Origin of the metallicity dependence of exoplanet host stars in the protoplanetary disc mass distribution

The probability of a star hosting a planet that is detectable in radial velocity surveys increases as P-pl(Z) proportional to (10(Z))(2), where Z is stellar metallicity. Models of planet formation by core accretion reproduce this trend, since the protoplanetary disc of a high-metallicity star has a high density of solids, and so forms planetary cores which accrete gas before the primordial gas disc dissipates. This paper considers the origin of the form of the metallicity dependence of P-pl(Z). We introduce a simple model in which detectable planets form when the mass of solid material in the protoplanetary disc, M-s, exceeds a critical value. In this model, the form of P-pl(Z) is a direct reflection of the distribution of protoplanetary disc masses, M-g, and the observed metallicity relation is reproduced if P(M-g > M'(g)) proportional to (M'(g))(-2). We argue that a protoplanetary disc's dust mass measured in submillimetre observations is a relatively pristine indicator of the mass available for planet-building, and find that the disc mass distribution derived from such observations is consistent with the observed P-pl(Z) if a solid mass M-s > 0.5M(J) is required to form detectable planets. Any planet formation model which imposes a critical solid mass for detectable planets to form would reproduce the observed metallicity relation, and core accretion models are empirically consistent with such a threshold criterion. While the outcome of planet formation in individual systems is debatable, we identify seven protoplanetary discs which, by rigid application of this criterion, would be expected to form detectable planets and may provide insight into the physical conditions required to form such planets. A testable prediction of the model is that the metallicity dependence should flatten both for Z > 0.5 dex and as more distant and lower mass planets are discovered. Further, combining this model with one in which the evolution of a star's debris disc is also influenced by the solid mass in its protoplanetary disc results in the prediction that debris discs detected around stars with planets should be more infrared luminous than those around stars without planets in tentative agreement with recent observations.