Physics and Chemistry of Planet Formation

Radiation-hydrodynamic shock simulations

Except for Beta Pictoris b, the mass-luminosity relation of young giant planets is today solely based on theoretical considerations. This theoretical relation is highly uncertain as it depends on assumptions about the post-formation entropy of the planetary interior: for a given planet mass, a hot start (high entropy) associated with a high luminosity results if the accretional energy is transported into the planet during formation. This scenario is traditionally associated with giant planet formation via gravitational instability. A cold start (low luminosity and entropy) in contrast results if the accretion energy is radiated away at an accretion shock. This is often associated with core accretion. In this project, we have conducted (Marleau et al. 2017, Marleau et al. 2019) detailed radiation-hydrodynamic simulations of the planetary accretion shock to obtain a grid of post shock entropies and radiative shock efficiencies. We have found rather high post-shock entropies which may point toward warm starts.

Figure caption: Physical loss efficiency of the radiative accretion shock as a function of shock Mach number. The limit of an efficiency = 0 corresponds to all incoming energy being absorbed into the planet, while an efficiency = 100% means that the kinetic energy of the gas is entirely radiated away. Note that even a high efficiency that is close to, but not exactly equal 100 % can already be a very efficient heating source for the planet. Figure from Marleau et al. (2017).

Luminosities at early times

The intrinsic luminosity of (young) giant planets is the observable quantities for direct imaging instruments like SPHERE or GPI. The plot shows the predicted luminosity as a function of time for different masses in combined formation and evolution calculations. On the left, "cold" accretion trough a supercritical accretion shock is assumed, where all shock energy is radiated away into space, leading to a low entropy state at the end of the formation phase (slightly after 1 Myrs). This is associated with low luminosities. On the right, no radiative losses at the accretion shock occur, leading to much higher luminosities at the beginning of the evolutionary phase ("hot" accretion leading to a "hot" start). Figure from Mordasini et al. (2012).

The analysis of the Kepler data shows that there is a depleted region in the distance-radius plane separating larger Sub-Neptune planets from smaller Super-Earth planets (Fulton et al. 2017). This valley can be explained as the consequence of the evaporative loss of primordial H/He envelopes, making it an “evaporation valley”. This feature was theoretically predicted before its discovery independently by several theoretical models, including ours. With the Bern evolution model COMPLETO21, we have shown (Jin & Mordasini 2018) that the observed gap not only agrees with the theoretical predictions, but that its locus even allows to constraint the bulk composition of the planets inside of a few 0.1 AU with radii less than about 1.6 Earth radii. It is found that the observed valley agrees with a predominantly Earth-like (silicate & iron) composition, but rules out a mainly ice-dominated composition. This result puts key constraints on formation models.

Figure caption: Comparison of the distribution of planets in the plane of orbital distance versus planetary radius in two numerical simulations (colored and black dots) with the observed frequency (yellow-brown contours, with dark colors indicating high occurrence, from Fulton et al. 2017). The diagonal evaporation valley is visible. Its locus agrees with rocky, Earth-like cores (left panel), but not with icy cores (right panel). Figure from Jin & Mordasini (2018).

Between the time when a planet forms, and the time when it is studied with observations, there are usually millions or even billions of years in between. To establish links between planet formation theory and observations it is therefore necessary to consider planetary evolution. A central research interest of the group is therefore the evolution of planets from their origins to present day. For this, the planetary properties resulting from formation are evolved over Gigayear timescales taking into account the effects of stellar irradiation, cooling and contraction of the gaseous envelope and solid core, atmospheric escape, bloating, radioactive decay, and – for massive objects – deuterium burning. This yields the most important characteristics of the planet like its radius, luminosity, atmospheric structure, spectrum, and magnitudes as a function of time. These results are compared with measurements of planetary radii (e.g. with Kepler, TESS, CHEOPS, or PLATO) and direct imaging (e.g., SPHERE).

Evolutionary models for low-mass planets (Coolow project)

Future observational facilities like the JWST and the ELT will be able to directly image young planets of lower masses than current instruments. Together with collaborators in Bern and at ETHZ, we (Linder et al. 2019) have therefore extended the classical cooling tracks (Burrows et al. 1997, Baraffe et al. 2003) to lower masses, and assessed the detectability of the planets. We have found that planets of 20, 100, and 1000 Earth masses are detectable with the JWST in the background-limited regime to ages of about 10, 100, and 1000 Myr, respectively.

Figure caption: Spectra of a 20 Earth-mass self-luminous planet as a function of time for cloud-free solar metallicity atmospheres together with the associated blackbody spectrum. The age is given in colors. The temperature of the blackbodies corresponds to the temperature of the planet at the given age. The grey dots show the background sensitivity limits of JWST/NIRCam, the black dots those of JWST/MIRI. Figure from Linder et al. (2019).

Deuterium Burning Planets

It is currently not clear where the upper mass limit for giant planet formation lies. In this project, the formation of massive companions with up to 23 Jovian masses by the core accretion mechanism was explored, including the effect of deuterium burning. Objects massive enough where found to burn deuterium, despite the presence of a solid core.

Figure caption: The plots show the radius und effective temperature as a function of time and mass at early times. The gas accretion is assumed to be "cold" (low entropy), so that the radius directly after the collapse is small (~1.2 Jovian radii). Due to the thermostatic nature of deuterium burning, the planets re-inflate. The conventional mass limit of about 13 Jovian masses for deuterium burning is only mildly affected by the fact that the planets form by core accretion. Figure from Mollière & Mordasini (2012).

The objective of this project is to develop a new population synthesis model in the framework of the gravitational instability/direct collapse model for giant planet formation. This activity is partially supported by the NCCR PlanetS new initiative “DIPSY” and is conducted in collaboration with the University of Zürich. This project is also linked to observations of young giant exoplanets by direct imagining and hydrodynamical simulations of protoplanetary disks

Figure caption: Surface density (color coded) as a function of time and distance from the star in an infalling and evolving circumstellar disk. The disk fragments six times, and the forming gas clump migrates into the star. From Schib et al. in prep.

As a member of the HARPS, SPHERE, and NIRPS consortia we are also directly involved in observational work besides the theoretical studies, with regular observational runs in Chile. Previous observational runs were also made at the Observatoire de Haute Provence (still with the 51 Peg b discovery instrument ELODIE) and observations of NEOs (Near Earth Objects) at Zimmerwald Observatory near Bern.

Detection of companions to HD 85390, HD 90156, and HD 103197

Figure caption: Radial velocity measured with HARPS as a function of orbital phase induced by the planetary companions around HD 85390 (Msini=42 MEarth, a=1.52 AU), HD 90156 (Msini=18 MEarth, a=0.25 AU), and HD 103197 (Msini=31 MEarth, a=0.25 AU). The host stars have spectral types ranging from K1V to G5V. These discoveries were made within the HARPS GTO survey. Reference: Mordasini, Mayor et al. (2011).

Intermediate mass planets and the timescale of gas accretion in runaway

Intermediate mass planets like HD 85390b or HD 103197b are interesting to constrain observationally the timescale of gas accretion at the beginning of the rapid gas runaway accretion phase. Theoretical models predict that at a total masse of ~30 MEarth, the rapid runaway accretion of gas sets in. If the accretion rate is very high in this phase, the transformation from a Neptunian type planet to a giant planet happens on a very short timescale. It is then unlikely that the gaseous disk disappears exactly during this short time interval, therefore it is predicted that planets in the intermediate mass regime (30-100 MEarth) are rare (so called planetary desert, Ida & Lin 2004). If the accretion rate is lower, a higher number of intermediate mass planets is in contrast predicted. Physically, the accretion rate depends on the local reservoir of gas, the gas capture cross section for a Bondi type accretion, and the viscous transport in the disk. The plot shows theoretical mass distributions from Neptunian to Jovian mass planets obtained from population synthesis calculations. The solid line shows a population where the gas accretion rate was only limited by the disk accretion rate if the planet has a mass higher than the local gas isolation mass. This means that the timescale of gas accretion is shorter, so that less intermediated mass planets are predicted. For the dotted line, the limit was always used (lower rates), so that only a minor deficit of intermediate mass planets results. Reference: Mordasini et al. (2011).

The animation shows a numerical simulation of the formation and subsequent evolution of Jupiter forming in orbit around the Sun. The planet forms via the core accretion mechanism, meaning that first a solid core made of ices and rocks forms and then a gaseous envelope of H/He is accreted from the circumstellar disk. The planetary embryo starts with a mass of approximately 0.6 Earth masses and is fixed at an orbital distance of 5.2 AU.

At the top, the numerical timestep and the elapsed time is shown. The top left panel contains the basic planetary properties (like the total, core, and envelope mass, radius, luminosity, accretion rates). The other panels of the top row show the mass M, radius R, and luminosity L as a function of time. In the mass panel, the yellow, red, and green line are the total, core, and envelope mass, respectively. In the radius panel, the total (yellow), core (red), and capture radius (green) are shown. The luminosity panel contains the total (yellow) and accretion shock luminosity (green).

The other panels show the radial structure of the gaseous envelope. The middle row shows the pressure P, temperature T, density rho, and contained mass M. The values of these quantities at the core-envelope interface are also shown. The bottom row shows the specific entropy S and opacity kappa as a function of radius. The pressure-temperature profile and the stability parameter Gamma are also given. Thick lines show convective layers.

The simulation starts at about 0.15 Mio years. In the first ~0.3 Mio years, the solid core forms until the isolation mass of about 6 Earth masses is reached. Then gas is accreted so that the core and envelope mass become equal at 2.1 Mio years at the so-called crossover point. Shortly afterwards, the gas accretion rate increases rapidly (runaway gas accretion), leading to a strong raise of the luminosity and mass. At 2.4 Mio years, the gas accretion rate hits the disk-limited value, so that the planet detaches from the nebula. The radius collapses rapidly from ~100 Jovian radii to initially about 2 Jovian radii. The luminosity reaches a maximum corresponding to about 0.3% of the luminosity of the Sun. Gas accretion continues at a high rate until the final mass of 320 Earth masses corresponding to ~1 Jovian mass is reached. The planet now simply cools and contracts at constant mass. At about 4.5 Gyrs, the radius has decreased to 1.01 Jovian radius, while the luminosity is about 1.11 Jovian intrinsic luminosities. The simulation ends at an age of 17 Gyrs.

The passage of a body (meteoroid, asteroid, comet, planetesimal) through the atmosphere of a planet or protoplanet can be described by the processes of gravity, gas drag, thermal ablation, flattening and aerodynamic fragmentation.

Collision of comet Shoemaker-Levy 9 with Jupiter (1994)

The impact of comet Shoemaker-Levy 9 into Jupiter allowed to test theoretical models for the interaction of massive bodies with planetary atmospheres. The plot shows the energy deposition per unit length as a function of height relative to the 1 bar level in Jupiter. The curves show the result for impactors with initial radii ranging between 0.24 to and 2.4 km. As expected, more massive bodies penetrate deeper. The black line shows for comparison the result of the hydrodynamic simulations of Mac Low & Zahnle 1994. Reference: Mordasini (2005)

Fate of planetesimals in the envelope of a growing protoplanet

During the nebular phase, already low-mass protoplanets (M~MEarth) can accrete an envelope of hydrogen and helium. The presence of this envelope has three important consequence for the further evolution of the planet: First, the capture radius of the core for planetesimals becomes (significantly) larger than the radius of the solid core itself due to drag-assisted capture. This speeds up the accretion of in particular small planetesimals or pebbles. Second, the place in the envelope where planetesimals deposit their energy influences the luminosity in the interior. Third, the place in the envelope where the planetesimals deposit their mass is important for the final size of the core, and the enrichment of the envelope. This enrichment can be compared with observations both in the Solar System, but also for extrasolar planets like GJ1214b which possibly has a very metal rich atmosphere. The plot shows as a function of envelope mass of the growing protoplanet, and initial size of the impacting planetesimals the fate of the planetesimals. Planetesimals to the right of the solid line penetrate down to the solid core, while impactors to the left get completely ablated in the envelope. Reference: Mordasini et al. (2005).

The gravitational interaction of a protoplanet with its host disk (gas or planetesimals) can lead to the migration of the planet due to angular momentum exchange. Depending on the mass of the planet and the properties of the disk, in- or outward migration results. The calculation of the resulting net torque is however non-trivial, because the torque exerted by the inner disk, and the one exerted by the outer disk typically nearly cancel.

Convergence zones: traps for migrating planets

Early descriptions of type I migration due to the gas disk assumed that the protoplanetary nebula is locally isothermal (e.g., Tanaka et al. 2002). In the last few years, progress was made, and this (often incorrect) assumption was dropped. These updated migration rates have the interesting implication that they lead to convergence zones, i.e. locations in the nebula towards which planets migrate both from the in- and the outside. Therefore, these convergence zones act as traps for migrating planets. In the figure, the background colors shows the regions of inward (blue) and outward (green) migration as a function of time in an evolving protoplanetary disk. The lines show the migration tracks of seven growing planetary embryo inserted at different starting locations. Colors indicate migration regimes. Blue: locally isothermal. Green: adiabatic, unsaturated. Red: adiabatic, saturated. Brown: type II. The outer four planets migrate towards the outer convergence zone, but leave it again after some time, as they grow to a mass where the corotation torque saturates (disappears). The three inner planets which are caught in the inner convergence zone in contrast remain in it until the end of the simulation. Reference: Dittkrist et al. (2014).

Evolution of protoplanetary disks

The direction and timescale of orbital migration is very sensitive to the properties of the protoplanetary disk, like its vertical scale height or the radial gradients of the temperature and surface density. In order to model planetary migration, one therefore also needs to model the evolution of the disk. The plot shows the temporal evolution of the surface density of gas as a function of distance from the star. The initial state at t=0 corresponds to the top curve, while the end state at 2 Myrs is shown by the bottom line. At this time, the total disk mass has fallen to 0.00001 solar masses. The disk evolves under the action of viscosity modeled with a constant alpha prescription, and internal and external photoevaporation. External photoevaporation erodes the disk from outside-in, while internal photoevaporation by the host star "burns" at late times a hole in the disk at about 6 AU. Reference: Mordasini et al. (2012)