Radio pulsations from the $\gamma$-ray millisecond pulsar PSR J2039-5617

The predicted nature of the candidate redback pulsar 3FGL\,J2039.6$-$5618 was recently confirmed by the discovery of $\gamma$-ray millisecond pulsations (Clark et al. 2020, hereafter Paper\,I), which identify this $\gamma$-ray source as \msp. We observed this object with the Parkes radio telescope in 2016 and 2019. We detect radio pulsations at 1.4\,GHz and 3.1\,GHz, at the 2.6ms period discovered in $\gamma$-rays, and also at 0.7\,GHz in one 2015 archival observation. In all bands, the radio pulse profile is characterised by a single relatively broad peak which leads the main $\gamma$-ray peak. At 1.4\,GHz we found clear evidence of eclipses of the radio signal for about half of the orbit, a characteristic phenomenon in redback systems, which we associate with the presence of intra-binary gas. From the dispersion measure of $24.57\pm0.03$\,pc\,cm$^{-3}$ we derive a pulsar distance of $0.9\pm 0.2$\,kpc or $1.7\pm0.7$\,kpc, depending on the assumed Galactic electron density model. The modelling of the radio and $\gamma$-ray light curves leads to an independent determination of the orbital inclination, and to a determination of the pulsar mass, qualitatively consistent to the results in Paper\,I.