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 $\mu\rm m$ data using plots tracing the relation at $1.2\times
1.2$-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 $a=0.59\pm 0.13$ (140 MHz) and
$a=0.75\pm 0.10$ (1365 MHz). Both relations have a scatter of $\sigma = 0.3~\rm
dex$. If we restrict ourselves to areas of young CREs ($\alpha > -0.65$; $I_\nu
\propto \nu^\alpha$), the relation becomes almost linear at both frequencies
with $a\approx 0.9$ 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.
