Eta-Earth Revisited II: Deriving a Maximum Number of Earth-Like Habitats in the Galactic Disk

In Lammer et al. (2024), we defined Earth-like habitats (EHs) as rocky exoplanets within the habitable zone of complex life (HZCL) on which Earth-like N2-O2-dominated atmospheres with minor amounts of CO2 can exist, and derived a formulation for estimating the maximum number of EHs in the galaxy given realistic probabilistic requirements that have to be met for an EH to evolve. In this study, we apply this formulation to the galactic disk by considering only requirements that are already scientifically quantifiable. By implementing literature models for star formation rate, initial mass function, and the mass distribution of the Milky Way, we calculate the spatial distribution of disk stars as functions of stellar mass and birth age. For the stellar part of our formulation, we apply existing models for the galactic habitable zone and evaluate the thermal stability of nitrogen-dominated atmospheres with different CO2 mixing ratios inside the HZCL by implementing the newest stellar evolution and upper atmosphere models. For the planetary part, we include the frequency of rocky exoplanets, the availability of surface water and subaerial land, and the potential requirement of hosting a large moon by evaluating their importance and implementing these criteria from minima to maxima values as found in the scientific literature. We also discuss further factors that are not yet scientifically quantifiable but may be requirements for EHs to evolve. Based on such an approach, we find that EHs are relatively rare by obtaining plausible maximum numbers of 2.5 - 2.4 + 71.6 × 10 5 and 0.6 - 0.59 + 27.1 × 10 5 planets that can potentially host N2-O2-dominated atmospheres with maximum CO2 mixing ratios of 10% and 1%, respectively, implying that, on average, a minimum of ∼ 10 3 - 10 6 rocky exoplanets in the HZCL are needed for 1 EH to evolve. The actual number of EHs, however, may be substantially lower than our maximum ranges since several requirements with unknown occurrence rates are not included in our model (e.g., the origin of life, working carbon-silicate and nitrogen cycles); this also implies extraterrestrial intelligence (ETI) to be significantly rarer still. Our results illustrate that not every star can host EHs nor can each rocky exoplanet within the HZCL evolve such that it might be able to host complex animal-like life or even ETIs. The Copernican Principle of Mediocrity therefore cannot be applied to infer that such life will be common in the galaxy.

Thermal conductivity of Fe-Si alloys and thermal stratification in Earth's core

Light elements in Earth's core play a key role in driving convection and influencing geodynamics, both of which are crucial to the geodynamo. However, the thermal transport properties of iron alloys at high-pressure and -temperature conditions remain uncertain. Here we investigate the transport properties of solid hexagonal close-packed and liquid Fe-Si alloys with 4.3 and 9.0 wt % Si at high pressure and temperature using laser-heated diamond anvil cell experiments and first-principles molecular dynamics and dynamical mean field theory calculations. In contrast to the case of Fe, Si impurity scattering gradually dominates the total scattering in Fe-Si alloys with increasing Si concentration, leading to temperature independence of the resistivity and less electron-electron contribution to the conductivity in Fe-9Si. Our results show a thermal conductivity of ∼100 to 110 W⋅m^-1⋅K^-1 for liquid Fe-9Si near the topmost outer core. If Earth's core consists of a large amount of silicon (e.g., > 4.3 wt %) with such a high thermal conductivity, a subadiabatic heat flow across the core-mantle boundary is likely, leaving a 400- to 500-km-deep thermally stratified layer below the core-mantle boundary, and challenges proposed thermal convection in Fe-Si liquid outer core.

Quantitative estimates of average geomagnetic axial dipole dominance in deep geological time

A defining characteristic of the recent geomagnetic field is its dominant axial dipole which provides its navigational utility and dictates the shape of the magnetosphere. Going back through time, much less is known about the degree of axial dipole dominance. Here we use a substantial and diverse set of 3D numerical dynamo simulations and recent observation-based field models to derive a power law relationship between the angular dispersion of virtual geomagnetic poles at the equator and the median axial dipole dominance measured at Earth's surface. Applying this relation to published estimates of equatorial angular dispersion implies that geomagnetic axial dipole dominance averaged over 10⁷-10⁹ years has remained moderately high and stable through large parts of geological time. This provides an observational constraint to future studies of the geodynamo and palaeomagnetosphere. It also provides some reassurance as to the reliability of palaeogeographical reconstructions provided by palaeomagnetism.

Reconciliation of Experiments and Theory on Transport Properties of Iron and the Geodynamo

We measure the electrical resistivity of hcp iron up to ∼170  GPa and ∼3000  K using a four-probe van der Pauw method coupled with homogeneous flattop laser heating in a DAC, and compute its electrical and thermal conductivity by first-principles molecular dynamics including electron-phonon and electron-electron scattering. We find that the measured resistivity of hcp iron increases almost linearly with temperature, and is consistent with our computations. The results constrain the resistivity and thermal conductivity of hcp iron to ∼80±5  μΩ cm and ∼100±10  W m^{-1} K^{-1}, respectively, at conditions near the core-mantle boundary. Our results indicate an adiabatic heat flow of ∼10±1  TW out of the core, supporting a present-day geodynamo driven by thermal and compositional convection.

Electronic correlations and transport in iron at Earth's core conditions

The transport properties of iron under Earth’s inner core conditions are essential input for the geophysical modelling but are poorly constrained experimentally. Here we show that the thermal and electrical conductivity of iron at those conditions remains high even if the electron-electron-scattering (EES) is properly taken into account. This result is obtained by ab initio simulations taking into account consistently both thermal disorder and electronic correlations. Thermal disorder suppresses the non-Fermi-liquid behavior of the body-centered cubic iron phase, hence, reducing the EES; the total calculated thermal conductivity of this phase is 220 Wm⁻¹ K⁻¹ with the EES reduction not exceeding 20%. The EES and electron-lattice scattering are intertwined resulting in breaking of the Matthiessen’s rule with increasing EES. In the hexagonal close-packed iron the EES is also not increased by thermal disorder and remains weak. Our main finding thus holds for the both likely iron phases in the inner core.

The heat and electrical conductivity of Earth’s core matter represent key input quantities for geophysical models of the Earth’s core evolution and geodynamo. Here, the authors show how the scattering due to interactions between electrons has a relatively weak impact on the electrical and thermal conductivities of iron at core conditions.

Low thermal conductivity of iron-silicon alloys at Earth's core conditions with implications for the geodynamo

Earth’s core is composed of iron (Fe) alloyed with light elements, e.g., silicon (Si). Its thermal conductivity critically affects Earth’s thermal structure, evolution, and dynamics, as it controls the magnitude of thermal and compositional sources required to sustain a geodynamo over Earth’s history. Here we directly measured thermal conductivities of solid Fe and Fe–Si alloys up to 144 GPa and 3300 K. 15 at% Si alloyed in Fe substantially reduces its conductivity by about 2 folds at 132 GPa and 3000 K. An outer core with 15 at% Si would have a conductivity of about 20 W m⁻¹ K⁻¹, lower than pure Fe at similar pressure–temperature conditions. This suggests a lower minimum heat flow, around 3 TW, across the core–mantle boundary than previously expected, and thus less thermal energy needed to operate the geodynamo. Our results provide key constraints on inner core age that could be older than two billion-years.

Thermal conductivity of Earth’s core affects Earth’s thermal structure, evolution and dynamics. Based on thermal conductivity measurements of iron–silicon alloys at high pressure and temperature conditions, the authors here propose Earth’s inner core could be older than previously expected.

Electronic correlations in dense iron: from moderate pressure to Earth's core conditions

We discuss the role of dynamical many-electron effects in the physics of iron and iron-rich solid alloys under applied pressure on the basis of recent ab initio studies employing the dynamical mean-field theory (DMFT). We review in detail two particularly interesting regimes: first, a moderate pressure range up to 60 GPa and, second, the ultra-high pressure of about 360 GPa expected inside the solid inner core of Earth. Electronic correlations in iron under the moderate pressure of several tens GPa are discussed in the first section. DMFT-based methods predict an enhancement of electronic correlations at the pressure-induced body-centered cubic α to hexagonal close-packed [Formula: see text] phase transition. In particular, the electronic effective mass, scattering rate and electron-electron contribution to the electrical resistivity undergo a step-wise increase at the transition point. One also finds a significant many-body correction to the [Formula: see text]-Fe equation of state, thus clarifying the origin of discrepancies between previous DFT studies and experiment. An electronic topological transition is predicted to be induced in [Formula: see text]-Fe by many-electron effects; its experimental signatures are analyzed. The next section focuses on the geophysically relevant pressure-temperature regime of the Earth's inner core (EIC) corresponding to the extreme pressure of 360 GPa combined with temperatures up to 6000 K. The three iron allotropes ([Formula: see text], [Formula: see text] and face-centered-cubic [Formula: see text]) previously proposed as possible stable phases at such conditions are found to exhibit qualitatively different many-electron effects as evidenced by a strongly non-Fermi-liquid metallic state of [Formula: see text]-Fe and an almost perfect Fermi liquid in the case of [Formula: see text]-Fe. A recent active discussion on the electronic state and transport properties of [Formula: see text]-Fe at the EIC conditions is reviewed in details. Estimations for the dynamical many-electron contribution to the relative phase stability are presented. We also discuss the impact of a Ni admixture, which is expected to be present in the core matter. We conclude by outlining some limitation of the present DMFT-based framework relevant for studies of iron-base systems as well as perspective directions for further development.

Paramagnetic-to-Diamagnetic Transition in Dense Liquid Iron and Its Influence on Electronic Transport Properties

The electrical σ and thermal conductivity λ of liquid iron are calculated with spin-polarized density-functional-theory-based simulations over a significant pressure and temperature range using the Kubo-Greenwood formalism. We show that a paramagnetic state is stable in the liquid up to high temperatures at ambient pressure and that the discrepancy between experimental results and spin-degenerate simulations for σ and λ of more than 30% are reduced to within 10% with lower values resulting from the spin-polarized simulations. Along the 3700 K isotherm, we explore the persistence of magnetic fluctuations toward high densities, and beyond 20-50 GPa the liquid becomes diamagnetic, which suggests the existence of a continuous paramagnetic-to-diamagnetic transition. This transition exerts a significant influence on the physical properties of liquid iron, especially on σ and λ, and is potentially of high relevance for dynamo processes in Mercury and Mars.

Measurement of Preheat Due to Nonlocal Electron Transport in Warm Dense Matter

This Letter presents a novel approach to study electron transport in warm dense matter. It also includes the first x-ray Thomson scattering (XRTS) measurement from low-density CH foams compressed by a strong laser-driven shock at the OMEGA laser facility. The XRTS measurement is combined with velocity interferometry (VISAR) and optical pyrometry (SOP) providing a robust measurement of thermodynamic conditions in the shock. Evidence of significant preheat contributing to elevated temperatures reaching 17.5-35 eV in shocked CH foam is measured by XRTS. These measurements are complemented by abnormally high shock velocities observed by VISAR and early emission seen by SOP. These results are compared to radiation hydrodynamics simulations that include first-principles treatment of nonlocal electron transport in warm dense matter with excellent agreement. Additional simulations confirm that the x-ray contribution to this preheat is negligible.

Subcritical Thermal Convection of Liquid Metals in a Rapidly Rotating Sphere

Planetary cores consist of liquid metals (low Prandtl number Pr) that convect as the core cools. Here, we study nonlinear convection in a rotating (low Ekman number Ek) planetary core using a fully 3D direct numerical simulation. Near the critical thermal forcing (Rayleigh number Ra), convection onsets as thermal Rossby waves, but as Ra increases, this state is superseded by one dominated by advection. At moderate rotation, these states (here called the weak branch and strong branch, respectively) are smoothly connected. As the planetary core rotates faster, the smooth transition is replaced by hysteresis cycles and subcriticality until the weak branch disappears entirely and the strong branch onsets in a turbulent state at Ek<10^{-6}. Here, the strong branch persists even as the thermal forcing drops well below the linear onset of convection (Ra=0.7Ra_{crit} in this study). We highlight the importance of the Reynolds stress, which is required for convection to subsist below the linear onset. In addition, the Péclet number is consistently above 10 in the strong branch. We further note the presence of a strong zonal flow that is nonetheless unimportant to the convective state. Our study suggests that, in the asymptotic regime of rapid rotation relevant for planetary interiors, thermal convection of liquid metals in a sphere onsets through a subcritical bifurcation.

Local magnetic moments in iron and nickel at ambient and Earth's core conditions

Some Bravais lattices have a particular geometry that can slow down the motion of Bloch electrons by pre-localization due to the band-structure properties. Another known source of electronic localization in solids is the Coulomb repulsion in partially filled d or f orbitals, which leads to the formation of local magnetic moments. The combination of these two effects is usually considered of little relevance to strongly correlated materials. Here we show that it represents, instead, the underlying physical mechanism in two of the most important ferromagnets: nickel and iron. In nickel, the van Hove singularity has an unexpected impact on the magnetism. As a result, the electron-electron scattering rate is linear in temperature, in violation of the conventional Landau theory of metals. This is true even at Earth's core pressures, at which iron is instead a good Fermi liquid. The importance of nickel in models of geomagnetism may have therefore to be reconsidered.

Direct measurement of thermal conductivity in solid iron at planetary core conditions

The conduction of heat through minerals and melts at extreme pressures and temperatures is of central importance to the evolution and dynamics of planets. In the cooling Earth's core, the thermal conductivity of iron alloys defines the adiabatic heat flux and therefore the thermal and compositional energy available to support the production of Earth's magnetic field via dynamo action. Attempts to describe thermal transport in Earth's core have been problematic, with predictions of high thermal conductivity at odds with traditional geophysical models and direct evidence for a primordial magnetic field in the rock record. Measurements of core heat transport are needed to resolve this difference. Here we present direct measurements of the thermal conductivity of solid iron at pressure and temperature conditions relevant to the cores of Mercury-sized to Earth-sized planets, using a dynamically laser-heated diamond-anvil cell. Our measurements place the thermal conductivity of Earth's core near the low end of previous estimates, at 18-44 watts per metre per kelvin. The result is in agreement with palaeomagnetic measurements indicating that Earth's geodynamo has persisted since the beginning of Earth's history, and allows for a solid inner core as old as the dynamo.

PALEOMAGNETISM. A Hadean to Paleoarchean geodynamo recorded by single zircon crystals

Knowing when the geodynamo started is important for understanding the evolution of the core, the atmosphere, and life on Earth. We report full-vector paleointensity measurements of Archean to Hadean zircons bearing magnetic inclusions from the Jack Hills conglomerate (Western Australia) to reconstruct the early geodynamo history. Data from zircons between 3.3 billion and 4.2 billion years old record magnetic fields varying between 1.0 and 0.12 times recent equatorial field strengths. A Hadean geomagnetic field requires a core-mantle heat flow exceeding the adiabatic value and is suggestive of plate tectonics and/or advective magmatic heat transport. The existence of a terrestrial magnetic field before the Late Heavy Bombardment is supported by terrestrial nitrogen isotopic evidence and implies that early atmospheric evolution on both Earth and Mars was regulated by dynamo behavior.