The origin of the high metallicity of close-in giant exoplanets II. The nature of the sweet spot for accretion

Context. The composition of gas giant planets reflects their formation and evolution history. Revealing the origin of the high heavy-element masses in giant exoplanets is an objective of planet formation theories. Planetesimal accretion during the phase of planetary migration could lead to the delivery of heavy elements into gas giant planets. In our previous paper, we used dynamical simulations and showed that planetesimal accretion during planetary migration occurs in a rather narrow region of the protoplanetary disk, which we refer to as the "sweet spot" for accretion. Aims. Our understanding of the sweet spot, however, is still limited. The location of the sweet spot within the disk and how it changes as the disk evolves were not investigated in detail. The goal of this paper is to reveal the nature of the sweet spot using analytical calculations and to investigate the role of the sweet spot in determining the composition of gas giant planets. Methods. We analytically derived the required conditions for the sweet spot. Then, using the numerical integration of the orbits of planetesimals around a migrating planet, we compared the derived equations with the numerical results. Results. We find that the conditions required for the sweet spot can be expressed by the ratio of the aerodynamic gas damping timescale of the planetesimal orbits to the planetary migration timescale. If the planetary migration timescale depends on the surface density of disk gas inversely, the location of the sweet spot does not change with the disk evolution. We expect that the planets observed inner to the sweet spot include a much greater amount of heavy elements than the planets outer to the sweet spot. The mass of planetesimals accreted by the protoplanet in the sweet spot depends on the amount of planetesimals that are shepherded by mean motion resonances. Our analysis suggests that tens Earth-masses of planetesimals can be shepherded into the sweet spot without planetesimal collisions. However, as more planetesimals are trapped into mean motion resonances, collisional cascade can lead to fragmentation and the production of smaller planetesimals. This could affect the location of the sweet spot and the population of small objects in planetary systems. Conclusions. We conclude that the composition of gas giant planets depends on whether the planets crossed the sweet spot during their formation. Constraining the metallicity of cold giant planets, which are expected to be beyond the sweet spot, with future observations would reveal key information for understanding the origin of heavy elements in giant planets.