Internal Structure of Giant and Icy Planets: Importance of Heavy Elements and Mixing

TL;DR

This paper reviews current understanding of giant and icy planet interiors, emphasizing heavy elements and mixing, and discusses implications for planet formation, evolution, and exoplanet studies.

Contribution

It provides a comprehensive summary of how heavy elements influence planetary structure, formation, and evolution, integrating Solar System and exoplanet perspectives.

Findings

Heavy elements significantly affect planetary internal structure.

Models of giant planets incorporate heavy element distribution.

Exoplanet observations help refine understanding of giant planet interiors.

Abstract

In this chapter we summarize current knowledge of the internal structure of giant planets. We concentrate on the importance of heavy elements and their role in determining the planetary composition and internal structure, in planet formation, and during the planetary long-term evolution. We briefly discuss how internal structure models are derived, present the possible structures of the outer planets in the Solar System, and summarise giant planet formation and evolution. Finally, we introduce giant exoplanets and discuss how they can be used to better understand giant planets as a class of planetary objects.

Tables

Table 1. Table 1: Basic Characteristics of the Outer Planets taken from Guillot & Gautier (2014) and NASA website: http://ssd.jpl.nasa.gov/?gravity_fields_op. Jupiter’s gravitational field is taken from Folkner et al., 2017.

Physical Property	Jupiter	Saturn	Uranus	Neptune
\svhline Distance to Sun (au)	5.204	9.582	19.201	30.047
Mass (1024 kg)	1898.13$\pm$0.19	568.319$\pm$0.057	86.8103$\pm$0.0087	102.410$\pm$0.010
Mean Radius (km)	69911$\pm$6	58232$\pm$6	25362$\pm$7	24622$\pm$16
Mean Density (g/cm3)	1.3262$\pm$0.0004	0.6871$\pm$0.0002	1.270$\pm$0.001	1.638$\pm$0.004
$J_2\times {10}^6$	14696.51$\pm$0.272	16290.71$\pm$0.27	3510.68$\pm$0.70	3408.43$\pm$4.50
$J_4\times {10}^6$	-586.62$\pm$0.36	-935.83$\pm$2.77	-34.17$\pm$1.30	-33.40$\pm$2.90
$J_6\times {10}^6$	34.24$\pm$0.24	86.14$\pm$9.64	—-	—-
Effective Temperature (K)	124.4$\pm$0.3	95.0$\pm$0.4	59.1$\pm$0.3	59.3$\pm$0.8
1-bar Temperature (K)	165$\pm$5	135$\pm$5	76$\pm$2	72$\pm$2

Equations

$$\frac{\partial }{\partial m}\frac{4\pi }{3}{r}^3=\frac{1}{\rho },$$

$$\frac{\partial p}{\partial m}=-\frac{Gm}{4\pi r^4}+\frac{{\omega }^2}{6\pi r}+\frac{GM}{4\pi R^{3}r}{\phi }_{\omega },$$

$$\frac{\partial T}{\partial m}={\nabla }_T\frac{\partial p}{\partial m},$$

$$\frac{\partial u}{\partial t}+p\frac{\partial }{\partial t}\frac{1}{\rho }=q-\frac{\partial L}{\partial m},$$

$$J_{2\ell }=-\frac{1}{M{a}^{2\ell }}\int r^{\prime 2\ell }{P}_{2\ell }(\cos {\theta }^{\prime })\rho (r^{\prime },{\theta }^{\prime })d^3{r}^{\prime }$$

Full text

1 Introduction

Characterisation of the outer planets in the Solar System has been one of the major objectives in planetary science since decades. Throughout the years significant progress has been made, both in theory and observations. We now have a much better understanding of the behaviour of hydrogen and other elements at high pressures and temperatures, and the physical processes that govern the planetary structure. The various spacecrafts that have visited (and are currently visiting) the outer planets in the Solar System, Jupiter, Saturn, Uranus, and Neptune, provide us with constraints on the gravitational fields, rotation periods, and atmospheric compositions of the planets that can be used by structure models. In parallel, the discovery of giant planets around other stars (giant exoplanets) provides an opportunity to study the diversity in giant planet composition, which can be used to better understand giant planet formation.

Despite the great progress in planetary modelling in the last few decades there are still several open questions regarding the nature of Jupiter, Saturn, Uranus and Neptune. Many review chapters have been written recently on giant planet interiors (e.g., Fortney & Nettelmann, 2010, Guillot & Gautier, 2014; Baraffe et al., 2014, Militzer et al., 2016) and this chapter aims to be somewhat complementary to those. Our chapter is organised as follows. First, we discuss the interiors of the Solar System’s gas giant planets (Jupiter and Saturn) and icy planets (Uranus and Neptune). Second, we discuss the standard formation mechanism of giant planets and how it is linked to their composition. Finally, we provide an outlook on the compositions of giant exoplanets.

2 Giant Planet Structure

2.1 Making an interior model

Information on the interiors of the giant planets in the Solar System is typically derived from theoretical structure models which are designed to fit the observed physical data of the planets, such as their gravitational fields, masses, internal rotations, and radii. The physical properties used by interiors models of the outer planets are listed in Table 1. The planetary interior is modelled by using the following structure equations which include the mass conservation, hydrostatic, thermodynamic, and energy conservation equations:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $P$ is the pressure, $\rho$ is the density, $m$ is the mass, $r$ is the radius and $G$ is the gravitational constant. The temperature gradient $\nabla_{T}$ depends on the process by which the internal heat is transported. The last equations is the only equation that is time ($t$) dependent and is used for modelling the planetary evolution. $u$ is the internal energy, $q$ is an energy source that is typically assumed to be zero for planets, and $L$ is the intrinsic luminosity.

In order to account for rotation, the hydrostatic equation (Eq. 2) includes additional terms which depend on $\omega$, the spin rate, $M$ the total mass of the planet, $R$ the total radius and $\phi_{\omega}$ is a function of the radius, internal density and spin rate (see Guillot 2005). For a non-spinning planet, $\phi_{\omega}=\omega=0$. For a spinning planet, this equation is valid in the limit of a barotropic fluid and a solid-body rotation. The radius is then considered as a mean volumetric radius. In that case, we can obtain constraints on the internal density distribution by measuring the departure of the planet’s gravity field from sphericity. These are expressed in the form of the gravitational moments, even functions of the radius, $r$, and the colatitude, $\theta$ (see e.g., Guillot 2005, Hubbard 2013):

[TABLE]

where $a$ is the equatorial radius, and $P_{n}$ is the $n$th-order Legendre polynomial. Interior models are constructed to fit the mass (essentially $J_{0}$) and as many of the $J_{2\ell}$’s as have been measured. Although each higher order $J$ gives additional information on $\rho(r)$. The density distribution correspond to a hydrostatic configuration when the contribution of dynamical effects (e.g., winds) on the gravitational moments are not included.

Unfortunately, there is no unique solution for the internal structure of a planet. The inferred structure depends on the model assumptions and the equations of state (EoSs) used by the modeller. The main uncertainties in structure models are linked to the following assumptions/setups: (i) number of layers (ii) the composition and distribution of heavy elements (iii) heat transport mechanism, and (iv) rotation period and the dynamical contribution of winds (e.g., differential rotation).

Since the gas giant planets (Jupiter and Saturn) consist of mainly hydrogen and helium, their modelling relies on the EoS of hydrogen, helium, and their mixture. The major uncertainty concerning the EoS of hydrogen is in the region of 0.5-10 Mbar, where hydrogen undergoes a transition from a molecular phase to a metallic phase. The EoS of helium in the relevant pressure region is simpler since helium ionization requires larger pressures and a phase transition is not expected to occur. The difficulty with calculating the EoS of helium, however, is due to the separation of helium droplets from the hydrogen-helium mixture (e.g, Fortney & Hubbard, 2003; Stevenson & Salpeter, 1977a,b). The EoS for the heavier elements (metals, rocks, ices) have generally received somewhat less attention than those for hydrogen and helium. Despite the difficulty, there have been substantial advances in high-pressure experiments and ab initio calculations of EoSs of hyrogen and helium and of of heavier materials, as well as on the miscibility properties, for water, ammonia, rock, and iron. Detailed description on EoSs and interior modeling can be found in Saumon & Guillot (2004), Baraffe et al. (2014), Fortney & Nettelmann (2010), Militzer et al. (2016), Miguel et al. (2016), Fortney et al. (2016).