Red supergiant stars in IC 1613 and metallicity-dependent mixing length in the evolutionary model

We report a spectroscopic study on red supergiant stars (RSGs) in the irregular dwarf galaxy IC 1613 in the Local Group. We derive the effective temperatures ($T_{\mathrm{e}\mathrm{f}\mathrm{f}}$) and metallicities of 14 RSGs by synthetic spectral fitting to the spectra observed with the MMIRS instrument on the MMT telescope for a wavelength range from 1.16 $\mu$m to 1.23 $\mu$m. A weak bimodal distribution of the RSG metallicity centered on the [Fe/H]=$-0.65$ is found, which is slightly lower than or comparable to that of the Small Magellanic Cloud (SMC). There is no evidence for spatial segregation between the metal rich ([Fe/H]$>-0.65$) and poor ([Fe/H]$<-0.65$) RSGs throughout the galaxy. The mean effective temperature of our RSG sample in IC 1613 is higher by about 250 K than that of the SMC. However, no correlation between $T_{\mathrm{e}\mathrm{f}\mathrm{f}}$ and metallicity within our RSG sample is found. We calibrate the convective mixing length (${\alpha }_{\mathrm{M}\mathrm{L}\mathrm{T}}$) by comparing stellar evolutionary tracks with the RSG positions on the HR diagram, finding that models with ${\alpha }_{\mathrm{M}\mathrm{L}\mathrm{T}}=2.2-2.4H_P$ can best reproduce the effective temperatures of the RSGs in IC 1613 for both Schwarzschild and Ledoux convection criteria. This result supports our previous study that a metallicity dependent mixing length is needed to explain the RSG temperatures observed in the Local Group, but we find that this dependency becomes relatively weak for RSGs having a metallicity equal to or less than the SMC metallicity.