Planetary systems in the light of asteroseismology: metallicity threshold for the planetary systems and age-metallicity relation
We compiled data for 127 hosts (plus six candidates) and used them as constraints to construct interior models of the hosts using the {\small MESA} code. Two significant conclusions emerge from these models. First, except for a few stars, the hosts' metallicity ($Z_0$) is greater than 0.007. This suggests a possible suppression of the occurrence of planets below $Z_0\approx0.007$. Second, it concerns how chemical evolution unfolds in the galactic disc. For a given $Z_0$ value, considering the oldest stars, there is a linear relationship between $Z_0$ and age ($t_9$). This line is around $t_9=13.4$ Gyr at $Z_0=0$, a value consistent with the age of the Galaxy. The linear relationship continues until around $t_9=6$ Gyr, and the maximum value of $Z_0$ remains constant between $t_9=2-6$ Gyr. We further modelled 12 hosts classified as red clump (RC) stars in the literature, explicitly accounting for mass loss along the red giant branch. These models highlight the critical role of mass-loss assumptions in determining the initial masses and ages of RC hosts, and their implications for the survival and evolution of close-in planets. Another key outcome of this study is the discovery of the relationship between $Z_0$ and the observed metallicity ($Z_{\rm s}$) for the hosts. We obtain a useful expression for $Z_0$, the input parameter for the models, as a function of stellar mass, radius, and $Z_{\rm s}$. This expression can be used to estimate $Z_0$ based on the reduced surface metallicity due to microscopic diffusion. We also derive an expression for planetary mass relative to the orbital semimajor axis and host mass. This expression may indicate a mass distribution near the inner disc where these planets formed, except for hot-Jupiters. Planet radii appear to depend on the planet's mass and irradiation energy, as well as the orbital period.