Lower-order zonal gravitational coefficients caused by zonal circulations inside gaseous planets: Convective flows and numerical comparison between modeling approaches

To infer the internal equilibrium structure of a gaseous planet, especially the equation of state(EOS) and size of its inner core,requires accurate determination of lower-order zonal gravitational coefficients. Modeling of the gravitational signature associated with deep zonal circulation depends critically upon reliable subtraction of the dynamical components from totally derived gravitational coefficients. In the era of the Juno mission and the Grand Finale phase of the Cassini mission, it is timely and necessary to revisit and examine the so-called ‘Thermal Wind Equation(TWE)’, which has been extensively utilized to diagnose the dynamical parts of the gravitational fields measured by the two spacecrafts. TWE treats as negligible a few terms in the full equation of balance. However, the self-gravitational anomaly of the distorted fluid, unlike oblateness effects of solid-body rotation, is not a priori minor and thus should not be neglected in the name of approximation. Another equation, the ‘Thermal Gravitational Wind Equation(TGWE)’, includes this important additional term;we compare it with the TWE and show that physically the TGWE models a fundamentally different balance from the TWE and delivers numerical results considerably different from models based on the TWE. We conclude that the TWE balance cannot be relied upon to produce realistic convection models. Only after the TGWE balance is obtained can the relative importance of terms be assessed.The calculations we report here are based on two types of zonal circulations that are produced by realistically possible convections inside planets, instead of being constructed or assumed.

Relation between Mass and Radius of Exoplanets Distinguished by their Density

The formation of the solar system has been studied since the 18th century and received a boost in 1995 with the discovery of the first exoplanet,51 Pegasi b.The investigations increased the number of confirmed planets to about5400 to date.The possible internal structure and composition of these planets can be inferred from the relationship between planet mass and radius,M-R.We have analyzed the M-R relation of a selected sample of iron-rock and ice-gas planets using a fractal approach to their densities.The application of fractal theory is particularly useful to define the physical meaning of the proportionality constant and the exponent in an empirical M-R power law in exoplanets,but this does not necessarily mean that they have an internal fractal structure.The M-R relations based on this sample are M=(1.46±0.08)R^(2.6±0.2)for the rocky population(3.6≤ρ≤14.3 g cm^(-3)),with 1.5≤M≤39M_(⊕),and M=(0.27±0.04)R^(2.7±0.2)for ice-gas planets(0.3≤ρ≤2.1 g cm^(-3))with 5.1≤M≤639 M_(⊕)(or■2 M_(J))and orbital periods greater than 10 days.Both M-R relations have in their density range a great predictive power for the determination of the mass of exoplanets and even for the largest icy moons of the solar system.The average fractal dimension of these planets is D=2.6±0.1,indicating that these objects likely have a similar degree of heterogeneity in their densities and a nearly similar composition in each sample.The M-R diagram shows a"gap"between ice-gas and iron-rock planets.This gap is a direct consequence of the density range of these two samples.We empirically propose an upper mass limit of about 100 M_(⊕),so that an M-R relation for ice-gas planets in a narrow density range is defined by M∝R^(3).

Atmospheric regimes and trends on exoplanets and brown dwarfs

A planetary atmosphere is the outer gas layer of a planet. Besides its scientific significance among the first and most accessible planetary layers observed from space, it is closely connected with planetary formation and evolution, surface and interior processes, and habitability of planets. Current theories of planetary atmospheres were primarily obtained through the studies of eight large planets, Pluto and three large moons(Io, Titan, and Triton) in the Solar System. Outside the Solar System, more than four thousand extrasolar planets(exoplanets) and two thousand brown dwarfs have been confirmed in our Galaxy, and their population is rapidly growing. The rich information from these exotic bodies offers a database to test, in a statistical sense, the fundamental theories of planetary climates. Here we review the current knowledge on atmospheres of exoplanets and brown dwarfs from recent observations and theories. This review highlights important regimes and statistical trends in an ensemble of atmospheres as an initial step towards fully characterizing diverse substellar atmospheres, that illustrates the underlying principles and critical problems.Insights are obtained through analysis of the dependence of atmospheric characteristics on basic planetary parameters. Dominant processes that influence atmospheric stability, energy transport, temperature, composition and flow pattern are discussed and elaborated with simple scaling laws. We dedicate this review to Dr. Adam P. Showman(1968–2020) in recognition of his fundamental contribution to the understanding of atmospheric dynamics on giant planets, exoplanets and brown dwarfs.

Delay of planet formation at large radius and the outward decrease in mass and gas content of Jovian planets

A prominent observation of the solar system is that the mass and gas content of Jovian planets decrease outward with orbital radius, except that, in terms of these properties, Neptune is almost the same as Uranus. In previous studies, the solar nebula was assumed to preexist and the formation process of the solar nebula was not considered. It was therefore assumed that planet formation at different radii started at the same time in the solar nebula. We show that planet formation at different radii does not start at the same time and is delayed at large radii. We suggest that this delay might be one of the factors that causes the outward decrease in the masses of Jovian planets. The nebula starts to form from its inner part because of the inside-out collapse of its progenitorial molecular cloud core. The nebula then expands outward due to viscosity. Material first reaches a small radius and then reaches a larger radius, so planet formation is delayed at the large radius. The later the material reaches a planet＇s location, the less time it has to gain mass and gas content. Hence, the delay tends to cause the outward decrease in mass and gas content of Jovian planets. Our nebula model shows that the material reaches Jupiter, Saturn, Uranus and Neptune at t = 0.40, 0.57, 1.50 and 6.29 × 10^6 yr, respectively. We discuss the effects of time delay on the masses of Jovian planets in the framework of the core accretion model of planet formation. Saturn＇s formation is not delayed by much time relative to Jupiter so that they both reach the rapid gas accretion phase and become gas giants. However, the delay in formation of Uranus and Neptune is long and might be one of the factors that cause them not to reach the rapid gas accretion phase before the gas nebula is dispersed. Saturn has less time to go through the rapid gas accretion, so Saturn＇s mass and gas content are significantly less than those of Jupiter.

A Linear Operator Method to Compute the Normal Modes with Rotation of any Asymmetric 3D Planet with Pure Vector Spherical Harmonics

In order to compute the free core nutation of the terrestrial planets, such as Earth and Mars, the Moon and lower degree normal modes of the Jovian planets, we propose a linear operator method(LOM). Generalized surface spherical harmonics(GSSHs) are usually applied to the elliptical models with a stress tensor, which cannot be expressed in vector spherical harmonics explicitly. However, GSSHs involve complicated math. LOM is an alternative to GSSHs,whereas it only deals with the coupling fields of the same azimuthal order m, as is the case when a planet model is axially symmetric and rotates about that symmetry axis. We extend LOM to any asymmetric 3D model. The lower degree spheroidal modes of the Earth are computed to validate our method, and the results agree very well with what is observed. We also compute the normal modes of a two-layer Saturn model as a simple application.