Context. The evolution of stars and planets is mostly controlled by the properties of their atmosphere. This is particularly true in the case of exoplanets close to their stars, for which one has to ...account both for an (often intense) irradiation flux, and from an intrinsic flux responsible for the progressive loss of the inner planetary heat. Aims. The goals of the present work are to help understanding the coupling between radiative transfer and advection in exoplanetary atmospheres and to provide constraints on the temperatures of the deep atmospheres. This is crucial in assessing whether modifying assumed opacity sources and/or heat transport may explain the inflated sizes of a significant number of giant exoplanets found so far. Methods. I use a simple analytical approach inspired by Eddington's approximation for stellar atmospheres to derive a relation between temperature and optical depth valid for plane-parallel static grey atmospheres which are both transporting an intrinsic heat flux and receiving an outer radiation flux. The model is parameterized as a function of mean visible and thermal opacities, respectively. Results. The model is shown to reproduce relatively well temperature profiles obtained from more sophisticated radiative transfer calculations of exoplanetary atmospheres. It naturally explains why a temperature inversion (stratosphere) appears when the opacity in the optical becomes significant compared to that in the infrared. I further show that the mean equivalent flux (proportional to T4) is conserved in the presence of horizontal advection on constant optical depth levels. This implies with these hypotheses that the deep atmospheric temperature used as outer boundary for the evolution models should be calculated from models pertaining to the entire planetary atmosphere, not from ones that are relevant to the day side or to the substellar point. In these conditions, present-day models yield deep temperatures that are ~1000 K too cold to explain the present size of planet HD 209458b. An tenfold increase in the infrared to visible opacity ratio would be required to slow the planetary cooling and contraction sufficiently to explain its size. However, the mean equivalent flux is not conserved anymore in the presence of opacity variations, or in the case of non-radiative vertical transport of energy: The presence of clouds on the night side or a downward transport of kinetic energy and its dissipation at deep levels would help making the deep atmosphere hotter and may explain the inflated sizes of giant exoplanets.
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Context. For up to a few millions of years, pebbles must provide a quasi-steady inflow of solids from the outer parts of protoplanetary disks to their inner regions. Aims. We wish to understand how a ...significant fraction of the pebbles grows into planetesimals instead of being lost to the host star. Methods. We examined analytically how the inward flow of pebbles is affected by the snow line and under which conditions dust-rich (rocky) planetesimals form. When calculating the inward drift of solids that is due to gas drag, we included the back-reaction of the gas to the motion of the solids. Results. We show that in low-viscosity protoplanetary disks (with a monotonous surface density similar to that of the minimum-mass solar nebula), the flow of pebbles does not usually reach the required surface density to form planetesimals by streaming instability. We show, however, that if the pebble-to-gas-mass flux exceeds a critical value, no steady solution can be found for the solid-to-gas ratio. This is particularly important for low-viscosity disks (α< 10-3) where we show that inside of the snow line, silicate-dust grains ejected from sublimating pebbles can accumulate, eventually leading to the formation of dust-rich planetesimals directly by gravitational instability. Conclusions. This formation of dust-rich planetesimals may occur for extended periods of time, while the snow line sweeps from several au to inside of 1 au. The rock-to-ice ratio may thus be globally significantly higher in planetesimals and planets than in the central star.
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Context. Heavy elements, even though they are a smaller constituent, are crucial to understand the formation history of Jupiter. Interior models are used to determine the amount of heavy elements in ...the interior of Jupiter, but this range is still subject to degeneracies because of the uncertainties in the equations of state. Aims. Before Juno mission data arrive, we present optimized calculations for Jupiter that explore the effect of different model parameters on the determination of the core and the mass of heavy elements of Jupiter. We compare recently published equations of state. Methods. The interior model of Jupiter was calculated from the equations of hydrostatic equilibrium, mass, and energy conservation, and energy transport. The mass of the core and heavy elements was adjusted to match the observed radius and gravitational moments of Jupiter. Results. We show that the determination of the interior structure of Jupiter is tied to the estimation of its gravitational moments and the accuracy of equations of state of hydrogen, helium, and heavy elements. Locating the region where helium rain occurs and defining its timescale is important to determine the distribution of heavy elements and helium in the interior of Jupiter. We show that the differences found when modeling the interior of Jupiter with recent EOS are more likely due to differences in the internal energy and entropy calculation. The consequent changes in the thermal profile lead to different estimates of the mass of the core and heavy elements, which explains differences in recently published interior models of Jupiter. Conclusions. Our results help clarify the reasons for the differences found in interior models of Jupiter and will help interpreting upcoming Juno data.
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Context. The classical planetesimal accretion scenario for the formation of planets has recently evolved with the idea that pebbles, centimeter- to meter-sized icy grains migrating in protoplanetary ...disks, can control planetesimal and/or planetary growth. Aims. We investigate how pebble accretion depends on disk properties and affects the formation of planetary systems. Methods. We construct analytical models of pebble accretion onto planetary embryos that consistently account for the mass and orbital evolution of the pebble flow and reflect disk structure. Results. We derive simple formulas for pebble accretion rates in the so-called settling regime for planetary embryos that are more than 100 km in size. For relatively smaller embryos or in outer disk regions, the accretion mode is three-dimensional (3D), meaning that the thickness of the pebble flow must be taken into account, and resulting in an accretion rate that is independent of the embryo mass. For larger embryos or in inner regions, the accretion is in a two-dimensional (2D) mode, i.e., the pebble disk may be considered infinitely thin. We show that the radial dependence of the pebble accretion rate is different (even the sign of the power-law exponent changes) for different disk conditions such as the disk heating source (viscous heating or stellar irradiation), drag law (Stokes or Epstein, and weak or strong coupling), and in the 2D or 3D accretion modes. We also discuss the effect of the sublimation and destruction of icy pebbles inside the snow line. Conclusions. Pebble accretion easily produces a large diversity of planetary systems. In other words, to infer the results of planet formation through pebble accretion correctly, detailed prescriptions of disk evolution and pebble growth, sublimation, destruction and migration are required.
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Context.
The equation of state for hydrogen and helium is fundamental for studying stars and giant planets. It has been shown that because of interactions at atomic and molecular levels, the ...behaviour of a mixture of hydrogen and helium cannot be accurately represented by considering these elements separately.
Aims.
This paper aims at providing a simple method to account for interactions between hydrogen and helium in interior and evolution models of giant planets.
Methods.
Using on the one hand ab initio simulations that involve a system of interacting hydrogen and helium particles and pure equations of state for hydrogen and helium on the other, we derived the contributions in density and entropy of the interactions between hydrogen and helium particles.
Results.
We show that relative variations of up to 15% in density and entropy arise when non-ideal mixing is accounted for. These non-ideal mixing effects must be considered in interior models of giant planets based on accurate gravity field measurements, particularly in the context of variations in the helium-to-hydrogen ratio. They also affect the mass-radius relation of exoplanets. We provide a table that contains the volume and entropy of mixing as a function of pressure and temperature. This table is to be combined with pure hydrogen and pure helium equations of state to obtain an equation of state that self-consistently includes mixing effects for any hydrogen and helium mixing ratio and may be used to model the interior structure and evolution of giant planets to brown dwarfs.
Conclusions.
Non-linear mixing must be included in accurate calculations of the equations of state of hydrogen and helium. Uncertainties on the equation of state still exist, however. Ab initio calculations of the behaviour of the hydrogen-helium mixture in the megabar regime for various compositions should be performed in order to gain accuracy.
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How deep do Saturn's zonal winds penetrate below the cloud level has been a decades‐long question, with important implications not only for the atmospheric dynamics but also for the interior density ...structure, composition, magnetic field, and core mass. The Cassini Grand Finale gravity experiment enables answering this question for the first time, with the premise that the planet's gravity harmonics are affected not only by the rigid body density structure but also by its flow field. Using a wide range of rigid body interior models and an adjoint based optimization for the flow field using thermal wind balance, we calculate the flow structure below the cloud level and its depth. We find that with a wind profile, largely consistent with the observed winds, when extended to a depth of around 8,800 km, all the gravity harmonics measured by Cassini are explained. This solution is in agreement with considerations of angular momentum conservation and is consistent with magnetohydrodynamics constraints.
Plain Language Summary
Observations show strong east‐west flows at the cloud level of Saturn. These winds are strongest at the equatorial regions, reaching up to 400 m/s, about 4 times stronger than tornado strength winds on Earth. Yet until now we had no knowledge on how deep these winds penetrate into the interior of the gas giant. The gravity experiment executed during the Grand Finale stage (May–August 2017) of the NASA Cassini mission helps answering this question. It is well established that any large‐scale motion of the fluid would have a signature in the density distribution and therefore in the planet gravity field. If we can estimate the internal structure and shape of the planet, we might be able to decipher the depth of the winds from its signal in the gravity measurements. Moreover, the rigid‐body and flow contribution to gravity field are entangled together, therefore it is necessary to use a wide range of rigid‐body models in order to define the wind‐induced gravity signal. In this work we propose a solution to the problem. We find that the gravity measurements can be explained with a flow pattern, similar to that observed at the cloud level, penetrating to depths of more than 8,000 km into the planet interior. This has important implications not only for the atmospheric dynamics but also for the interior density structure, composition, magnetic field, and core mass.
Key Points
Cassini gravity measurements point to deep differential flows within Saturn
Using a wide range of rigid body internal structure models, the required wind‐induced gravity signal is defined
With a conservatively modified cloud‐level wind and an optimized vertical profile, extended to a depth of 8,800km, all gravity measurements are explained
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We know that giant planets played a crucial role in the making of our Solar System. The discovery of giant planets orbiting other stars is a formidable opportunity to learn more about these objects, ...what their composition is, how various processes influence their structure and evolution, and most importantly how they form. Jupiter, Saturn, Uranus, and Neptune can be studied in detail, mostly from close spacecraft flybys. We can infer that they are all enriched in heavy elements compared to the Sun, with the relative global enrichments increasing with distance to the Sun. We can also infer that they possess dense cores of varied masses. The intercomparison of presently characterized extrasolar giant planets shows that they are also mainly made of hydrogen and helium, but that they either have significantly different amounts of heavy elements, have had different orbital evolutions, or both. Hence, many questions remain and need to be answered to make significant progress on the origins of planets.
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The Juno spacecraft has measured Jupiter's low‐order, even gravitational moments, J2–J8, to an unprecedented precision, providing important constraints on the density profile and core mass of the ...planet. Here we report on a selection of interior models based on ab initio computer simulations of hydrogen‐helium mixtures. We demonstrate that a dilute core, expanded to a significant fraction of the planet's radius, is helpful in reconciling the calculated Jn with Juno's observations. Although model predictions are strongly affected by the chosen equation of state, the prediction of an enrichment of Z in the deep, metallic envelope over that in the shallow, molecular envelope holds. We estimate Jupiter's core to contain a 7–25 Earth mass of heavy elements. We discuss the current difficulties in reconciling measured Jn with the equations of state and with theory for formation and evolution of the planet.
Plain Language Summary
The Juno spacecraft has measured Jupiter's gravity to unprecedented precision. We present models of the planet's interior structure, which treat the hydrogen‐helium mixture using computer simulations of the material. We demonstrate that dilute core, with the heavy elements dissolved in hydrogen and expanded outward through a portion of the planet, may be helpful for explaining Juno's measurements.
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We present a one-dimensional model of the formation and viscous evolution of protoplanetary disks. The formation of the early disk is modeled as the result of the gravitational collapse of an ...isothermal molecular cloud. The disk's viscous evolution is integrated according to two parameterizations of turbulence: the classical α representation and a β parameterization, representative of non-linear turbulence driven by the keplerian shear. We apply the model to DM Tau and GM Aur, two classical T-Tauri stars with relatively well-characterized disks, retrieving the evolution of their surface density with time. We perform a systematic Monte-Carlo exploration of the parameter space (i.e. values of the α-β parameters, and of the temperature and rotation rate in the molecular cloud) to find the values that are compatible with the observed disk surface density distribution, star and disk mass, age and present accretion rate. We find that the observations for DM Tau require $0.001<\alpha<0.1$ or $2\times 10^{-5}<\beta<5\times 10^{-4}$. For GM Aur, we find that the turbulent viscosity is such that $4\times 10^{-4}<\alpha<0.01$ or $2\times 10^{-6}<\beta<8\times 10^{-5}$. These relatively large values show that an efficient turbulent diffusion mechanism is present at distances larger than ~$10\,$AU. This is to be compared to studies of the variations of accretion rates of T-Tauri stars versus age that mostly probe the inner disks, but also yield values of $\alpha\sim 0.01$. We show that the mechanism responsible for turbulent diffusion at large orbital distances most probably cannot be convection because of its suppression at low optical depths.
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The shape of the two gas giants, Jupiter and Saturn, is determined primarily by their rotation rate, and interior density distribution. It is also affected by their zonal winds, causing an anomaly of ...O(10 km) at low latitudes. However, uncertainties in the observed cloud‐level wind and the polar radius, translate to an uncertainty in the shape with the same order of magnitude. The Juno (Jupiter) and Cassini (Saturn) missions gave unprecedented accurate gravity measurements, constraining better the uncertainty in the wind structure. Using an accurate shape calculation, and a joint optimization, given both gravity and radio‐occultation measurements, we calculate the possible range of dynamical height for both planets. We find that for Saturn there is an excellent match to the radio‐occultation measurements, while at Jupiter such a match is not achieved. This may point to deviations from a barotropic flow above the cloud level, which might be tested with the forthcoming radio‐occultation measurements by Juno.
Plain Language Summary
The shape of the gaseous planets Jupiter and Saturn, are predominantly set by their rotation rate, but are also affected by the density structure (manifested in the planet's gravity field), and the winds at the planet's outer surface. For both Jupiter and Saturn, the gravity fields have been measured to high accuracy by NASA's Juno and Cassini missions, respectively. This, together with the observed zonal winds, allows an accurate calculation of their shapes. Further constraints can be obtained from radio‐occultation measurements, which give radially dependent profiles of density for specific spatial locations. Here, we propose a new method for calculating the shape of the gas giants, based on an optimization of the wind latitudinal profile, decay structure, and the polar radius, given both gravity and radio‐occultation measurements. We use thermal wind balance to relate the wind to the gravity measurements, and a shape model to relate the wind and polar radius to the radio‐occultation measurements. We find that for Saturn there is a good match between the calculated shape and the radio‐occultation measurements, while for Jupiter, no such correlation exists. We expect the new radio‐occultation measurements to be performed by Juno, to help resolve the shape of Jupiter with a better accuracy.
Key Points
The shapes of Jupiter and Saturn are calculated by jointly fitting their gravity and radio‐occultation measurements
Saturn's shape has a good match to the radio‐occultation measurements, while Jupiter's shape does not
The upcoming Juno radio‐occultation experiment might give better constraints on the shape of Jupiter
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