The Ribaucour families of discrete R-congruences

While a generic smooth Ribaucour sphere congruence admits exactly two envelopes, a discrete R-congruence gives rise to a 2-parameter family of discrete enveloping surfaces. The main purpose of this paper is to gain geometric insights into this ambiguity. In particular, discrete R-congruences that are enveloped by discrete channel surfaces and discrete Legendre maps with one family of spherical curvature lines are discussed.

Understanding Jupiter’s deep interior: the effect of a dilute core

Context. The Juno spacecraft has significantly improved the accuracy of low-order even gravitational harmonics. It has been demonstrated that a dilute core is helpful to interpret Juno’s gravity measurements. However, introducing a dilute core adds a new degree of freedom to Jupiter’s interior models in addition to the uncertainties in the equations of state for hydrogen and helium. Aims. We present four-layer structure models for Jupiter where a dilute core region is added above a central compact core of rocks. The effect of the dilute core on the structure and composition of Jupiter is investigated in detail. Combined with current knowledge of Jupiter’s composition and thermal state, we aim to obtain information on the dilute core. Also, we investigate the effect of equations of state for hydrogen and helium on the predictions of the core mass and heavy element abundance. Methods. In the four-layer structure model, the heavy element abundances in the outer two envelopes and the mass of the compact core were adjusted to reproduce Jupiter’s equatorial radius as well as Juno’s gravity observations. Different dilute core configurations were constructed in terms of its size and composition and different equations of state for hydrogen and helium were used in interior structure calculations. Optimized calculations were then performed to investigate the effect of dilute cores and equations of state on Jupiter’s internal structure and composition. Results. It is found that the absolute values of J6 and J8 tend to decrease as helium becomes more depleted in the dilute core region. Most interior structure calculations seem to prefer an inward decrease of the helium mass fraction from the metallic envelope to the dilute core region. We also show that the core mass and heavy element abundance in Jupiter are dependent upon the rock-to-ice ratio in the dilute core region, the temperature jump from the molecular to metallic envelope, and the equations of state for hydrogen and helium. The resulting heavy-element mass in the core is generally larger than the three-layer structure models owing to the heavy elements dissolved in the dilute core region, and the global heavy-element abundance is in good agreement with the available dilute-core predictions.

The Interiors of Jupiter and Saturn

Probing the interiors of the gaseous giant planets in our solar system is not an easy task. It requires a set of accurate measurements combined with theoretical models that are used to infer the planetary composition and its depth dependence. The masses of Jupiter and Saturn are 317.83 and 95.16 Earth masses (M⊕), respectively, and since a few decades, it has been known that they mostly consist of hydrogen and helium. The mass of heavy elements (all elements heavier than helium) is not well determined, nor are their distribution within the planets. While the heavy elements are not the dominating materials inside Jupiter and Saturn, they are the key to understanding the planets’ formation and evolutionary histories. The planetary internal structure is inferred from theoretical models that fit the available observational constraints by using theoretical equations of states (EOSs) for hydrogen, helium, their mixtures, and heavier elements (typically rocks and/or ices). However, there is no unique solution for determining the planetary structure and the results depend on the used EOSs as well as the model assumptions imposed by the modeler. Major model assumptions that can affect the derived internal structure include the number of layers, the heat transport mechanism within the planet (and its entropy), the nature of the core (compact vs. diluted), and the location (pressure) of separation between the two envelopes. Alternative structure models assume a less distinct division between the layers and /or a non-homogenous distribution of the heavy elements. The fact that the behavior of hydrogen at high pressures and temperatures is not perfectly known and that helium may separate from hydrogen at the deep interior add sources of uncertainty to structure models. In the 21st century, with accurate measurements of the gravitational fields of Jupiter and Saturn from the Juno and Cassini missions, structure models can be further constrained. At the same time, these measurements introduce new challenges for planetary modelers.