Calculation of Jupiter’s gravitational field: testing the Concentric Maclaurin Spheroid method

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Mars is expected to shed light on many unknowns of terrestrial planetary formation.

Gaseous planets, on the other hand, are a bit more challenging. First the formation process is less clear as the mass coming from accretion of planetesimals is unknown, contrary to terrestrial planets mainly formed by colliding rocks. But most importantly classical seismology is impossible to perform on these planets. On stars, asterosismology is a growing field. The pulsations of the Sun were discovered in 1960 by Robert Leighton [Leighton et al., 1962] and allowed the inference of the Sun’s structure in the last 50 years. Mainly, we can observe acoustic waves with sufficient magnitude to decompose them onto spherical harmonics up to orders superior to 1000. On Jupiter, the convection zone breaks the propagation of gravity waves, as in the sun, but the acoustic waves on this cooler and smaller object have a lower amplitude and lower frequency, which make them harder to separate from the atmo-spheric motions. Still, Gaulme et al. [2011] were able to provide the first observations of Jupiter’s pulsation. This work hasn’t been used yet to derive internal models of Jupiter because signal to noise ratios are still too low, and these observations are quite scarce and preliminary.
One of the remaining options to understand giant planets’ structure and formation is to study their gravitational field. If Jupiter was a homogeneous sphere, we could only learn its mass from the knowledge of its gravity. Fortu-nately, Jupiter is stratified in density and rotating hence the deviation of the gravity field from that of a homogeneous sphere can teach us lots about the composition of the planet. The first satellite observations were made 40 years ago, when the trajectories of Pioneer and Voyager allowed the evaluation of the Jovian gravity field (see notably Campbell and Synnott [1985]). Subsequent satellites have since studied the attraction of Jupiter in depth. Galileo provided better constraints on the atmospheric composition of the planet, especially with its entry probe that dived into the outer layers (Seiff et al. [1998] and von Zahn et al. [1998]). More recently, Juno has measured with unprecedented accuracy the gravity field and magnetic field of the planet (Bolton et al. [2017], Iess et al. [2018], Connerney et al. [2018]). Juno also provided pictures of the planet of tremendous quality, as displayed on Figure 1.1.
Such a precision on the gravity field of Jupiter is a tremendous opportunity for the astrophysical community. It allows us to refine appropriate models or refute previous underconstrained propositions. But it also reminds us of the huge degree of degeneracy in the models. No matter the precision, inferring Jupiter’s composition by solely observing its external gravity field yields a huge span of fitting models. Additionally, older models predicted a metallization transition with an entropy jump (see Guillot and Gautier [2014] for a review), but recent calculations with double diffusive convection in Leconte and Chabrier [2012] propose a gradient of entropy and heavy elements throughout the whole planet, eventually suppressing the need for a compact core in Jupiter. The different proposed physical processes increase yet further the degeneracy.
In this context, we have been interested in understanding Jupiter with the most up to date physical and math-ematical developments. Mathematically speaking, William Hubbard in Hubbard [2012] and Hubbard [2013] has developed a powerful method to calculate the gravitational potential of a barotropic, hydrostatically balanced celes-tial body (said otherwise, a giant planet). Although its numerical cost is quite expensive, the level of adaptability of this method makes it an outstanding tool for understanding giant planets. Our first contribution was to study the limitations of this method in the context of Jupiter, to assess that its level of accuracy is sufficient with the extremely precise data of Juno. As reported in Debras and Chabrier [2018], we have highlighted the conditions under which the method can be safely used, and our knowledge of Jupiter improved.
Physically speaking, the advances in numerical quantum molecular dynamics (e.g., Soubiran and Militzer [2016], Mazzola et al. [2018] and other references throughout the thesis), plasma experiments (e.g., Loubeyre et al. [1985], Pépin et al. [2017], Celliers et al. [2018], etc.) and transport processes from numerical simulations and analytical expectations (e.g., Rosenblum et al. [2011], Moll et al. [2017], Leconte and Chabrier [2012], etc.) allow us to under-stand the conditions prevailing in the interior of Jupiter with much more subtlety. Based on these recent results, we have worked on deriving new interior structures of the planet, constrained by the knowledge of theoretical physics. Our work, submitted for revision in the Astrophysical Journal, shows that the structure of Jupiter is more complex than previously thought. However, lots of questions remain unanswered, and our knowledge of the interior of Jupiter will without a doubt improve in the fortchoming years.
Nonetheless, this broad picture of the structure of Jupiter and the understanding of the physical processes at stake in the planet have a fundamental, contemporary application: extrasolar planets. Since the first discovery of an exoplanet in 1995 [Mayor and Queloz, 1995], more than 3000 planets have been detected ( and there are still more than 8000 candidates yet to be confirmed. Among them, a few percent are gas giants, with sizes and masses ranging from sub-Neptune to super-Jovian, and orbits, eccentricities, radius and inclinations scattered across the parameter space. As we have seen, although 6 satellites have observed Jupiter from up close, our knowledge of the planet remains quite poor. In that regard, deeper understanding of extrasolar planets is a difficult, observationally limited task. Although comparison can be misleading when conditions are extremely different, it seems that using the solar system planets as templates for exoplanet structure is one of the only few ways of gaining further information. Of course, it is also needed to explore and exploit as much as possible the physics we can assess from the sole observations of these planets, reminding that to date, a direct image of a planet is actually one saturated pixel on a state of the art camera, with a 8 meters telescope.
Most exoplanets have been detected by two methods: radial velocities and transits. The imprecision from the transit, not able to distinguish a planet from a distant binary for example, requires a validation by radial velocities to confirm the existence of a planet. This seems constraining, but it has the advantage of providing complementary information on extrasolar planets. On the one hand, the radial velocity technique aims at observing the motion of the star around the centre of gravity of the star+planet system. It allows to estimate the mass of the planet (given the mass of the star and an unknown inclination) as well as its semi major axis and eccentricity. The more massive the planet, the easier the detection is.
On the other hand, in order to observe a transit, the planet has to cross our line of sight between its star and the Earth. As the planet passes in front of its star, we can observe a dip in the luminosity of the star, which gives an estimate of the radius of the planet given the radius of the star. Therefore, the bigger the planet and the shorter its orbital period, hence the closer it is to its star, the easier the detection. At the dawn of exoplanetary science, the biggest, closest to their stars planets were therefore the first to be observed. At the time this thesis was written, we are approaching the limit were a planet strictly analogous to the Earth could be discovered (which happens to be the most difficult targets to observe). We show in Figure 1.2 the fraction of planet per star, as we picture it to date, as well as the diversity of observed planets by Kepler. Clearly, the solar system is not enough to fully understand extrasolar planets.
The first detection notably was amongst a class of planets that doesn’t exist in the solar system: hot Jupiters. These are Jupiter-mass planets about 10 times closer to their star than Mercury is to the Sun. Because of this proximity, they are supposed to be tidally locked (see the review of Baraffe et al. [2010]) as the Moon is to Earth, thus always presenting the same side to the star. They also seem to have a much bigger radius than predicted by our understanding of planetary evolution. This issue of inflated radius is still a puzzle for the scientific community (see notably Guillot and Showman [2002], Batygin and Stevenson [2010] and Tremblin et al. [2017] and chapter 5).
What is really interesting with the transit detection method is that when the planet passes in front of the star the light of the star is affected by the composition of the planet. At the limbs, where the optical depth is low enough to let the incoming light from the star escape to our telescopes, molecules of the atmosphere of the planet can absorb a fraction of the energy. Therefore by comparing the spectroscopic signature of the star alone and the observed spectrum when the planet transits, one can obtain clues on the chemical composition of the atmosphere of the planet. It also becomes possible in some cases to observe the emission spectra of the planet during the secondary transit, just before the planet passes behind its stars. Instead of having a dip in the luminosity there is an increase in the received flux because of the reflection of the starlight by the planet. A comparative study of the composition of 10 hot Jupiters, as observed by transit techniques, has been performed in Sing et al. [2016]. Even with such a low statistical sample, it is quite obvious that there is a tremendous diversity of hot Jupiters.
The new data provided by exoplanetary observations demand new theories or numerical experiments in order to interpret them. Specifically, it is needed to predict what the atmosphere of hot Jupiters should be like. From meterological research, the numerical method used to study the evolution of an atmosphere is called a Global Cir-culation Models (GCM). First dedicated to weather prediction on Earth [Phillips, 1956], these models solve the equations of motion and of conservation (mass, energy, momentum) of an atmosphere to understand both the local scale for weather forecasting and the global scale for climate studies. These GCM are now adapted for astrophysics purposes, first on the solar system (on Venus typically [Kalnay de Rivas, 1975]) or to go back in time on Earth. In 2002, Showman and Guillot [2002] used for the first time a GCM on a hot Jupiter, leading the way to more than fifteen years of numerical predictions of exoplanet atmospheres. Many scientific teams across the world followed this pioneering work (Rauscher and Menou [2012], Perna et al. [2012], Dobbs-Dixon and Agol [2013], Heng et al. [2011] for example) and today there are multiples GCM used for exoplanetary science, not only dedicated to hot Jupiters. The university of Exeter works with the Unified Model (UM), their own adaptation of the Met Office GCM first presented in Mayne et al. [2014b] and used throughout this thesis.
These GCMs provide a framework for interpreting the observational data, but do not allow for a comprehensive grasp of the physical phenomena at stake. Additional understanding has to be provided by simpler, physically motivated studies. One of the main features obtained by GCM for hot Jupiter is the presence of an equatorial superrotation: at the equator the winds are going faster than the solid body rotation of the planet, up to a few kilometers per second. As we have shown in Mayne et al. [2017], this feature, first predicted by Showman and Guillot [2002], is largely insensitive to the equations and conditions used in the GCM as long as the planet is tidally locked and intensively irradiated.
The first comprehensive theory explaining this superrotation has been provided by Showman and Polvani [2011], later completed by Tsai et al. [2014] and Komacek and Showman [2016]. They rely on the propagation and in-teraction of stationary forced waves with the mean flow, setting up an initial circulation that eventually leads to superrotation. The complementarity of these three papers, from the simple 2D shallow water theory in Showman and Polvani [2011], to the 3D wave resonance study of Tsai et al. [2014] and the analytical orders of magnitude estimates of Komacek and Showman [2016], allows for a global coherent understanding of superrotation in hot Jupiters. However, there are still a few uncertainties, notably regarding the viability of the proposed mechanism with an accurate treatment of the radiative transfer and in the limits of low dissipation. In this context, we have also worked on enlightening the remaining areas of the physical explanation for superrotation on hot Jupiters. We have tackled this issue with the use of ECLIPS3D, a linear solver we have developed, and analytical arguments on both the dissipation of the waves excited by the stellar insolation and the momentum transfers they imply. This work, complemented by the numerical study performed in Mayne et al. [2017], is about to be submitted to Astronomy and Astrophysics.
Globally, in this thesis, I conduct a tour of many physical processes of prime importance in giant planets. Starting from the deep interior of Jupiter to the extreme conditions of the irradiated atmosphere of gaseous exoplanets, and notably HD209458b, I study the structure, equilibrium and disequilibrium properties of giant planets. In the context of Juno and the ever accelerating field of exoplanetary science, this manuscript is encapsulated in a thrilling context that brings new vision on the properties of gas giants. Jupiter, by its own, possesses more than 90% of the angular momentum of the solar system and is twice more massive than the other planets of the solar system altogether. Eventually, a better understanding of gas giants, starting 5 astronomical units from the Earth and going as far as a few hundreds of light years, will yield valuable knowledge on the formation and evolution of planetary systems across the galaxy.

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Outline of the thesis

This manuscript is divided into 8 chapters, starting with this introduction. Chapter 2 details the theoretical back-ground of the thesis, and notably the wave mechanisms relevant for atmosperic dynamics as well as the theory of convection in the deep interior of Jupiter. Then, chapter 3 provides a thorough study of the concentric Maclaurin spheroid method, which allowed me to derive new internal structures of Jupiter in chapter 4, based on the ther-modynamical properties of hydrogen and helium. This work is the only one to date to match all the observational data, and we provide a thorough discussion on the physical assumptions of the models.
Chapter 5 is devoted to the transition from the core of Jupiter to the atmospheres of hot Jupiters, notably by explaining the issue of the inflated radius and by introducing global circulation models. In chapter 6 is detailed the linear code that I developed for the study of waves, instabilities and steady circulations in planetary atmospheres. As this chapter is mainly a benchmark of the code, it can be skipped at first reading. Combining theoretical calculations, global circulation models and ECLIPS3D, chapter 7 focuses on the understanding of the spin up of superrotation in hot Jupiters. This chapter extands prior works on the superrotation of hot Jupiters with the consideration of the time dependent linear solution and vertical momentum transfers. Finally, chapter 8 concludes the thesis, and details the perspectives for further studies.

Equations of motion: Navier-Stokes equations

This section is largely inspired by Vallis [2006] where a full derivation of the equations can be found.

Eulerian and Lagrangian viewpoints

The notion of velocity in fluid dynamics is not obvious. When looking at a point object, its speed simply is the infinitesimal variation of its position with respect to time. For a solid, Newton laws are applied to its center of gravity and the inertia equation provides constraints on the equilibrium or global motion of an object. Its thermo-dynamical quantities can depend on the position in the solid, but the dynamical quantities are, on average, common to the whole object. The question is: how to define dynamical quantities for a fluid?
There are two ways of doing so. The first is to consider that an observer is in a fixed frame of reference and is looking at the properties of the fluid at a definite point of space. The quantities are then evolving with the fluid passing by and the observer sees the fluid as a field rather than an ensemble of particles.
The other approach is to consider that the fluid is an ensemble of parcels with their own properties that are moving together. In this Lagrangian description, we are following the evolution of a specific piece of the fluid within the surrounding flow. Therefore, the Lagrangian derivative, D/Dt represents the evolution with time of any quantity of the parcel, supposed constant within the parcel, as the parcel moves within the fluid. If an Eulerian observer was to follow the parcel, they would observe that a quantity q (temperature, density, …) has an intrinsic change, ∂q/∂t plus an evolution due to the advection of the parcel within the fluid. The Lagrangian derivative can therefore be connected to the Eulerian derivative through:
where #v» = #v»(x, y, z, t) is the velocity field of the fluid. Although most of this thesis is written from a Eulerian approach, this distinction is important to bear in mind when interpreting the results (convection within Jupiter, notably, is easier to understand phenomenologically from a Lagrangian point of view).

Equations of hydrodynamics

The description of motion for a fluid rely on three conservation laws, leading to a set of 5 equations. First in a closed system the mass must be conserved, which leads to the equation of continuity. This conservation implies that the change in the density of a parcel of fluid must be equal to the transport of density by the flow, so that there is no mass source or sink (we neglect any non conservation of mass due to nuclear effects). This gives:
The second conserved quantity is the momentum. As expressed in Newton’s second law, the derivative of the momentum of a system with respect to time is equal to the forces applied on this system. With a fluid dynamics approach we obtain:
where p is the pressure, ν the viscosity and F the external forces (per unit mass). This states that the change of the momentum of a parcel is determined by the forces applied to the parcel: the pressure gradient, the viscous effects and any other external force applied to the system (gravity mostly for a planet). This conservation law actually contains three equations as the velocity vector has three scalar components. At the first order, |∇2 #v»| can be estimated as |v| divided by the square of a characteristic length over which v changes. Calling Lv this length we have |ν∇2 #v»| ≈ νv/L2v = v/τv where τv = L2v /ν is a characteristic time of dissipation for the velocity. Diffusion by viscosity, which in the Navier Stokes equation acts as redistributing the momentum within the fluid, can also be seen as a friction which opposes the emergence of strong velocities on a timescale τv (which is not constant within the fluid). We will consider this approximation in the final chapters of this thesis.

Table of contents :

1 Introduction 
2 Definition of the basic concepts 
2.1 Equations of motion: Navier-Stokes equations
2.1.1 Eulerian and Lagrangian viewpoints
2.1.2 Equations of hydrodynamics
2.1.3 The planetary case: rotating fluid under gravity
2.1.4 Beta-plane approximation and shallow water equations
2.1.5 Importance of waves in the dynamics
2.2 Structure of a giant planet
2.2.1 The gravitational moments
2.2.2 Why are giant planets convective?
2.2.3 Hydrostatic balance
2.3 Short summary
3 Calculation of Jupiter’s gravitational field: testing the Concentric Maclaurin Spheroid method 
3.1 Brief overview of the CMS method
3.2 Evaluation of the uncertainties of the CMS method
3.2.1 Analytical evaluation
3.2.2 Spacing as a power of k
3.2.3 Exponential spacing of the layers
3.3 Numerical calculations
3.3.1 How to match Juno’s error bars
3.3.2 Importance of the first layers
3.3.3 1 bar radius and external radius
3.4 Calculations with a realistic equation of state
3.4.1 Impact of the high atmospheric layers on the CMS method
3.4.2 Errors arising from a (quasi) linear repartition
3.4.3 Intrinsic uncertainties on the Jupiter models
3.5 Taking the 1 bar level as the outer boundary condition
3.5.1 Irreducible errors due to the high atmosphere region (less than 1 bar)
3.5.2 Error from the finite number of spheroids
3.6 Conclusion
4 New models of the interior of Jupiter 
4.1 A brief history of Jupiter
4.2 Method
4.2.1 Concentric MacLaurin Spheroids
4.2.2 Equations of state
4.2.3 Galileo constraints on the composition
4.3 Simple benchmark models
4.3.1 Homogeneous adiabatic gaseous envelope
4.3.2 A region of compositional and entropy variation within the planet
4.4 Locally inward decreasing Z-abundance in the gaseous envelope
4.4.1 Inward decreasing abundance of heavy elements in some part of the outer envelope
4.4.2 Constraints from the evolution
4.5 Models with at least 4 layers and an entropy discontinuity in the gaseous envelope
4.5.1 No entropy discontinuity
4.5.2 Entropy discontinuity in the gaseous envelope
4.6 Discussion
4.6.1 Hydrogen pressure metallization and H/He phase separation
4.6.2 Layered convection
4.6.3 External impacts, atmospheric dynamical effects
4.6.4 Magnetic field
4.6.5 Evolution
4.6.6 Does the observed outer condition lie on an adiabat?
4.7 Conclusion
5 From the core of Jupiter to the atmosphere of hot Jupiters
5.1 Introduction
5.2 The inflated radius: a connection between atmospheric dynamics and the interior structure
5.2.1 Observations
5.2.2 Kinetic or ohmic dissipation
5.2.3 Advection of potential temperature
5.3 Global Circulation Models
5.3.1 Common simplifications
5.3.2 The Unified Model
5.3.3 A step further: clouds and disequilibrium chemistry
5.4 A deep and robust feature: superrotation
5.5 Conclusion
6.1 Introduction
6.2 The algorithm
6.2.1 Linearised equations
6.2.2 Boundary conditions
6.2.3 Energy equation
6.2.4 Method of solution
6.2.5 Maximum Resolution
6.3 Benchmarking
6.3.1 Initial atmospheric rest state
6.3.2 Steady state circulation: unstable jet
6.3.3 Baroclinic instability
6.3.4 Rossby-Haurwitz waves
6.3.5 Linear steady circulation with drag
6.4 Conclusion
7 Equatorial dynamics of hot Jupiters 
7.1 Introduction
7.2 Notations and 2D shallow water equation and solution
7.2.1 Theoretical framework
7.2.2 Non linear accelerations from the linear steady state
7.2.3 Time dependent solutions
7.3 Insensitivity of Matsuno-Gill to the differential heating
7.4 Wave propagation and dissipation
7.4.1 Decay time of damped waves
7.4.2 The particular case of Kelvin waves
7.4.3 Short summary
7.5 Transition to superrotation
7.5.1 Shape of the linear steady states
7.5.2 Order of magnitude analysis
7.5.3 3D GCM simulations with various forcings
7.6 Conclusion
8 Conclusion and perspectives


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