Bayesian Evolution Models for Jupiter With Helium Rain and Double-Diffusive Convection

Abstract

Hydrogen and helium demix when sufficiently cool, and this bears on the evolution of all giant planets at large separations at or below roughly a Jupiter mass. We model the thermal evolution of Jupiter, including its evolving helium distribution following results of ab initio simulations for helium immiscibility in metallic hydrogen. After 4 Gyr of homogeneous evolution, differentiation establishes a thin helium gradient below 1 Mbar that dynamically stabilizes the fluid to convection. The region undergoes overstable double-diffusive convection (ODDC), whose weak heat transport maintains a superadiabatic temperature gradient. With a generic parameterization for the ODDC efficiency, the models can reconcile Jupiter's intrinsix flux, atmospheric helium content, and radius at the age of the solar system if the Lorenzen et al. H-He phase diagram is translated to lower temperatures. We cast the evolutionary models in an MCMC framework to explore tens of thousands of evolutionary sequences, retrieving probability distributions for the total heavy element mass, the superadiabaticity of the temperature gradient due to ODDC, and the phase diagram perturbation. The adopted SCvH-I equation of state favors inefficient ODDC such that a thermal boundary layer is formed, allowing the molecular envelope to cool rapidly while the deeper interior actually heats up over time. If the overall cooling time is modulated with an additional free parameter to imitate the effect of a colder or warmer EOS, the models favor those that are colder than SCvH-I. In this case the superadiabaticity is modest and a warming or cooling deep interior are equally likely. Subject headings: planets and satellites: physical evolution -planets and satellites: interiors -planets and satellites: individual (Jupiter) -methods: statistical 1. INTRODUCTION Cool giant planets are relics of the protoplanetary systems from which they formed in the sense that they do not fuse protons, and they are well-bound enough that even hydrogen does not escape appreciably over tens of billions of years. Their thermal evolution is thus relatively simple, and understanding it empowers us to use the present states of giant planets to learn about their history and formation. The open questions about planet formation thus motivate a comprehensive theory of giant planet evolution, which will continue to be driven heavily by our own, well-studied giant planets, Jupiter and Saturn.A Henyey-type stellar evolution calculation for a Jupitermass object was first performed by Graboske et al. (1975), who showed that a convective, homogeneous sphere of fluid hydrogen and helium could cool to Jupiter's observed luminosity over roughly the right timescale, and noted that among all model inputs, the equation of state (EOS) and superadiabaticity of the temperature gradient have the strongest influence on the overall cooling time.These two fundamental physical inputs are closely related. The EOS (paired with a hydrostatic model) is necessary to translate the planet's tangible properties...

Evidence for a Dichotomy in the Interior Structures of Jupiter and Saturn from Helium Phase Separation

“…Although superadiabaticity resulting from some flavor of double-diffusive convection in metallic regions possessing helium gradients does modulate the cooling time for Jupiter and Saturn, it is not required in all cases. In particular, good solutions for Jupiter do require R ρ > 0, consistent with the expectation from earlier modeling efforts (e.g., Hubbard et al 1999;Fortney & Hubbard 2003;Fortney et al 2011;Nettelmann et al 2015;Mankovich et al 2016) that Jupiter required some mechanism (non-adiabatic interiors or otherwise) to speed its evolution rather than prolong it 2 . However, the new models find interiors that are closer to adiabatic (R ρ closer to zero) due to the major improvement in our understanding of the internal heat flow of Jupiter in light of results from Cassini (Li et al 2018).…”

“…The parameter estimation performed in this work provides statistically meaningful distributions of model parameters, estimating for instance that based on Jupiter's helium depletion, the true phase diagram is warmer than the most current ab initio phase diagram (Schöttler & Redmer 2018) by (539 ± 23) K (1σ uncertainty) at the ≈ 2 Mbar pressures that it predicts for the onset of helium immiscibility in metallic hydrogen. For comparison, Jupiter models built on a previous generation of phase diagram that assumed ideal hydrogen-helium mixing entropy predicted the necessary temperature offset to be between 200 and 300 K, in the opposite direction (Nettelmann et al 2015;Mankovich et al 2016). These findings imply a rather precise prediction for Saturn's atmospheric helium abundance summarized in Figure 14.…”

“…Assuming that hydrogen-helium immiscibility is the mechanism responsible for Jupiter's atmospheric helium depletion, the Galileo abundance measurement imposes a stringent constraint for discerning among viable, if still uncertain, phase diagrams. It is for this reason that, following Nettelmann et al (2015) and Mankovich et al (2016), we introduce a degree of freedom via an additive temperature offset ∆T phase modulating the overall temperature of the phase curves applied in this work, relative to SR18. This parameter allows us to explore a more general space of phase diagrams, with larger ∆T phase values leading to more pronounced differentiation, smaller values leading to less, and ∆T phase = 0 recovering the SR18 phase curves as published.…”

“…same simple, generic model as in Mankovich et al (2016). The temperature gradient is allowed to take on superadiabatic values in helium gradient regions, the value of the superadiabaticity assumed to be proportional to the magnitude of the helium gradient there following…”

“…As described in Section 2.3, the phase diagram temperature offset ∆T phase controls the temperatures of the hydrogen-helium phase curves and thus dictates the overall amount of helium differentiation. Finally the density ratio R ρ sets the superadiabaticity of the temperature profile in regions with continuous helium gradients, providing an additional degree of freedom in setting the rate of cooling from the planet's surface by limiting the flux emerging from the metallic interior (e.g., Nettelmann et al 2015;Mankovich et al 2016).…”