = 115 and 189 MHz with the CRe spectra by Orlando (2018; green line) and Padovani et al. (2018; violet line) for a slab of thickness 1 pc, with an angular resolution θ_{b} = 4′. Lower panel:brightness temperature spectral index β as a functionof B_{⊥} (same colour coding as in the upper panel). The hatched region highlights typical values of β at intermediate and high latitudes (Mozdzen et al. 2017). 
3.1 Uniform slab
To show theimportance of an accurate modelling of the CRe spectrum, we compute the brightness temperature, T_{ν}, for a slab with a fixed, spatially uniform component of the magnetic field B_{⊥}, varying from 2 to 20 μG as specified in Sect. 2, exposed to a flux of CRe, j_{e}(E), given by the Orlando (2018) and Padovani et al. (2018) spectra. For illustration, we select the frequency range ν = 115–189 MHz, with a frequency resolution of 183 kHz and assume an angular resolution of θ_{b} = 4′. These parameters are representative of the LOFAR High Band Antenna observations carried out by Jelić et al. (2015). We assume a thickness of the slab of 1 pc, although the brightness temperature can be easily scaled for any value. We then compute the spectral index, β, through a linear fit of logT_{ν} versus logν for each B_{⊥} value.
The brightness temperatures generated by the two realisations of the CRe spectrum are on average within ≃ 25% of each other (upper panel of Fig. 4), of the order of a few to several K. Assuming a synchrotron polarisation fraction of p ≈ 70% (more details in Sect. 4.3), the polarised intensity would be of the same order as T_{ν}. This is exactly the amount of diffuse polarised emission observed by LOFAR in the range 100–200 MHz (e.g. Jelić et al. 2014, 2015; Van Eck et al. 2017). These simple arguments support the scenario suggested by Van Eck et al. (2017) and Bracco et al. (2020) where parsecscale neutral clouds in the diffuse ISM could significantly contribute to the Faradayrotated synchrotron polarisation observed with LOFAR. However, if we compare the value of β (lower panelof Fig. 4) with typical values derived from observations at intermediate and high Galactic latitudes (Mozdzen et al. 2017, shaded strip in the lower panel of Fig. 4), it is evident that in this example only the spectrum by Orlando (2018) is consistent with the observations, and only for values of B_{⊥} of the order of 6–9 μG, while the one by Padovani et al. (2018) requires unrealistically high values of B_{⊥}. As this example illustrates, the constraint imposed by the highly accurate determination of the spectral index β allowed by current instruments is a strong motivation to further and thoroughly model the total and polarised synchrotron emission of the diffuse ISM.
3.2 Numerical simulations
We now compute synthetic observations of synchrotron emission adopting stateoftheart numerical simulations of the diffuse, magnetised, multiphase, turbulent, and neutral atomic ISM. Our approach here is not to consider the simulations described below as true representations of any given LOS in the diffuse ISM, but rather as a laboratory with which to study realistic physical conditions of the multiphase medium where synchrotron emission may originate. In particular, we perform numerical simulations of the diffuse ISM using the RAMSES code (Teyssier 2002; Fromang et al. 2006), a gridbased solver with adaptive mesh refinement (Berger & Oliger 1984), and a fully treated tree data structure (Khokhlov 1998).
The gas density of the medium we consider is typically dominated by neutral hydrogen that can be traced via the 21 cm emission line (Heiles & Troland 2003a,b; Murray et al. 2015, 2018). This line is usually decomposed into several Gaussian components (Kalberla & Haud 2018; Marchal et al. 2019) associated to distinct gas phases in pressure balance: a dense cold neutral medium, CNM, with temperature and density T ≃ 50 K and n_{H} ≃ 50 cm^{−3}, respectively, immersed in a diffuse warm neutral medium, WNM, with T ≃ 8000 K and n_{H} ≃ 0.3 cm^{−3}, and a third intermediate unstable phase, with temperature comprised between those of the CNM and the WNM (e.g. Wolfire et al. 2003; Bracco et al. 2020). Field (1965) and Field et al. (1969) pointed out that the microphysical processes of heating and cooling naturally lead to two thermally stable phases (CNM and WNM) and one thermally unstable phase coexisting in a range of thermal pressure. Through the thermal processes of condensation and evaporation, and with the help of turbulent transport and turbulent mixing, the diffuse matter can flow from one stable state to the other.
The local diffuse matter in our Galaxy is simulated over a box of 50 pc with periodic boundary conditions, using a fixed grid of 128^{3} pixels, corresponding to an effective resolution of 0.39 pc. The initial state is characterised by a homogeneous density n_{H} = 1.5 cm^{−3}, a temperature T = 8000 K, and a uniform magnetic field B_{0} = B_{0}ê_{x}. We consider two snapshots of the simulation, one with a standard average magnetic field strength, B_{0} = 4 μG (hereafter, the ‘weak field’ case), and one with a stronger B_{0} = 20 μG (hereafter, the ‘strong field’ case). The gas evolves under the joint influence of turbulence, magnetic field, radiation field, and thermal instability, and separates in three different phases: CNM, WNM, and unstable. A turbulent forcing is applied to mimic the injection of mechanical energy into the diffuse ISM. Following Schmidt et al. (2009) and Federrath et al. (2010), this forcing, modelled by an acceleration term in the momentum equation, is driven through a pseudospectral method. The turbulent acceleration parameter is set to 2.77 × 10^{−3} kpc Myr^{−2}. Using classical notation, the dimensionless compressible parameter for the turbulent forcing modes, ζ_{0}, ranges from pure solenoidal modes (ζ_{0} = 1) to pure compressible modes (ζ_{0} = 0). We set ζ_{0} = 0.5.
The matter is assumed to be illuminated on all sides by an isotropic spectrum of UV photons set to the standard interstellar radiation field (Habing 1968). To correctly describe the thermal state of the diffuse ISM,we have included the heating induced by the photoelectric effect on interstellar dust grains and secondary electrons produced during cosmicray propagation, and the cooling induced by emission of Lymanα photons, the finestructure lines of OI and CII, and the recombination of electrons onto grains. All these processes, described in Appendix B of Bellomi et al. (2020), are modelled with the analytical formulae given by Wolfire et al. (2003).
These simulations represent two different scenarios: in the weak field case, the magnetic field has a turbulent component of the same order as its mean component. By contrast, in the strong field case, the magnetic field is mostly directed along the xaxis, that is, its projection is mainly contained in the xy and zx planes. The advantage of using simulations is that we can rotate each snapshot according to the three axes and calculate the quantities of interest integrated along three different LOSs, thus increasing our statistics. Table 2 summarises the ranges of the strength of the magnetic field in the plane perpendicular to a given LOS and their median values (marked by a superscript tilde) for the two snapshots under consideration. Specifically, for the LOS i, we computethe minimum and maximum values of the magnetic field strength, , in the plane of the sky (POS) jk. Subscripts i, j, and k follow the cyclic permutation of Cartesian coordinates (ijk) = {xyz, yzx, zxy}.
Minimum and maximum values of the strength of the magnetic field in the plane perpendicular to the LOS i, , and its median value for the two cases of weak and strong field.
4 Results
In contrast to typical angular resolutions of earlier facilities, such as the 5° of Guzmán et al. (2011), nowadays, LOFAR provides a significantly higher resolution, typically up to 4′ at frequencies of 115−189 MHz for observations of Galactic diffuse emission (Jelić et al. 2014, 2015; Van Eck et al. 2017). In the following, we show our results at the resolution of6.7′. The latter value corresponds to the spatial resolution of the simulations (0.39 pc) if the simulation snapshots are placed at a distance of 200 pc. We focus on the frequency range 115−189 MHz, but the same conclusions apply to higher frequencies (see Sect. 5.2 and Appendix A).
4.1 Brightness temperature maps
Figure 5 shows the brightness temperature maps at a frequency ν = 130 MHz for the two snapshots described in Sect. 3.2, obtained with the CRe spectrum of Orlando (2018)^{3}. These images are derived by integrating along the three different LOSs (x, y, z) over the length of the snapshot (50 pc), resulting in temperature maps identified by the respective POSs (yz, zx, xy). As anticipated by the upper panel of Fig. 4, the temperature is higher where B_{⊥} is larger, reaching a maximum in the xy and zx POSs of the strong field case (right column of Fig. 5; see also the values of in Table 2), while the smallest temperature values are in the yz POS of the strong field case. Temperature maps are more inhomogeneous in the yz POS of both snapshots. That is because, in these planes, B_{⊥} has a significant turbulent component. Therefore, we introduce the ratio between the standard deviation and the median value of B_{⊥}, , to quantify the relative turbulent component of B_{⊥}. This quantity, the median values of which are reported in Table 3, is useful for the discussion on β and on the polarisation fraction in the following two sections.
Fig. 5 Brightness temperature maps (colour scale) computed with the Orlando (2018) spectrum, at ν = 130 MHz and with resolution 6.7′ for the three POSs (three rows, labelled xy, yz, and zx) of the weak and strong field cases (left and right column, respectively). 
4.2 Brightness temperature spectral index
We compute the brightness temperature spectral index, β, for each LOS, by a linear fit of logT_{ν} versus logν for the three POSs of the two snapshots, in the frequency range 115–189 MHz with a frequency resolution of 183 kHz as in the LOFAR dataset by Jelić et al. (2015). In Fig. 6, we show the results in the form of bivariate and marginal distributions (see also Appendix B for the maps of β). From the inspection of Fig. 6, it is evident that in no case, at low frequencies, is β equal to − 2.5, as would follow from the assumption of constant s = −2. As shown in Sect. 2.2, the same consideration holds in the highfrequency regime (ν ≳ 400 MHz), where the assumption of constant s = −3, and hence β = −3, turns out to be incorrect. Figure 6 also shows that is less negative in the POSs xy and zx, where is larger (see also Tables 2 and 3). This is a consequence of what is shown in Fig. 3: for a given frequency, the larger the value of B_{⊥} at a given position along the LOS, the smaller the median value of the energy range determining its synchrotron emissivity. Therefore, the corresponding values of s, and hence of β, increase. Finally, an important aspect to note is that the dispersion of β values around its median depends weakly on μ. Indeed, in both yz POSs, where μ reaches the largest values, the interquartile range (i.e. the difference between the third and the first quartile) is only 0.04 and 0.06 for the weak and the strong cases, respectively. For the sake of completeness, in Appendix A we show the bivariate distributions of μ and β for higher frequency ranges (467−672 MHz and 833−1200 MHz), which are used in Sect. 5.2. The conclusions are the same as for the 115−189 MHz range.
Fig. 6 Bivariate distribution of the ratio between the standard deviation of B_{⊥} and its median value, μ, and the brightness temperature spectral index, β, computed in the frequency range 115−189 MHz for each LOS for the weak and strong field case (upper and lower panel, respectively), at a resolution of 6.7′. The POSs are identified by the three different colours displayed in the legend. Isodensity contours are plotted at 5, 30, 50, and 75% levels. The histograms above and to the right of the main plots show the marginal distribution of the two quantities. The median values are marked by horizontal and vertical lines and are listed in Table 3. 
Medians of the ratio between the standard deviation of B_{⊥} and its median value, , of the brightness temperature spectral index, and of the polarisation fraction at a resolution of 6.7′ for the three POSs and the two cases of weak and strong field^{(a)}.
4.3 Polarisation fraction
The local polarisation fraction, that is, the polarisation fraction based on the local emissivities for an energyindependent value of s is (10)
(see, e.g. Rybicki & Lightman 1986), which, for s = −2 and −3, gives p = 69 and 75%, respectively. If the orientation and the strength of B_{⊥} do not vary along the LOS, Eq. (10) would also give the polarisation fraction in the POS. To quantify the limitations of the above assumption, we calculate the expected polarisation fraction (Eq. (8)) at the reference frequency ν = 130 MHz for the two simulations and at the resolution of 6.7′ in the threePOSs. Similarly to Fig. 6, in Fig. 7 we show the results in the form of bivariate and marginal distributions, and we find a similar correlation of μ with p as we found with β: as μ increases, the spread of values of p around its median value is even larger than that of β (see Table 3).
Furthermore, except for the POSs xy and zx in the strong field case, the median value is much lower than the theoretical value of ≃ 70% of the local polarisation fraction, which is the maximum value of the observed polarisation fraction. Such depolarisation effects in our study are mainly caused by the tangling of the turbulent component of the magnetic field along the LOS. This effect has been extensively reported in the literature in both the radio (e.g. Gaensler et al. 2011) and in the submillimetre domains (e.g. Planck Collaboration Int XX 2015). Moreover, radio synchrotron polarisation can be severely affected by other mechanisms that drastically reduce the amount of detectable polarised emission, such as Faraday rotation and beam depolarisation (see, e.g. Sokoloff et al. 1998; Haverkorn et al. 2004). We note that Faraday rotation at a few hundred MHz cannot be neglected from an observational point of view. However, in this study we choose not to detail this process and to only focus on the emission mechanism of synchrotron radiation with an energydependent spectral slope. Faraday rotation will be the focus of a followup paper that will investigate the complexity of the ionisation degree of the multiphase diffuse ISM, a key ingredient to correctly model the effect of Faraday rotation.
Fig. 7 Bivariate distribution of the ratio between the standard deviation of B_{⊥} and its median value, μ, and the polarisation fraction, p, computed at 130 MHz for each LOS for the weak and strong field cases (upper and lower panel, respectively), at a resolution of 6.7′. The POSs are identified by the three different colours displayed in the legend. Isodensity contours are plotted at 5, 30, 50, and 75% levels. The histograms above and to the right of the main plots show the marginal distribution of the two quantities. The median values are marked by horizontal and vertical lines and are listed in Table 3. 
Fig. 8 Histograms of the spectral index β (upper panels) and polarisation fraction p (lower panels) for the weak and strong field cases (left and right column, respectively). Solid (empty) histograms refer to a resolution of 6.7′ (5°). The POSs are identified by the three different colours displayed in the legend. 
5 Discussion
5.1 Effect of angular resolution on β and p
In this study we assumed the typical angular resolution of observations carried out with the most recent facilities, such as LOFAR. Here we show the distributions of the spectral index and the polarisation fraction obtained from lower resolution observations such as those shownin Guzmán et al. (2011). This latter paper presented an allsky Galactic radio emission map, convolved to a common 5° resolution, combining observations obtained with different telescopes such as the Parkes 64 m, the Jodrell Bank MkI and MkIA, and the Effelsberg 100 m (Haslam et al. 1981, 1982), the Maipú Radio Astronomy Observatory 45MHz array (Alvarez et al. 1997), and the Japanese Middle and Upper atmosphere radar array (MU radar; Maeda et al. 1999). Figure 8 shows the comparison of the distributions of β and p (upper and lower panels, respectively) at the resolutions of 6.7′ and 5° (filled and empty histograms, respectively). Despite the increase in resolution of LOFAR, the median value of β does not change appreciably (see upper panels). However, what does change is the spread of the distribution, because at a lower resolution it is not possible to pick up the finer variations of the spectral index. More interesting is how the polarisation fraction distributions differ in the two cases. The lower panels show the beam depolarisation effect mentioned in Sect. 4.3: where μ is larger, a lower resolution clearly results in a lower p.
5.2 A lookup plot for B_{⊥}
At any given position along a LOS, the synchrotron emission is jointly determined by the local CRe spectrum and the local value of B_{⊥}. As we show in Sect. 3.1, thanks to the range of β obtained from observations, it is possible to constrain the CRe spectrum at ~GeV energies (see also Fig. 3). This makes it possible to reduce the uncertainly on the spectralshape, an appreciable improvement over the assumption of an energyindependent s. One should also consider that the latest generation of telescopes, such as LOFAR and in the near future SKA, reach resolutions up to at least three orders of magnitude higher than previous instruments. This means that the uncertainties on the values of β derived from observations will be reduced and it will be possible to obtain β variations over smaller fields of view. Finally, in Sect. 4.2 we show that variations of B_{⊥} along the LOS do not affect the estimates of β. This featureis particularly relevant for reducing the uncertainty in the estimation of the average value of B_{⊥} along a LOS. Given these premises, we suggest a method for estimating ⟨B_{⊥}⟩, the average value of B_{⊥}, along a given LOS.
We adopt the Orlando (2018) CRe spectrum and assume ν in the range 10 MHz−20 GHz and ⟨B_{⊥}⟩ in the range 0.5−30 μG. We compute the corresponding emissivities (see Eqs. (1)) integrated along the LOS for 1 pc^{4}, and compute . By inverting the relation for β, in Fig. 9 we show the values of ⟨B_{⊥}⟩ expected for each couple (ν, β).
We then consider three different ranges of frequencies: 115−189 MHz with a resolution of 183 kHz, 467−672 MHz with a resolution of 507 kHz, and 833−1200 MHz with a resolution of 908 kHz^{5}, labelled in Fig. 9 as low (), mid (), and high () frequency range, respectively. Following the procedure described in Sect. 4.2, for each POS of the two snapshots, we calculate the temperature maps as a function of frequency at a resolution of 6.7′ and we extract the β value for each LOS and for each of the three frequency intervals, , , and , by a linear fit of logT_{ν} versus log ν. We show in Fig. 9 the values for each POS by estimating the errors using the first and third quartiles for each frequency interval. We note that, as the POSs xy and zx for both snapshots have the same , in the figure we show four values instead of six. The values of for the frequency range are listed in Table 3, while those for the and ranges are reported in Table A.1.
Figure 9 shows that, for each POS, the estimates of in the three frequency intervals correspond to the same ⟨B_{⊥}⟩, which also agree with the respective value of listed in Table 1. This confirms the consistency of our procedure.
This plot shows that there is a preferred range of frequencies that can be conveniently used to estimate ⟨B_{⊥}⟩ along a LOS. This range corresponds approximately to 100 MHz−5 GHz, the frequency interval where s, and hence β, varies the most and where the isocontours of ⟨B_{⊥}⟩ are more separated. Conversely, at frequencies that are too low (high), the CRe spectrum flattens out if ⟨B_{⊥}⟩ is too high (low), and the estimate of ⟨B_{⊥}⟩ becomes more uncertain. From this figure it can be concluded that, in order to have a more precise estimate of ⟨B_{⊥}⟩, it is advisable to simultaneously observe in narrow frequency ranges with high spectral resolution (as our , , and intervals) in order to have independentβ estimates that should follow a specific isocontour of B_{⊥}.
Fig. 9 Lookup plot with which to estimate the average value of B_{⊥} along a LOS, ⟨B_{⊥}⟩, for a given frequency ν and brightness temperature spectral index β assuming the CRe spectrum by Orlando (2018). Black and white isocontours show specific values of ⟨B_{⊥}⟩ with labels in μG. Yellow symbols indicate estimates of ⟨B_{⊥}⟩ obtained from the values of in the frequency ranges 115−189 MHz, 467−672 MHz, and 833−1200 MHz (labelled as , , and , respectively)at a common resolution of 6.7′ for the different POSs of the two simulation snapshots (circles: POS yz in the strongfield case; squares: POS yz in the weak field case; diamonds: POS xy and zx in the weak field case; triangles: POS xy and zx in the strong field case). Rectangles at the bottom of the plot show the frequency ranges of the low and high band antenna of LOFAR (LBA and HBA, respectively), the low and mid frequency bands of SKA (SKA1Low and SKA1Mid, respectively), and SKA pathfinders and precursors: the Hydrogen Epoch of Reionisation Array (HERA); the Murchison Widefield Array (MWA); the Karoo Array Telescope (MeerKAT); and the Australian SKA Pathfinder (ASKAP). 
6 Summary
We carried out a quantitative study to understand the consequences of an energydependent CRe spectral slope for the interpretation of observations of synchrotron emission in the diffuse and magnetised ISM. We focused in particular on metre wavelengths that can currently be observed with stateoftheart facilities such as LOFAR and in the near future with SKA. At frequencies lower (higher) than ≃ 400 MHz a constant spectral slope s = −2 (s = −3) is often assumed, mainly to avoid timeconsuming calculations in analytical models and numerical simulations. As a consequence, one should also expect a constant value for the brightness temperature spectral index, β, related to s by β = (s − 3)∕2 (Rybicki & Lightman 1986). However, metrewavelength observations show that β is not constant across the Galaxy, taking on values quite different from −2.5 (corresponding to s = −2), varying between about − 2.7 and −2.1 (Guzmán et al. 2011).
For typical magnetic field strengths expected in the diffuse ISM (≃ 2−20 μG), the electrons that mostly determine the synchrotron emission at frequencies between about 100 MHz and 10 GHz have energies in the range ≃100 MeV−50 GeV. It is precisely at these energies that the spectral slope shows the largest variations. For example, for the CRe spectrum described in Orlando (2018), representative of intermediate Galactic latitudes including most of the local emission within about 1 kpc around the Sun, in this energy range s varies between about − 3.2 and − 1.8.
In order to understand the effect of an energydependent spectral slope at a quantitative level, we first considered a slab with a fixed, spatially uniform B_{⊥} exposed to a flux of CRe and we showed that, thanks to highprecision observational estimates of β at low frequencies (Mozdzen et al. 2017), it is possible to discard some realisations of the CRe spectrum that would imply unrealistically high values of B_{⊥}. We then used two snapshots of 3D MHD simulations (Bellomi et al. 2020) with different median magnetic field strength, and studied the synchrotron emission according to the CRe spectrum by Orlando (2018). We computed the distribution of β for three frequency ranges, at low, mid, and high frequencies (115−189 MHz, 467−672 MHz, and 833−1200 MHz, respectively). We show that the assumption of an energyindependent s is not justified and leads to nonnegligible biases in the interpretation of the observed spectral index distributions. In particular, we find that β becomes less negative as B_{⊥} increases and that the dispersion of the distribution of spectral index values around its median weakly depends on how much B_{⊥} varies along the LOS. This property is of special relevance, because once a CRe spectrum is assumed, the uncertainty on the expected average value of B_{⊥} for a given LOS, ⟨B_{⊥}⟩, is reduced. We then presented a lookup plot that makes it possible to estimate ⟨B_{⊥}⟩ given β values obtained from observations in one or more frequency intervals. More precisely, we suggest repeating observationsin narrow frequency intervals with high spectral resolution in order to have independent estimates of β that should lie on the same ⟨B_{⊥}⟩ isocontour in the lookup plot.
Finally,we computed the expected polarisation fraction, p, finding that it is expected to decrease as the turbulence of the magnetic field along the LOS increases, deviating noticeably from the maximum value of ≃ 70%. The dispersion of p around its median value is larger than that of β as a consequence of the turbulent tangling of the magnetic field lines along the LOS. The analysis of this depolarisation effect and the consequences of Faraday rotation are deferred to a subsequent paper.
Acknowledgements
The authors wish to thank the referee, Katia Ferrière, for her careful reading of the manuscript and insightful comments that considerably helped to improve the paper. M.P. thanks Tommaso Grassi for useful discussions. A.B. acknowledges the support from the European Union’s Horizon 2020 research and innovation program under the Marie Sk?odowskaCurie Grant agreement No. 843008 (MUSICA). V.J. acknowledges support by the Croatian Science Foundation for the project IP2018012889 (LowFreqCRO).
Appendix A Bivariate distributions of μ and β at high frequencies
Following the procedure outlined in Sect. 4.2, we consider two higher frequency ranges, 467−672 MHz and 833−1200 MHz (labelled and , respectively), for the calculation of β in the three POSs of the two simulation snapshots. Figure A.1 shows the bivariate distribution of μ and β for these two frequency ranges, comparing them with the distribution for the range 115−189 MHz (labelled ) described inSect. 4.2. In order to increase clarity and to avoid overlapping distributions, we only show isodensity contours plotted at 50 and 80%. As the frequency range increases, the β distributions shift towards more negative values. This can be explained by looking at Fig. 3 where we see that as the frequency increases, the energies determining the synchrotron emissivity are higher and higher. Higher energies correspond to more negative values of s, and therefore of β. For completeness, Table A.1 shows the values of for the three POSs of the two snapshots considered in the frequency intervals and .
Fig. A.1 Bivariate distribution of the ratio between the standard deviation of B_{⊥} and its median value, μ, and the brightness temperature spectral index, β, computed in the frequency ranges 115−189 MHz, 467−672 MHz, and 833−1200 MHz (labelled , , and , respectively) for each LOS for the weak and strong field case (left and right panel, respectively), at a resolution of 6.7′. The POSs are identified by the three different colours displayed in the legend. Isodensity contours are plotted at 50 and 80%. 
Medians of the brightness temperature spectral index for the three POSs and the two cases of weak and strong field at a resolution of 6.7′.
Appendix B Brightness temperature spectral index maps
Here we show the spectral index maps obtained for the two simulations described in Sect. 3.2. These maps have been derived considering the same frequency range and frequency resolution of LOFAR observations by Jelić et al. (2015), namely ν = 115−189 MHz and Δν = 183 kHz, respectively. In the POS where μ ≪ 1, namely the xy and zx POSs of the strong field case, β shows a constant value. Moreover, the greater the turbulent component of the field, the greater the variations exhibited by β on small scales.The latter can in principle be resolved by observations with LOFAR. From these maps, the histograms displayed in Fig. 6 were produced.
Fig. B.1 Brightness temperature spectral index maps for the three POSs (three rows) of the weak and strong field case (left and right column, respectively) for a resolution of 6.7′. The three panels of each column share the same colour bar at the top. 
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We note that CRe in the energy range 100 MeV–50 GeV are affected by energy losses such as bremsstrahlung only after crossing column densities ≳ 10^{25} cm^{−2} (see, e.g. Padovani et al. 2009, 2018), much larger than those typical of the diffuse medium. Therefore, we compute synchrotron emissivities without accounting for any attenuation of the CRe spectrum.
We verified that both the effect of synchrotron selfabsorption (Ginzburg & Syrovatskii 1965) and the TsytovichRazin effect (Ginzburg & Syrovatskii 1964) can be neglected for the ISM under consideration.
All Tables
Ranges of energy E and spectral slope s for two valuesof B_{⊥} and for a set of frequencies.
Minimum and maximum values of the strength of the magnetic field in the plane perpendicular to the LOS i, , and its median value for the two cases of weak and strong field.
Medians of the ratio between the standard deviation of B_{⊥} and its median value, , of the brightness temperature spectral index, and of the polarisation fraction at a resolution of 6.7′ for the three POSs and the two cases of weak and strong field^{(a)}.
Medians of the brightness temperature spectral index for the three POSs and the two cases of weak and strong field at a resolution of 6.7′.
All Figures
Fig. 1 CRe spectrum according to Orlando (2018; green line) and Padovani et al. (2018; violet line). The inset shows the ratio of the spectra (O = Orlando, P = Padovani; solid black line). Data: Voyager 1 (orange squares; Cummings et al. 2016); Voyager 2 (grey squares; Stone et al. 2019); FermiLAT (magenta triangles; Ackermann et al. 2010); Pamela (redcircles; Adriani et al. 2011); AMS02 (blue squares; Aguilar et al. 2014). 

In the text 
Fig. 2 Spectral slope vs. energy of the CRe spectra by Orlando (2018; green line) and Padovani et al. (2018; violet line) in the energy range relevant for our study (100 MeV−50 GeV). The inset shows the difference Δs between the spectral slope of the two spectra. 

In the text 
Fig. 3 Left panel, upper plot: integrand of the specific emissivity, dε_{ν}∕dE, as a functionof the energy E, normalised to its maximum value and summed over the two polarisations (Eqs. (1)), computed at 100 MHz, 1 GHz, and 10 GHz (green, orange, and violet line, respectively), for B_{⊥} = 2 μG. The coloured area below each curve corresponds to 95% of the total emissivity. Right plot: Same as the upper plot, as a function of the CRe spectral slope s. Main plot: Spectral slope as a function of the energy for the CRe spectrum by Orlando (2018; black solid line). Coloured lines show the range of energies (and the corresponding spectral slopes) contributing to 95% of the specific emissivity. Right panel: same as left panel, but for B_{⊥} = 20 μG. 

In the text 
Fig. 4 Upper panel: brightness temperature T_{ν} as a functionof B_{⊥} computed atν = 115 and 189 MHz with the CRe spectra by Orlando (2018; green line) and Padovani et al. (2018; violet line) for a slab of thickness 1 pc, with an angular resolution θ_{b} = 4′. Lower panel:brightness temperature spectral index β as a functionof B_{⊥} (same colour coding as in the upper panel). The hatched region highlights typical values of β at intermediate and high latitudes (Mozdzen et al. 2017). 

In the text 
Fig. 5 Brightness temperature maps (colour scale) computed with the Orlando (2018) spectrum, at ν = 130 MHz and with resolution 6.7′ for the three POSs (three rows, labelled xy, yz, and zx) of the weak and strong field cases (left and right column, respectively). 

In the text 
Fig. 6 Bivariate distribution of the ratio between the standard deviation of B_{⊥} and its median value, μ, and the brightness temperature spectral index, β, computed in the frequency range 115−189 MHz for each LOS for the weak and strong field case (upper and lower panel, respectively), at a resolution of 6.7′. The POSs are identified by the three different colours displayed in the legend. Isodensity contours are plotted at 5, 30, 50, and 75% levels. The histograms above and to the right of the main plots show the marginal distribution of the two quantities. The median values are marked by horizontal and vertical lines and are listed in Table 3. 

In the text 
Fig. 7 Bivariate distribution of the ratio between the standard deviation of B_{⊥} and its median value, μ, and the polarisation fraction, p, computed at 130 MHz for each LOS for the weak and strong field cases (upper and lower panel, respectively), at a resolution of 6.7′. The POSs are identified by the three different colours displayed in the legend. Isodensity contours are plotted at 5, 30, 50, and 75% levels. The histograms above and to the right of the main plots show the marginal distribution of the two quantities. The median values are marked by horizontal and vertical lines and are listed in Table 3. 

In the text 
Fig. 8 Histograms of the spectral index β (upper panels) and polarisation fraction p (lower panels) for the weak and strong field cases (left and right column, respectively). Solid (empty) histograms refer to a resolution of 6.7′ (5°). The POSs are identified by the three different colours displayed in the legend. 

In the text 
Fig. 9 Lookup plot with which to estimate the average value of B_{⊥} along a LOS, ⟨B_{⊥}⟩, for a given frequency ν and brightness temperature spectral index β assuming the CRe spectrum by Orlando (2018). Black and white isocontours show specific values of ⟨B_{⊥}⟩ with labels in μG. Yellow symbols indicate estimates of ⟨B_{⊥}⟩ obtained from the values of in the frequency ranges 115−189 MHz, 467−672 MHz, and 833−1200 MHz (labelled as , , and , respectively)at a common resolution of 6.7′ for the different POSs of the two simulation snapshots (circles: POS yz in the strongfield case; squares: POS yz in the weak field case; diamonds: POS xy and zx in the weak field case; triangles: POS xy and zx in the strong field case). Rectangles at the bottom of the plot show the frequency ranges of the low and high band antenna of LOFAR (LBA and HBA, respectively), the low and mid frequency bands of SKA (SKA1Low and SKA1Mid, respectively), and SKA pathfinders and precursors: the Hydrogen Epoch of Reionisation Array (HERA); the Murchison Widefield Array (MWA); the Karoo Array Telescope (MeerKAT); and the Australian SKA Pathfinder (ASKAP). 

In the text 
Fig. A.1 Bivariate distribution of the ratio between the standard deviation of B_{⊥} and its median value, μ, and the brightness temperature spectral index, β, computed in the frequency ranges 115−189 MHz, 467−672 MHz, and 833−1200 MHz (labelled , , and , respectively) for each LOS for the weak and strong field case (left and right panel, respectively), at a resolution of 6.7′. The POSs are identified by the three different colours displayed in the legend. Isodensity contours are plotted at 50 and 80%. 

In the text 
Fig. B.1 Brightness temperature spectral index maps for the three POSs (three rows) of the weak and strong field case (left and right column, respectively) for a resolution of 6.7′. The three panels of each column share the same colour bar at the top. 

In the text 
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