arXiv subjects

Michiel Lambrechts

Publications and source records attributed to Michiel Lambrechts.

At least 37 records · Page 2Linked to original sources

arXiv2024

Dust ring and gap formation by gas flow induced by low-mass planets embedded in protoplanetary disks $\rm II$. Time-dependent model

The observed dust rings and gaps in protoplanetary disks could be imprints of forming planets. Even low-mass planets in the one-to-ten Earth-mass regime, that do not yet carve deep gas gaps, can generate such dust rings and gaps by driving a radially-outwards gas flow, as shown in previous work. However, understanding the creation and evolution of these dust structures is challenging due to dust drift and diffusion, requiring an approach beyond previous steady state models. Here we investigate the time evolution of the dust surface density influenced by the planet-induced gas flow, based on post-processing three-dimensional hydrodynamical simulations. We find that planets larger than a dimensionless thermal mass of $m=0.05$, corresponding to 0.3 Earth mass at 1 au or 1.7 Earth masses at 10 au, generate dust rings and gaps, provided that solids have small Stokes numbers} (${\rm St}\lesssim10^{-2}$) and that the disk midplane is weakly turbulent ($α_{\rm diff}\lesssim10^{-4}$). As dust particles pile up outside the orbit of the planet, the interior gap expands with time, when the advective flux dominates over diffusion. Dust gap depths range from a factor a few, to several orders of magnitude, depending on planet mass and the level of midplane particle diffusion. We construct a semi-analytic model describing the width of the dust ring and gap, and then compare it with the observational data. We find that up to 65\% of the observed wide-orbit gaps could be explained as resulting from the presence of a low-mass planet, assuming $α_{\rm diff}=10^{-5}$ and ${\rm St}=10^{-3}$. However, it is more challenging to explain the observed wide rings, which in our model would require the presence of a population of small particles (${\rm St=10^{-4}}$). Further work is needed to explore the role of pebble fragmentation, planet migration, and the effect of multiple planets.

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arXiv2024

How the presence of a giant planet affects the outcome of terrestrial planet formation simulations

The architecture and masses of planetary systems in the habitable zone could be strongly influenced by outer giant planets, if present. We investigate here the impact of outer giants on terrestrial planet formation, under the assumption that the final assembly of the planetary system is set by a giant impact phase. Utilizing a state-of-the-art N-body simulation software, GENGA, we interpret how the late stage of terrestrial planet formation results in diversity within planetary systems. We design two global model setups: in the first we place a gas giant on the outer side of planetesimals and embryos disk, while the other only has planetesimals and embryos but no giant. For the model including the outer giant, we study the effect of different giant initial masses, in the range 1.0-3.0 Jupiter mass, and orbital radii, in the range 2.0-5.8 AU.We also study the influence of different initial positions of planetesimals and embryos on the results. Our N-body simulation time is approximately 50 Myr. The results show that the existence of outer giant will promote the interaction between planetesimals and embryos, making the orbits of the formed terrestrial planets more compact, but placing the giant planet too close to the planetesimals and embryos disk suppresses the formation of massive rocky planets. In addition, under the classical theory, where planetary embryos and planetesimals collide to form terrestrial planets, our results show that the presence of a giant planet actually decreases the gap complexity of the inner planetary system.

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arXiv2024

Thermal processing of primordial pebbles in evolving protoplanetary disks

During protoplanetary disk formation, dust grains located in the outer disk retain their pristine icy composition, while solids in the inner stellar-heated disk undergo volatile loss. This process may have left a fossil record in Solar System material showing different nucelosynthetic imprints that have been attributed to different degrees of thermal processing. However, it remains unclear how a large mass fraction of thermally-processed inner-disk pebbles is produced and how these grains are subsequently transported throughout the disk. In this work we numerically investigate the evolution in time of a two-component pebble disk, consisting of pristine pebbles and those that underwent ice sublimation. We find that stellar outbursts exceeding 1000 times the solar luminosity are efficient in thermally altering, through ice sublimation, a large mass fraction of pebbles (around 80%). After the establishment of this initial radial dust composition gradient throughout the disk, the subsequent mixing and inward drift of pristine outer-disk pebbles alter the inner disk bulk composition from processed to more unprocessed in time. Therefore, if processed pebbles without ice mantles have an isotopic composition similar to ureilite meteorites from the inner Solar System, inner-disk minor bodies forming from the early pebble flux (<1Myr) will be isotopically ureilite-like, while later-formed bodies will be increasingly admixed with the signature of the late incoming CI chondrite-like unprocessed pebbles. This appears to be largely consistent with the trend seen between the accretion age of different meteoric classes and their different stable isotope composition anomalies (in $μ$54Cr, $μ$48Ca, $μ$30Si, $μ$58Ni), but further work may be needed to explain the role of isotopically anomalous refractory inclusions and anomaly trends in other elements.

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arXiv2024

Formation of wide-orbit giant planets in protoplanetary disks with a decreasing pebble flux

The presence of distant protoplanets may explain the observed gaps in the dust emission of protoplanetary disks. Here, we derive a novel analytical model to describe the temporal decay of the pebble flux through a protoplanetary disk as the result of radial drift. This has allowed us to investigate the growth and migration of distant protoplanets throughout the lifespan of the disk. We find that Moon-mass protoplanets that formed early on can grow to their pebble isolation mass, between approximately $20$ and $80\,M_{\oplus}$, within less than $1\,\mathrm{Myr,}$ in the $20$ to $80\,\mathrm{AU}$ region around solar-like stars. The subsequent fast migration in the early stages of gas accretion, after pebble accretion ends, transports these giant planets into their final orbits at $<\,$$10\,\mathrm{AU}$. However, our pebble decay model allows us to include a new pathway that may trigger the transition from pebble accretion to gas accretion after the pebble flux has decayed substantially. With this pebble decay pathway, we show that it is also possible to form gas giants beyond $10\,\mathrm{AU}$. The occurrence of these wide-orbit gas giants should be relatively low, since their core must attain sufficient mass to accrete gas before the pebble flux decays, while avoiding excessive migration. Since these gas giants do not reach the pebble isolation mass, their heavy element content is typically less than $10\,M_{\oplus}$. Our results imply that the observed gaps in protoplanetary disks could be caused by distant protoplanets that reached the pebble isolation mass and then migrated, while gas giants in wide orbits, such as PDS 70 b and c, accreted their gas after the decay in the pebble flux.

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arXiv2023

JWST reveals excess cool water near the snowline in compact disks, consistent with pebble drift

Previous analyses of mid-infrared water spectra from young protoplanetary disks observed with the Spitzer-IRS found an anti-correlation between water luminosity and the millimeter dust disk radius observed with ALMA. This trend was suggested to be evidence for a fundamental process of inner disk water enrichment, used to explain properties of the Solar System 40 years ago, in which icy pebbles drift inward from the outer disk and sublimate after crossing the snowline. Previous analyses of IRS water spectra, however, were uncertain due to the low spectral resolution that blended lines together. We present new JWST-MIRI spectra of four disks, two compact and two large with multiple radial gaps, selected to test the scenario that water vapor inside the snowline is regulated by pebble drift. The higher spectral resolving power of MIRI-MRS now yields water spectra that separate individual lines, tracing upper level energies from 900 K to 10,000 K. These spectra clearly reveal excess emission in the low-energy lines in compact disks, compared to the large disks, demonstrating an enhanced cool component with $T \approx$ 170-400 K and equivalent emitting radius $R_{\rm{eq}}\approx$ 1-10 au. We interpret the cool water emission as ice sublimation and vapor diffusion near the snowline, suggesting that there is indeed a higher inwards mass flux of icy pebbles in compact disks. Observation of this process opens up multiple exciting prospects to study planet formation chemistry in inner disks with JWST.

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arXiv2023

Disc population synthesis: Decrease in the solid mass reservoir through pebble drift

Surveys of star-forming regions reveal that the dust mass of protoplanetary discs decreases by several orders of magnitude on a timescale of a few million years. This decrease in the mass budget of solids is likely due to the gas-drag-induced radial drift of mm-sized solids, called pebbles. However, quantifying the evolution of this dust component in young stellar clusters is difficult due to the inherent large spread in stellar masses and formation times. Therefore, we aim to model the collective evolution of a cluster to investigate the effectiveness of radial drift in clearing the discs of mm-sized particles. We use a protoplanetary disc model that numerically solves for disc formation, and the viscous evolution and photoevaporative clearing of the gas component, while also including the drift of particles limited in size by fragmentation. We find that discs are born with dust masses between 50 Earth masses and 1000 Earth masses, for stars with, respectively, masses between 0.1 solar masses and 1 solar masses. The majority of this initial dust reservoir is typically lost through drift before photoevaporation opens a gap in the gas disc for models both with and without strong X-ray-driven mass loss rates. We conclude that the decrease in time of the mass locked in fragmentation-limited pebbles is consistent with the evolution of dust masses and ages inferred from nearby star-forming regions when assuming viscous evolution rates corresponding to mean gas disc lifetimes between 3 Myr and 8 Myr.

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arXiv2023

The Effect of Dust Evolution and Traps on Inner Disk Water Enrichment

Substructures in protoplanetary disks can act as dust traps that shape the radial distribution of pebbles. By blocking the passage of pebbles, the presence of gaps in disks may have a profound effect on pebble delivery into the inner disk, crucial for the formation of inner planets via pebble accretion. This process can also affect the delivery of volatiles (such as H$_2$O) and their abundance within the water snow line region (within a few au). In this study, we aim to understand what effect the presence of gaps in the outer gas disk may have on water vapor enrichment in the inner disk. Building on previous work, we employ a volatile-inclusive multi-Myr disk evolution model that considers an evolving ice-bearing drifting dust population, sensitive to dust-traps, which loses its icy content to sublimation upon reaching the snow line. We find that vapor abundance in the inner disk is strongly affected by fragmentation velocity (v$_{\rm f}$) and turbulence, which control how intense vapor enrichment from pebble delivery is, if present, and how long it may last. Generally, for disks with low to moderate turbulence ($α$ $\le$ 1 $\times$ 10$^{-3}$) and for a range of v$_{\rm f}$, radial location, and gap depth (especially that of the innermost gaps), can significantly alter enrichment. Shallow inner gaps may continuously leak material from beyond it, despite the presence of additional deep outer gaps. We finally find that the for realistic v$_{\rm f}$ ($\le$ 10 m s$^{-1}$), presence of gaps is more important than planetesimal formation beyond the snow line in regulating pebble and volatile delivery into the inner disk.

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arXiv2023

Planet Formation Theory in the Era of ALMA and Kepler: from Pebbles to Exoplanets

Our understanding of planet formation has been rapidly evolving in recent years. The classical planet formation theory, developed when the only known planetary system was our own Solar System, has been revised to account for the observed diversity of the exoplanetary systems. At the same time, the increasing observational capabilities of the young stars and their surrounding disks bring new constraints on the planet formation process. In this chapter, we summarize the new information derived from the exoplanets population and the circumstellar disks observations. We present the new developments in planet formation theory, from dust evolution to the growth of planetary cores by accretion of planetesimals, pebbles, and gas. We review the state-of-the-art models for the formation of diverse planetary systems, including the population synthesis approach which is necessary to compare theoretical model outcomes to the exoplanet population. We emphasize that the planet formation process may not be spatially uniform in the disk and there are preferential locations for the formation of planetesimals and planets. Outside of these locations, a significant fraction of solids is not growing past the pebble-sizes. The reservoir of pebbles plays an important role in the growth of planetary cores in the pebble accretion process. The timescale of the emergence of massive planetary cores is an important aspect of the present models and it is likely that the cores within one disk form at different times. In addition, there is growing evidence that the first planetary cores start forming early, during the circumstellar disk buildup process.

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arXiv2023

A potential site for wide-orbit giant planet formation in the IM Lup disk

The radial transport, or drift, of dust has taken a critical role in giant planet formation theory. However, it has been challenging to identify dust drift pile ups in the hard-to-observe inner disk. We find that the IM Lup disk shows evidence that it has been shaped by an episode of dust drift. Using radiative transfer and dust dynamical modeling we study the radial and vertical dust distribution. We find that high dust drift rates exceeding 110 M_earth/Myr are necessary to explain both the dust and CO observations. Furthermore, the bulk of the large dust present in the inner 20 au needs to be vertically extended, implying high turbulence alpha_z > 10^{-3} and small grains (0.2-1 mm). We suggest that this increased level of particle stirring is consistent with the inner dust-rich disk undergoing turbulence triggered by the vertical shear instability. The conditions in the IM Lup disk imply that giant planet formation through pebble accretion is only effective outside 20 au. If such an early, high turbulence inner region is a natural consequence of high dust drift rates, then this has major implications for understanding the formation regions of giant planets including Jupiter and Saturn.

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arXiv2022

Formation of pebbles in (gravito-)viscous protoplanetary disks with various turbulent strengths

Aims. Dust plays a crucial role in the evolution of protoplanetary disks. We study the dynamics and growth of initially sub-$μm$ dust particles in self-gravitating young protoplanetary disks with various strengths of turbulent viscosity. We aim to understand the physical conditions that determine the formation and spatial distribution of pebbles when both disk self-gravity and turbulent viscosity can be concurrently at work. Methods. We perform the thin-disk hydrodynamics simulations of self-gravitating protoplanetary disks over an initial time period of 0.5 Myr using the FEOSAD code. Turbulent viscosity is parameterized in terms of the spatially and temporally constant $α$-parameter, while the effects of gravitational instability on dust growth is accounted for by calculating the effective parameter $α_{\rm GI}$. We consider the evolution of dust component including momentum exchange with gas, dust self-gravity, and also a simplified model of dust growth. Results. We find that the level of turbulent viscosity strongly affects the spatial distribution and total mass of pebbles in the disk. The $α=10^{-2}$ model is viscosity-dominated, pebbles are completely absent, and dust-to-gas mass ratio deviates from the reference 1:100 value no more than by 30\% throughout the disk extent. On the contrary, the $α=10^{-3}$ model and, especially, the $α=10^{-4}$ model are dominated by gravitational instability. The effective parameter $α+α_{\rm GI}$ is now a strongly varying function of radial distance. As a consequence, a bottle neck effect develops in the innermost disk regions, which makes gas and dust accumulate in a ring-like structure. Abridged.

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arXiv2022

Natural separation of two primordial planetary reservoirs in an expanding solar protoplanetary disk

Meteorites display an isotopic composition dichotomy between non-carbonaceous (NC) and carbonaceous (CC) groups, indicating that planetesimal formation in the solar protoplanetary disk occurred in two distinct reservoirs. The prevailing view is that a rapidly formed Jupiter acted as a barrier between these reservoirs. We show a fundamental inconsistency in this model: if Jupiter is an efficient blocker of drifting pebbles, then the interior NC reservoir is depleted by radial drift within a few hundred thousand years. If Jupiter lets material pass it, then the two reservoirs will be mixed. Instead, we demonstrate that the arrival of the CC pebbles in the inner disk is delayed for several million years by the viscous expansion of the protoplanetary disk. Our results support that Jupiter formed in the outer disk (>10 AU) and allowed a considerable amount of CC material to pass it and become accreted by the terrestrial planets.

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arXiv2021

Hot Jupiters, cold kinematics: High phase space densities of host stars reflect an age bias

Context. The birth environments of planetary systems are thought to influence planet formation and orbital evolution, through external photoevaporation and stellar flybys. Recent work has claimed observational support for this, in the form of a correlation between the properties of planetary systems and the local Galactic phase space density of the host star. In particular, Hot Jupiters are found overwhelmingly around stars in regions of high phase space density, which may reflect a formation environment with high stellar density. Aims. We instead investigate whether the high phase space density may have a galactic kinematic origin: Hot Jupiter hosts may be biased towards being young and therefore kinematically cold, because tidal inspiral leads to the destruction of the planets on Gyr timescales, and the velocity dispersion of stars in the Galaxy increases on similar timescales. Methods. We use 6D positions and kinematics from Gaia for the Hot Jupiter hosts and their neighbours, and construct distributions of the phase space density. We investigate correlations between the stars' local phase space density and peculiar velocity. Results. We find a strong anticorrelation between the phase space density and the host star's peculiar velocity with respect to the Local Standard of Rest. Therefore, most stars in "high-density" regions are kinematically cold, which may be caused by the aforementioned bias towards detecting Hot Jupiters around young stars before the planets' tidal destruction. Conclusions. We do not find evidence in the data for Hot Jupiter hosts preferentially being in phase space overdensities compared to other stars of similar kinematics, nor therefore for their originating in birth environments of high stellar density.

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arXiv2021

Streaming instability of multiple particle species II -- Numerical convergence with increasing particle number

The streaming instability provides an efficient way of overcoming the growth barriers in the initial stages of the planet formation process. Considering the realistic case of a particle size distribution, the dynamics of the system is altered compared to the outcome of single size models. In order to understand the outcome of the multi-species streaming instability in detail, we perform a large parameter study in terms of particle number, particle size distribution, particle size range, initial metallicity and initial particle scale height. We study vertically stratified systems and determine the metallicity threshold for filament formation. We compare these with a system where the initial particle distribution is unstratified and find that its evolution follows that of its stratified counterpart. We find that change in particle number does not result in significant variation in the efficiency and timing of filament formation. We also see that there is no clear trend for how varying the size distribution in combination with particle size range changes the outcome of the multi-species streaming instability. Finally, we find that an initial metallicity of $Z_{\rm{init}}=0.005$ and $Z_{\rm{init}}=0.01$ both result in similar critical metallicity values for the start of filament formation. Our results show that the inclusion of a particle size distribution into streaming instability simulations, while changing the dynamics as compared to mono-disperse systems, does not result in overall unfavorable conditions for solid growth. We attribute the sub-dominant role of multiple species to the high-density conditions in the midplane, conditions under which also linear stability analysis predict little difference between single and multiple species.

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arXiv2021

Formation of planetary systems by pebble accretion and migration: Hot super-Earth systems from breaking compact resonant chains

At least 30\% of main sequence stars host planets with sizes of between 1 and 4 Earth radii and orbital periods of less than 100 days. We use N-body simulations including a model for gas-assisted pebble accretion and disk--planet tidal interaction to study the formation of super-Earth systems. We show that the integrated pebble mass reservoir creates a bifurcation between hot super-Earths or hot-Neptunes ($\lesssim15M_{\oplus}$) and super-massive planetary cores potentially able to become gas giant planets ($\gtrsim15M_{\oplus}$). Simulations with moderate pebble fluxes grow multiple super-Earth-mass planets that migrate inwards and pile up at the inner edge of the disk forming long resonant chains. We follow the long-term dynamical evolution of these systems and use the period ratio distribution of observed planet-pairs to constrain our model. Up to $\sim$95\% of resonant chains become dynamically unstable after the gas disk dispersal, leading to a phase of late collisions that breaks the original resonant configurations. Our simulations naturally match observations when they produce a dominant fraction ($\gtrsim95\%$) of unstable systems with a sprinkling ($\lesssim5\%$) of stable resonant chains (the Trappist-1 system represents one such example). Our results demonstrate that super-Earth systems are inherently multiple (${\rm N\geq2}$) and that the observed excess of single-planet transits is a consequence of the mutual inclinations excited by the planet--planet instability. In simulations in which planetary seeds are initially distributed in the inner and outer disk, close-in super-Earths (abridged).

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arXiv2021

A pebble accretion model for the formation of the terrestrial planets in the Solar System

Pebbles of millimeter sizes are abundant in protoplanetary discs around young stars. Chondrules inside primitive meteorites - formed by melting of dust aggregate pebbles or in impacts between planetesimals - have similar sizes. The role of pebble accretion for terrestrial planet formation is nevertheless unclear. Here we present a model where inwards-drifting pebbles feed the growth of terrestrial planets. The masses and orbits of Venus, Earth, Theia (which later collided with the Earth to form the Moon) and Mars are all consistent with pebble accretion onto protoplanets that formed around Mars' orbit and migrated to their final positions while growing. The isotopic compositions of Earth and Mars are matched qualitatively by accretion of two generations of pebbles, carrying distinct isotopic signatures. Finally, we show that the water and carbon budget of Earth can be delivered by pebbles from the early generation before the gas envelope became hot enough to vaporise volatiles.

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arXiv2020

Dust clearing by radial drift in evolving protoplanetary discs

Recent surveys have revealed that protoplanetary discs typically have dust masses that appear to be insufficient to account for the high occurrence rate of exoplanet systems. We demonstrate that this observed dust depletion is consistent with the radial drift of pebbles. Using a Monte Carlo method we simulate the evolution of a cluster of protoplanetary discs, using a 1D numerical method to viscously evolve each gas disc together with the radial drift of dust particles that have grown to 100 $μ$m in size. For a 2 Myr old cluster of stars, we find a slightly sub-linear scaling between the gas disc mass and the gas accretion rate ($M_\mathrm{g}\propto\dot{M}^{0.9}$). However, for the dust mass we find that evolved dust discs have a much weaker scaling with the gas accretion rate, with the precise scaling depending on the age at which the cluster is sampled and the intrinsic age spread of the discs in the cluster. Ultimately, we find that the dust mass present in protoplanetary disc is on the order of 10-100 Earth masses in 1-3 Myr old star-forming regions, a factor of 10 to 100 depleted from the original dust budget. As the dust drains from the outer disc, pebbles pile up in the inner disc and locally increase the dust-to-gas ratio by a factor of up to 4 above the initial value. In these high dust-to-gas ratio regions we find conditions that are favourable for planetesimal formation via the streaming instability and subsequent growth by pebble accretion. We also find the following scaling relations with stellar mass within a 1-2 Myr old cluster: a slightly super-linear scaling between the gas accretion rate and stellar mass ($\dot{M}\propto M_\star^{1.4}$), a slightly super-linear scaling between the gas disc mass and the stellar mass ($M_\mathrm{g}\propto M_\star^{1.4}$) and a super-linear relation between the dust disc mass and stellar mass ($M_\mathrm{d}\propto M_\star^{1.4-4.1}$).

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arXiv2020

Influence of grain growth on the thermal structure of protoplanetary discs

The thermal structure of a protoplanetary disc is regulated by the opacity that dust grains provide. However, previous works have often considered simplified prescriptions for the dust opacity in hydrodynamical disc simulations, e.g. by considering only a single particle size. In the present work we perform 2D hydrodynamical simulations of protoplanetary discs where the opacity is self-consistently calculated for the dust population, taking into account the particle size, composition and abundance. We first compare simulations using single grain sizes to two different multi-grain size distributions at different levels of turbulence strengths, parameterized through the $α$-viscosity, and different gas surface densities. Assuming a single dust size leads to inaccurate calculations of the thermal structure of discs, because the grain size dominating the opacity increases with orbital radius. Overall the two grain size distributions, one limited by fragmentation only and the other determined from a more complete fragmentation-coagulation equilibrium, give similar results for the thermal structure. We find that both grain size distributions give less steep opacity gradients that result in less steep aspect ratio gradients, in comparison to discs with only micrometer sized dust. Moreover, in the discs with a grain size distribution, the innermost outward migration region is removed and planets embedded is such discs experience lower migration rates. We also investigate the dependency of the water iceline position on the alpha-viscosity, the initial gas surface density at 1 AU and the dust-to-gas ratio and find $r_{ice} \propto α^{0.61} Σ_{g,0}^{0.8} f_{DG}^{0.37}$ independently of the distribution used. The inclusion of the feedback loop between grain growth, opacities and disc thermodynamics allows for more self-consistent simulations of accretion discs and planet formation.

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arXiv2020

Pebble-driven Planet Formation around Very Low-mass Stars and Brown Dwarfs

We conduct a pebble-driven planet population synthesis study to investigate the formation of planets around very low-mass stars and brown dwarfs, in the (sub)stellar mass range between $0.01 \ M_{\odot}$ and $0.1 \ M_{\odot}$. Based on the extrapolation of numerical simulations of planetesimal formation by the streaming instability, we obtain the characteristic mass of the planetesimals and the initial masses of the protoplanets (largest bodies from the planetesimal size distributions), in either the early self-gravitating phase or the later non-self-gravitating phase of the protoplanetary disk evolution. We find that the initial protoplanets form with masses that increase with host mass, orbital distance and decrease with disk age. Around late M-dwarfs of $0.1 \ M_{\odot}$, these protoplanets can grow up to Earth-mass planets by pebble accretion. However, around brown dwarfs of $0.01 \ M_{\odot}$, planets do not grow larger than Mars mass when the initial protoplanets are born early in self-gravitating disks, and their growth stalls at around $0.01$ Earth-mass when they are born late in non-self-gravitating disks. Around these low mass stars and brown dwarfs, we find no channel for gas giant planet formation because the solid cores remain too small. When the initial protoplanets form only at the water-ice line, the final planets typically have ${\gtrsim} 15\%$ water mass fraction. Alternatively, when the initial protoplanets form log-uniformly distributed over the entire protoplanetary disk, the final planets are either very water-rich (water mass fraction ${\gtrsim}15\%$) or entirely rocky (water mass fraction ${\lesssim}5\%$).

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