Nearby galaxies in LoTSS-DR2: insights into the non-linearity of the radio-SFR relation

Context. Cosmic rays and magnetic fields are key ingredients in galaxy evolution, regulating both stellar feedback and star formation. Their properties can be studied with low-frequency radio continuum observations, free from thermal contamination. Aims. We define a sample of 76 nearby (< 30 Mpc) galaxies, with rich ancillary data in the radio continuum and infrared from the CHANG-ES and KINGFISH surveys, which will be observed with the LOFAR Two-metre Sky Survey (LoTSS) at 144 MHz. Methods. We present maps for 45 of them as part of the LoTSS data release 2 (LoTSS-DR2), where we measure integrated flux densities and study integrated and spatially resolved radio spectral indices. We investigate the radio-SFR relation, using star-formation rates (SFR) from total infrared and H $\alpha$ + 24-$\mu$m emission. Results. The radio-SFR relation at 144 MHz is clearly super-linear with $L_{144} \propto SFR^{1.4-1.5}$. The mean integrated radio spectral index between 144 and $\approx$1400 MHz is $\langle \alpha\rangle = -0.56 \pm 0.14$, in agreement with the injection spectral index for cosmic ray electrons (CRE). However, the radio spectral index maps show a variation of spectral indices with flatter spectra associated with star-forming regions and steeper spectra in galaxy outskirts and, in particular, in extra-planar regions. We found that galaxies with high star-formation rates (SFR) have steeper radio spectra; we find similar correlations with galaxy size, mass, and rotation speed. Conclusions. Galaxies that are larger and more massive are better electron calorimeters, meaning that the CRE lose a higher fraction of their energy within the galaxies. This explains the super-linear radio-SFR relation, with more massive, star-forming galaxies being radio bright. We propose a semi-calorimetric radio-SFR relation, which employs the galaxy mass as a proxy for the calorimetric efficiency.