Here, we develop scaling laws for (1) the distribution of impact-induced heat within the mantle and (2) shape of the impact-induced melt based on more than 100 smoothed particle hydrodynamic (SPH) simulations. We use Legendre polynomials to describe these scaling laws and determine their coefficients by linear regression, minimizing the error between our model and SPH simulations. The input parameters are the impact angle $θ$ ($0^{\circ}, 30^{\circ}, 60^{\circ}$, and $90^{\circ}$), total mass $M_T$ ($1M_{\rm Mars}-53M_{\rm Mars}$, where $M_{\rm Mars}$ is the mass of Mars), impact velocity $v_{\rm imp}$ ($v_{\rm esc} - 2v_{\rm esc}$, where $v_{\rm esc}$ is the mutual escape velocity), and impactor-to-total mass ratio $γ$ ($0.03-0.5$). We find that the equilibrium pressure at the base of a melt pool can be higher (up to $\approx 80 \%$) than those of radially-uniform global magma ocean models. This could have a significant impact on element partitioning. These melt scaling laws are publicly available on GitHub ($\href{https://github.com/mikinakajima/MeltScalingLaw}{https://github.com/mikinakajima/MeltScalingLaw}$).
Magma oceans are a common result of the high degree of heating that occurs during planet formation. It is thought that almost all of the large rocky bodies in the Solar System went through at least one magma ocean phase. In this paper, we review some of the ways in which magma ocean models for the Earth, Moon, and Mars match present day observations of mantle reservoirs, internal structure, and primordial crusts, and then we present new calculations for the oxidation state of the mantle produced during the magma ocean phase. The crystallization of magma oceans likely leads to a massive mantle …
The water content of magma oceans is widely accepted as a key factor that determines whether a terrestrial planet is habitable. Water ocean mass is determined as a result not only of water delivery and loss, but also of water partitioning among several reservoirs. Here we review our current understanding of water partitioning among the atmosphere, magma ocean, and solid mantle of accreting planetary embryos and protoplanets just after giant collisions. Magma oceans are readily formed in planetary embryos and protoplanets in their accretion phase. Significant amounts of water are partitioned in…
Intense tidal heating within Io produces active volcanism on the surface, and its internal structure has long been a subject of debate. A recent reanalysis of the Galileo magnetometer data suggested the presence of a high melt fraction layer with $>$50~km thickness in the subsurface region of Io. Whether this layer is a ``magmatic sponge'' with interconnected solid or a rheologically liquid ``magma ocean'' would alter the distribution of tidal heating and would also influence the interpretation of various observations. To this end, we explore the steady state of a magmatic sponge and estimate …
The energy associated with giant impacts is large enough to generate global magma oceans during Earth's accretion. However, geochemical evidence requiring a terrestrial magma ocean is scarce. Here we present evidence for at least two separate magma ocean outgassing episodes on Earth based on the ratio of primordial 3He to 22Ne in the present-day mantle. We demonstrate that the depleted mantle 3He/22Ne ratio is at least 10 while a more primitive mantle reservoir has a 3He/22Ne ratio of 2.3 to 3. The 3He/22Ne ratios of the mantle reservoirs are higher than possible sources of terrestrial volatil…
The Dawn mission confirms predictions that the asteroid 4 Vesta is differentiated with an iron-rich core, a silicate mantle and a basaltic crust, and confirms Vesta as the parent body of the HED meteorites. To better understand its early evolution, we perform numerical calculations of the thermo-chemical evolution adopting new data obtained by the Dawn mission such as mass, bulk density and size of the asteroid. We have expanded the model of Neumann et al. (2012) that includes accretion, compaction, melting and associated changes of material properties and partitioning of 26Al, advective heat …
Massive steam and CO$_2$ atmospheres have been proposed for magma ocean outgassing of Earth and terrestrial planets. Yet formation of such atmospheres depends on volatile exchange with the molten interior, governed by volatile solubilities and redox reactions. We determine the evolution of magma ocean--atmosphere systems for a range of oxygen fugacities, C/H ratios and hydrogen budgets that include redox reactions for hydrogen (H$_2$--H$_2$O), carbon (CO--CO$_2$), methane (CH$_4$), and solubility laws for H$_2$O and CO$_2$. We find that small initial budgets of hydrogen, high C/H ratios, and o…
Recent observations of the potentially habitable planets TRAPPIST-1 e, f, and g suggest that they possess large water mass fractions of possibly several tens of wt% of water, even though the host star's activity should drive rapid atmospheric escape. These processes can photolyze water, generating free oxygen and possibly desiccating the planet. After the planets formed, their mantles were likely completely molten with volatiles dissolving and exsolving from the melt. In order to understand these planets and prepare for future observations, the magma ocean phase of these worlds must be underst…
We demonstrate that the deep volatile storage capacity of magma oceans has significant implications for the bulk composition, interior and climate state inferred from exoplanet mass and radius data. Experimental petrology provides the fundamental properties on the ability of water and melt to mix. So far, these data have been largely neglected for exoplanet mass-radius modeling. Here, we present an advanced interior model for water-rich rocky exoplanets. The new model allows us to test the effects of rock melting and the redistribution of water between magma ocean and atmosphere on calculated …
The magma ocean concept was first conceived to explain the geology of the Moon, but hemispherical or global oceans of silicate melt could be a widespread "lava world" phase of rocky planet accretion, and could persist on planets on short-period orbits around other stars. The formation and crystallization of magma oceans could be a defining stage in the assembly of a core, origin of a crust, initiation of tectonics, and formation of an atmosphere. The last decade has seen significant advances in our understanding of this phenomenon through analysis of terrestrial and extraterrestrial samples, p…
Despite its small size, the asteroid 4 Vesta has been completely differentiated to core and mantle. Its composition is similar to howardite--eucrite--diogenite (HED) meteorites of which the detailed petrology is known. Therefore, Vesta is a good target for understanding the differentiation of terrestrial planets. A new differentiation model for crust formation has been developed by taking magma ocean fluid dynamics, chemical equilibrium, the presence of $^{26}$Al, and cooling into consideration with a special focus on crystal separation. The role of crystal size, thickness of the conductive li…
Terrestrial planets are thought to experience episode(s) of large-scale melting early in their history. Fractionation during magma-ocean freezing leads to unstable stratification within the related cumulate layers due to progressive iron enrichment upward, but the effects of incremental cumulate overturns during MO crystallization remain to be explored. Here, we use geodynamic models with a moving-boundary approach to study convection and mixing within the growing cumulate layer, and thereafter within the fully crystallized mantle. For fractional crystallization, cumulates are efficiently stir…
In this paper we propose an analytic function for the spherical albedo values of airless and near-airless magma ocean planets (AMOPs). We generated 2-D fractal surfaces with varying compositions onto which we individually threw 10,000 light rays. Using an approximate form of the Fresnel equations we measured how much of the incident light was reflected. Having repeated this algorithm on varying surface roughnesses we find the spherical albedo as a function of the Hurst exponent, the geochemical composition of the magma, and the wavelength. As a proof of concept, we used our model on Kepler-10b…
Low-mass M stars are plentiful in the Universe and often host small, rocky planets detectable with the current instrumentation. Recently, seven small planets have been discovered orbiting the ultracool dwarf TRAPPIST-1\cite{Gillon16,Gillon17}. We examine the role of electromagnetic induction heating of these planets, caused by the star's rotation and the planet's orbital motion. If the stellar rotation and magnetic dipole axes are inclined with respect to each other, induction heating can melt the upper mantle and enormously increase volcanic activity, sometimes producing a magma ocean below t…
The magma ocean (MO) is a crucial stage in the build-up of terrestrial planets. Its solidification and the accompanying outgassing of volatiles set the conditions for important processes occurring later or even simultaneously, such as solid-state mantle convection and atmospheric escape. To constrain the duration of a global-scale Earth MO we have built and applied a 1D interior model coupled alternatively with a grey H2O/CO2 atmosphere or with a pure H2O atmosphere treated with a line-by-line model described in a companion paper by Katyal et al. (2019). We study in detail the effects of sever…
A sufficient amount of water is required at the surface to develop water oceans. A significant fraction of water, however, remains in the mantle during magma ocean solidification, and thus the existence of water oceans is not guaranteed even for exoplanets located in the habitable zone. To discuss the likelihood of ocean formation, we built two models to predict the rate of mantle degassing during the magma ocean stage and the subsequent solid-state convection stage. We find that planets with low H$_2$O/CO$_2$ ratios would not have a sufficient amount of surface water to develop water oceans i…
The evolution of Earth's early atmosphere and the emergence of habitable conditions on our planet are intricately coupled with the development and duration of the magma ocean phase during the early Hadean period (4 to 4.5 Ga). In this paper, we deal with the evolution of the steam atmosphere during the magma ocean period. We obtain the outgoing longwave radiation using a line-by-line radiative transfer code GARLIC. Our study suggests that an atmosphere consisting of pure H$_{2}$O, built as a result of outgassing extends the magma ocean lifetime to several million years. The thermal emission as…
Astronomers have discovered a handful of exoplanets with rocky bulk compositions but orbiting so close to their host star that the surface of the planet must be at least partially molten. It is expected that the dayside of such "lava planets" harbors a rock vapor atmosphere that flows quickly towards the airless nightside -- this partial atmosphere is critical to the interpretation of lava planet observations, but transports negligible heat towards the nightside. As a result, the surface temperature of the magma ocean may range from 3000~K near the sub-stellar point down to 1500~K near the day…
During accretion, the young rocky planets are so hot that they become endowed with a magma ocean. From that moment, the mantle convective thermal flux control the cooling of the planet and an atmosphere is created by outgassing. This atmosphere will then play a key role during this cooling phase. Studying this cooling phase in details is a necessary step to explain the great diversity of the observed telluric planets and especially to assess the presence of surface liquid water. We used here a radiative-convective 1D atmospheric model (H2O, CO2) to study the impact of the Bond albedo on the ev…
We create the first quantitative model for the early lunar atmosphere, coupled with a magma ocean crystallization model. Immediately after formation, the moon's surface was subject to a radiative environment that included contributions from the early Sun, a post-impact Earth that radiated like a mid-type M dwarf star, and a cooling global magma ocean. This radiative environment resulted in a largely Earth-side atmosphere on the Moon, ranging from $\sim$10$^4$ to $\sim$10$^2$ pascals, composed of heavy volatiles (Na and SiO). This atmosphere persisted through lid formation and was additionally …
The Moon-forming giant impact extensively melts and partially vaporizes the silicate Earth and delivers a substantial mass of metal to Earth's core. Subsequent evolution of the magma ocean and overlying atmosphere has been described by theoretical models but observable constraints on this epoch have proved elusive. Here, we report calculations of the primordial atmosphere during the magma ocean and water ocean epochs and forge new links with observations to gain insight into the behavior of volatiles on the early Earth. As Earth's magma ocean crystallizes, it outgasses the bulk of the volatile…
Various theories have been proposed to explain the Moon's current inclined orbit. We test the viability of these theories by reconstructing the thermal-orbital history of the Moon. We build on past thermal-orbital models and incorporate the evolution of the lunar figure including a fossil figure component. Obliquity tidal heating in the lunar magma ocean would have produced rapid inclination damping, making it difficult for an early inclination to survive to the present-day. An early inclination is preserved only if the solid-body of the early Moon were less dissipative than at present. If ins…
Anorthosites that comprise the bulk of the lunar crust are believed to have formed during solidification of a Lunar Magma Ocean (LMO) in which these rocks would have floated to the surface. This early flotation crust would have formed a thermal blanket over the remaining LMO, prolonging solidification. Geochronology of lunar anorthosites indicates a long timescale of LMO cooling, or re-melting and re-crystallization in one or more late events. To better interpret this geochronology, we model LMO solidification in a scenario where the Moon is being continuously bombarded by returning projectile…
Magma ocean interactions can explain JWST observations of the sub-Neptune TOI-270 d
School of Earth and Space Exploration, Arizona State University（亚利桑那州立大学地球与空间探索学院）；Department of Astronomy, University of Maryland（马里兰大学天文学系）；Space Telescope Science Institute（太空望远镜科学研究所）；Department of Astronomy & Astrophysics, University of Chicago（芝加哥大学天文学与天体物理学系）；Kapteyn Astronomical Institute, University of Groningen（格罗宁根大学卡普坦天文研究所）；Earth and Planets Laboratory, Carnegie Institution for Science（卡内基科学学会地球与行星实验室）
We present the thermal evolution and emergent spectra of solidifying terrestrial planets along with the formation of steam atmospheres. The lifetime of a magma ocean and its spectra through a steam atmosphere depends on the orbital distance of the planet from the host star. For a type-I planet, which is formed beyond a certain critical distance from the host star, the thermal emission declines on a timescale shorter than approximately $10^6$ years. Therefore, young stars should be targets when searching for molten planets in this orbital region. In contrast, a type-II planet, which is formed i…
Internal redox reactions may irreversibly alter the mantle composition and volatile inventory of terrestrial and super-Earth exoplanets and affect the prospects for atmospheric observations. The global efficacy of these mechanisms, however, hinges on the transfer of reduced iron from the molten silicate mantle to the metal core. Scaling analysis indicates that turbulent diffusion in the internal magma oceans of sub-Neptunes can kinetically entrain liquid iron droplets and quench core formation. This suggests that the chemical equilibration between core, mantle, and atmosphere may be energetica…
Planetary atmospheres are commonly thought to result from the efficient outgassing of cooling magma oceans. During this stage, vigorous convective motions in the molten interior are believed to rapidly transport the dissolved volatiles to shallow depths where they exsolve and burst at the surface. This assumption of efficient degassing and atmosphere formation has important implications for planetary evolution, but has never been tested against fluid dynamics considerations. Yet, during a convective cycle, only a finite fraction of the magma ocean can reach the shallow depths where volatiles e…
Metal-Silicate Segregation During Planetary Accretion: Limited Iron Emulsification through intermediate Impacts
School of Science and Engineering, Tulane University（杜兰大学科学与工程学院）；Department of Physics, University of Louisiana at Lafayette（路易斯安那拉斐特大学物理系）；School of Earth and Atmospheric Sciences, Queensland University of Technology（昆士兰科技大学地球与大气科学学院）
The impact heat accumulated during the late stage of planetary accretion can melt a significant part or even the entire mantle of a terrestrial body, giving rise to a global magma ocean. [...] Assuming fractional crystallization of the magma ocean, dense cumulates are produced close to the surface, largely due to iron enrichment in the evolving magma ocean liquid (Elkins-Tanton et al., 2003). A gravitationally unstable mantle thus forms, which is prone to overturn. We investigate the cumulate overturn and its influence on the thermal evolution of Mars using mantle convection simulations in 2D …
When the extent of protoplanetary melting approached magma ocean (MO)-like conditions, alloy melts efficiently segregated from the silicates to form metallic cores. The nature of the MO of a differentiating protoplanet, i.e., internal or external MO (IMO or EMO), not only determines the abundances of life-essential volatiles like nitrogen (N) and carbon (C) in its core and mantle reservoirs but also the timing and mechanism of volatile loss. Whether the earliest formed protoplanets had IMOs or EMOs is, however, poorly understood. Here we model equilibrium N and C partitioning between alloy and…
Terrestrial planets with large water inventories are likely ubiquitous and will be among the first Earth-sized planets to be characterized with upcoming telescopes. It has previously been argued that waterworlds-particularly those possessing more than 1% H$_2$O-experience limited melt production and outgassing due to the immense pressure overburden of their overlying oceans, unless subject to high internal heating. But an additional, underappreciated obstacle to outgassing on waterworlds is the high solubility of volatiles in high-pressure melts. Here, we investigate this phenomenon and show t…
