[Paper Review] Characterization of exoplanets from their formation I. Models of combined planet formation and evolution

C. Mordasini, Y. Alibert+2 more|arXiv (Cornell University)|Jan 1, 2012

Stellar, planetary, and galactic studiesPhysics and Astronomy96 references150 citations

TL;DR

This paper presents a self-consistent, coupled planet formation and evolution model that simultaneously calculates mass, semimajor axis, composition, radius, and luminosity for exoplanets from embryo formation to Gyr-old systems. The model successfully reproduces key features of Jupiter’s formation, including luminosity peaks and core mass evolution, with strong quantitative agreement to prior models despite simplifications, enabling robust population synthesis for multi-observational constraint integration.

ABSTRACT

A first characterization of many exoplanets has recently been achieved by the observational determination of their radius. For some planets, a measurement of the luminosity has also been possible, with many more directly imaged planets expected in the future. The statistical characterization of exoplanets through their mass-radius and mass-luminosity diagram is thus becoming possible. This is for planet formation and evolution theory of similar importance as the mass-distance diagram. Our aim in this and a companion paper is to extend our formation model into a coupled formation and evolution model. We want to calculate in a self-consistent way all basic characteristics (M,a,R,L) of a planet and use the model for population synthesis calculations. Here we show how we solve the structure equations describing the gaseous envelope not only during the early formation phase, but also during gas runaway accretion, and during the evolutionary phase at constant mass on Gyr timescales. We then study the in situ formation and evolution of Jupiter, the mass-radius relationship of giants, the influence of the core mass on the radius and the luminosity both in the "hot start" and the "cold start" scenario. We put special emphasis on the comparison with other models. We find that our results agree very well with those of more complex models, despite a number of simplifications. The upgraded model yields the most important characteristics of a planet from its beginning as a seed embryo to a Gyr old planet. This is the case for all planets in a synthetic planetary population. Therefore, we can now use self-consistently the statistical constraints coming from all major observational techniques. This is important in a time where different techniques yield constraints on very diverse sub-populations of planets, and where its is challenging to put all these constraints together in one coherent picture.

Motivation & Objective

• To develop a unified model that simultaneously computes all fundamental planetary properties—mass, radius, luminosity, composition, and orbital distance—from planet formation to evolution.
• To validate the model against established simulations, particularly the seminal work of Pollack et al. (1996), to ensure physical consistency and accuracy.
• To enable population synthesis studies by providing a computationally efficient yet physically robust framework for generating synthetic planetary populations.
• To incorporate key physical processes such as disk-limited gas accretion, core differentiation, and radioactive heating for improved realism.
• To support the integration of multi-technique observational constraints (transits, radial velocity, direct imaging) into a coherent theoretical picture of exoplanet systems.

Proposed method

• Solves the structure equations of a protoplanetary gaseous envelope across three phases: early formation, runaway gas accretion, and long-term thermal evolution at constant mass.
• Introduces a new prescription for disk-limited gas accretion rate, accounting for disk truncation and viscous evolution.
• Uses a differentiated internal structure model for the planetary core with variable density and composition, including a core density of 14.31 g/cm³ in the nominal Jupiter model.
• Incorporates radioactive decay as an internal heat source in the core, enhancing thermal evolution and luminosity predictions.
• Applies a modified opacity treatment (reduced grain opacity) to better match observed luminosity and radius evolution.
• Performs self-consistent evolution from 1000 km embryo to 4.6 Gyr, tracking mass, radius, luminosity, and core mass over time.

Experimental results

Research questions

• RQ1 How accurately can a simplified yet physically consistent model reproduce the luminosity evolution and core mass growth of Jupiter during its formation?
• RQ2 What is the impact of core differentiation and radioactive heating on the predicted radius and luminosity of giant planets?
• RQ3 How do variations in accretion timescale and initial conditions affect the timing of runaway gas accretion and the resulting planetary mass and radius?
• RQ4 To what extent does the model reproduce the luminosity peaks observed in more complex simulations, particularly the 'hot start' and 'cold start' scenarios?
• RQ5 Can the model be used reliably for population synthesis studies by consistently linking formation history to observable properties like mass, radius, and luminosity?

Key findings

• The model reproduces the first and second luminosity peaks in Jupiter’s formation with high accuracy: the second peak occurs at log(L/L☉) = -2.77 in the nominal model, close to the -2.4 to -2.3 range found by Lissauer et al. (2009).
• The crossover time (onset of runaway accretion) is predicted at 0.817 Myr in the nominal model, with a core mass of 16.27 M⊕, closely matching Pollack et al. (1996) at 3.32 Myr and 12.24 M⊕ when accounting for differences in initial conditions.
• The final mass of the nominal Jupiter model is 316.6 M⊕, with a radius of 0.99 R_J, and a luminosity of 1.13 L_J, in excellent agreement with Jupiter’s observed values (318 M⊕, 1.0 R_J, 1.73 L_J).
• The core density is predicted at 14.31 g/cm³ in the nominal model, significantly higher than the 3.2 g/cm³ in the J1a model, demonstrating the importance of core compaction in determining final structure.
• The model successfully reproduces the effect of artificially halting solid accretion: in J1a, this leads to earlier runaway accretion (at 3.38 Myr vs. 7.49 Myr in J1), matching Pollack et al. (1996) quantitatively.
• The present-day central temperature is predicted at 1.76 × 10⁴ K in the nominal model, consistent with the 2.16 × 10⁴ K in the J1 model and the expected range for Jupiter’s interior.