Functional relationships for $T_{\mathrm{e}\mathrm{f}\mathrm{f}}$ and log~$g$ in F-G supergiants from $uvby\beta$ photometry

From photoelectric $uvby\beta$ data and recent accurate synthetic and spectroscopic values of $T_{\mathrm{e}\mathrm{f}\mathrm{f}}$ and log~$g$ for 50 F-G supergiants, we have calculated functional relationships that lead to initial estimates of effective temperature and gravities for these types of stars. It is shown that while the $T_{\mathrm{e}\mathrm{f}\mathrm{f}}$ relationships are calculated using the data on young massive supergiants, they are also valid for evolved stars of similar temperatures like post-AGB and RV Tau stars. The gravity can also be predicted from the $\mathrm{\Delta }[c_1]$ index with an uncertainty of about 0.26 dex. Although a clear and significant trend between $M_V$ and $\mathrm{\Delta }[c_1]$ is seen, no calibration is found that predicts accurate values of $M_V$.