New insights into the interstellar medium of the dwarf galaxy IC 10: connection between magnetic fields, the radio--infrared correlation and star formation

We present the highest sensitivity and angular resolution study at 0.32 GHz of the dwarf irregular galaxy IC\,10, observed using the Giant Metrewave Radio Telescope, probing $\sim45$ pc spatial scales. We find the galaxy-averaged radio continuum spectrum to be relatively flat, with a spectral index $\alpha = -0.34\pm0.01$ ($S_\nu \propto \nu^\alpha$), mainly due to a high contribution from free--free emission. At 0.32 GHz, some of the H{\sc ii} regions show evidence of free--free absorption as they become optically thick below $\sim0.41$ GHz with corresponding free electron densities of $\sim11-22~\rm cm^{-3}$. After removing the free--free emission, we studied the radio--infrared relations on 55, 110 and 165 pc spatial scales. We find that on all scales the non-thermal emission at 0.32 and 6.2 GHz correlates better with far-infrared (FIR) emission at $70\,\mu$m than mid-infrared emission at $24\,\mu$m. The dispersion of the radio--FIR relation arises due to variations in both magnetic field and dust temperature, and decreases systematically with increasing spatial scale. The effect of cosmic ray transport is negligible as cosmic ray electrons were only injected $\lesssim5$ Myr ago. The average magnetic field strength ($B$) of $12~\mu$G in the disc is comparable to that of large star-forming galaxies. The local magnetic field is strongly correlated with local star formation rate ($\mathrm{SFR}$) as $B \propto \mathrm{SFR}^{0.35\pm0.03}$, indicating a star-burst driven fluctuation dynamo to be efficient ($\sim10$ per cent) in amplifying the field in IC\,10. The high spatial resolution observations presented here suggest that the high efficiency of magnetic field amplification and strong coupling with SFR likely sets up the radio--FIR correlation in cosmologically young galaxies.