Rotation measure structure functions with higher-order stencils as a probe of small-scale magnetic fluctuations and its application to the Small and Large Magellanic Clouds

Magnetic fields and turbulence are important components of the interstellar medium (ISM) of star-forming galaxies. It is challenging to measure the properties of the small-scale ISM magnetic fields (magnetic fields at scales smaller than the turbulence driving scale). Using numerical simulations, we demonstrate how the second-order rotation measure (RM, which depends on thermal electron density, ne, and magnetic field, b) structure function can probe the properties of small-scale b. We then apply our results to observations of the Small and Large Magellanic Clouds (SMC and LMC). First, using Gaussian random b, we show that the characteristic scale, where the RM structure function flattens is approximately equal to the correlation length of b. We also show that computing the RM structure function with a higher-order stencil (more than the commonly-used two-point stencil) is necessary to accurately estimate the slope of the structure function. Then, using Gaussian random b and lognormal ne with known power spectra, we derive an empirical relationship between the slope of the power spectrum of b, ne, and RM. We apply these results to the SMC and LMC and estimate the following properties of small-scale b: correlation length (160 ± 21 pc for the SMC and 87 ± 17 pc for the LMC), strength (14 ± 2 μG for the SMC and 15 ± 3 μG for the LMC), and slope of the magnetic power spectrum (-1.3 ± 0.4 for the SMC and -1.6 ± 0.1 for the LMC). We also find that ne is practically constant over the estimated b correlation scales.