Calibrating the relation of low-frequency radio continuum to star formation rate at 1 kpc scale with LOFAR

Context. Radio continuum (RC) emission in galaxies allows us to measure star formation rates (SFRs) unaffected by extinction due to dust, of which the low-frequency part is uncontaminated from thermal (free-free) emission. Aims. We calibrate the conversion from the spatially resolved 140 MHz RC emission to the SFR surface density (ΣSFR) at 1 kpc scale. Radio spectral indices give us, by means of spectral ageing, a handle on the transport of cosmic rays using the electrons as a proxy for GeV nuclei. Methods. We used recent observations of three galaxies (NGC 3184, 4736, and 5055) from the LOFAR Two-metre Sky Survey (LoTSS), and archival LOw-Frequency ARray (LOFAR) data of NGC 5194. Maps were created with the facet calibration technique and converted to radio ΣSFR maps using the Condon relation. We compared these maps with hybrid ΣSFR maps from a combination of GALEX far-ultraviolet and Spitzer 24 μm data using plots tracing the relation at the highest angular resolution allowed by our data at 1.2 × 1.2 kpc ² resolution. Results. The RC emission is smoothed with respect to the hybrid ΣSFR owing to the transport of cosmic-ray electrons (CREs) away from star formation sites. This results in a sublinear relation (ΣSFR)RC [(ΣSFR)hyb] ^a , where a = 0.59 ± 0.13 (140 MHz) and a = 0.75 ± 0.10 (1365 MHz). Both relations have a scatter of σ = 0.3 dex. If we restrict ourselves to areas of young CREs (α > -0.65; Iν ν ^α ), the relation becomes almost linear at both frequencies with a 0.9 and a reduced scatter of σ = 0.2 dex. We then simulate the effect of CRE transport by convolving the hybrid ΣSFR maps with a Gaussian kernel until the RC-SFR relation is linearised; CRE transport lengths are l = 1-5 kpc. Solving the CRE diffusion equation, assuming dominance of the synchrotron and inverse-Compton losses, we find diffusion coefficients of D = (0.13-1.5) × 10 ²⁸ cm ² s ^-1 at 1 GeV. Conclusions. A RC-SFR relation at 1.4 GHz can be exploited to measure SFRs at redshift z 10 using 140 MHz observations.