arXiv subjects

Gennaro D'Angelo

Publications and source records attributed to Gennaro D'Angelo.

At least 19 recordsLinked to original sources

arXiv2026

Feeding the Circumplanetary Disk: 3D Simulations of Dust Filtration and Accretion in PDS 70 c

How circumplanetary disks (CPDs) capture and retain solids is central to constraining the timescale for formation of rocky satellites and interpreting submillimeter continuum observations around giant planets. Here, we investigate the transport of gas and dust from the circumstellar disk into the CPD, using PDS 70 as our fiducial example. Based on observation-driven parameters, we perform high-resolution 3D adaptive mesh refinement (AMR) hydrodynamic simulations including a multifluid dust component to study dust accretion onto planets with masses of 1 $M_\mathrm{J}$ and 2.5 $M_\mathrm{J}$. We find that the pressure maximum at the gap edge imposes strong, size-dependent dust filtration, drastically lowering the solid content of the accreting flow. The net dust-to-gas mass ratio of the material accreting onto the planet is reduced by roughly two orders of magnitude relative to the outer disk. Only small grains ($\lesssim 61 μ$m for the 1 $M_\mathrm{J}$ case and $\lesssim 10 μ$m for the 2.5 $M_\mathrm{J}$ case) are able to accrete efficiently onto the CPD. Despite this filtering, we show that a continuous inflow of small grains can still deliver sufficient mass to build the observed PDS 70 c CPD or a Galilean-like satellite system within a few million years. Because more massive planets more effectively prevent the accretion of large grains, the dust that reaches the CPD is dominated by small particles with low millimeter-wave opacities. Consequently, in the absence of grain growth, interpreting millimeter continuum measurements of CPDs around massive giant planets may require invoking larger total dust masses than typically assumed.

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arXiv2025

Gas and dust dynamics in $γ$ Cephei-type disks

(Abridged) Giant planets are observed orbiting the primary stars of close binary systems. Such planets may have formed in compact circumprimary disks, under conditions much different than those around single stars. To quantify the effects of the strong gravitational perturbations exerted on circumprimary disk material, the 3D dynamics of gas and dust in orbit around the primary star of a compact and eccentric binary was modeled by using gamma Cephei as a representative of a class of close binary systems. Circumprimary gas was approximated as an Eulerian viscous and compressible fluid. Dust grains were modeled as Lagrangean particles, subjected to gravity and drag forces. Some models also include a giant planet. Models show that spiral density waves excited around pericenter passage propagate toward the inner disk boundary, through a few pressure scaleheights from the midplane, inducing radial and vertical mixing in the gas. However, perturbations imparted to gas, both in terms of eccentricity and precession, are far weaker than previously estimated by 2D simulations. Models predict small eccentricities, <~ 0.03, and slow retrograde precession. The addition of a giant planet does not change the low eccentricity state of the disk. The parameters applied to the disk would lead to the formation of a planet many times the mass of Jupiter, in agreement with some observations. Micron to mm-size dust grains are well coupled to the gas, resulting in similar dynamics and statistically similar distributions of orbital elements. The planet only affects the dust distributions locally. In agreement with results of recent 2D models, the lifetime of an isolated circumprimary disk would be brief, ~1e5 years, due to its compact nature, requiring a long-term external supply of mass to allow for the in situ formation of a giant planet.

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arXiv2025

Planetary waves can activate resonant drag instabilities in 3D dusty gaseous discs

Resonant Drag Instabilities (RDIs) in protoplanetary discs are driven by the aerodynamic back-reaction of dust on gas and occur when the relative dust-gas motion resonate with a wave mode intrinsic to the gas fluid. Axisymmetric models indicate that the RDI generates filamentary perturbations, leading to grain clumping and planetesimal formation. Motivated by these findings, we investigate the dust-gas interaction in a non-axisymmetric inviscid protoplanetary disc with an embedded low-mass planet ($M_{\mathrm{p}}\in[0.3, 3] M_\oplus$, here $M_\oplus$ is the Earth mass). We conduct global 3D high-resolution two-fluid simulations, with the dust being parametrized by the Stokes number $\mathrm{St}\in[0.01,0.5]$. We find that planetary waves (PWs; also known as Rossby waves), which propagate along the downstream separatrices of the horseshoe region, resonate with the streaming motion and trigger the RDI. The consequent development of a global-scale filamentary dust distribution does not sensitively depend on the Stokes number, nor does it depend on the fast dust settling that takes place in an inviscid disc. The rapid onset of this instability, which is comparable to the dynamical orbital time-scale, suppresses the formation of asymmetric structures in the dust in the vicinity of the planet (such as dust voids and filaments). Additionally, we find that the dust feedback enables buoyancy resonances in an otherwise non-buoyant (globally isothermal) disc. Therefore, our results provide the first numerical evidence of RDIs generation driven by planetary waves.

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arXiv2025

Formation and Evolution Simulations of Saturn, Including Composition Gradients and Helium Immiscibility

The formation of Saturn is modeled by detailed numerical simulations according to the core-nucleated accretion scenario. Previous models are enhanced to include the dissolution of accreting planetesimals, composed of water ice, rock, and iron, in the gaseous envelope of the planet, leading to a non-uniform composition with depth. The immiscibility of helium in metallic hydrogen layers is also considered. The calculations start at a mass $0.5$ Earth masses and are extended to the present day. At 4.57 Gyr, the model, proceeding outwards, has the following structure: (i) a central core composed of $100$% heavy elements and molecules, (ii) a region with decreasing heavy element mass fraction, down to a value of $0.1$, (iii) a layer of uniform composition with the helium mass fraction $Y$ enhanced over the primordial value, (iv) a helium rain region with a gradient in $Y$, (v) an outer convective, adiabatic region with uniform composition in which $Y$ is reduced from the primordial value, and (vi) the very outer layers where cloud condensation of the heavy elements occurs. Models of the distribution of heavy elements as a function of radius are compared with those derived to fit the observations of the Cassini mission, with rough qualitative agreement. The helium mass fraction in Saturn's outer layers is estimated to be around $20$%. Models are found which provide good agreement with Saturn's intrinsic luminosity and radius.

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arXiv2025

Dust supply to slose binary systems

Context. Binary systems can be born surrounded by circumbinary discs. The gaseous discs surrounding either of the two stellar companions can have their life extended by the supply of mass arriving from the circumbinary disc. Aims. The objective of this study is to investigate the gravitational interactions exerted by a compact and eccentric binary system on the circumbinary and circumprimary discs, and the resulting transport of gas and solids between the disc components. Methods. We assume that the gas in the system behaves as a fluid and model its evolution by means of high resolution hydrodynamical simulations. Dust grains are modeled as Lagrangian particles that interact with the gas and the stars. Results. Models indicate that significant fluxes of gas and dust proceed from the circumbinary disc toward the circumprimary disc. For the applied system parameters, grains of certain sizes are segregated outside the tidal gap generated by the stars. Consequently, the size distribution of the transported dust is not continuous but it presents a gap in the mm size range. In close binaries, the lifetime of an isolated circumprimary disc is found to be short, approximately 10 5 years, because of its small mass. However, because of the influx of gas from beyond the tidal gap, the disc around the primary star can survive much longer, about 10 6 years, as long as gas accretion from the circumbinary disc continues. The supply of solids and the extended lifetime of a circumbinary disc also aids in the possible formation of giant planets. Compared to close binary systems without a circumbinary disc, we expect a higher frequency of single- or multiple- planet systems. Additionally, a planetesimal or debris belt can form in proximity of the truncation radius of the circumprimary disc and/or around the location of the exterior edge of the tidal gap.

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arXiv2024

Planet Formation by Gas-Assisted Accretion of Small Solids

We compute the accretion efficiency of small solids, with radii 1 cm $\le$ Rs $\le$ 10 m, on planets embedded in gaseous disks. Planets have masses 3 $\le$ Mp $\le$ 20 Earth masses (Me) and orbit within 10 AU of a solar-mass star. Disk thermodynamics is modeled via three-dimensional radiation-hydrodynamic calculations that typically resolve the planetary envelopes. Both icy and rocky solids are considered, explicitly modeling their thermodynamic evolution. The maximum efficiencies of 1 $\le$ Rs $\le$ 100 cm particles are generally $\lesssim$ 10%, whereas 10 m solids tend to accrete efficiently or be segregated beyond the planet's orbit. A simplified approach is applied to compute the accretion efficiency of small cores, with masses Mp $\le$ 1 Me and without envelopes, for which efficiencies are approximately proportional to Mp^(2/3). The mass flux of solids, estimated from unperturbed drag-induced drift velocities, provides typical accretion rates dMp/dt $\lesssim$ 1e-5 Mearth/yr. In representative disk models with an initial gas-to-dust mass ratio of 70-100 and total mass of 0.05-0.06 Msun, solids' accretion falls below 1e-6 Mearth/yr after 1-1.5 million years (Myr). The derived accretion rates, as functions of time and planet mass, are applied to formation calculations that compute dust opacity self-consistently with the delivery of solids to the envelope. Assuming dust-to-solid coagulation times of approximately 0.3 Myr and disk lifetimes of approximately 3.5 Myr, heavy-element inventories in the range 3-7 Me require that approximately 90-150 Me of solids cross the planet's orbit. The formation calculations encompass a variety of outcomes, from planets a few times the Earth mass, predominantly composed of heavy elements, to giant planets. The peak luminosities during the epoch of solids' accretion range from $\approx$ 1e-7 to $\approx$ 1e-6 times the solar luminosity.

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arXiv2022

Mixing of Condensable Constituents with H-He During the Formation & Evolution of Jupiter

Simulations of Jupiter's formation are presented that incorporate mixing of H-He with denser material entering the planet as solids. Heavy compounds and gas mix substantially when the planet becomes roughly as massive as Earth, because incoming planetesimals can fully vaporize. Supersaturation of vaporized silicates causes the excess to sink as droplets, but water remains at higher altitudes. Because the mean molecular weight decreases rapidly outward, some of the compositional inhomogeneities produced during formation can survive for billions of years. After 4.57 Gyr, our Jupiter model retains compositional gradients; proceeding outwards one finds: i) an inner heavy-element core, the outer part derived from hot supersaturated rain-out; ii) a composition-gradient region, containing most of the heavy elements, where H-He abundance increases outward, reaching about 0.9 mass fraction at 0.3 of the radius, with silicates enhanced relative to water in the lower parts and depleted in the upper parts; iii) a uniform composition region (neglecting He immiscibility) that is enriched over protosolar and contains most of the planet's mass; and iv) an outer region where cloud formation (condensation) of heavy constituents occurs. This radial compositional profile has heavy elements more broadly distributed than predicted by classical formation models, but less diluted than suggested by Juno-constrained gravity models. The compositional gradients in the region containing the bulk of the heavy elements prevent convection, in both our models and those fitting current gravity, resulting in a hot interior where much of the accretion energy remains trapped.

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arXiv2021

Second-generation dust in planetary systems: The case of HD 163296

Observations indicate that large, dust-laden protoplanetary discs are common. Some features, like gaps, rings and spirals, suggest they may host young planets, which can excite the orbits of nearby leftover planetesimals. Energetic collisions among these bodies can lead to the production of second-generation dust. Grains produced by collisions may have a dynamical behaviour different from that of first-generation, primordial dust out of which planetesimals and planets formed. We aim to study these differences for the HD 163296 system and determine whether dynamical signatures in the mixture of the two dust populations can help separate their contributions. We use three-dimensional (3-D) hydrodynamic models to describe the gaseous disc with three, Saturn- to Jupiter-mass, embedded planets. Dust grains, of sizes 1um-1mm, are treated as Lagrangean particles with resolved thermodynamics and mass loss. Initial disc and planet configurations are derived from observation-based work, which indicates low gas viscosity. The 3-D approach also allows us to detect the formation of vortices induced by Rossby waves, where dust becomes concentrated and may contribute to planetesimal formation. We find that the main differences in the dynamical behaviour of first- and second-generation dust occur in the vertical distribution. The two populations have similar distributions around the disc mid-plane, although second-generation dust shows longer residence times close to the radial locations of the planets' gas gaps. Sedimentation rates of um-size grains are comparable to or lower than the production rates by planetesimals' collisions, making this population potentially observable. These outcomes can be extended to similar systems harbouring giant planets.

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arXiv2020

Growth of Jupiter: Formation in Disks of Gas and Solids and Evolution to the Present Epoch

[Abridged] The formation of Jupiter is modeled via core-nucleated accretion, and the planet's evolution is simulated up to the present epoch. The growth from a small embryo until gas accretion overtakes solids' accretion was presented by D'Angelo et al. (Icarus 2014, 241, 298). Those calculations followed the formation for $4\times 10^{5}$ years, until the heavy-element and H/He masses were $M_{Z}\approx 7.3$ and $M_{XY}\approx 0.15$ Earth's masses ($M_{\oplus}$), respectively, and $dM_{XY}/dt\approx dM_{Z}/dt$. The calculation is continued through the phase when $M_{XY}=M_{Z}$, at which age, about $2.4\times 10^{6}$ years, the planet mass is $M_{p}\approx 20\,M_{\oplus}$. About $9\times 10^{5}$ years later, $M_{p}$ is approximately $60\,M_{\oplus}$ and $M_{Z}\approx 16\,M_{\oplus}$. Around this epoch, the contraction of the envelope dictates gas accretion rates a few times $10^{-3}\,M_{\oplus}$ per year, initiating the regime of disk-limited accretion, when the planet's evolution is tied to disk's evolution. Growth is continued by constructing simplified models of accretion disks that evolve through viscous diffusion, winds, and accretion on the planet. Jupiter's formation ends after $\approx 3.4$-$4.2$ Myr, when nebula gas disperses. The young Jupiter is $4.5$-$5.5$ times as voluminous as it is presently and thousands of times as luminous, $\sim 10^{-5}\,L_{\odot}$. The heavy-element mass is $\approx 20\,M_{\oplus}$. The evolution proceeds through the cooling and contraction phase, in isolation except for solar irradiation. After $4570$ Myr, radius and luminosity of the planet are within $10$% of current values. During formation, and soon thereafter, the planet exhibits features, e.g., luminosity and effective temperature, that may probe aspects of the latter stages of formation, if observable. These possibly distinctive features, however, seem to disappear within a few tens of Myr.

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arXiv2020

Dust distribution around low-mass planets on converging orbits

Super-Earths can form at large orbital radii and migrate inward due to tidal interactions with the circumstellar disk. In this scenario, convergent migration may occur and lead to the formation of resonant pairs of planets. We explore the conditions under which convergent migration and resonance capture take place, and what dynamical consequences can be expected on the dust distribution surrounding the resonant pair. We combine hydrodynamic planet--disk interaction models with dust evolution calculations to investigate the signatures produced in the dust distribution by a pair of planets in mean-motion resonances. We find that convergent migration takes place when the outer planet is the more massive. However, convergent migration also depends on the local properties of the disk, and divergent migration may result as well. For similar disk parameters, the capture in low degree resonances (e.g., 2:1 or 3:2) is preferred close to the star where the resonance strength can more easily overcome the tidal torques exerted by the gaseous disk. Farther away from the star, convergent migration may result in capture in high degree resonances. The dust distribution shows potentially observable features typically when the planets are trapped in a 2:1 resonance. In other cases, with higher degree resonances (e.g., 5:4 or 6:5) dust features may not be sufficiently pronounced to be easily observable. The degree of resonance established by a pair of super-Earths may be indicative of the location in the disk where capture occurred. There can be significant differences in the dust distribution around a single super-Earth and a pair of super-Earths in resonance.

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arXiv2020

Capture and migration of Jupiter and Saturn in mean motion resonance in a gaseous protoplanetary disc

We study the dynamical evolution of Jupiter and Saturn embedded in a gaseous, solar-nebula-type disc by means of hydrodynamics simulations with the FARGO2D1D code. We study the evolution for different initial separations of the planets' orbits, Delta a_SJ , to investigate whether they become captured in mean motion resonance (MMR) and the direction of the subsequent migration of the planet (inward or outward). We also provide an assessment of the planet's orbital dynamics at different epochs of Saturn's growth. We find that the evolution of initially compact orbital configurations is dependent on the value of Delta a_SJ . This implies that an evolution as that proposed in the Grand Tack model depends on the precise initial orbits of Jupiter and Saturn and on the timescales for their formation. Capture in the 1:2 MMR and inward or (nearly) stalled migration are highly favoured. Within its limits, our work suggests that the reversed migration, associated with the resonance capture of Jupiter and Saturn, may be a low probability evolutionary scenario, so that other planetary systems with giant planets are not expected to have experienced a Grand Tack-like evolutionary path.

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arXiv2018

Circumstellar dust distribution in systems with two planets in resonance

We investigate via numerical modeling the effects of two planets locked in resonance, and migrating outward, on the dust distribution of the natal circumstellar disk. We aim to test whether the dust distribution exhibits peculiar features arising from the interplay among the gravitational perturbations of the planets in resonance, the evolution of the gas, and its influence on the dust grains' dynamics. We focus on the 3:2 and 2:1 resonance, where the trapping may be caused by the convergent migration of a Jupiter- and Saturn-mass planet, preceding the common gap formation and ensuing outward (or inward) migration. Models show that a common gap also forms in the dust component -- similarly to what a single, more massive planet would generate -- and that outward migration leads to a progressive widening of the dust gap and to a decoupling from the gas gap. As the system evolves, a significantly wider gap is observed in the dust distribution, which ceases to overlap with the gas gap in the inner disk regions. At the outer edge of the gas gap, outward migration of the planets produces an over-density of dust particles, which evolve differently in the 3:2 and 2:1 resonances. For the 3:2, the dust trap at the gap's outer edge is partly efficient and a significant fraction of the grains filters through the gap. For the 2:1 resonance, the trap is more efficient and very few grains cross the gap, while the vast majority accumulate at the outer edge of the gap.

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arXiv2018

New Formation Models for the Kepler-36 System

Formation of the planets in the Kepler-36 system is modeled by detailed numerical simulations according to the core-nucleated accretion scenario. The standard model is updated to include the dissolution of accreting rocky planetesimals in the gaseous envelope of the planet, leading to substantial enrichment of the envelope mass in heavy elements and a non-uniform composition with depth. For Kepler-36 c, models involving in situ formation and models involving orbital migration are considered. The results are compared with standard formation models. The calculations include the formation (accretion) phase, as well as the subsequent cooling phase, up to the age of Kepler-36 (7 Gyr). During the latter phase, mass loss induced by stellar XUV radiation is included. In all cases, the results fit the measured mass, 7.84 M$_\oplus$, and radius, 3.68 R$_\oplus$, of Kepler-36 c. Two parameters are varied to obtain these fits: the disk solid surface density at the formation location, and the "efficiency" factor in the XUV mass loss rate. The updated models are hotter and therefore less dense in the silicate portion of the planet and in the overlying layers of H/He, as compared with standard models. The lower densities mean that only about half as much H/He is needed to be accreted to fit the present-day mass and radius constraints. For Kepler-36 b, an updated in situ calculation shows that the entire H/He envelope is lost, early in the cooling phase, in agreement with observation.

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arXiv2018

Formation of Giant Planets

Giant planets are tens to thousands of times as massive as the Earth, and many times as large. Most of their volumes are occupied by hydrogen and helium, the primary constituents of the protostellar disks from which they formed. Significantly, the solar system giants are also highly enriched in heavier elements relative to the Sun, indicating that solid material participated in their assembly. Giant planets account for most of the mass of our planetary system and of those extrasolar planetary systems in which they are present. Therefore, giant planets are primary actors in determining the orbital architectures of planetary systems and, possibly, in affecting the composition of terrestrial planets. This Chapter describes the principal route that, according to current knowledge, can lead to the formation of giant planets, the core nucleated accretion model, and an alternative route, the disk instability model, which may lead to the formation of planetary-mass objects on wide orbits.

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arXiv2018

Witnessing Planetary Systems in the Making with the Next Generation Very Large Array

The discovery of thousands of exoplanets over the last couple of decades has shown that the birth of planets is a very efficient process in nature. Theories invoke a multitude of mechanisms to describe the assembly of planets in the disks around pre-main-sequence stars, but observational constraints have been sparse on account of insufficient sensitivity and resolution. Understanding how planets form and interact with their parental disk is crucial also to illuminate the main characteristics of a large portion of the full population of planets that is inaccessible to current and near-future observations. This White Paper describes some of the main issues for our current understanding of the formation and evolution of planets, and the critical contribution expected in this field by the Next Generation Very Large Array.

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arXiv2016

In Situ and Ex Situ Formation Models of Kepler 11 Planets

We present formation simulations of the six Kepler 11 planets. Models assume either in situ or ex situ assembly, the latter with migration, and are evolved to the estimated age of the system, 8 Gyr. Models combine detailed calculations of both the gaseous envelope and the condensed core structures, including accretion of gas and solids, of the disk's viscous and thermal evolution, including photo-evaporation and disk-planet interactions, and of the planets' evaporative mass loss after disk dispersal. Planet-planet interactions are neglected. Both sets of simulations successfully reproduce measured radii, masses, and orbital distances of the planets, except for the radius of Kepler 11b, which loses its entire gaseous envelope shortly after formation. Gaseous (H+He) envelopes account for < 18% of the planet masses, and between 35 and 60% of the planet radii. In situ models predict a very massive inner disk, whose solids' surface density (sigma_Z) varies from over 1e4 to 1e3 g/cm2 at stellocentric distances 0.1 < r < 0.5 AU. Initial gas densities would be in excess of 1e5 g/cm2 if solids formed locally. Given the high disk temperatures (> 1000 K), planetary interiors can only be composed of metals and highly refractory materials. Sequestration of hydrogen by the core and subsequent outgassing is required to account for the observed radius of Kepler 11b. Ex situ models predict a relatively low-mass disk, whose initial sigma_Z varies from 10 to 5 g/cm2 at 0.5 < r < 7 AU and whose initial gas density ranges from 1e3 to 100 g/cm2. All planetary interiors are expected to be rich in H2O, as core assembly mostly occurs exterior to the ice condensation front. Kepler 11b is expected to have a steam atmosphere, and H2O is likely mixed with H+He in the envelopes of the other planets. Results indicate that Kepler 11g may not be more massive than Kepler 11e.

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arXiv2015

Capture and Evolution of Planetesimals in Circumjovian Disks

We study the evolution of planetesimals in evolved gaseous disks, which orbit a solar-mass star and harbor a Jupiter-mass planet at a_p~5AU. The gas dynamics is modeled with a three-dimensional hydrodynamics code that employes nested-grids and achieves a resolution of one Jupiter's radius in the circumplanetary disk. The code models solids as individual particles. Planetesimals are subjected to gravitational forces by the star and the planet, drag force by the gas, disruption via ram pressure, and mass loss through ablation. The mass evolution of solids is calculated self-consistently with their temperature, velocity, and position. We consider icy and icy/rocky bodies of radius 0.1-100km, initially deployed on orbits around the star within a few Hill radii (Rhill) of the planet's orbit. Planetesimals are scattered inward, outward, and toward disk regions of radius r>>a_p. Scattering can relocate significant amounts of solids, provided that regions |r-a_p|~ 3Rhill are replenished with planetesimals. Scattered bodies can be temporarily captured on planetocentric orbits. Ablation consumes nearly all solids at gas temperatures > ~220K. Super-keplerian rotation around and beyond the outer edge of the gas gap can segregate < ~0.1km bodies, producing solid gap edges at size-dependent radial locations. Capture, break-up, and ablation of solids result in a dust-laden circumplanetary disk with low surface densities of km-size planetesimals, implying relatively long timescales for satellite formation. After a giant planet acquires most of its mass, accretion of solids is unlikely to alter significantly its heavy-element content. The luminosity generated by solids' accretion can be of a similar order of magnitude to the contraction luminosity.

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arXiv2014

Growth of Jupiter: Enhancement of Core Accretion by a Voluminous Low-Mass Envelope

We present calculations of the early stages of the formation of Jupiter via core nucleated accretion and gas capture. The core begins as a seed body of about 350 kilometers in radius and orbits in a swarm of planetesimals whose initial radii range from 15 meters to 50 kilometers. The evolution of the swarm accounts for growth and fragmentation, viscous and gravitational stirring, and for drag-assisted migration and velocity damping. During this evolution, less than 9% of the mass is in planetesimals smaller than 1 kilometer in radius; < ~25% is in planetesimals with radii between 1 and 10 kilometers; and < ~7% is in bodies with radii larger than 100 kilometers. Gas capture by the core substantially enhances the size-dependent cross-section of the planet for accretion of planetesimals. The calculation of dust opacity in the planet's envelope accounts for coagulation and sedimentation of dust particles released as planetesimals are ablated. The calculation is carried out at an orbital semi-major axis of 5.2 AU and the initial solids' surface density is 10 g/cm^2 at that distance. The results give a core mass of nearly 7.3 Earth masses (Mearth) and an envelope mass of approximately 0.15 Mearth after about 4e5 years, at which point the envelope growth rate surpasses that of the core. The same calculation without the envelope yields a core of only about 4.4 Mearth.

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