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K. G. Kislyakova

Publications and source records attributed to K. G. Kislyakova.

At least 19 recordsLinked to original sources

First Resolution of a Main Sequence G-Star Astrosphere Using Chandra

We report resolution of a halo of X-ray line emission surrounding the Zero Age Main Sequence (ZAMS) G8.5V star HD 61005 by Chandra ACIS-S. Located only 36.4 pc distant, HD 61005 is young (approx. 100 Myr), x-ray bright (300 times Solar), observed with nearly edge-on geometry, and surrounded by Local Interstellar Medium (LISM) material denser than in the environ of the Sun. HD 61005 is known to harbor large amounts of circumstellar dust in a dense ecliptic plane full of mm-sized particles plus attached, extended wing like structures full of micron sized particles, which are evidence for a strong LISM-dust disk interaction. These properties aided our ability to resolve the 220 au wide astrosphere of HD61005, the first ever observed for a main sequence G-star. The observed x-ray emission morphology is roughly spherical, as expected for an astrospheric structure dominated by the host star. The Chandra spectrum of HD 61005 is a combination of a hard stellar coronal emission (T=8 MK) at Lx = 6 x10e29 erg per sec, plus an extended halo contribution at Lx = 1x10e29 erg per sec dominated by charge exchange (CXE) lines, such as those of OVIII and NeIX. The Chandra CXE x-ray morphology does not track the planar dust morphology but does extend out roughly to where the base of the dust wings begins. We present a toy model of x-ray emission produced by stellar wind (SW)-LISM CXE interactions, similar to the state of the young Sun when it was approximately 100 Myrs old (Guinan and Engle 2007), and transiting through an approximately 1000 times denser part of the interstellar medium (ISM) such as a Giant Molecular Cloud (Stern 2003, Opher and Loeb 2024).

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Long-term X-ray variability of the multiple-planet host L 98-59: Hints of an activity cycle

High-energy irradiation in X-rays and UV (XUV) can transform the planetary atmospheres through photoevaporation and photochemistry. This is more crucial for M stars, whose habitable zones for Earth-like planets are located within a few percent of an AU. Transiting exoplanets around M dwarfs offer the opportunity to study their characteristics and habitability conditions. L 98-59 is an M3 dwarf hosting six Earth-like planets, with two of them in the habitable zone of the star. X-ray observations made in 2020 and 2021 detected significant flares above a quiescent luminosity of 4-10 x 10^26 erg/s. We present the results from two short XMM-Newton observations of L 98-59, which are part of a monitoring survey to detect long-term X-ray variability and activity cycles. In October 2024 the X-ray quiescent luminosity of the star was about 5.9 x 10^25 erg/s, and it was about 6.3 x 10^26 erg/s in February 2025. We speculate that in late 2024 the star had a minimum of activity; in 2021 the star was near a maximum of an activity cycle, and in 2025 it was at the middle of the cycle. We suggest a coarse estimate of the period of about 2 years and a peak-to-peak amplitude of about 10, which is the highest among the stars with a known X-ray cycle other than the Sun. We also infer that even the outer planet in the habitable zone, L 98-59f, is exposed to an X-ray dose between 100 and 1600 times the X-ray irradiation of the Earth in the XMM band.

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Detecting and characterising the magnetic field of exoplanets

Magnetic fields play a crucial role in planetary evolution and habitability. While the intrinsic magnetic fields of solar system planets are relatively well understood, the magnetic properties of exoplanets remain largely unconstrained, despite their potential ubiquity. Detecting exoplanetary magnetic fields is essential to advancing our understanding of planetary habitability beyond the solar system. This paper focuses on two promising spectropolarimetric techniques for detecting magnetic fields in hot exoplanets: direct detection through polarization signatures in the He I 1083 nm triplet and indirect detection via star-planet magnetic interactions manifesting as stellar hot spots. The direct method is particularly suited to close-in gas giants, leveraging the Hanle and Zeeman effects to detect low-amplitude magnetic fields. The indirect method can apply to both giant and low-mass planets by identifying magnetic connectivity-induced features in the stellar atmosphere. Although the interpretation of current detections remain tentative, upcoming high-resolution spectropolarimetric capabilities in the UV and near-infrared, particularly with future missions like HWO, promise to enable definitive measurements of exoplanetary magnetic fields. These advancements will open new avenues for probing the magnetic environments of exoplanets and their implications for atmospheric retention and habitability.

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X-ray detection of astrospheres around three main-sequence stars and their mass-loss rates

Stellar winds of cool main sequence stars are very difficult to constrain observationally. One way to measure stellar mass loss rates is to detect soft X-ray emission from stellar astrospheres produced by charge exchange between heavy ions of the stellar wind and cold neutrals of the interstellar medium (ISM) surrounding the stars. Here we report detections of charge-exchange induced X-ray emission from the extended astrospheres of three main sequence stars, 70 Ophiuchi, epsilon Eridani, and 61 Cygni based on analysis of observations by XMM-Newton. We estimate the corresponding mass loss rates to be 66.5 +- 11.1, 15.6 +- 4.4, and 9.6 +- 4.1 times the solar mass loss rate for 70 Ophiuchi, epsilon Eridani, and 61 Cygni, respectively, and compare our results to the hydrogen wall method. We also place upper limits on the mass loss rates of several other main sequence stars. This method has potential utility for determining the mass loss rates from X-ray observations showing spatial extension beyond a coronal point source.

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Non-thermal motions and atmospheric heating of cool stars

The magnetic processes associated with the non-thermal broadening of optically thin emission lines appear to carry enough energy to heat the corona and accelerate the solar wind. We investigate whether non-thermal motions in cool stars exhibit the same behaviour as on the Sun by analysing archival stellar spectra taken by the Hubble Space Telescope, and full-disc Solar spectra taken by the Interface Region Imaging Spectrograph. We determined the non-thermal velocities by measuring the excess broadening in optically thin emission lines formed in the stellar atmosphere; the chromosphere, the transition region and the corona. Assuming the non-thermal broadening is caused by the presence of Alfvén waves, we also determined the associated wave energy densities. Our results show that, with a non-thermal velocity of $\sim$23 kms$^{-1}$ the Sun-as-a-star results are in very good agreement with values obtained from spatially-resolved solar observations. The non-thermal broadening in our sample show correlation to stellar rotation, with the strength of the non-thermal velocity decreasing with decreasing rotation rate. Finally, the non-thermal velocity in cool Sun-like stars varies with atmospheric height or temperature of the emission lines, and peaks at transition region temperatures. This points towards a solar-like Alfvén wave driven heating in stellar atmospheres. However, the peak is at a lower temperature in some cool stars suggesting that, other magnetic process such as flaring events could also dominate.

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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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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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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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Modeling the Ly$α$ transit absorption of the hot Jupiter HD 189733b

Hydrogen-dominated atmospheres of hot exoplanets expand and escape due to the intense heating by the X-ray and extreme ultraviolet (XUV) irradiation of their host stars. Excess absorption of neutral hydrogen has been observed in the Ly$α$ line during transits of several close-in exoplanets, indicating such extended atmospheres. For the hot Jupiter HD 189733b, this absorption shows temporal variability. Variations in stellar XUV emission and/or stellar wind conditions have been invoked to explain this effect. We apply a 1D hydrodynamic upper atmosphere model and a 3D MHD stellar wind flow model to study the effect of variations of the stellar XUV and wind conditions on the neutral hydrogen distribution, including the production of energetic neutral atoms (ENAs), and the related Ly$α$ transit signature. We obtain comparable, albeit slightly higher Ly$α$ absorption as observed in 2011 with a stellar XUV flux of $1.8\times10^4$ erg cm$^{-2}$ s$^{-1}$, rather typical activity conditions for this star. Flares similar to the one observed 8 h before the transit are unlikely to have caused a significant modulation of the transit signature. The resulting Ly$α$ absorption is dominated by atmospheric broadening, whereas the contribution of ENAs is negligible, as they are formed inside the bow shock from decelerated wind ions that are heated to high temperatures. Thus, within our modeling framework and assumptions, we find an insignificant dependence on the stellar wind parameters. Since the transit absorption can be modeled with typical stellar XUV and wind conditions, it is possible that the non-detection of the absorption in 2010 was affected by less typical stellar activity conditions, such as a very different magnitude and/or shape of the star's spectral XUV emission, or temporal/spatial variations in Ly$α$ affecting the determination of the transit absorption.

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Stellar Driven Evolution of Hydrogen-Dominated Atmospheres from Earth-Like to Super-Earth-Type Exoplanets

In the present chapter we present the results of evolutionary studies of exoplanetary atmospheres. We mostly focus on the sub- to super-Earth domain, although these methods are applicable to all types of exoplanets. We consider both thermal and nonthermal loss processes. The type of thermal loss mechanism depends on so-called escape parameter $β$, which is the ratio of the gravitational energy of a particle to its thermal energy. While $β$ is decreasing, an exoplanet switches from classical Jeans to modified Jeans and finally to blow-off escape mechanisms. During blow-off the majority of the atmospheric particles dispose of enough energy to escape the planet's gravity field. This leads to extreme gas losses. Although nonthermal losses never exceed blow-off escape, they are of significant importance for planets with relatively weak Jeans-type escape. From the diversity of nonthermal escape mechanisms, in the present chapter we focus on ion pickup and discuss the importance of other loss mechanisms. The general conclusion of the chapter is, that escape processes strongly shape the evolution of the exoplanets and determine, if the planet loses its atmosphere due to erosion processes or, on the contrary, stays as mini-Neptune type body, which can probably not be considered as a potential habitat as we know it.

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The solar wind from a stellar perspective: how do low-resolution data impact the determination of wind properties?

Alfvén-wave-driven 3D magnetohydrodynamic (MHD) models, which are increasingly used to predict stellar wind properties, contain unconstrained parameters and rely on low-resolution stellar magnetograms. We explore the effects of the input Alfvén wave energy flux and the surface magnetogram on the wind properties predicted by the Alfvén Wave Solar Model (AWSoM). We lowered the resolution of two solar magnetograms during solar cycle maximum and minimum using spherical harmonic decomposition. The Alfvén wave energy was altered based on non-thermal velocities determined from a far ultraviolet (FUV) spectrum of the solar twin 18 Sco. Additionally, low-resolution magnetograms of three solar analogues were obtained using Zeeman Doppler imaging (ZDI). Finally, the simulated wind properties were compared to Advanced Composition Explorer (ACE) observations. AWSoM simulations using well constrained input parameters taken from solar observations can reproduce the observed solar wind mass and angular momentum loss rates. The resolution of the magnetogram has a small impact on the wind properties and only during cycle maximum. However, variation in Alfvén wave energy influences the wind properties irrespective of the solar cycle activity level. Furthermore, solar wind simulations carried out using the low-resolution magnetogram of the three stars instead of the solar magnetogram could lead to an order of a magnitude difference in the simulated wind properties. The choice in Alfvén energy has a stronger influence on the wind output compared to the magnetogram resolution. The influence could be even stronger for stars whose input boundary conditions are not as well constrained as those of the Sun. Unsurprisingly, replacing the solar magnetogram with a stellar magnetogram could lead to completely inaccurate solar wind properties, and should be avoided in solar and stellar wind simulations.

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Solar XUV and ENA-driven water loss from early Venus' steam atmosphere

The influence of the hydrogen hydrodynamic upper atmosphere escape, driven by the solar soft X-ray and extreme ultraviolet radiation (XUV) flux, on an expected magma ocean outgassed steam atmosphere of early Venus is studied. By assuming that the young Sun was either a weak or moderate active young G star, we estimated the water loss from a hydrogen dominated thermosphere due to the absorption of the solar XUV flux and the precipitation of solar wind produced energetic hydrogen atoms (ENAs). The production of ENAs and their interaction with the hydrodynamic extended upper atmosphere, including collision-related feedback processes, have been calculated by means of Monte Carlo models. ENAs that collide in the upper atmosphere deposit their energy and heat the surrounding gas mainly above the main XUV energy deposition layer. It is shown that precipitating ENAs modify the thermal structure of the upper atmosphere, but the enhancement of the thermal escape rates caused by these energetic hydrogen atoms is negligible. Our results also indicate that the majority of oxygen arising from dissociated H$_2$O molecules is left behind during the first 100 Myr. It is thus suggested that the main part of the remaining oxygen has been absorbed by crustal oxidation.

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Global 3D hydrodynamic modeling of in-transit Lyα absorption of GJ436b

Using a global 3D, fully self-consistent, multi-fluid hydrodynamic model, we simulate the escaping upper atmosphere of the warm Neptune GJ436b, driven by the stellar XUV radiation impact and gravitational forces and interacting with the stellar wind. Under the typical parameters of XUV flux and stellar wind plasma expected for GJ436, we calculate in-transit absorption in Lyα and find that it is produced mostly by Energetic Neutral Atoms outside of the planetary Roche lobe, due to the resonant thermal line broadening. At the same time, the influence of radiation pressure has been shown to be insignificant. The modelled absorption is in good agreement with the observations and reveals such features as strong asymmetry between blue and red wings of the absorbed Lyα line profile, deep transit depth in the high velocity blue part of the line reaching more than 70%, and the timing of early ingress. On the other hand, the model produces significantly deeper and longer egress than in observations, indicating that there might be other processes and factors, still not accounted, that affect the interaction between the planetary escaping material and the stellar wind. At the same time, it is possible that the observational data, collected in different measurement campaigns, are affected by strong variations of the stellar wind parameters between the visits, and therefore, they cannot be reproduced altogether with the single set of model parameters.

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The Kepler-11 system: evolution of the stellar high-energy emission and {initial planetary} atmospheric mass fractions

The atmospheres of close-in planets are strongly influenced by mass loss driven by the high-energy (X-ray and extreme ultraviolet, EUV) irradiation of the host star, particularly during the early stages of evolution. We recently developed a framework to exploit this connection and enable us to recover the past evolution of the stellar high-energy emission from the present-day properties of its planets, if the latter retains some remnants of their primordial hydrogen-dominated atmospheres. Furthermore, the framework can also provide constraints on planetary initial atmospheric mass fractions. The constraints on the output parameters improve when more planets can be simultaneously analysed. This makes the Kepler-11 system, which hosts six planets with bulk densities between 0.66 and 2.45g cm^{-3}, an ideal target. Our results indicate that the star has likely evolved as a slow rotator (slower than 85\% of the stars with similar masses), corresponding to a high-energy emission at 150 Myr of between 1-10 times that of the current Sun. We also constrain the initial atmospheric mass fractions for the planets, obtaining a lower limit of 4.1% for planet c, a range of 3.7-5.3% for planet d, a range of 11.1-14% for planet e, a range of 1-15.6% for planet f, and a range of 4.7-8.7% for planet g assuming a disc dispersal time of 1 Myr. For planet b, the range remains poorly constrained. Our framework also suggests slightly higher masses for planets b, c, and f than have been suggested based on transit timing variation measurements. We coupled our results with published planet atmosphere accretion models to obtain a temperature (at 0.25 AU, the location of planet f) and dispersal time of the protoplanetary disc of 550 K and 1 Myr, although these results may be affected by inconsistencies in the adopted system parameters.

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Modelling atmospheric escape and MgII near-ultraviolet absorption of the highly irradiated hot Jupiter WASP-12b

We present two-dimensional multi-fluid numerical modelling of the upper atmosphere of the hot Jupiter WASP-12b. The model includes hydrogen chemistry, and self-consistently describes the expansion of the planetary upper atmosphere and mass loss due to intensive stellar irradiation, assuming a weakly magnetized planet. We simulate the planetary upper atmosphere and its interaction with the stellar wind (SW) with and without the inclusion of tidal force and consider different XUV irradiation conditions and SW parameters. With the inclusion of tidal force, even for a fast SW, the escaping planetary material forms two streams, propagating towards and away from the star. The atmospheric escape and related mass loss rate reaching the value of 10^12 gs^-1 appear to be mostly controlled by the stellar gravitational pull. We computed the column density and dynamics of MgII ions considering three different sets of SW parameters and XUV fluxes. The simulations enable to compute the absorption at the position of the Mg h line and to reproduce the times of ingress and egress. In case of a slow SW and without accounting for tidal force, the high orbital velocity leads to the formation of a shock approximately in the direction of the planetary orbital motion. In this case, mass loss is proportional to the stellar XUV flux. At the same time, ignoring of tidal effects for WASP-12b is a strong simplification, so the scenario with a shock, altogether is an unrealistic one.

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Extreme hydrodynamic losses of Earth-like atmospheres in the habitable zones of very active stars

Aims. In this letter, we calculate for the first time the full transonic hydrodynamic escape of an Earth-like atmosphere. We consider the case of an Earth-mass planet with an atmospheric composition identical to that of the current Earth orbiting at 1 AU around a young and very active solar mass star. Methods. To model the upper atmosphere, we used the Kompot Code, which is a first-principles model that calculates the physical structures of the upper atmospheres of planets, taking into account hydrodynamics and the main chemical and thermal processes taking place in the upper atmosphere of a planet. This model enabled us to calculate the 1D vertical structure of the atmosphere using as input the high-energy spectrum of a young and active Sun. Results. The atmosphere has the form of a transonic hydrodynamic Parker wind, which has an outflow velocity at the upper boundary of our computational domain that exceeds the escape velocity. The outflowing gas is dominated by atomic nitrogen and oxygen and their ion equivalents and has a maximum ionization fraction of 20%. The mass outflow rate is found to be 1.8x10^9 g s^-1, which would erode the modern Earth's atmosphere in less than 0.1 Myr. Conclusions. This extreme mass loss rate suggests that an Earth-like atmosphere cannot form when the planet is orbiting within the habitable zone of a very active star. Instead, such an atmosphere can only form after the activity of the star has decreased to a much lower level. This happened in the early atmosphere of the Earth, which was likely dominated by other gases such as CO2. Since the time it takes for the activity of a star to decay is highly dependent on its mass, this is important for understanding possible formation timescales for planets orbiting low-mass stars.

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Transit Ly-$α$ signatures of terrestrial planets in the habitable zones of M dwarfs

We modeled the transit signatures in the Lya line of a putative Earth-sized planet orbiting in the HZ of the M dwarf GJ436. We estimated the transit depth in the Lya line for an exo-Earth with three types of atmospheres: a hydrogen-dominated atmosphere, a nitrogen-dominated atmosphere, and a nitrogen-dominated atmosphere with an amount of hydrogen equal to that of the Earth. We calculated the in-transit absorption they would produce in the Lya line. We applied it to the out-of-transit Lya observations of GJ 436 obtained by the HST and compared the calculated in-transit absorption with observational uncertainties to determine if it would be detectable. To validate the model, we also used our method to simulate the deep absorption signature observed during the transit of GJ 436b and showed that our model is capable of reproducing the observations. We used a DSMC code to model the planetary exospheres. The code includes several species and traces neutral particles and ions. At the lower boundary of the DSMC model we assumed an atmosphere density, temperature, and velocity obtained with a hydrodynamic model for the lower atmosphere. We showed that for a small rocky Earth-like planet orbiting in the HZ of GJ436 only the hydrogen-dominated atmosphere is marginally detectable with the STIS/HST. Neither a pure nitrogen atmosphere nor a nitrogen-dominated atmosphere with an Earth-like hydrogen concentration in the upper atmosphere are detectable. We also showed that the Lya observations of GJ436b can be reproduced reasonably well assuming a hydrogen-dominated atmosphere, both in the blue and red wings of the Lya line, which indicates that warm Neptune-like planets are a suitable target for Lya observations. Terrestrial planets can be observed in the Lya line if they orbit very nearby stars, or if several observational visits are available.

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Effective induction heating around strongly magnetized stars

Planets that are embedded in the changing magnetic fields of their host stars can experience significant induction heating in their interiors caused by the planet's orbital motion. For induction heating to be substantial, the planetary orbit has to be inclined with respect to the stellar rotation and dipole axes. Using WX~UMa, for which the rotation and magnetic axes are aligned, as an example, we show that for close-in planets on inclined orbits, induction heating can be stronger than the tidal heating occurring inside Jupiter's satellite Io; namely, it can generate a surface heat flux exceeding 2\,W\,m$^{-2}$. An internal heating source of such magnitude can lead to extreme volcanic activity on the planet's surface, possibly also to internal local magma oceans, and to the formation of a plasma torus around the star aligned with the planetary orbit. A strongly volcanically active planet would eject into space mostly SO$_2$, which would then dissociate into oxygen and sulphur atoms. Young planets would also eject CO$_2$. Oxygen would therefore be the major component of the torus. If the O{\sc i} column density of the torus exceeds $\approx$10$^{12}$\,cm$^{-2}$, the torus could be revealed by detecting absorption signatures at the position of the strong far-ultraviolet O{\sc i} triplet at about 1304\,Å. We estimate that this condition is satisfied if the O{\sc i} atoms in the torus escape the system at a velocity smaller than 1--10\,km\,s$^{-1}$. These estimates are valid also for a tidally heated planet.

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