arXiv subjects

H. Lammer

Publications and source records attributed to H. Lammer.

At least 19 recordsLinked to original sources

arXiv2026

The impact of electron precipitation on Earth's thermospheric NO production and the drag of LEO satellites

We investigate the response of space weather events on Earth's upper atmosphere over the polar regions by studying their effect on the drag of the CHAMP and GRACE satellites. Increasing solar activity that results in heating and the expansion of the upper atmosphere threatens low Earth orbit (LEO) satellites. Auroral events are closely related to the stellar energy deposition of solar EUV radiation and precipitating energetic electrons, which influence photochemical processes such as the production of nitric oxide (NO) in the upper atmosphere. To study the production of NO molecules and their influence on the thermospheric structure and satellite drag, we first model Earth's background thermosphere with the 1D upper atmosphere model Kompot by considering the incident X-ray, EUV, and IR radiation during selected space weather events. To investigate the effect of electron precipitation in the production of NO molecules in the polar thermosphere, we apply a Monte Carlo model accounting for the stochastic nature of collisional scattering of auroral electrons in collisions with the surrounding N$_2$-O$_2$ atmosphere, including the production of suprathermal N atoms. The observed effect of the atmospheric drag on CHAMP and GRACE during the two studied events indicates that a sporadic enhancement of NO molecule production in the polar thermosphere and its IR-cooling capability, which counteracts thermospheric expansion and can lead to an ``overcooling'' with decreased density after the space weather event, can have a protective effect on LEO satellites. Their production efficiency, however, is highly dependent on the energy flux of the precipitating electrons. Our results have direct implications for empirical satellite orbit prediction models, as our simulations highlight the need to consider precipitation-induced NO production to improve the predictive power of these models.

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arXiv2024

CAI formation in the early Solar System

Ca-Al-rich inclusions (CAIs) are the oldest dated solid materials in the solar system, found as light-coloured crystalline ingredients in meteorites. Their formation time is commonly associated with age zero of the Solar System. Yet, the physical and chemical processes that once led to the formation of these sub-millimetre to centimetre-sized mineral particles in the early solar nebula are still a matter of debate. This paper proposes a pathway to form such inclusions during the earliest phases of disc evolution. We combine 1D viscous disc evolutionary models with 2D radiative transfer, equilibrium condensation, and new dust opacity calculations. We show that the viscous heating associated with the high accretion rates in the earliest evolutionary phases causes the midplane inside of about 0.5 au to heat up to limiting temperatures of about 1500-1700 K, but no further. These high temperatures force all refractory material components of the inherited interstellar dust grains to sublimate - except for a few Al-Ca-Ti oxides such as Al2O3, Ca2Al2SiO7, and CaTiO3. Once the Mg-Fe silicates are gone, the dust becomes more transparent and the heat is more efficiently transported to the disc surface, which prevents any further warm-up. This thermostat mechanism keeps these minerals above their annealing temperature for hundreds of thousands of years, which creates large, pure and crystalline particles. These particles are dragged out by the viscously spreading disc. Beyond about 0.5 au, the silicates re-condense on the Ca-Al-rich particles, adding an amorphous silicate matrix. We estimate that this mechanism to produce CAIs works during the first 50000 years of disc evolution. These particles then continue to move outward and populate the entire disc up to radii of about 50 au, before, eventually, the accretion rate subsides, the disc cools, and the particles start to drift inwards.

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arXiv2023

On the required mass for exoplanetary radio emission

The detection of radio emission from an exoplanet would constitute the best way to determine its magnetic field. Indeed, the presence of a planetary magnetic field is a necessary condition for radio emission via the Cyclotron Maser Instability. The presence of a magnetic field is, however, not sufficient. At the emission site, the local cyclotron frequency has to be sufficiently high compared to the local plasma frequency. As strong stellar insolation on a low-mass planet can lead to an extended planetary atmosphere, the magnetospheric plasma frequency depends on the planetary mass, its orbital distance, and its host star. We show that an extended planetary atmosphere can quench the radio emission. This seems to be true, in particular, for an important fraction of the planets less massive than approximately two Jupiter masses and with orbital distances below $\sim$0.2 AU. Most of the best candidates suggested by radio scaling laws lie in this parameter space. Taking this effect quenching into account will have important implications for the target selection of observation campaigns. At the same time, this effect will have consequences for the interpretation of observational data.

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arXiv2022

Modification of the radioactive heat budget of Earth-like exoplanets by the loss of primordial atmospheres

The initial abundance of radioactive heat producing isotopes in the interior of a terrestrial planet are important drivers of its thermal evolution and the related tectonics and possible evolution to an Earth-like habitat. The moderately volatile element K can be outgassed from a magma ocean into H$_2$-dominated primordial atmospheres of protoplanets with assumed masses between 0.55-1.0$ M_{\rm Earth}$ at the time when the gas disk evaporated. We estimate this outgassing and let these planets grow through impacts of depleted and non-depleted material that resembles the same $^{40}$K abundance of average carbonaceous chondrites until the growing protoplanets reach 1.0 $M_{\rm Earth}$. We examine different atmospheric compositions and, as a function of pressure and temperature, calculate the proportion of K by Gibbs Free Energy minimisation using the GGChem code. We find that for H$_2$-envelopes and for magma ocean surface temperatures that are $\ge$ 2500 K, no K condensates are thermally stable, so that outgassed $^{40}$K can populate the atmosphere to a great extent. However, due to magma ocean turn-over time and the limited diffusion of $^{40}$K into the upper atmosphere, from the entire $^{40}$K in the magma ocean only a fraction may be available for escaping into space. The escape rates of the primordial atmospheres and the dragged $^{40}$K are further simulated for different stellar EUV-activities with a multispecies hydrodynamic upper atmosphere evolution model. Our results lead to different abundances of heat producing elements within the fully grown planets which may give rise to different thermal and tectonic histories of terrestrial planets and their habitability conditions.

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arXiv2022

Global 3D simulation of the upper atmosphere of HD189733b and absorption in metastable HeI and Lyα lines

A 3D fully self-consistent multi-fluid hydrodynamic aeronomy model is applied to simulate the hydrogen-helium expanding upper atmosphere of the hot Jupiter HD189733b, and related absorption in the Lya line and the 10830 A line of metastable helium. We studied the influence of a high-energy stellar flux, stellar wind, and Lya cooling to reproduce the available observations. We found that to fit the width of the absorption profile in 10830 A line the escaping upper atmosphere of planet should be close to the energy limited escape achieved with a significantly reduced Lya cooling at the altitudes with HI density higher than 3*10^6 cm^-3. Based on the preformed simulations, we constrain the helium abundance in the upper atmosphere of HD189733b by a rather low value of He/H~0.005. We show that under conditions of a moderate stellar wind similar to that of the Sun the absorption of Lya line takes place mostly within the Roche lobe due to thermal broadening at a level of about 7%. At an order of magnitude stronger wind, a significant absorption of about 15% at high blue shifted velocities of up to 100 km/s is generated in the bowshock region, due to Doppler broadening. These blue shifted velocities are still lower than those (~200 km/s) detected in one of the observations. We explain the differences between performed observations, though not in all the details, by the stellar activity and the related fluctuations of the ionizing radiation (in case of 10830 A line), and stellar wind (in case of Lya line).

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arXiv2022

Non-thermal escape of the Martian CO$_2$ atmosphere over time: constrained by Ar isotopes

The ion escape of Mars' CO$_2$ atmosphere caused by its dissociation products C and O atoms is {simulated} from present time to $\sim 4.1$ billion years ago (Ga) by {numerical models of the upper atmosphere and its interaction with the solar wind}. The planetward-scattered pick-up ions are used for sputtering estimates of exospheric particles including $^{36}$Ar and $^{38}$Ar isotopes. Total ion escape, sputtering and photochemical escape rates are compared. For solar EUV fluxes $\geq$\,3 times that of today's Sun (earlier than $\sim 2.6$ Ga) ion escape becomes the dominant atmospheric non-thermal loss process until thermal escape takes over during the pre-Noachian eon (earlier than $\sim 4.0\,-\,4.1$ Ga). If we extrapolate the total escape of CO$_2$-related dissociation products back in time until $\sim$ 4.1 Ga we obtain a {maximum} theoretical equivalent to CO$_2$ partial pressure of more than {$\sim 0.4$ bar through non-thermal escape during quiet solar wind conditions}. {However, surface-atmosphere interaction and/or extreme solar events such as frequent CMEs could have increased this value even further. By including the surface as a sink, up to 0.9\,bar, or even up to 1.8\,bar in case of hidden carbonate reservoirs, could have been present at 4.1\,Ga} The fractionation of $^{36}$Ar/$^{38}$Ar isotopes through sputtering and volcanic outgassing from its initial chondritic value of 5.3, as measured in the 4.1 billion years old Mars meteorite ALH 84001, until the present day, however, can be reproduced for assumed CO$_2$ partial pressures of {$\sim0.01 -- 0.4$\,bar without, and $\sim0.4 -- 1.8$\,bar including surface sinks, and} depending on the cessation time of the Martian dynamo (assumed between 3.6\,--\,4.0 Ga) - if atmospheric sputtering of Ar started afterwards.

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arXiv2022

The Exosphere as a Boundary: Origin and Evolution of Airless Bodies in the Inner Solar System and Beyond Including Planets with Silicate Atmospheres

In this review we discuss all the relevant solar/stellar radiation and plasma parameters and processes that act together in the formation and modification of atmospheres and exospheres that consist of surface-related minerals. Magma ocean degassed silicate atmospheres or thin gaseous envelopes from planetary building blocks, airless bodies in the inner Solar System, and close-in magmatic rocky exoplanets such as CoRot-7b, HD219134b and 55 Cnc e are addressed. The depletion and fractionation of elements from planetary embryos, which act as the building blocks for protoplanets are also discussed. In this context the formation processes of the Moon and Mercury are briefly reviewed. The Lunar surface modification since its origin by micrometeoroids, plasma sputtering, plasma impingement as well as chemical surface alteration and the search of particles from the early Earth's atmosphere that were collected by the Moon on its surface are also discussed. Finally, we address important questions on what can be learned from the study of Mercury's environment and its solar wind interaction by MESSENGER and BepiColombo in comparison with the expected observations at exo-Mercurys by future space-observatories such as the JWST or ARIEL and ground-based telescopes and instruments like SPHERE and ESPRESSO on the VLT, and vice versa.

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arXiv2021

The impact of intrinsic magnetic field on the absorption signatures of elements probing the upper atmosphere of HD209458b

The signs of an expanding atmosphere of HD209458b have been observed with far-ultraviolet transmission spectroscopy and in the measurements of transit absorption by metastable HeI. These observations are interpreted using the hydrodynamic and Monte-Carlo numerical simulations of various degree of complexity and consistency. At the same time, no attempt has been made to model atmospheric escape of a magnetized HD209458b, to see how the planetary magnetic field might affect the measured transit absorption lines. This paper presents the global 3D MHD self-consistent simulations of the expanding upper atmosphere of HD209458b interacting with the stellar wind, and models the observed HI (Lya), OI (1306 A), CII (1337 A), and HeI (10830 A) transit absorption features. We find that the planetary dipole magnetic field with the equatorial surface value of Bp = 1 G profoundly changes the character of atmospheric material outflow and the related absorption. We also investigate the formation of planetary magnetosphere in the stellar wind and show that its size is more determined by the escaping atmosphere flow rather than by the strength of magnetic field. Fitting of the simulation results to observations enables constraining the stellar XUV flux and He abundance at ~10 erg cm2/s (at 1 a.u.) and He/H=0.02, respectively, as well as setting an upper limit for the dipole magnetic field of Bp<0.1 G on the planetary surface at the equator. This implies that the magnetic dipole moment of HD209458b should be less than 0.06 of the Jovian value.

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arXiv2021

The young Sun's XUV-activity as a constraint for lower CO$_2$-limits in the Earth's Archean atmosphere

Despite their importance for determining the evolution of the Earth's atmosphere and surface conditions, the evolutionary histories of the Earth's atmospheric CO$_2$ abundance during the Archean eon and the Sun's activity are poorly constrained. In this study, we apply a state-of-the-art physical model for the upper atmosphere of the Archean Earth to study the effects of different atmospheric CO$_2$/N$_2$ mixing ratios and solar activity levels on the escape of the atmosphere to space. We find that unless CO$_2$ was a major constituent of the atmosphere during the Archean eon, enhanced heating of the thermosphere by the Sun's strong X-ray and ultraviolet radiation would have caused rapid escape to space. We derive lower limits on the atmospheric CO$_2$ abundance of approximately 40\% at 3.8~billion years ago, which is likely enough to counteract the faint young Sun and keep the Earth from being completely frozen. Furthermore, our results indicate that the Sun was most likely born as a slow to moderate {rotating young G-star} to prevent rapid escape, putting essential constraints on the Sun's activity evolution throughout the solar system's history. In case that there were yet unknown cooling mechanisms present in the Archean atmosphere, this could reduce our CO$_2$ stability limits, and it would allow a more active Sun.

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arXiv2021

Formation of Venus, Earth and Mars: Constrained by isotopes

We discuss the current state of knowledge of terrestrial planet formation from the aspects of different planet formation models and isotopic data from 182Hf-182W, U-Pb, lithophile-siderophile elements, 48Ca/44Ca isotope samples from planetary building blocks, 36Ar/38Ar, 20Ne/22Ne, 36Ar/22Ne isotope ratios in Venus' and Earth's atmospheres, the expected solar 3He abundance in Earth's deep mantle and Earth's D/H sea water ratios that shed light on the accretion time of the early protoplanets. Accretion scenarios that can explain the different isotope ratios, including a Moon-forming event after ca. 50 Myr, support the theory that the bulk of Earth's mass (>80%) most likely accreted within 10-30 Myr. From a combined analysis of the before mentioned isotopes, one finds that proto-Earth accreted 0.5-0.6 MEarth within the first ~4-5 Myr, the approximate lifetime of the protoplanetary disk. For Venus, the available atmospheric noble gas data are too uncertain for constraining the planet's accretion scenario accurately. However, from the available Ar and Ne isotope measurements, one finds that proto-Venus could have grown to 0.85-1.0 MVenus before the disk dissipated. Classical terrestrial planet formation models have struggled to grow large planetary embryos quickly from the tiniest materials within the typical lifetime of protoplanetary disks. Pebble accretion could solve this long-standing time scale controversy. Pebble accretion and streaming instabilities produce large planetesimals that grow into Mars-sized and larger planetary embryos during this early accretion phase. The later stage of accretion can be explained well with the Grand-Tack, annulus or depleted disk models. The relative roles of pebble accretion and planetesimal accretion/giant impacts are poorly understood and should be investigated with N-body simulations that include pebbles and multiple protoplanets.

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arXiv2021

Did Mars possess a dense atmosphere during the first ~400 million years?

It is not yet entirely clear whether Mars began as a warm and wet planet that evolved towards the present-day cold and dry body or if it always was cold and dry with just some sporadic episodes of liquid water on its surface. An important clue into this question can be gained by studying the earliest evolution of the Martian atmosphere and whether it was dense and stable to maintain a warm and wet climate or tenuous and susceptible to strong atmospheric escape. We discuss relevant aspects for the evolution and stability of a potential early Martian atmosphere. This contains the solar EUV flux evolution, the formation timescale and volatile inventory of the planet including volcanic degassing, impact delivery and removal, the loss of a catastrophically outgassed steam atmosphere, atmosphere-surface interactions, and thermal and non-thermal escape processes affecting any secondary atmosphere. While early non-thermal escape at Mars before 4 billion years ago (Ga) is poorly understood, particularly in view of its ancient intrinsic magnetic field, research on thermal escape processes indicate that volatile delivery and volcanic degassing cannot counterbalance the strong thermal escape. Therefore, a catastrophically outgassed steam atmosphere of several bars of CO2 and H2O, or CO and H2 for reduced conditions, could have been lost within just a few million years (Myr). Thereafter, Mars likely could not build up a dense secondary atmosphere during its first ~400 Myr but might only have possessed an atmosphere sporadically during events of strong volcanic degassing, potentially also including SO2. This indicates that before ~4.1 Ga Mars indeed might have been cold and dry. A denser CO2- or CO-dominated atmosphere, however, might have built up afterwards but must have been lost later-on due to non-thermal escape processes and sequestration into the ground.

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arXiv2021

Simulation of 10830 Å absorption with a 3D hydrodynamic model reveals the solar He abundance in upper atmosphere of WASP-107b

Transmission spectroscopy of WASP-107b revealed 7-8% absorption at the position of metastable HeI triplet at 10830 Å in Doppler velocity range of [-20; 10] km/s, which is stronger than that measured in other exoplanets. With a dedicated 3D self-consistent hydrodynamic multi-fluid model we calculated the expanding upper atmosphere of WASP-107b and reproduced within the observations accuracy the measured HeI absorption profiles, constraining the stellar XUV flux to (6-10) erg cm-2 s-1 at 1 a.u., and the upper atmosphere helium abundance He/H to 0.075-0.15. The radiation pressure acting on the metastable HeI atoms was shown to be an important factor affecting the shape of the absorption profiles. Its effect is counterbalanced by the processes of collisional depopulation of the HeI metastable state. Altogether, the observed HeI absorption in WASP-107b can be interpreted with the expected reasonable parameters of the stellar-planetary system and appropriate account of the electron and atom impact processes.

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arXiv2020

Loss and fractionation of noble gas isotopes and moderately volatile elements from planetary embryos and early Venus, Earth and Mars

Here we discuss the current state of knowledge on how atmospheric escape processes can fractionate noble gas isotopes and moderately volatile rock-forming elements that populate primordial atmospheres, magma ocean related environments, and catastrophically outgassed steam atmospheres. Variations of isotopes and volatile elements in different planetary reservoirs keep information about atmospheric escape, composition and even the source of accreting material. We summarize our knowledge on atmospheric isotope ratios and discuss the latest evidence that proto-Venus and Earth captured small H$_2$-dominated primordial atmospheres that were lost by EUV-driven hydrodynamic escape after the disk dispersed. All relevant thermal and non-thermal atmospheric escape processes that can fractionate various isotopes and volatile elements are discussed. Erosion of early atmospheres, crust and mantle by large planetary impactors are also addressed. Further, we discuss how moderately volatile elements such as the radioactive heat producing element $^{40}$K and other rock-forming elements such as Mg can also be outgassed and lost from magma oceans that originate on large planetary embryos and accreting planets. Outgassed elements escape from planetary embryos with masses that are $\geq$\,M$_{\rm Moon}$ directly, or due to hydrodynamic drag of escaping H atoms originating from primordial- or steam atmospheres at more massive embryos. We discuss how these processes affect the final elemental composition and ratios such as K/U, Fe/Mg of early planets and their building blocks. Finally, we review modeling efforts that constrain the early evolution of Venus, Earth and Mars by reproducing their measured present day atmospheric $^{36}$Ar/$^{38}$Ar, $^{20}$Ne/$^{22}$Ne noble gas isotope ratios and the role of isotopes on the loss of water and its connection to the redox state on early Mars.

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arXiv2020

Nitrogen Atmospheres of the Icy Bodies in the Solar System

This brief review will discuss the current knowledge on the origin and evolution of the nitrogen atmospheres of the icy bodies in the solar system, particularly of Titan, Triton and Pluto. An important tool to analyse and understand the origin and evolution of these atmospheres can be found in the different isotopic signatures of their atmospheric constituents. The $^{14}$N/$^{15}$N ratio of the N$_2$-dominated atmospheres of these bodies serve as a footprint of the building blocks from which Titan, Triton and Pluto originated and of the diverse fractionation processes that shaped these atmospheres over their entire evolution. Together with other measured isotopic and elemental ratios such as $^{12}$C/$^{13}$C or Ar/N these atmospheres can give important insights into the history of the icy bodies in the solar system, the diverse processes that affect their N$_2$-dominated atmospheres, and the therewith connected solar activity evolution. Titan's gaseous envelope most likely originated from ammonia ices with possible contributions from refractory organics. Its isotopic signatures can yet be seen in the - compared to Earth - comparatively heavy $^{14}$N/$^{15}$N ratio of 167.7, even though this value slightly evolved over its history due to atmospheric escape and photodissociation of N$_2$. The origin and evolution of Pluto's and Triton's tenuous nitrogen atmospheres remain unclear, even though it might be likely that their atmospheres originated from the protosolar nebula or from comets. An in-situ space mission to Triton such as the recently proposed Trident mission, and/or to the ice giants would be a crucial cornerstone for a better understanding of the origin and evolution of the icy bodies in the outer solar system and their atmospheres in general.

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arXiv2020

Hydrogen dominated atmospheres on terrestrial mass planets: evidence, origin and evolution

The discovery of thousands of highly irradiated, low-mass, exoplanets has led to the idea that atmospheric escape is an important process that can drive their evolution. Of particular interest is the inference from recent exoplanet detections that there is a large population of low mass planets possessing significant, hydrogen dominated atmospheres, even at masses as low as $\sim 2$~M$_\oplus$. The size of these hydrogen dominated atmospheres indicates the the envelopes must have been accreted from the natal protoplanetary disc. This inference is in contradiction with the Solar System terrestrial planets, that did not reach their final masses before disc dispersal, and only accreted thin hydrogen dominated atmospheres. In this review, we discuss the evidence for hydrogen dominated atmospheres on terrestrial mass ($\lesssim$ 2~M$_\oplus$) planets. We then discuss the possible origins and evolution of these atmospheres with a focus on the role played by hydrodynamic atmospheric escape driven by the stellar high-energy emission (X-ray and EUV; XUV).

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arXiv2020

Escape and evolution of Titan's N$_2$ atmosphere constrained by $^{14}$N/$^{15}$N isotope ratios

We apply a 1D upper atmosphere model to study thermal escape of nitrogen over Titan's history. Significant thermal escape should have occurred very early for solar EUV fluxes 100 to 400 times higher than today with escape rates as high as $\approx 1.5\times 10^{28}$ s$^{-1}$ and $\approx 4.5\times 10^{29}$ s$^{-1}$, respectively, while today it is $\approx 7.5\times 10^{17}$ s$^{-1}$. Depending on whether the Sun originated as a slow, moderate or fast rotator, thermal escape was the dominant escape process for the first 100 to 1000 Myr after the formation of the solar system. If Titan's atmosphere originated that early, it could have lost between $\approx 0.5 - 16$ times its present atmospheric mass depending on the Sun's rotational evolution. We also investigated the mass-balance parameter space for an outgassing of Titan's nitrogen through decomposition of NH$_3$-ices in its deep interior. Our study indicates that, if Titan's atmosphere originated at the beginning, it could have only survived until today if the Sun was a slow rotator. In other cases, the escape would have been too strong for the degassed nitrogen to survive until present-day, implying later outgassing or an additional nitrogen source. An endogenic origin of Titan's nitrogen partially through NH$_3$-ices is consistent with its initial fractionation of $^{14}$N/$^{15}$N $\approx$ 166 - 172, or lower if photochemical removal was relevant for longer than the last $\approx$ 1,000 Myr. Since this ratio is slightly above the ratio of cometary ammonia, some of Titan's nitrogen might have originated from refractory organics.

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arXiv2020

Global 3D hydrodynamic modeling of absorption in Lyα and He 10830 A lines at transits of GJ3470b

Warm Neptune GJ3470b has been recently observed in 23S-23P transition of metastable helium, yielding absorption of about 1% in Doppler velocity range of [-40; 10] km/s. Along with previous detection of absorption in Lyα with depth of 20-40% in the blue and red wings of the line, it offers a complex target for simulation and testing of the current models. Obtained results suggest that absorption in both these lines comes from interaction of expanding upper planetary atmosphere with stellar plasma wind, allowing to constrain the stellar plasma parameters and the helium abundance in planet atmosphere.

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arXiv2020

Evolution of the Earth's Polar Outflow From Mid-Archean to Present

The development of habitable conditions on Earth is tightly connected to the evolution of its atmosphere which is strongly influenced by atmospheric escape. We investigate the evolution of the polar ion outflow from the open field line bundle which is the dominant escape mechanism for the modern Earth. We perform Direct Simulation Monte Carlo (DSMC) simulations and estimate the upper limits on escape rates from the Earth's open field line bundle starting from three gigayears ago (Ga) to present assuming the present-day composition of the atmosphere. We perform two additional simulations with lower mixing ratios of oxygen of 1% and 15% to account for the conditions shortly after the Great Oxydation Event (GOE). We estimate the maximum loss rates due to polar outflow three gigayears ago of $3.3 \times10^{27}$ s$^{-1}$ and $2.4 \times 10^{27}$ s$^{-1}$ for oxygen and nitrogen, respectively. The total integrated mass loss equals to 39% and 10% of the modern atmosphere's mass, for oxygen and nitrogen, respectively. According to our results, the main factors that governed the polar outflow in the considered time period are the evolution of the XUV radiation of the Sun and the atmosphere's composition. The evolution of the Earth's magnetic field plays a less important role. We conclude that although the atmosphere with the present-day composition can survive the escape due to polar outflow, a higher level of CO$_2$ between 3.0 and 2.0~Ga is likely necessary to reduce the escape.

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