Why take the square root? An assessment of interstellar magnetic field strength estimation methods

The magnetic field strength in interstellar clouds can be estimated indirectly by using the spread of dust polarization angles ($\delta \theta$). The method developed by Davis 1951 and by Chandrasekhar and Fermi 1953 (DCF) assumes that incompressible magnetohydrodynamic (MHD) fluctuations induce the observed dispersion of polarization angles, deriving $B\propto 1/\delta \theta$ (or, $\delta \theta \propto M_{A}$, in terms of the Alfv\'{e}nic Mach number). However, observations show that the interstellar medium (ISM) is highly compressible. Recently, Skalidis & Tassis 2021 (ST) relaxed the incompressibility assumption and derived instead $B\propto 1/\sqrt{\delta \theta}$ ($\delta \theta \propto M_{A}^2$). We explored what the correct scaling is in compressible and magnetized turbulence with numerical simulations. We used 26 magnetized, ideal-MHD numerical simulations with different types of forcing. The range of $M_{A}$ and sonic Mach numbers $M_{s}$ explored are $0.1 \leq M_{A} \leq 2.0$ and $0.5 \leq M_{s} \leq 20$. We created synthetic polarization maps and tested the assumptions and accuracy of the two methods. The synthetic data have a remarkable consistency with the $\delta \theta \propto M_{A}^{2}$ scaling, which is inferred by ST, while the DCF scaling fails to follow the data. The ST method shows an accuracy better than $50\%$ over the entire range of $M_{A}$ explored; DCF performs adequately only in the range of $M_{A}$ for which it has been optimized through the use of a "fudge factor". For low $M_{A}$, DCF is inaccurate by factors of tens. The assumptions of the ST method reflect better the physical reality in clouds with compressible and magnetized turbulence, and for this reason the method provides a much better estimate of the magnetic field strength over the DCF method.