The Mass-Metallicity Relation for Giant Planets

Exoplanet discoveries of recent years have provided a great deal of new data for studying the bulk compositions of giant planets. Here we identify 47 transiting giant planets ($20 M_\oplus < M < 20 M_{\mathrm{J}}$) whose stellar insolation is low enough ($F_* < 2\times10^8\; \text{erg}\; \text{s}^{-1}\; \text{cm}^{-2}$, or roughly $T_\text{eff} < 1000$) that they are not affected by the hot Jupiter radius inflation mechanism(s). We compute a set of new thermal and structural evolution models and use these models in comparison with properties of the 47 transiting planets (mass, radius, age) to determine their heavy element masses. A clear correlation emerges between the planetary heavy element mass $M_z$ and the total planet mass, approximately of the form $M_z \propto \sqrt{M}$. This finding is consistent with the core accretion model of planet formation. We also study how stellar metallicity [Fe/H] affects planetary metal-enrichment and find a weaker correlation than has been previously reported from studies with smaller sample sizes. We confirm a strong relationship between the planetary metal-enrichment relative to the parent star $Z_{\rm planet}/Z_{\rm star}$ and the planetary mass, but see no relation in $Z_{\rm planet}/Z_{\rm star}$ with planet orbital properties or stellar mass. The large heavy element masses of many planets ($>50$ $M_{\oplus}$) suggest significant amounts of heavy elements in H/He envelopes, rather than cores, such that metal-enriched giant planet atmospheres should be the rule. We also discuss a model of core-accretion planet formation in a one-dimensional disk and show that it agrees well with our derived relation between mass and $Z_{\rm planet}/Z_{\rm star}$.