Asteroseismology of blue stragglers in $\omega$ Centauri: the case of five double-mode radial SX Phoenicis pulsators

We present the first extensive seismic modelling of SX Phe stars in the stellar system $\omega$ Cen. First, using the new values of reddening $E(B-V)$ and distance modulus $(m-M)_V$, and bolometric corrections from Kurucz model atmospheres, we determine the effective temperatures and luminosities of all SX Phe variables in $\omega$ Cen with available $(B-V)$ colours. Next, we carefully select SX Phe stars that have a frequency ratio strongly suggesting excitation of two radial modes, and, in addition, their preliminary pulsational models have the values of $(T_{\rm eff},~ L)$ consistent with observational determinations. For five double-mode radial pulsators, we perform an extensive seismic modeling using the Bayesian analysis based on Monte Carlo simulations. We study the effect of opacity tables and helium abundance. With the OPAL data and $Y=0.30$, we obtained masses in the range (1.0,~1.2) M$_{\odot}$, metallicity $Z\in(0.0007,~0.0029)$ and the age of about (1.9,~3.8) Gyr. The OP and OPLIB seismic models have always higher metallicites, sometimes outside the allowed range for $\omega$ Cen. In the case of three stars, we find seismic models within the observed range of $(T_{\rm eff},~L)$ with all three opacity tables. In the case of two stars, with the highest metallicity, seismic models computed with the OP and OPLIB tables are located far outside the observed error box. The OPAL seismic models follow the age$-$metallicity relation known for $\omega$ Cen from the literature.