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Tidal Downsizing model. I. Numerical methods: saving giant planets from tidal disruptions

Tidal Downsizing (TD) is a recently developed planet formation theory that supplements the classical Gravitational disc Instability (GI) model with planet migration inward and tidal disruptions of GI fragments in the inner regions of the disc. Numerical methods for a detailed population synthesis of TD planets are presented here. As an example application, the conditions under which GI fragments collapse faster than they migrate into the inner $a\sim$ few AU disc are considered. It is found that most gas fragments are tidally or thermally disrupted unless (a) their opacity is $\sim 3$ orders of magnitude less than the interstellar dust opacity at metallicities typical of the observed giant planets, or (b) the opacity is high but the fragments accrete large dust grains (pebbles) from the disc. Case (a) models produce very low mass solid cores ($M_{\rm core} < 0.1$ Earth masses) and follow a negative correlation of giant planet frequency with host star metallicity. In contrast, case (b) models produce massive solid cores, correlate positively with host metallicity and explain naturally while giant gas planets are over-abundant in metals.