With the increasing number of exoplanets discovered, statistical properties of the population of exoplanets as a whole become unique constraints on planet formation models. However, planetary systems are formed in protoplanetary discs with a diversity of characteristics (e.g. mass, composition, lifetime, etc...). Moreover, even if the different processes at work during planet formation are studied with detailled numerical models, it is not always clear what is the resulting effect of the combination of these processes. The goal of population synthesis is therefore twofold: 1. computing the formation of a planetary system from the stage of protoplanetary disc up to the stage of full mature planetary systems, and 2. use these end-to-end formation models for a variety of initial conditions (in particular related to the properties of the protoplanetary disc) to infer the statistics of formed planetary systems. By nature, such end-to-end simulation are difficult to carry out in a complete and detailed manner. Hence, assumptions have to be made in order to make the problem tractable. The models developed in our team take into a account a number of different physical processes, namely: the structure and evolution of the protoplanetary disc, both for the gaseous and the solid phase, the mass growth of the forming planets, by accretion of solids and gas, the interactions between the forming planets and the disc, leading to processes like planetary migration, excitation of planetesimals, etc... , and the interactions between planets, that can themselves be split into gravitational interactions on one side, and competition for accretion of gas and solids on the other side.
Population synthesis models are based on two main components. The first component is a planet formation and evolution model, which provides, starting from a protoplanetary disk, the properties of the planetary system that can be formed in it. These properties can be the ones of the system at the end of the disc lifetime (in the case of formation models) or after a few Gyr (in the case of formation-evolution models). The second component is a distribution of the properties of protoplanetary discs, in particular their mass, thermal structure, lifetime and composition.
From these two components, a large number of planetary systems are computed, each assuming different properties of the disc. The distribution of these properties is taken at random, following some observed distribution. The ensemble of planetary systems obtained this way constitutes the theoretical planetary population. By taking into account the observational bias related to a given observational technique (for example by selecting planets producing a Doppler semi-amplitude larger than a given threshold), one obtains the theoretically observable population. Finally, this population is compared to the actually observed population (taking into account only planets detectable with the same observational bias). By trying to match the two population, one can gain knownledge on the physics at work during planet formation.
In case the two populations match, the full theoretical population can be ‘observed’ with another technique (e.g. with Transit) in order to predict what another observational method would yield.
One of the main outcome of population synthesis models is the planetary initial mass function (PIMF), namely the distribution of planetary masses. Our models predict a lage fraction of low mass planets (Earth to Super-Earth to Neptune mass range), and small number of planets between Neptune and Saturn, and a local maximum in the planetary abundance around the mass of Jupiter.
Figure 1 (Benz et al. 2014) shows a comparison of the observed, de-biased mass function found by the HARPS high-precision radial velocity survey (gray line with error bars, Mayor et al. 2011) and the mass function predicted by our population syntheses. One can see the roughly speaking flat mass function in the giant planet region, the strong increase towards small masses, and the break in the mass function at about 30 Earth masses. This corresponds to the transition from solid-dominated to gas-dominated planets that undergo gas runaway accretion. A similar basic structure is also found in the observational data, indicating that core accretion is able to recover important aspects of the planetary mass accretion process.
Population synthesis, coupled with models of the chemistry in protoplanetary disks, allow to determine the composition of planets of different type (mass, semi-major axis). The two plots presented here show the Fe/Si and the Mg/Si ratio in planets as computed by our planet formation models. A strong correlation between the obtained ratios and the ones in the star is noticed, meaning that it is possible, to a good approximation, to assume that the planetary and stellar refractory compositions are equal. This has strong implications in internal structure retrieval models developed for example in the context of CHEOPS.
See also our work on the composition and structure of planets for this.