SALT spectroscopy of the HMXB associated with the LMC supernova remnant MCSNR\,J0513$-$6724

We report the results of optical \'echelle spectroscopy with the Southern African Large Telescope (SALT) of the mass donor star BSDL\,923 in the neutron star (NS) high-mass X-ray binary XMMU\,J051342.6$-$672412 associated with the LMC supernova remnant (SNR) MCSNR\,J0513$-$6724. We found that BSDL\,923 is a B0.7\,III star with double peaked emission lines originating in a circumbinary disk-like structure. This classification and the presence of double-peaked emission lines imply that BSDL\,923 is a Be star. Modelling with the stellar atmosphere code {\sc fastwind} was used to derive the effective temperature $T_{\rm eff}=27\pm1$\,kK, surface gravity $\log g=3.22\pm0.10$, projected rotational velocity $v\sin i\approx100\pm45 \, \kms$, colour excess $E(B-V)=0.53\pm0.05$\,mag, and luminosity $\log(L_*/\lsun)=5.46\pm0.10$ of BSDL\,923, as well as to show that the surface of this star is polluted with $\alpha$-elements (O, Mg and Si) from the supernova ejecta. We found also that the NS is orbiting BSDL\,923 in an eccentric ($e=0.158\pm0.061$) orbit with the orbital period of $1.280\pm0.006$\,d and the semi-major axis of $17\pm3 \, \rsun$, and the radius of BSDL\,923 is $25\pm5 \,\rsun$. We speculate that the NS is embedded in the atmosphere of BSDL\,923 either because it was kicked at birth towards this star or because of inflation of BSDL\,923 caused by the energy input from the supernova blast wave. Using long-slit spectroscopy with SALT, we searched for possible signs of the SNR shell in the 2D spectrum, but did not find them. This lack of detection is consistent with the young age ($\approx4^{+2} _{-1}$\,kyr) of MCSNR\,J0513$-$6724, implying that it is still in the adiabatic (non-radiative) phase.