Robust magnetic field estimates in star-forming galaxies with the equipartition formula in the absence of equipartition

TL;DR

This study tests the reliability of the classical equipartition formula for estimating galactic magnetic field strengths using high-resolution CR-MHD simulations with a two-moment CR transport model. By generating synthetic synchrotron maps and comparing the inferred equipartition fields to the true magnetic fields, the authors derive a theoretical, weak dependence of B_eq/B_sim on the CR-to-B energy density ratio and demonstrate that the formula remains practical for face-on projections. In edge-on configurations, the method biases magnetic-field estimates, overestimating in halos and underestimating in disks, underscoring that equipartition does not imply true energy equipartition locally. The results inform the interpretation of radio-derived magnetic fields, suggesting B_eq is a useful estimator for field strength in many cases but should be used with caution for magnetic pressure and CR pressure inferences, particularly in spatially complex, multiphase ISM environments.

Abstract

The equipartition model is widely used to estimate magnetic field strength from synchrotron intensity in radio galaxies, yet the validity of its underlying assumptions remains uncertain. Using an Arepo simulation which incorporates a two-moment cosmic ray (CR) transport scheme and a multiphase interstellar medium, we compare magnetic fields inferred from synthetic synchrotron emission maps with the true fields in the simulation. Starting from the derivation of the equipartition formula, we find that the deviation between the equipartition magnetic field and the true magnetic field depends only weakly on the ratio of the magnetic to the CR energy density. In practice, for both face-on and edge-on projections, the equipartition model slightly overestimates the total synchrotron-weighted magnetic field with mean offsets of 32% (0.17 dex) and 36% (0.2 dex), even though the energy equipartition does not hold locally. Beyond these average offsets, a clear trend emerges in edge-on projections that the model underestimates the field in the disk and overestimates it in the halo. Our results demonstrate that the validity of the equipartition model depends only weakly on the strict fulfillment of energy equipartition, and that the equipartition model remains a practical method for estimating magnetic field strengths in face-on projection maps based on our CR-magnetohydrodynamics simulation.

Figures (18)

• Figure 1: Left: Magnetic field lines color-coded by magnetic field strength. Middle: Synchrotron intensity shown in the background, with magnetic field directions derived from the Stokes Q and U parameters overlaid in the foreground. The effect of Faraday rotation is not included. Right: Radio spectral slope map calculated between 140 MHz and 1.4 GHz. The spectral index steepens from approximately $\alpha_\nu = 0.55$ (corresponding to $\alpha_\mathrm{e} = 2.1$, our $\alpha_{\rm inj}$) in the spiral arm to $\alpha_\nu = 0.9$ (corresponding to $\alpha_\mathrm{e} = 2.8$) in the inter-arm region. The middle and right panel are calculated with projected synchrotron intensity using a depth of 1 kpc along the line of sight.
• Figure 2: Comparison of the true magnetic field strength from the simulation (upper left) with the equipartition magnetic field (upper right), and the true CR energy density (lower left) with the equipartition energy density (lower right). Both the true magnetic field and the CR energy density are weighted by synchrotron emission. The equipartition magnetic field is derived by inserting the synchrotron intensity and radio spectral index into Equation \ref{['eq:equipartition']}, assuming $K_0 = 100$, $l = 1$, and $c_4 = 1$. All panels show projections integrated over $\pm 0.5$ kpc above and below the disk ($|z|$<0.5 kpc). This comparison demonstrates that while the equipartition magnetic field can partially recover the true magnetic field morphology (due to the weak dependence of the field ratio on the energy density ratio), the CR energy density calculated from the equipartition magnetic field fails to accurately trace the CR energy density.
• Figure 3: Correlation between the magnetic field ratio and the energy density ratio in individual computational cells within the disk ($|x| < 10$ kpc, $|y| < 10$ kpc, and $|z| < 2.5$ kpc). The magnetic field strength is computed using the field components in the plane of the galaxy. The equipartition magnetic field in each cell is calculated using the synchrotron intensity, $I_{\rm syn}=L_{\rm syn}/4\pi (\pi r_{\rm cell}^2)$, where $r_{\rm cell}$ is the radius of each gas cell, and the corresponding $\alpha_\nu$, assuming $K_0 = 100$, $c_4 = 1$, and $l = r_{\rm cell}$. The best-fit line across the cells is shown in red, with a slope of 0.28 and an intercept of $-0.08$. This result is consistent with our analytical derivation and confirms that the magnetic field ratio depends only weakly on the energy density ratio.
• Figure 4: Left column: Ratio of the magnetic field in the simulation weighted by synchrotron emissivity ($\langle B_{\rm sim}\rangle_{\rm syn}$) to the field inferred from mock observation based on the equipartition assumption beck_revised_2005, assuming $K_0=100$, $c_4=1$, and $l=1$ kpc. Right column: Synchrotron-weighted average of the log of the ratio of magnetic to CR energy density. Middle column: Pixel-by-pixel correlation between the corresponding maps in the left and right columns, with best-fit lines overlaid. The upper panels show results weighted by synchrotron emission, while the lower panels show results weighted by volume. Weighting by volume rather than synchrotron emission reduces both the slope and the intercept of the correlation. All maps are computed using a projection depth of 1 kpc along the line of sight.
• Figure 5: Ratio of the synchrotron-weighted average $B$-field ($\langle B_{\rm sim}\rangle_{\rm syn}$) and the volume-weighted average $B$-field ($\langle B_{\rm sim}\rangle_{\rm vol}$) over a length of 1 kpc along the line of sight. $\langle B_{\rm sim}\rangle_{\rm syn}$ is in general larger than $\langle B_{\rm sim}\rangle_{\rm vol}$, and is much larger than $B_{\rm vol}$ around the star-forming region. We note that the $\langle B_{\rm sim}\rangle_{\rm vol}$ ratio is computed only within the disk and therefore is not biased by contributions from the volume-filling halo.