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Machine Learning Applications to Metal-Silicate Equilibria and their Insights into Core Formation

An extensive number of studies have experimentally investigated how elements distribute between metal and silicate phases, to better constrain core-mantle chemical equilibrium. Here, we present a new database compiling all (to our knowledge) experimental data on liquid metal-silicate partitioning from 118 peer-reviewed publications. We applied various machine learning techniques to gain further insights into these partitioning equilibria and their dependencies. We performed a network analysis to investigate the relationship between experiments and partition coefficients, which enables visualizing gaps in the experimental dataset and biases related to varying experimental conditions and analytical setup. In addition, semi-empirical thermodynamic models are commonly used to extrapolate these chemical reactions to the wide range of pressure, temperature and compositional conditions of planetary differentiation. These models are based on linear regressions that assume continuous relationship between partition coefficients and experimental variables. Here, we considered random forest regressions, which are algorithms based on ensembles of decision trees and does not consider continuous effects of each variable. The application of this regression significantly improves the prediction of metal-silicate partitioning for several elements including Ni, Si and Cr. We will show how this new approach improves our understanding of elemental exchange between metal and silicate and their implications for the Earth’s core formation.

siderophile element↗
NASA NTRS1974

The internal structure of the convective envelope of the sun

An axisymmetric hydrodynamic model of the sun consisting of a core at radiant equilibrium and a convective envelope is studied with consideration of gravitational forces, rotation, turbulence effects, and electromagnetic forces. The dimensionless parameters that control the equations of plasma motion within the sun and the boundary conditions are investigated; estimates are given for these parameters on the basis of experimental data and the existing theoretical models. Stability conditions are derived for convective equilibrium, developed cellular convection, and developed turbulent convection with consideration of rotation and turbulent Reynolds stresses. These conditions characterize the possible physical state of the matter in the convective envelope. Using estimates of the dimensionless parameters and models of some of them, a method of successive approximations is proposed for solution of the problem as a whole.

Yavorskaya, I. M.↗
NASA NTRS1974

N-Player Stochastic Differential Games

Conditions are described which guarantee that the control strategies adopted by N players constitute an efficient solution, an equilibrium, or a core solution. The system dynamics are described by an Ito equation, and all players have perfect information. It was found that when the set of instantaneous joint costs and velocity vectors is convex, the conditions are necessary.

Varaiya, P.↗
NASA NTRS1976

N-player stochastic differential games

The paper presents conditions which guarantee that the control strategies adopted by N players constitute an efficient solution, an equilibrium, or a core solution. The system dynamics are described by an Ito equation, and all players have perfect information. When the set of instantaneous joint costs and velocity vectors is convex, the conditions are necessary.

Varaiya, P.↗
NASA NTRS1985

Exercise thermoregulation in men after 6 hours of immersion

The present investigation is concerned with thermoregulation at rest and during exercise after water-immersion deconditioning, giving particular attention to the effects of fluid shifts and negative water balance on sweat rate and rectal temperature. Six healthy males 20-35 years old were used in the experiments. Rectal and mean skin temperature, skin heat conductance, heart rate, and total body sweat rate were measured during 70 min of supine leg exercise at 50 percent of peak O2 uptake. The data were taken after a 6-h control period in air and after immersion to the neck in water (34.5 C) for 6 h after overnight food and fluid restriction. Attention is given to end exercise heart rates and data during exercise. The obtained results suggest that, compared with control responses, the equilibrium level of core temperature during submaximal exercise is regulated at a higher level after immersion.

Greenleaf, J. E.↗
NASA NTRS1969

ELAS: A general-purpose computer program for the equilibrium problems of linear structures. Volume 2: Documentation of the program

A general purpose digital computer program for the in-core solution of linear equilibrium problems of structural mechanics is documented. The program requires minimum input for the description of the problem. The solution is obtained by means of the displacement method and the finite element technique. Almost any geometry and structure may be handled because of the availability of linear, triangular, quadrilateral, tetrahedral, hexahedral, conical, triangular torus, and quadrilateral torus elements. The assumption of piecewise linear deflection distribution insures monotonic convergence of the deflections from the stiffer side with decreasing mesh size. The stresses are provided by the best-fit strain tensors in the least squares at the mesh points where the deflections are given. The selection of local coordinate systems whenever necessary is automatic. The core memory is used by means of dynamic memory allocation, an optional mesh-point relabelling scheme and imposition of the boundary conditions during the assembly time.

Utku, S.↗
NASA NTRS2004

On the Final Mass of Giant Planets

In the core accretion model of giant planet formation, when the core reaches critical mass, hydrostatic equilibrium is no longer possible and gas accretion ensues. If the envelope is radiative, the critical core mass is nearly independent of the boundary conditions and is roughly M(sub crit) ~ 10Mass of the Earth (with weak dependence on the rate of planetesimal accretion M(sub core) and the disk opacity k). Given that such a core may form at the present location of Jupiter in a time comparable to its Type I migration time (10(exp 5) - 10(exp 6) years) provided that the nebula was significantly enhanced in solids with respect to the MMSN and stall at this location in a weakly turbulent (alpha approximately less than 10(exp -4) disk, it may be appropriate to assume that such objects inevitably form and drive the evolution of late-phase T Tauri star disks. Here we investigate the final masses of giant planets in disks with one or more than one such cores. Although the presence of several planets would lead to Type II migration (due to the effective viscosity resulting from the planetary tidal torques), we ignore this complication for now and simply assume that each core has stalled at its location in the disk. Once a core has achieved critical mass, its gaseous accretion is governed by the given Kelvin-Helmholtz timescale.

Estrada, P. R.↗
NASA NTRS1992

A new class of g-modes in neutron stars

Because a neutron star is born hot, its internal composition is close to chemical equilibrium. In the fluid core, this implies that the ratio of the number densities of charged particles (protons and electrons) to neutrons is an increasing function of the mass density. This composition gradient stably stratifies the matter giving rise to a Brunt-Vaisala frequency N of about 500/s. Consequently, a neutron star core provides a cavity that supports gravity modes (g-modes). These g-modes are distinct from those previously identified with the thermal stratification of the surface layers and the chemical stratification of the crust. We compute the lowest-order, quadrupolar, g-modes for cold, Newtonian, neutron star models with M/solar M = 0.581 and M/solar M = 1.405, and show that the crustal and core g-modes have similar periods. We also discuss damping mechanisms and estimate damping rates for the core g-modes. Particular attention is paid to damping due to the emission of gravitational radiation.

Reisenegger, Andreas↗
NASA NTRS2012

Provenance and Concentration of Water in the Shergottite Mantle

The water content of the martian mantle is controversial. In particular, the role of water in the petrogenesis of the shergottites has been much debated. Although the shergottites, collectively, contain very little water [e.g., 1,2], some experiments have been interpreted to show that percent levels of water are required for the petrogenesis of shergottites such as Shergotty and Zagami [3]. In this latter interpretation, the general paucity of water in the shergottites and their constituent minerals is attributed to late-stage degassing. Y980459 (Y98) is a very primitive, perhaps even parental, martian basalt, with a one-bar liquidus temperature of approx.1400 C. Olivine is the liquidus phase, and olivine core compositions are in equilibrium with the bulk rock [e.g., 4]. Petrogenetically, therefore, Y98 has had a rather simple history and can potentially help constrain the role of water in martian igneous processes. In particular, once trapped, melt inclusions should not be affected by subsequent degassing.

Jones, J. H.↗
NASA NTRS1986

Fe-Mg-Mn relations of ureilite olivines and pyroxenes and the genesis of ureilites

Microprobe analyses of ureilite and pigeonite cores are studied. The Fe/Mn-Fe/Mg relationship in the olivine core is examined. It is observed that magnetic processs such as fractional crystallization and partial melting, and FeO reduction contribute to the olivine core composition. The study of the Mg/Mn and Fe/Mn distributions reveals that these two distributions are not in equilibrium in the olivine and pigeonite cores. The effect of a reducing agent, carbon, on the ureilite genesis is investigated. It is concluded that fractional crystallization and FeO reduction are the major processes of ureilite genesis.

Mittlefehldt, D. W.↗
NASA NTRS1987

Experiments pertaining to the formation and equilibration of planetary cores

The phase diagram of FeO was experimentally determined to pressures of 155 GPa and temperatures of 4000 K using shock wave and diamond-cell techniques. Researchers discovered a metallic phase of FeO at pressures greater than 70 GPa and temperatures exceeding 1000 K. The metallization of FeO at high pressures implies that oxygen can be present as the light alloying element of the Earth's outer core, in accord with the geochemical predictions of Ringwood. The high pressures necessry for this metallization suggest that the core has acquired its composition well after the initial stages of the Earth's accretion. The core forming alloy can react chemically with oxides such as those forming the mantle. The core and mantle may never have reached complete chemical equilibrium, however. If this is the case, the core-mantle boundary is likely to be a zone of active chemical reactions.

Jeanloz, Raymond↗
NASA NTRS1986

High-pressure metallization of FeO and implications for the earth's core

The phase diagram of FeO has been experimentally determined to pressures of 155 GPa and temperatures of 4000 K using shock-wave and diamond-cell techniques. A metallic phase of FeO is observed at pressures greater than 70 GPa and temperatures exceeding 1000 K. The metallization of FeO at high pressures implies that oxygen can be present as the light alloying element of the earth's outer core, in accord with the geochemical predictions of Ringwood (1977 and 1979). The high pressures necessary for this metallization suggest that the core has acquired its composition well after the initial stages of the earth's accretion. Direct experimental observations at elevated pressures and temperatures indicate that core-forming alloy can react chemically with oxides such as those forming the mantle. The core and mantle may never have reached complete chemical equilibrium, however. If this is the case, the core-mantle boundary is likely to be a zone of active chemical reactions.

Knittle, Elise↗
NASA NTRS1972

Evolution of stars with suppressed core convection

Stellar evolution on the upper main sequence was computed for models of stars with cores assumed to be in radiative equilibrium, up to the point of central helium ignition. The role of the Schonberg-Chandrasekhar limit for an isothermal core is found to be critical for the evolutionary tracks. Observational data are used to rule out the hypothesis of evolution with radiative cores (in upper main-sequence stars) and, by implication, of magnetic fields that are sufficiently strong to have suppressed the core convention.

Stothers, R. B.↗
NASA NTRS1973

Evolution of stars with suppressed core convection.

Stellar evolution on the upper main sequence has been computed for models of stars with cores assumed to be in radiative equilibrium, up to the point of central helium ignition. The role of the Schoenberg-Chandrasekhar limit for an isothermal core is found to be critical for the evolutionary tracks. Observational data are used to rule out the hypothesis of evolution with radiative cores (in upper main-sequence stars) and, by implication, of magnetic fields that are sufficiently strong to have suppressed the core convection.

Stothers, R.↗
NASA NTRS1987

Collapse of 9 solar mass stars

General relativistic hydrodynamical calculations of the collapse of O + Ne + Mg cores of a 9 solar mass star are reported. Collapse is induced by rapid electron captures as the O + Ne + Mg is burned to nuclear statistical equilibrium. The high entropy in the core after burning leads to a large abundance of free protons which readily capture electrons. This leads to large neutrino losses and a correspondingly small infalling homologous core. The hydrodynamic shock thus forms at a small mass point. The shock stalls before reaching the edge of the O + Ne + Mg core and thereby fails to produce a successful supernova explosion by the direct mechanism. No enhancement in the shock energy due to nuclear burning is found.

Baron, E.↗
NASA NTRS1982

Thermonuclear processes on accreting neutron stars - A systematic study

A series of model calculations for the evolution of the surface layers of an accreting neutron star is carried out. The neutron star mass, radius, core temperature, and surface magnetic field strength are systematically varied, as are the accretion rate onto the neutron star surface and the metallicity of the accreting matter, in order to determine the effects of these parameters on the properties of thermonuclear flashes in the surface layers and the emitted X-ray bursts that result from such flashes. The core temperatures required for thermal equilibrium are found to be approximately a factor of 2 lower than estimated in earlier work. Owing to the effects of the gravitational redshift, the emitted X-ray bursts have lower peak luminosities and longer durations than those calculated in the Newtonian approximation. The entrainment of hydrogen into helium flashes can cause the flashes to exhibit a rather wide range of observable effects and can decrease by a factor of more than 2 the ratio of persistent accretion-driven luminosity to time-averaged burst luminosity emitted by the neutron star.

Ayasli, S.↗
NASA NTRS2012

Constraints on the Parental Melts of Enriched Shergottites from Image Analysis and High Pressure Experiments

Martian basalts can be classified in at least two geochemically different families: enriched and depleted shergottites. Enriched shergottites are characterized by higher incompatible element concentrations and initial Sr-87/Sr-86 and lower initial Nd-143/Nd-144 and Hf-176/Hf-177 than depleted shergottites [e.g. 1, 2]. It is now generally admitted that shergottites result from the melting of at least two distinct mantle reservoirs [e.g. 2, 3]. Some of the olivine-phyric shergottites (either depleted or enriched), the most magnesian Martian basalts, could represent primitive melts, which are of considerable interest to constrain mantle sources. Two depleted olivine-phyric shergottites, Yamato (Y) 980459 and Northwest Africa (NWA) 5789, are in equilibrium with their most magnesian olivine (Fig. 1) and their bulk rock compositions are inferred to represent primitive melts [4, 5]. Larkman Nunatak (LAR) 06319 [3, 6, 7] and NWA 1068 [8], the most magnesian enriched basalts, have bulk Mg# that are too high to be in equilibrium with their olivine megacryst cores. Parental melt compositions have been estimated by subtracting the most magnesian olivine from the bulk rock composition, assuming that olivine megacrysts have partially accumulated [3, 9]. However, because this technique does not account for the actual petrography of these meteorites, we used image analysis to study these rocks history, reconstruct their parent magma and understand the nature of olivine megacrysts.

Collinet, M.↗
NASA NTRS2016

Platinum Partitioning at Low Oxygen Fugacity: Implications for Core Formation Processes

Highly siderophile elements (HSE = Au, Re, and the Pt-group elements) are tracers of silicate / metal interactions during planetary processes. Since most core-formation models involve some state of equilibrium between liquid silicate and liquid metal, understanding the partioning of highly siderophile elements (HSE) between silicate and metallic melts is a key issue for models of core / mantle equilibria and for core formation scenarios. However, partitioning models for HSE are still inaccurate due to the lack of sufficient experimental constraints to describe the variations of partitioning with key variable like temperature, pressure, and oxygen fugacity. In this abstract, we describe a self-consistent set of experiments aimed at determining the valence of platinum, one of the HSE, in silicate melts. This is a key information required to parameterize the evolution of platinum partitioning with oxygen fugacity.

Medard, E.↗