A theoretical framework of BL Her stars. I. Effect of metallicity and convection parameters on period-luminosity and period-radius relations

We present a new grid of convective BL Herculis models using the state-of-the-art 1D non-linear radial stellar pulsation tool MESA-RSP. We investigate the impact of metallicity and four sets of different convection parameters on multi-wavelength properties. Non-linear models were computed for periods typical for BL Her stars, i.e. $1\le \mathrm{P}(\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s})\le 4$ covering a wide range of input parameters - metallicity ($-2.0; $\mathrm{d}\mathrm{e}\mathrm{x}$ \leq $[\mathrm{F}\mathrm{e}\mathrm{/}\mathrm{H}]$ \leq 0.0; $\mathrm{d}\mathrm{e}\mathrm{x}$$), stellar mass (0.5M${}_{\odot }$-0.8M${}_{\odot }$), luminosity (50L${}_{\odot }$-300L${}_{\odot }$) and effective temperature (full extent of the instability strip; in steps of 50K). The total number of BL Her models with full-amplitude stable pulsations used in this study is 10280 across the four sets of convection parameters. We obtain their multiband ($UBVRIJHKL{L}^{\mathrm{\prime }}M$) light curves and derive new theoretical period-luminosity ($PL$), period-Wesenheit ($PW$) and period-radius ($PR$) relations at mean light. We find that the models computed with radiative cooling show statistically similar slopes for $PL$, $PW$ and $PR$ relations. Most empirical relations match well with the theoretical $PL$, $PW$ and $PR$ relations from the BL Her models computed using the four sets of convection parameters. However, $PL$ slopes of the models with radiative cooling provide a better match to empirical relations for BL Her stars in the LMC in the $H{K}_S$ bands. For each set of convection parameters, the effect of metallicity is significant in $U$ and $B$-bands and negligible in infrared bands, which is consistent with empirical results. No significant metallicity effects are seen in the $PR$ relations.