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The exoplanet field now abounds with new discoveries of planets and planetary systems. It is of great interest for the scientific community to understand the compositions and internal structures of these new planets outside our own solar system, and then infer their formation scenarios. In particular, the proper implementation of the equation-of-states of various matters into the calculation of internal structure is crucial. Thus, we have come up with this write-up that provides the detailed steps to turn an equation-of-state (EOS) into a mass-radius relation of exoplanets, and highlight the physics and chemistry involved in these calculations. It must be visualized!, Comment: For pedagogical purpose only (non-profit). Comments and suggestions welcome!

We explain the overall equilibrium-temperature-dependent trend in the exoplanet mass-radius diagram, using the escape mechanisms of hydrogen and relevant volatiles, and the chemical equilibrium calculation of molecular hydrogen (H$_2$) break-up into atomic hydrogen (H). We identify two Cosmic Hydrogen and Ice Loss Lines (CHILLs) in the mass-radius diagram. Gas disks are well known to disperse in ten million years. However, gas-rich planets may lose some or almost all gas on a much longer timescale. We thus hypothesize that most planets that are born out of a hydrogen-gas-dominated nebular disk begin by possessing a primordial H$_2$-envelope. This envelope is gradually lost due to escape processes caused by host-stellar radiation., Comment: (Version 3) Published in Icarus, Volume 414, 116033, 2024

Recent astronomical observations obtained with the Kepler and TESS missions and their related ground-based follow-ups revealed an abundance of exoplanets with a size intermediate between Earth and Neptune. A low occurrence rate of planets has been identified at around twice the size of Earth, known as the exoplanet radius gap or radius valley. We explore the geometry of this gap in the mass-radius diagram, with the help of a Mathematica plotting tool developed with the capability of manipulating exoplanet data in multidimensional parameter space, and with the help of visualized water equations of state in the temperature-density graph and the entropy-pressure graph. We show that the radius valley can be explained by a compositional difference between smaller, predominantly rocky planets and larger planets that exhibit greater compositional diversity including cosmic ices (water, ammonia, methane) and gaseous envelopes. In particular, among the larger planets, when viewed from the perspective of planet equilibrium temperature, the hot ones are consistent with ice-dominated composition without significant gaseous envelopes, while the cold ones have more diverse compositions, including various amounts of gaseous envelopes., Comment: 30 pages, 15 figures

We discuss major challenges in modeling giant impacts between planetary bodies, focusing on the equations of state (EOS). During the giant impact stage of planet formation, rocky planets are melted and partially vaporized. However, most EOS models fail to reproduce experimental constraints on the thermodynamic properties of the major minerals over the required phase space. Here, we present an updated version of the widely-used ANEOS model that includes a user-defined heat capacity limit in the thermal free energy term. Our revised model for forsterite (Mg$_2$SiO$_4$), a common proxy for the mantles of rocky planets, provides a better fit to material data over most of the phase space of giant impacts. We discuss the limitations of this model and the Tillotson equation of state, a commonly used alternative model., Comment: M-ANEOS source code public release (https://github.com/isale-code/M-ANEOS); forsterite EOS public release (https://github.com/ststewart/aneos-forsterite-2019); manuscript accepted (no changes): Stewart, S., et al. (accepted). In J. Lane, T. Germann, and M. Armstrong (Eds.), 21st Biennial APS Conference on Shock Compression of Condensed Matter (SCCM19). AIP Publishing

The radii and orbital periods of 4000+ confirmed/candidate exoplanets have been precisely measured by the Kepler mission. The radii show a bimodal distribution, with two peaks corresponding to smaller planets (likely rocky) and larger intermediate-size planets, respectively. While only the masses of the planets orbiting the brightest stars can be determined by ground-based spectroscopic observations, these observations allow calculation of their average densities placing constraints on the bulk compositions and internal structures. Yet an important question about the composition of planets ranging from 2 to 4 Earth radii still remains. They may either have a rocky core enveloped in a H2-He gaseous envelope (gas dwarfs) or contain a significant amount of multi-component, H2O-dominated ices/fluids (water worlds). Planets in the mass range of 10-15 Earth masses, if half-ice and half-rock by mass, have radii of 2.5 Earth radii, which exactly match the second peak of the exoplanet radius bimodal distribution. Any planet in the 2-4 Earth radii range requires a gas envelope of at most a few mass percentage points, regardless of the core composition. To resolve the ambiguity of internal compositions, we use a growth model and conduct Monte Carlo simulations to demonstrate that many intermediate-size planets are water worlds., Comment: PNAS link: https://www.pnas.org/content/116/20/9723 Complete data and mass-radius tables are available at: https://www.cfa.harvard.edu/~lzeng/planetmodels.html

Applying the survival function analysis to the planet radius distribution of the Kepler confirmed/candidate planets, we have identified two natural divisions of planet radius at 4 Earth radii and 10 Earth radii. These divisions place constraints on planet formation and interior structure model. The division at 4 Earth radii separates small exoplanets from large exoplanets above. When combined with the recently-discovered radius gap at 2 Earth radii, it supports the treatment of planets 2-4 Earth radii as a separate group, likely water worlds. For planets around solar-type FGK main-sequence stars, we argue that 2 Earth radii is the separation between water-poor and water-rich planets, and 4 Earth radii is the separation between gas-poor and gas-rich planets. We confirm that the slope of survival function in between 4 and 10 Earth radii to be shallower compared to either ends, indicating a relative paucity of planets in between 4-10 Earth radii, namely, the sub-Saturnian desert there. We name them transitional planets, as they form a bridge between the gas-poor small planets and gas giants. Accordingly, we propose the following classification scheme: (<2 Earth radii) rocky planets, (2-4 Earth radii) water worlds, (4-10 Earth radii) transitional planets, and (>10 Earth radii) gas giants., Comment: 19 pages, 8 figures, accepted for publication in MNRAS. arXiv admin note: text overlap with arXiv:1801.03993

The size distribution of exoplanets is a bimodal division into two groups: Rocky planet (<2 Earth radii) and water-rich planet (>2 Earth radii) with or without gaseous envelope., Comment: 2 pages, 4 figures, abstract publicly available since January 2017 (see https://www.hou.usra.edu/meetings/lpsc2017/pdf/1576.pdf) and presented at the 48th LPSC on March 23, 2017. Also See: http://adsabs.harvard.edu/abs/2017LPI....48.1576Z

The giant impact hypothesis remains the leading theory for lunar origin. However, current models struggle to explain the Moon's composition and isotopic similarity with Earth. Here we present a new lunar origin model. High-energy, high-angular momentum giant impacts can create a post-impact structure that exceeds the corotation limit (CoRoL), which defines the hottest thermal state and angular momentum possible for a corotating body. In a typical super-CoRoL body, traditional definitions of mantle, atmosphere and disk are not appropriate, and the body forms a new type of planetary structure, named a synestia. Using simulations of cooling synestias combined with dynamic, thermodynamic and geochemical calculations, we show that satellite formation from a synestia can produce the main features of our Moon. We find that cooling drives mixing of the structure, and condensation generates moonlets that orbit within the synestia, surrounded by tens of bars of bulk silicate Earth (BSE) vapor. The moonlets and growing moon are heated by the vapor until the first major element (Si) begins to vaporize and buffer the temperature. Moonlets equilibrate with BSE vapor at the temperature of silicate vaporization and the pressure of the structure, establishing the lunar isotopic composition and pattern of moderately volatile elements. Eventually, the cooling synestia recedes within the lunar orbit, terminating the main stage of lunar accretion. Our model shifts the paradigm for lunar origin from specifying a certain impact scenario to achieving a Moon-forming synestia. Giant impacts that produce potential Moon-forming synestias were common at the end of terrestrial planet formation., Comment: Accepted for publication in Journal of Geophysical Research: Planets. Main text: 44 pages, 24 figures. Supplement: 16 pages, 5 figures, 3 tables

Applying survival function analysis to the planet orbital period (P) and semi-major axis (a) distribution from the Kepler sample, we find that all exoplanets are uniformly distributed in (ln a) or (ln P), with an average inner cut-off of 0.05 AU to the host star. More specifically, this inner cut-off is 0.04 AU for rocky worlds (1-2 Earth radii) and 0.08 AU for water worlds (2-4 Earth radii). Moreover, the transitional planets (4-10 Earth radii) and gas giants (>10 Earth radii) have a change of slope of survival function at 0.4 AU from -1 to -1/2, suggesting a different statistical distribution uniform in \Sqrt[a] inside 0.4 AU, compared to small exoplanets (<4 Earth radii). This difference in distribution is likely caused by the difference in planet migration mechanism, and susceptibility to host stellar irradiation, for gas-poor (<4 Earth radii) versus gas-rich (>4 Earth radii) planets. Armed with this knowledge and combined with the survival function analysis of planet size distribution, we can make precise estimates of planet occurrence rate and predict the TESS mission yield., Comment: 11 pages, 3 figures, under review with AJ, comments are very welcome and much appreciated!

Applying the survival function analysis to the planet radius distribution of the Kepler exoplanet candidates, we have identified two natural divisions of planet radius at 4 Earth radii and 10 Earth radii. These divisions place constraints on planet formation and interior structure model. The division at 4 Earth radii separates small exoplanets from large exoplanets above. When combined with the recently-discovered radius gap at 2 Earth radii, it supports the treatment of planets 2-4 Earth radii as a separate group, likely water worlds. Thus, for planets around solar-type FGK main-sequence stars, we argue that 2 Earth radii is the separation between water-poor and water-rich planets, and 4 Earth radii is the separation between gas-poor and gas-rich planets. We confirm that the slope of survival function in between 4 and 10 Earth radii to be shallower compared to either ends, indicating a relative paucity of planets in between 4-10 Earth radii, namely, the sub-Saturnian desert there. We name them transitional planets, as they form a bridge between the gas-poor small planets and gas giants. Accordingly, we propose the following classification scheme: (<2 Earth radii) rocky planets, (2-4 Earth radii) water worlds, (4-10 Earth radii) transitional planets, and (>10 Earth radii) gas giants., Comment: 10 pages, 2 figures, under review with MNRAS, comments are very welcome and much appreciated!

Exoplanets smaller than Neptune are numerous, but the nature of the planet populations in the 1-4 Earth radii range remains a mystery. The complete Kepler sample of Q1-Q17 exoplanet candidates shows a radius gap at ~ 2 Earth radii, as reported by us in January 2017 in LPSC conference abstract #1576 (Zeng et al. 2017). A careful analysis of Kepler host stars spectroscopy by the CKS survey allowed Fulton et al. (2017) in March 2017 to unambiguously show this radius gap. The cause of this gap is still under discussion (Ginzburg et al. 2017; Lehmer & Catling 2017; Owen & Wu 2017). Here we add to our original analysis the dependence of the radius gap on host star type., Comment: 2pages, 1 figure, RNAAS 2017

In the past few years, the number of confirmed planets has grown above 2000. It is clear that they represent a diversity of structures not seen in our own solar system. In addition to very detailed interior modeling, it is valuable to have a simple analytical framework for describing planetary structures. Variational principle is a fundamental principle in physics, entailing that a physical system follows the trajectory which minimizes its action. It is alternative to the differential equation formulation of a physical system. Applying this principle to planetary interior can beautifully summarize the set of differential equations into one, which provides us some insight into the problem. From it, a universal mass-radius relation, an estimate of error propagation from equation of state to mass-radius relation, and a form of virial theorem applicable to planetary interiors are derived., Comment: accepted by ApJ

This work aims at exploring the scaling relations among rocky exoplanets. With the assumption of internal gravity increasing linearly in the core, and staying constant in the mantle, and tested against numerical simulations, a simple model is constructed, applicable to rocky exoplanets of core mass fraction (CMF) $\in 0.1\sim0.4$ and mass $\in 0.1 \sim 10 M_{\oplus}$. Various scaling relations are derived: (1) core radius fraction $\rm CRF \approx \sqrt{CMF}$, (2) Typical interior pressure $P_{\text{typical}} \sim g_{\rm{s}}^2$ (surface gravity squared), (3) core formation energy $E_{\rm diff} \sim \frac{1}{10} E_{\rm grav}$ (the total gravitational energy), (4) effective heat capacity of the mantle $C_{\rm p}\approx \left( \frac{M_{\rm{p}}}{M_{\oplus}} \right) \cdot 7 \cdot 10^{27}$ J K$^{-1}$, and (5) the moment of inertia $I\approx \frac{1}{3} \cdot M_{\rm{p}} \cdot R_{\rm{p}}^2$. These scaling relations, though approximate, are handy for quick use owing to their simplicity and lucidity, and provide insights into the interior structures of rocky exoplanets. As examples, this model is applied to several planets including Earth, GJ 1132b, Kepler-93b, and Kepler-20b, and made comparison with the numerical method., Comment: accepted by ApJ

Laser Ablation ICP-MS (LA-ICP-MS) has been widely accepted as a microanalytical technique for in-situ trace (ppb) elemental analysis on the micron scale in a variety of geologic materials. LA-ICP-MS (single or multi-collector) provides both elemental and isotopic measurements critical for a wide range of geological research by generating a fine grained aerosol (nm scale) during the laser ablation event and delivering that aerosol to the ICP ion source of the mass spectrometer via an inert carrier gas. LA-ICP-MS, however, suffers from limitations in analyzing high ionization potential elements as well as elements subject to atmospheric and argon based interferences. LA-ICP-MS also has limitations in analyzing major elements due to detector saturation. An alternative laser ablation technique, Laser Induced Breakdown Spectroscopy (LIBS), employs an optical spectrometer integrated into the laser ablation system that analyzes the laser induced plasma at the sample surface across the entire optical spectrum for emission lines of every element in the periodic table. Elements that are difficult or impossible to measure with LA-ICP-MS are now possible to analyze with LIBS down to low ppm levels with CCD and/or ICCD detection. We introduce a new laser based technique, “Tandem LA-LIBS”, that combines LA for ICP-MS and LIBS into one integrated laser ablation system. This system has the effect of expanding the elemental coverage and the dynamic range of the laser ablation experiment as measurements from ppb to % level matrix elements can now be analyzed in a single ablation experiment. We present both femtosecond and nanosecond Tandem LA-LIBS quantitative and qualitative data on wide range of geological materials for those elements that are difficult or impossible by traditional LA-ICP-MS techniques such as F, H, O, N, C, S, halogens, etc. We also demonstrate that the simultaneous measurement of trace, minor and major elements are now possible in a single laser ablation experiment with Tandem LA-LIBS technology.