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

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. We calibrate the conversion from the spatially resolved 140 MHz RC emission to the SFR surface density ($Sigma_{rm SFR}$) at 1 kpc scale. 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 $Sigma_{rm SFR}$ maps using the Condon relation. We compared these maps with hybrid $Sigma_{rm SFR}$ maps from a combination of GALEX far-ultraviolet and Spitzer 24 $murm m$ data using plots tracing the relation at $1.2times 1.2$-kpc$^2$ resolution. The RC emission is smoothed with respect to the hybrid $Sigma_{rm SFR}$ owing to the transport of cosmic-ray electrons (CREs). This results in a sublinear relation $(Sigma_{rm SFR})_{rm RC} propto [(Sigma_{rm SFR})_{rm hyb}]^{a}$, where $a=0.59pm 0.13$ (140 MHz) and $a=0.75pm 0.10$ (1365 MHz). Both relations have a scatter of $sigma = 0.3~rm dex$. If we restrict ourselves to areas of young CREs ($alpha > -0.65$; $I_ u propto u^alpha$), the relation becomes almost linear at both frequencies with $aapprox 0.9$ and a reduced scatter of $sigma = 0.2~rm dex$. We then simulate the effect of CRE transport by convolving the hybrid $Sigma_{rm SFR}$ 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, we find diffusion coefficients of $D=(0.13$-$1.5) times 10^{28} rm cm^2,s^{-1}$ at 1 GeV. A RC-SFR relation at $1.4$ GHz can be exploited to measure SFRs at redshift $z approx 10$ using $140$ MHz observations.