The overestimation of equipartition magnetic field strengths from synchrotron emission using synthetically observed galaxies

Our high-resolution cosmological simulations study the ”nut galaxy”, a Milky Way-like system forming in the centre of an approximately spherical zoom region. These initial conditions (ICs) were originally presented by Powell et al. (2011). The galaxy formation and evolution of the nut galaxy has been studied in detail over the last decade (Geen et al., 2012; Tillson et al., 2015; Kimm et al., 2017; Martin-Alvarez et al., 2018; Rodríguez Montero et al., 2023).

The nut galaxy is simulated in a cubic box with 12.5 comoving Mpc (cMpc) per side, with a spherical zoom region approximately $4.5$ cMpc across. The nut galaxy resides in a dark matter halo with virial mass $M_{\text{vir}}(z=0)\simeq 5\cdot 10^{11}\mathrm{M}_{\odot}$. The mass resolution for the dark matter ($m_{\text{DM}}$) and stellar components ($m_{*}$) are $m_{\text{DM}}\simeq 5\cdot 10^{4}\mathrm{M}_{\odot}$ and $m_{*}\simeq 5\cdot 10^{3}\mathrm{M}_{\odot}$, respectively. We allow AMR spatial refinement down to a minimum physical cell size $\sim 10$ pc (equivalent to $\sim 5$ pc for a particle-like treatment). At this resolution, nut showcases turbulent dynamo amplification (Martin-Alvarez et al., 2018, 2022a). All our simulations are generated assuming a WMAP5 cosmology (Dunkley et al., 2009).

To model the formation of a realistic galaxy, we include additional physical prescriptions briefly described in this section. We model gas cooling accounting for the metal content and thermodynamical properties of each gas cell. For temperatures above $10^{4}$ K we interpolate cloudy cooling tables (Ferland et al., 1998). Crucially for the formation of a realistic ISM, we also model cooling below $10^{4}$ K following Rosen & Bregman (1995). This specific modeling of gas cooling at low temperatures is necessary for a self-consistent treatment of the cold gas distribution within the ISM. We also include a uniform UV background, turned on at $z=10$, in order to model the impact of reionization (Haardt & Madau, 1996).

We model star formation through a magneto-thermo-turbulent (MTT) star formation prescription Kimm et al. (2017); Trebitsch et al. (2017); Martin-Alvarez et al. (2020). It allows star formation in cells at the highest level of refinement in which the gravitational pull is larger than the MTT pressure support, and employing a local star formation efficiency (Federrath & Klessen, 2012). We refer to Appendix B of Martin-Alvarez et al. (2020) for details on the implementation of MTT star formation.

All stellar particles in our simulation are allowed to generate supernova (SN) events following the mechanical stellar feedback prescription presented by Kimm & Cen (2014); Kimm et al. (2015). Each stellar particle injects mass, momentum and energy back to its hosting cell and its immediate neighbours. For further details on the models governing the number and behavior of these SN events, we refer to Martin-Alvarez et al. (2024).

In Figure 1, we offer a face-on view of two intrinsic properties of the simulated galaxy with ongoing star formation in dense gas clumps: the star formation rate surface density and the total gas surface number density. In this work, the galaxy is studied at redshift $z=0.95$. The system is a stable disk galaxy matching the properties of late-type spiral galaxies and has star formation predominantly taking place within a radius of 8 kpc.

We mainly focus on a model of magnetization where the galaxy is magnetized by ab-initio magnetic fields described by $B_{0}$, the comoving strength seeded uniformly at the start of the simulation ($z=499$). We focus on the $B_{0}\sim 10^{-11}$ G model (well-below the current Planck constraints of $10^{-9}$ (Planck Collaboration, 2015)). See Martin-Alvarez et al. (2024) for further details on this magnetized run of the galaxy as well as other $B_{0}$ models studied. We also have also reviewed the other models in Martin-Alvarez et al. (2024), and analyzed in detail a model where magnetic fields are sourced by magnetized SN feedback (i.e., injecting 1% of each SN in the form of magnetic energy; MBinj). While the results are not included here, they confirm the main findings of this work.

In order to quantitatively investigate the accurate recovery of magnetic field measurements from radio intensities using equations that assume CR equipartition, we test our analysis throughout by imposing energy equipartition between the magnetic field and CRs. We consider equipartition at the scale of individual cells, which typically corresponds to 10 - 20 pc in the ISM of the simulated galaxy. We also test the impact on the synchrotron intensities of two other cases of equipartition: equiparting at the scale of 100 pc and equiparting at 10 - 20 pc followed by the diffusion of the CR electrons out with a diffusion scale length of $\sim 250$ pc or a diffusion timescale of 100 Myr.

To compute equipartition at a given scale $\mathcal{L}$, we employ a similar approach to the ‘small scale turbulence computation’ employed by Martin-Alvarez et al. (2022a), described in their Section 2.3. For each cell, and for each separate component of the magnetic field, we measure its coherent component as its average in a sphere of radius $0.5\mathcal{L}_{\text{scale}}$. In this calculation we employ the full AMR grid reconstruction, subdividing cells intersecting the boundary of the sphere down to the highest level of refinement to capture their correct contribution down to our resolution limit. This per-scale computation is stored for each AMR cell, and the resulting magnetic fields $(B_{x},B_{y},B_{z})_{\mathcal{L}}$ are employed to compute the magnetic energy used for establishing equipartition at a scale $\mathcal{L}$.

An alternative scenario to consider is that where equipartition is only held at the smallest scales of the galaxy, and the CRs distribution is then smoothed by their diffusion to larger scales. To approximate this scenario, we compute the density of energy for CRs equipartition employing a similar method to that above. First, we compute the equipartition CR energy density on a per-cell level. Then, we compute its diffusive re-distribution assuming it follows the behaviour of CRs under isotropic diffusion. We assume a diffusion coefficient $\kappa=3\cdot 10^{25}\mathrm{cm}^{2}/\mathrm{s}$ (Seta & Beck, 2019), and diffusion on a timescale of 100 Myr. The resulting half-concentration radius is $\sim 120\,\mathrm{pc}$ (and $\mathcal{L}_{\text{scale}}\sim 240\,\mathrm{pc}$), where the final energy for equipartition with CRs in each cell receives contributions from other cells up to a distance of $\sim 1\,\mathrm{kpc}$. This energy is then store for each cell, and employed by ramses2polaris to establish diffused small-scale equipartition.

A critical consideration at this point is which component of CRs to consider in equipartition with the magnetic field energy. Throughout this work, we focus on the case of equipartition with the entire CR population (approximated by the protons, which dominate the CR energy budget) (Beck & Krause, 2005; Beck, 2015b; Seta & Beck, 2019). In Appendix A, we briefly consider the case of assumed equipartition between the CR electrons and the magnetic field, as employed by Reissl et al. (2019). We present the methodologies used in both cases below.

In order to probe how the equipartition assumption affects recovery of the magnetic field strengths when equipartition is not manifestly true, we prescribe a smooth distribution of CR electrons motivated by modeling of the Milky Way (Drimmel & Spergel, 2001; Page et al., 2007; Sun et al., 2008; Reissl et al., 2019). In this second model of the CR electron distribution, the number density $n_{\text{CR}}$ is set following the CR1 model in Reissl et al. (2019):

We consider two cases for the magnitude of the CR density at the center of the galaxy, which sets the amplitude of the density distribution (Model Exp1 and Exp2). In the first case (Model Exp1), $n_{\text{CR,0}}=1.74\times 10^{-4}\,\text{e}^{-}\,\mathrm{cm}^{-3}$, corresponding to the number density of CR electrons in the center of the Milky Way. This is obtained by extrapolating the measured CR number density in the Solar neighbourhood as determined in Sun et al. (2008) and Reissl et al. (2019), based on GeV electron energy spectra from Barwick et al. (1998) and Grimani et al. (2002). We select $R_{\text{gal}}=8$ kpc and $h_{\text{gal}}=1$ kpc to adjust the distribution to our simulated galaxy. This scale radius is also roughly consistent with the extent of the radiating CRs determined for the observed galaxy IC 342 with which we compare throughout (Graeve & Beck, 1988). While some observational studies have hinted at larger scale heights from synchrotron emission (Krause, 2014; Krause et al., 2018), we note that once a minimum $h_{\text{gal}}$ is surpassed, the total intensity is approximately insensitive to the choice of vertical scale, and is instead modulated by the vertical scale of the magnetic field. This, and further details of the chosen scale height are discussed in Appendix B of Martin-Alvarez et al. (2024). In our second case (Model Exp2), we set $n_{\text{CR,0}}=1.74\times 10^{-6}\,\text{e}^{-}\,\mathrm{cm}^{-3}$, a factor of 10² lower than in Model Exp1. We keep all other values of the model prescribed CR distribution the same. We use this case to probe a lower regime of CR number densities. Such number densities are closer to the values yielded by the equipartition assumptions, but with the smoother distribution expected from a continued diffusion of CRs at galactic scales over the lifetime of a galaxy.

For each combination of CR distribution model and power-law index $p$, we generate synthetic radio observations corresponding to Stokes parameter ($I$, $Q$, and $U$) maps at a wavelength of 6.2 cm and at a full-resolution approximately matching that of the simulation. We focus our study on the total intensity $I$ maps. These intensity maps are synthetically observed assuming a distance of 3.5 Mpc from the galaxy to the observer. This is to reflect the distance of the galaxy IC 342 analyzed in Beck (2015a). To further match real observations, we account for beam convolution effects by convolving this intensity map with a 2D Gaussian kernel. We choose a kernel size corresponding to a full-width half-max of 25 arcseconds (corresponding to 0.417 kpc), the beam size of the radio intensity observations included in Beck (2015a).

We first discuss our primary focus of equipartition between the CR protons and magnetic field energy. The equation used predominantly in the literature to convert observed synchrotron intensities into equipartition magnetic field strength measurements (e.g., Fletcher et al., 2011; Lacki & Beck, 2013; Basu & Roy, 2013; Krause et al., 2018) was presented in BK05. This BK05 equation is derived by combining the formula for synchrotron intensity with the CR number density equations derived using the treatment and assumptions outlined in Section 3.1.1, between Equation 2 through Equation 7 with two differences. The first difference is their inclusion of the subdominant CR electrons in the energy equipartition. To obtain the BK05 equation, one would have to modify Equation 3 to be $\epsilon_{\text{CR}}\approx\epsilon_{\text{CR,p}}(1+K_{0}^{-1})=\epsilon_{B}$ where the inverse of the constant $K_{0}$ has been included to modulate the contribution of CR electrons and is defined in Equation 2. We emphasize that this altered assumption has a below percent level impact on the equipartition magnetic field estimate for the assumed $K_{0}$ value. The second difference between the CR density treatment used to obtain the BK05 equation and our treatment is that we explicitly integrate between the minimum and maximum kinetic energies defined by the Lorentz factors in Equation 1. We discuss the motivations and implications of the minimum energy chosen in Section 6.1. In short, this term accounts for our assumption on the shape of the CR electron energy spectrum. We rewrite the BK05 equation as:

where $I_{\lambda}$ is the intensity measured at the given wavelength, $i$ is the inclination angle of the galaxy, $E_{\rm p}$ is the rest energy of the proton, $E_{\rm e}$ is the rest energy of the electron, $L$ is the approximate vertical thickness of the emitting galaxy disk, and $K_{0}$ is the ratio of proton to electron number densities at an energy $E_{0}$ (see Equation 2). We note that the power-law index $p$ is related to the synchrotron spectral index by $\alpha=(p-1)/2$.

We modify this equation to reflect equipartition between only the CR protons and the magnetic field and to reflect the energy bounds of the integral with $E_{\text{min}}$ and $E_{\text{max}}$ (see Equation 1). We do this by combining the modified equation we derived for CR electron density (Equation 8) with the synchrotron intensity formula found in equation A15 of BK05, which is largely identical to earlier synchrotron formulae found in works such as Pacholczyk (1970). This formula relates the intensity at a particular frequency to the number density of CR electrons, the magnetic field strength perpendicular to the line of sight, the frequency of observation, and the power-law index. Combining the formula with our CR electron density equation results in the following equation for the equipartition magnetic field strength:

To continue our measurement of magnetic field strength maps using Equation 12, we must set the values of the following constants. We set $\lambda$ = 6.2 cm, the wavelength corresponding to our synthetic intensity observation. We set inclination $i$ = 0 since in our analysis we view and measure the galaxy face-on. Note that small inclinations have a negligible impact on the overall synthetic intensity maps (Martin-Alvarez et al., 2024). We assume a $K_{0}$ of 100, motivated by the ratios of CR protons and electrons estimated for supernovae remnant shock fronts as well as by observed Galactic CR data (Bell, 1978). This number has been assumed in other works including but not limited to Bell (2004); Beck & Krause (2005); Basu & Roy (2013); Beck (2015b, a). We assume a full-thickness of the disk contributing to the intensity of $L$ = 1 kpc based on the thickness of the gas, stellar component, and magnetic field of this simulated galaxy (Martin-Alvarez et al., 2020, 2024). This value also matches the assumed thickness for similar spiral galaxies such as IC 342 (Beck, 2015a).

For our secondary case of equipartition only between the radiating CR electrons and the magnetic field (Appendix A), we begin with the equation presented in Reissl et al. (2019) that relates a magnetic field to the synchrotron emission produced when given a CR density. This equation governs how ramses2polaris and polaris calculate intensities from the intrinsic magnetic fields in the simulation. It is consistent up to the factor of $E_{\rm min}^{1-p}$ (discussed in Section 6.1) with the synchrotron emission formulae presented in Blumenthal & Gould (1970), Leung et al. (2011), and Pandya et al. (2016). The coefficient of total emission is:

where we have rewritten the equations to be explicitly expressed in terms of the CR electron kinetic energies. Here, $\theta$ is the angle between the magnetic field and the observer’s line-of-sight, and $\lambda_{\rm c}$ is the critical wavelength corresponding to the critical frequency $\nu_{\rm c}$. It is defined as

We combine this equation with the electron equipartition CR number density found in Equation 9 to derive the electron equipartition magnetic field in the plane of the disk galaxy at inclination $i$:

An important assumption made in all the observational analyses that follow these considerations is that $j_{\lambda}$ can be approximated as $\frac{I_{\lambda}}{L}$ where $L$ is the approximate thickness of the galaxy disk that dominates the emission. As with CR proton equipartition, we set $L$ = 1 kpc, $\lambda$ = 6.2 cm, and $i=0$ for the face-on galaxy.

In order to probe the validity of equipartition and its recovery of the intrinsic magnetic field profile, it is first instructive to consider that intrinsic profile and the information it summarizes about the galaxy. Depending on how the profiling is performed, radial profiles can probe different regions and phases of the galaxy. This is determined by the weighting employed for the resolution elements, which all feature different masses and volumes.

We explore three alternative weightings in Figure 2, each summarizing different aspects of the magnetic field distribution across the galaxy, with important interrelations between them. The first two are the standard density-weighted (using total gas density) and volume-weighted spherically-averaged profiles of the magnetic field, shown with dotted and dashed curves respectively. These two profiles were azimuthally averaged within concentric cylindrical annuli of 1 kpc vertical thickness to focus our measurement on the disk of the galaxy, where we align the $z$ component with the angular momentum vector of the disk. We note that volume-weighted profiles shift to lower magnetic field strength values with increasing vertical thickness included, while the density-weighted profile remains dominated by the high-density disk. The 1 kpc thickness is also consistent with the disk thickness assumed in our inference of the magnetic field from our synthetic observations (see Section 4.1), and encompasses well the gas thick disk (Martin-Alvarez et al., 2020). The third profile (solid curve) is the surface area-weighted azimuthal average of the magnetic field strength, measured in density-weighted projections into the 2D disk plane. This procedure is meant to reflect how the intensity profiles would be created observationally.

As expected, the density-weighted profile (Figure 2, dotted line) is noticeably higher and more irregular than the other two profiles. It typically features field strengths of $\sim 4\times 10^{-5}$ G, and reaches peak values of $10^{-4}$ G. The volume-weighted profile (dashed line) is relatively flat at all radii and is consistently weaker than the projected profile (solid line) at all radii. It typically features values of a few $10^{-6}$ G, with important contributions from the volume outside of the dense galactic disk. Note that the projected profile matches the density-weighted profile at the smallest radii where it is dominated by its line-of-sight density weighting and little area, allowing only for small surface variations. Conversely, at large radii, the projected profile matches the volume-weighted profile. This is due to the density variations being larger, with a vast dynamic range within each projected annuli.

The projected profile provides the most faithful interpretable approximation to what is probed by the “observed” synchrotron emission and we consider it as the “intrinsic” magnetic field strength profile moving forward.

Combining the intrinsic magnetic field from the simulation with a model for the distribution of CR electrons, we generate maps of synthetically observed synchrotron intensity including beam convolution effects (see Section 3). We examine the intensity produced by five cases of CR electron distribution. For each case and for a range of power-law indices $p$ describing the CR energy spectrum, we can produce synchrotron intensity maps at 6.2 cm (see Section 3). We show the resulting intensity maps obtained with $p=2.9$ for the five cases in Figure 3 along with the observations of a representative spiral galaxy, IC 342 by Beck (2015a), for qualitative comparison (upper leftmost panel). We compare to this galaxy throughout as it is nearby, nearly face-on, and is well-studied in the radio with high angular resolution (Beck, 2015a). Note that for IC 342 we show the observed total intensity, which includes a small contribution from the thermal intensity ($<15\%$). The first two CR electron cases (see the rightmost upper two panels of Figure 3) are the Model Exp cases of high and low prescribed CR electron distribution discussed in Section 3.2. The last three cases (see the lower three panels of Figure 3) are three versions of Model pEq, imposed equipartition between the CR protons and the magnetic field which results in the CR electron distribution derived in Section 3.1.1. The three equipartition cases include equipartition at the scale of 10 pc, equipartition at the scale of 100 pc, and equipartition at the scale of 10 pc with subsequent diffusion of the CRs out to a diffusion scale of $\sim$250 pc (see Section 3.1.1). The imposed equipartition cases result in clumpier intensity maps with regions of drastically lower intensity compared to the smoothness of the two prescribed CR distribution cases. The diffusion case has slightly smoother clumps of intensity, but even the large kernel of diffusion does not reproduce the smoothness seen in the prescribed CR distribution cases and the IC 342 data.

In Figure 4, we present the radial intensity profiles for our five CR electron cases, varying the power-law index between $p=2.2$ and $p=3.0$. The two cases of Milky Way-like prescribed CR electron distribution (upper two panels) result in similarly-shaped smooth and shallow curves, shifted to significantly lower intensities for higher power-law index $p$ values. This extreme variation arises from imposing a fixed $n_{\rm CR}$ distribution and varying $p$ in our synthetic intensity observation alone. For shallower, power-law slopes (i.e., lower $p$), the CR energy spectrum features a higher proportion of higher energy CR electrons that provide larger contributions to the intensity. In the last three cases of imposed equipartition (bottom panel), the profiles feature only small variations with $p$. This is because $p$ affects both the $n_{\rm CR}$ distribution as well as the synthetic emissivities. Notably, the three equipartition intensity profiles are overall jagged with large troughs of low intensity. The cases with equipartition at 100 pc and diffusion with a scale of 250 pc have slightly less severe deviations from their average, now approximately centered around the lower end of the 10 pc case. We also tested equipartition on scales of 1 kpc, which reproduced the behaviour of the other cases and resulted in even lower intensities (not shown). In other words, the price for smoothing the distributions of CR electrons under equipartition is even lower intensities.

On all three panels, we overplot the observed radial nonthermal intensity profile of the galaxy IC 342 as presented in Beck (2015a). We include the nonthermal intensity profile out to only 7 kpc, beyond which the spectral index for the separation of thermal and nonthermal components is not well determined (Beck, 2015a). We note that IC 342 is not entirely face-on and has an inclination angle of $i=31\deg$ which is accounted for in the profile. The observed profile of IC 342 is well matched both in terms of shape and intensity values by Model Exp2 ($n_{0}=1.74\times 10^{-6}\,{\rm cm}^{-3}$ and $p=2.4$), and especially by Model Exp1 ($n_{0}=1.74\times 10^{-4}\,{\rm cm}^{-3}$ and $p=2.9$), illustrating the degeneracy between $n_{\rm CR}$ and $p$ in the resulting intensities. The Model Exp1 ($n_{0}$, $p$) values provide a more realistic match to IC 342, as they reproduce the local Milky Way CR electron properties (see Section 3.2). For the equipartition cases, the observed profile is in noticeable disagreement with the steep intensity troughs and overall low intensities seen in the three imposed cases studied. This suggests that true equipartition at scales of 10 pc, 100 pc, 1 kpc (not shown) and even with diffusion out to 250 pc, is not able to produce the smooth intensity distributions of observed galactic synchrotron profiles. This result is consistent with the findings of Stepanov et al. (2014), Seta et al. (2018), and Seta & Beck (2019), indicating that equipartition does not hold on scales below 100 pc and even 1 kpc. It also reinforces the claim by Stepanov et al. (2014) that equipartition leads to fluctuations in the synchrotron intensity far exceeding what is observed, suggesting smoother and less strongly varying CR density distributions in such galaxies. We note that depending on the details of the treatment of the CR electron energy spectrum in the equipartition assumption, the intensities have some freedom to move to higher values (closer to the IC 342 observation). However, the favored expectation is that in reality imposing equipartition would result in even lower intensities that are even more discrepant with observations. This is discussed further in Section 6.1.

In Figure 5, we summarize the intensity information contained in the profiles as functions of the CR electron density $n_{\text{CR}}$ and power-law index $p$. Each vertical series of squares in Figure 5 displays the magnitude of the galaxy’s intensity profiles for the two Model Exp cases. The four vertical squares correspond to four wide radial bins between 0 and 10 kpc within which the intensity profile is averaged, with the color of each square corresponding to the average intensity within that radial bin. The $n_{\text{CR}}$ value for each radial bin (i.e., each square) is determined by averaging its values in a similar manner. As $n_{\text{CR}}$ decreases with radius, the top square of a vertical series corresponds to the innermost radial bin, and the bottom square to the outermost. The vertical series of circles displays the same information but for Model pEq(10pc), where we impose CR equipartition at a scale of 10 pc. To obtain the values of $n_{\text{CR}}$ in radial bins for Model pEq(10pc), we generate maps of the resulting $n_{\text{CR}}$ from equipartition, compute their radial profiles, and then average their number density using the same radial bins as described above. Throughout the rest of this work, we focus on the Model pEq(10pc), which leads to the closest intensities to the observations. We stress that the qualitative differences between the equipartition cases do not meaningfully impact our later results.

The smooth background colors in Figure 5 display the theoretical prediction for the intensity in this parameter space by assuming a vertical ISM slab with uniform magnetic field perpendicular to the line-of-sight, and calculating Equation 13 for each point in $n_{\text{CR}}$ and $p$ space. The uniform magnetic field employed for these predictions ($\sim 4\mu$G), corresponding to the average intrinsic projected magnetic field of the simulated galaxy within 8 kpc ($B$, delineated in Figure 2 as the horizontal solid line). The near perfect agreement between the theoretical background and the square points from the simulated galaxy intensity is due to the smooth and shallowly decreasing nature of the intensity profiles in the cases of prescribed equipartition (Model Exp1 and Model Exp2) (see Figure 4). We include a white dashed curve showing the theoretical expectation for the unique $n_{\text{CR}}$ values defined by CR equipartition in Equation 8 as a function of $p$, and a fixed intrinsic $B$ (set to the same average projected $B$ within 8 kpc).

The black star in Figure 5 indicates estimates of the Milky Way $p$ and $n_{\text{CR}}$ in the Solar vicinity (Sun et al., 2008). Note that the Milky Way might possess an intrinsic magnetic field slightly different from the value we employ here from our simulated galaxy, and is included in the figure to provide observational context for the expected number density and power-law index of CRs in the ISM of galaxies. The Milky Way total magnetic field strength unperturbed by the draping around the heliosphere has been modeled to be around 2.9 $\mu$G using a combination of MHD simulations of the solar wind and local ISM and observations of NASA’s Interstellar Boundary Explorer, and matched to Voyager data (Zirnstein et al., 2016). This magnetic field value, if used to compute the background and white dashed line expected by equipartition, would actually shift them slightly such that equipartition corresponds to even lower intensities.

Both the theoretical background and our synthetic intensity measurements (Figure 5) demonstrate how for a given intrinsic magnetic field, there are sizeable regions in ($n_{\text{CR}}$, $p$) parameter space that are effectively ruled out due to their drastic deviation from the observed radio intensities in comparable systems. E.g., are the 6.2 cm nonthermal intensities of IC 342 at $\sim 10^{-6}$ Jy/arcsec^-2 (Beck, 2015a), M51 at $\sim 10^{-4}$ Jy/arcsec^-2 (Fletcher et al., 2011), and NGC 6946 at $\sim 10^{-3}$ Jy/arcsec^-2 (Beck, 2007). Even for our reasonably strong magnetic field at $\sim 4$ $\mu$G (see Figure 2 and Martin-Alvarez et al. (2020)), the intensities predicted by the imposed equipartition case (circular points) are at least an order of magnitude lower than those observed in similar galaxies. We do not display points corresponding to the 100 pc, 1 kpc and diffusion out to 250 pc equipartition cases, for the sake of readability, but we note their intensities are even lower than the 10 pc case. As the intrinsic magnetic field is reasonably large, these results suggest that CR equipartition produces CR densities that are not sufficient to generate the radio intensities observed.

The disagreement between observations and the intensities resulting from imposing equipartition as well as the shape of their radial profiles suggest that this equipartition scenario (at a range of scales) is disfavored by both theoretical predictions and synthetic observations of our simulations. This finding complements the claims of Seta et al. (2018) and Seta & Beck (2019) which assert that CR energy densities and magnetic field energy densities are statistically independent and uncorrelated at scales smaller than approximately 100 pc, making it unlikely for equipartition to be valid at smaller scales. We have shown further that even at scales of 100 pc and above, or even in a diffusion out to 250 pc scenario, equipartition is unable to match observed intensities. As a result, the assumption of CR energy equipartition to compute magnetic field strengths from the observed intensities is likely inaccurate, and could lead to over or underestimation of the magnetic field strengths. We review this in the next section.

In this section, we examine the implications of applying the equipartition assumptions when the observed galaxy does not fulfill such an assumption. Specifically, we focus on its impact on the estimation of magnetic field strengths (perpendicular to the line-of-sight), studied as a function of the intrinsic CR distribution, the CR power-law index $p$, and the different ISM environments of the galaxy. Using the synthetically observed intensity maps discussed in Section 5.2 along with Equation 12, we obtain inferred magnetic field strength maps for each case of prescribed CR distribution and imposed equipartition CR distribution, varying the power-law index $p$ across our studied range. Figure 6 displays the azimuthally-averaged radial profiles obtained from these inferred magnetic field strength maps, alongside the intrinsic magnetic field profiles discussed in Section 5.1.

The 10 pc imposed equipartition case (third row) serves as a best-case scenario test for reconstructing the intrinsic magnetic field, as it explores the situation when the equipartition assumption is exactly true. As expected, reversing the assumption employed to generate the emission provides a reasonable recovery of the intrinsic magnetic field. The accuracy of this recovery is made more evident in the second and third panels of the third row which show the probability distribution functions of the magnetic field strength pixel values that contribute to the profile in four large radial bins. These distributions are almost entirely overlapping for each corresponding radial bin. The slight overestimation by the equipartition magnetic field strength compared to the intrinsic projected field strength is expected due to the non-linear relationship between synchrotron intensity $I_{\lambda}$ and $B_{\rm tot,\perp}$, modulated by line-of-sight fluctuations in $B_{\rm tot,\perp}$ (Seta & Beck, 2019; Beck, 2015b). Another assumption that contributes to the lack of exact recovery is the assumption of the thickness of the emitting region $L$, assumed to be a constant 1 kpc throughout. This nevertheless close recovery indicates that the magnetic field probed by synchrotron emission can be physically interpreted as the line-of-sight density-weighted magnetic field. This result holds despite, and independently of, the demonstrated failure of the equipartition assumption in generating realistic synchrotron intensities.

The upper rows of Figure 6 display our intrinsic CR configurations following a smooth distribution of the exponential form (Equation 10), capable of matching the observed intensity profiles for realistic $n_{\rm CR}$ values. However, such CR distributions face significant overestimation of the magnetic field strength with respect to the intrinsic projected profile due to incorrectly applying the equipartition assumption (top and central rows of Figure 6). This overestimation strongly depends on the assumed $p$. The rightmost panels of these rows show an inferred magnetic field with considerably narrow distributions in the four radial bins that does not match the spread of the intrinsic magnetic field (central column). Synchrotron emission is the combined effect of the CR electron density, CR spectral energy distribution, and the magnetic field configuration. The overestimation found when the equipartition assumption is applied to infer the magnetic field is driven by the fact that the same emission can be generated by multiple combinations of magnetic field strengths and CR electron densities. The equipartition assumption simply fixes them to a specific, but not necessarily true, combination. Assuming equipartition when it is not valid results, not in a simple offset in the magnetic field, but in a different profile shape altogether. Equipartition will always result in a magnetic field strength profile proportional to a specific power of the intensity profile for a constant assumed power-law index (see Equation 3). Therefore depending on the details of the CRs distribution, this assumption may limit the ability of observations to recover important features of the magnetic field across different parts of galaxies.

The observationally inferred total magnetic field strength in IC 342 using the equipartition assumption is 15 $\mu$G averaged within the inner $\sim$3 kpc and 13 $\mu$G within $\sim$6 kpc (Beck, 2015a). Beck (2015a) also indicates that these total field strengths are weaker than the spiral galaxy NGC 6946. These strengths are significantly higher than the magnetic fields intrinsic to our simulated galaxy, and higher than the field inferred in the case of true equipartition. However, the magnetic field strengths measured in IC 342 and NGC 6946 are consistent with some of the magnetic field strength profiles inferred assuming equipartition in our cases of prescribed smooth exponential CR electron distributions (where equipartition is manifestly not true). As a result, for underlying smooth CR distributions that do not fulfill equipartition, applying this assumption to infer a magnetic field strength can lead to measurements that match those observed, but which are actually revealed to be significantly overestimated when compared to the intrinsic magnetic field of the system.

We summarize the overestimation (or underestimation) of the inferred galactic magnetic field strengths (from the equipartition assumption) as a function of the power-law index $p$ and the CR electron densities present in the galaxy in Figure 7. The colors on the plot indicate the ratio of the inferred to intrinsic magnetic field $B_{\rm inf}$/$B_{\rm int}$, to quantitatively illustrate how the overestimation or underestimation varies with $p$ and $n_{\text{CR}}$. As described for Figure 5 in Section 5.2, the square and circle points represent radially binned profile values for the two cases of Model Exp and Model pEq(10pc), respectively. In this figure, the profile that is binned in four wide radial bins is the ratio of the inferred to intrinsic profiles. The horizontal and vertical positions of the points are the same as in Figure 5. The background colors in Figure 7 are the theoretical expectation for $B_{\rm inf}$/$B_{\rm int}$ for a given $p$ and $n_{\text{CR}}$. This ratio is determined using Equation 13, and Equation 11 in a vertical ISM slab 1 kpc thick with a uniform intrinsic $B$. As in Figure 5, the white line represents the unique $n_{\text{CR}}$ values defined by CR equipartition in Equation 8 for a set of $p$ values and a given intrinsic $B$. The intrinsic $B$ for the theoretical background and equipartition line are set to be the average intrinsic projected magnetic field of the simulated galaxy within 8 kpc (the straight solid lines in Figure 6).

As expected, the theoretical equipartition line exactly traces the theoretical $B_{\rm inf}$/$B_{\rm int}$ of unity, at the center of the white-colored region, as CR electron densities equal to the values necessitated by CR equipartition will exactly recover the provided $B$ field. The same happens for the simulation points with imposed equipartition (circles; see also Figure 6). Both the background prediction and the Model Exp measurements (square points) show large regions of parameter space far from true equipartition, leading to overestimation of the $B$ field by factors of 10 or even 50. This largest overestimation can occur even when the CR densities are set to the values modeled for the Milky Way, particularly at low $p$ and at large radii (lowest square points in the vertical series). For higher $p$ and CR electron densities comparable to those modelled locally for the Milky Way, assuming equipartition can result in an overestimation of $\sim$ 2 or 3.

We explore how this overestimation (or underestimation) of the magnetic field strength from the application of the equipartition assumption varies spatially across different regions of the galaxy in Figure 8. We show a selection of three representative power-law indices, and our three main CR distribution models. The figure shows maps of $B_{\rm inf}$/$B_{\rm int}$ (where the intrinsic $B$ maps are convolved in the same manner as the intensity maps, see Section 3). To understand how these ratios relate to the emitted intensity, we overlay on these maps contours of synchrotron intensity.

In the top and central rows, the highest overestimation occurs in lower intensity inter-arm regions. Notably, some of the regions emitting the highest intensities (which also correspond to dense clumps with the highest intrinsic magnetic fields; $B\sim 10^{-4}$ G) either do not overestimate (first row and second row, first two panels) or even underestimate the $B$ field (second row, third column). This effect of the intrinsic $B$ in Equation 12 is discussed further in Section 5.3.1.

For the case of true equipartition, $B_{\rm inf}$/$B_{\rm int}$ is near unity in all regions, as expected. Some inner regions have the magnetic field strength only slightly underestimated whereas in some outer regions it is only slightly overestimated. Minor deviations are attributed to line-of-sight fluctuations in the magnetic field strength.

Figure 8 illustrates how, if the distribution of CRs in galaxies is akin to the smooth exponential modeled for the Milky Way (Equation 10), then star-forming clumps and spiral arms in general are significantly less overestimated (and even underestimated) than low intensity inter-arm regions, where this may reach up to two dex overestimation (depending on $p$). Extreme over- and underestimation could potentially be alleviated by assuming spatially variable $p$, motivated by CR electron energy losses: lower in dense and star-forming regions, and higher in the inter-arms. This is further discussed in Section 6.2.

We discuss the treatment of the spectral shape of the CR energy spectrum, and its bounding energies. In this work, we assumed one of the most common assumptions employed to extract information from synchrotron observations: that the energy spectrum of CR electrons is well-described by a power-law (Equation 5). In their integration of this spectrum, BK05 implicitly take its lower bound to be $\gamma_{\rm min}=2$ (i.e., $E_{\rm min}=E_{\rm e}$). Here, we set the lower bound on our energy integral to be $\gamma_{\rm min}=5$ (Equation 1; i.e., $E_{\rm min}=4E_{\rm e}$).

This minimum kinetic energy intervenes in two parts of our analysis, the synchrotron intensity calculated with polaris, and the equipartition equation for the CR electron density. The intensity is negligibly affected by this choice of $E_{\rm min}$, as polaris corrects for this minimum energy with an additional factor $E_{\rm min}^{1-p}$ ($\gamma_{\rm min}^{1-p}$ in their notation; see Appendix A in Reissl et al. (2019)). Intensities at $\lambda=6.2\,\mathrm{cm}$ are dominated by the contribution of higher energy electrons due to $\gamma_{\rm min}^{2}\ll\lambda/\lambda_{\rm c}$ for $\gamma_{\rm min}=5$ and below (Reissl et al., 2019; Pandya et al., 2016).

The shape of the CR electron energy spectrum is significantly more important for determining the CR electron density resulting from the energy equipartition between the CR protons and the magnetic field (see Equation 5 and Section 3.1.1). As the number density of electrons matters in the derivation of the equipartition formulae, it is important to accurately capture the energy contained even within the lower energy end of the CR electron energy spectrum. In Figure 10, we show various possible treatments of the energy spectrum $N_{\rm CR}=\frac{dn_{\rm CR}}{dE}$ displayed as a function of the kinetic energy ($E$) to motivate our fiducial choices. The first is the commonly assumed (e.g., BK05 or this work) power-law form of the energy spectrum. A power-law tail towards high energies is highly motivated for such nonthermal distributions, as they are observed in synchrotron-emitting plasmas, solar winds, and particle-in-cell simulations of reconnection and shock acceleration (Pandya et al., 2016).

However, at lower energies, the abrupt transition from abundant CR electrons to little or none below kinetic energy $E=E_{\rm min}\sim E_{\rm e}$ might be unrealistic. Instead, a rollover to a flatter distribution at lower electron energies is expected due to several mechanisms. If not in the vicinity of the CR acceleration sites, multiple energy loss mechanisms (e.g., Coulomb losses, ionization losses) will lead to shallower slopes at the low energy end of the spectrum (Ip & Axford, 1985). This end might already be flatter after CR electrons are accelerated and propagated through multiple shocks from supernova remnants and a turbulent ISM environment (Bell, 1978; Korpi et al., 1999). Furthermore, these low energy electrons have a lower probability to escape from acceleration sites such as supernovae remnants, leading to a deficit with respect to the power-law scaling (Ohira et al., 2012).

A $\kappa$ distribution function-spectrum (Vasyliunas, 1968) addresses these problems and captures such a turn-over into a thermal distribution at low energies while maintaining the high $E$ power-law behaviour. This distribution can be expressed (Xiao, 2006; Pandya et al., 2016) as follows:

We also include a model from Langner et al. (2001) that extends to low CR electron energies. This model is based on Galactic polar synchrotron data and was constrained to match high energy electron energy spectrum observations while exceeding the solar modulated Pioneer 10 intensities from the outer heliosphere.

We further include in Figure 10 observational data for the the CR electron energy spectrum. At the low energy end, we show measurements from Voyager 1 spacecraft between the end of 2012 and 2015, after the spacecraft crossed the heliopause (TET and HET telescopes as circles and triangles, respectively; Cummings et al., 2016). For the high energy end where modulation from the heliopause is minimal, data points correspond to the Alpha Magnetic Spectrometer (AMS) mission on the International Space Station, (Aguilar et al. 2014; see also Cummings et al. (2016)).

We employ this observational data to fit the $\kappa$ model to the Voyager and AMS data in log-space, resulting to $p=1.8$ and $w=2.1$. We fix the displayed power-law to this $p$ index to match the $\kappa$ fit. Both are normalized to the same $N_{\rm CR}$ value at $E_{\rm max}$, well in the power-law regime. A direct fit of the power-law to the observed data results would result in unrealistic $p<2$ (see BK05 and Seta & Beck (2019) for details on the implications of lower power-law indices), while still failing to capture the high-energy regime.

To illustrate how the $\kappa$ with the low energies turn-over better reproduces the observational data, we show the log-space residuals highlighting the distance between the different models and the observational data in the lower left panel of Figure 10. This panel shows how the power-law describes well the high energy behavior but fails to capture the lower energy end, significantly overestimating the number density of CR electrons.

In the derivations of the equipartition CR electron number density, the integral of the CR electron energy spectrum shape at lower energies has significant impact. In the upper right panel of Figure 10, we consider $n_{\rm CR,X}$, the integral of each $N_{\rm CR}=\frac{dn_{\rm CR}}{dE}$ curve between $E_{\rm min}$ and an $E_{\rm max}=300E_{\rm e}$. We vary the $p$ index to review how the integrals would extrapolate for spectra that have different power-law slopes, but maintain the normalization fixed at $E_{\rm max}$ as well as the $w$ fixed at $w_{\rm fit}$ for simplicity. While the units of $n_{\rm CR}$ are arbitrary here, the ratio of the curves indicates the factor by which these different spectra would change the equipartition CR electron densities, and therefore the intensities resulting from our equipartition models. Importantly, the power-law integrated down to $E_{\rm min}=E_{\rm e}$ leads to significantly higher $n_{\rm CR}$ than the $\kappa$ model integrated to the same $E_{\rm min}$. This is due to the power-law assuming an increasingly large number of electrons all the way down to that minimum energy. We emphasize that the $\kappa$ distribution is a significantly better fit to the lower energy data from direct CR spectrum observations, and that the spectrum is also theoretically expected to deviate from a power-law towards low energies.

By integrating the power-law down to a $E_{\rm min}=4E_{\rm e}$, we conduct a conservative treatment of the uncertain energy spectrum integral, as it follows exactly the mean of the power-law and the $\kappa$ model (upper right panel, Figure 10). This is emphasized in the lower right panel of Figure 10 where we take a ratio of those two extreme cases to our power-law model with $E_{\rm min}=4E_{\rm e}$. This plot indicates the factors by which our CR electron number densities (and therefore the synchrotron intensities) would be modified for our equipartition models (Model pEq) if a power-law down to $E_{\rm min}=E_{\rm e}$ or a $\kappa$ model fit to the Milky Way observational data were assumed instead. This factor is roughly an order of magnitude higher or lower for those two models respectively. In other words, if the electron spectrum follows the $\kappa$ model fit, the equipartition intensities would be around a factor of 5 - 15 lower in the lowest panel of Figure 4, bringing them to even more tension with the observed intensities of IC 342. If the electron spectrum does indeed continue as a power-law all the way until $E_{\rm min}=E_{\rm e}$ and sharply drops, the equipartition cases will shift upwards by around a factor of 5 - 15, bringing them in better agreement with the observed intensities. Despite this, we note that the profiles will still present an appearance significantly less smooth than that of the observations. Furthermore, the 100 pc equipartition, 1 kpc equipartition, and diffused equipartition to 250 pc would still have intensities considerably lower than those observed. From this, one could assume an uncertainty in our predicted intensities of $\sim 1$ dex. However, we favour intensities comparable to our teal line model, or even lower as suggested by the $\kappa$ model. Finally, the Langner et al. (2001) model and its integral also favors the use of $E_{\rm min}=4E_{\rm e}$ or higher.

Our treatment of the CR electron energy spectrum and our claim that the integral of the spectrum is (conservatively) lower than the integral taken all the way to $E_{\rm min}=E_{\rm e}$ in BK05 is accounted for in our equipartition $B$ field equations through the factor of $E_{\rm min}^{1-p}$ (see Equation 12). Along these lines, different assumptions about the energy spectrum would affect the magnetic field recovered using the equipartition assumption (by effectively changing that factor), varying our values for the overestimation of the magnetic field strength by $\sim 1.5-2$ in our Exp Models. Assuming a realistic $\kappa$ distribution matching the observational measurements would increase overestimation. Alternatively, enforcing a power-law without any spectral flattening down to $E_{\rm min}=E_{\rm e}$ would decrease the overestimation by artificially decreasing the energy budget available for high energy electrons. The above mentioned factors would need to be included in order to take the factors of magnetic field overestimation we have shown to be possible in this work and interpret them through the lens of the standard full power-law assumption or the observationally motivated $\kappa$ distribution.

We emphasize that the uncertainty in the low-energy end of the electron spectrum, particularly in other galaxies, and its apparent impacts on the magnetic fields inferred through equipartition provides a further reason for caution in its usage. In cases where it is applied, it would also benefit from incorporating alternative spectral distributions (such as the $\kappa$ model) into said assumption.

An assumption that we have made throughout this work is that the power-law index $p$ of the CR electrons and protons (and therefore the synchrotron spectral index $\alpha=(p-1)/2$) remains constant throughout the galaxy. This assumption might be valid on average but it does not generally hold, especially when considering specific sub-regions of galaxies, as discussed in Beck (2007) and Beck (2015a). In particular, the synchrotron spectral index is observed to be lower in star-forming regions, and higher in the inter-arm space. This is likely the result of losses sustained by the CR electrons as they diffuse out from acceleration sites (e.g., Tabatabaei et al. (2007)). Spatial variability in the spectral index has also been observed in the Milky way (Krachmalnicoff et al., 2018; de la Hoz et al., 2023).

By showing our synthetic observations and the comparisons of inferred and intrinsic magnetic field for a range of power-law indices, we avoid fixing to a single $p$ index value. In Figure 8, the reader could interpret the factors of overestimation in the Model Exp cases considering the power-law index they expect in different regions. This may allow the avoidance of extreme over- and underestimation: assuming a lower power-law index in dense star-forming regions and a higher power-law index in the more diffuse inter-arm regions. We caution however that the picture of how and where the power-law steepens and flattens can be more complex when multiple effects are at play. Regions with stronger magnetic fields would also lead to higher synchrotron losses, steepening the local profile with respect to the diffuse regions with weaker magnetic fields. Future work should address in more detail how reasonable models for varying the power-law index spatially affect the overestimation in observed magnetic field strength spatial distributions and radial profiles.

Spatially-varying the power-law index could also affect the amplitude and smoothness of the Model Exp intensity profiles in Figure 4, especially if dramatic deviations from the average profile take place. However, averaging effects may dampen such deviations, particularly under various prescribed CR electron distributions. Note that, due to the negligible variation of the Model pEq intensity profiles with the power-law index (see Figure 4), spatial variations of $p$ are likely to have only a minor effect on the shape of those resulting intensity profiles and therefore would not aid in bringing them to the smoothness of the observations.

While we have assumed a perfect knowledge of the intrinsic power-law index when recovering the magnetic field strength through the equipartition assumption, this is not necessarily the case in observations. The impact of assuming an incorrect $p$ is still to be determined in future work, and motivates new instruments with enhanced resolution that may be capable of recovering resolved synchrotron spectral indices. This encompasses the assumption of a constant power-law index for the entire galaxy when in reality this index actually varies throughout the system.

Finally, while measurements of the CR proton and electron energy spectra by Voyager 1 in Cummings et al. (2016) support the use of the same power-law index for electrons and protons, these data might only be representative of our Local ISM. This is implicitly required by the equipartition formulae presented here, but effects such as e.g., different energy loss mechanisms may invalidate such an assumption as well as the assumptions inherent in $K_{0}$ (see e.g, Lacki & Beck (2013) for a discussion of this in the context of starburst galaxies).

The discussed caveats and assumptions indicate that further consideration should be taken into account when measuring magnetic fields from synchrotron observations. In particular, they illustrate how further refinement of the techniques employed for recovering magnetic field information from observations is required. This is particularly urgent in the advent of new facilities such as the Square Kilometer Array (SKA), poised to revolutionize our understanding of astrophysical magnetic fields (Heald et al., 2020).