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Antoine Strugarek

Publications and source records attributed to Antoine Strugarek.

At least 19 recordsLinked to original sources

Active nests at solar maximum: How nested flux emergence dominated flaring activity and structured the heliosphere

Multi-viewpoint observations from near-Earth satellites and ESA's Solar Orbiter enable a global view of the Sun. This continuous coverage is essential for characterising active nests. We investigated how the formation of active nests during the maximum of solar cycle 25 affected the Sun's flaring activity and large-scale magnetic field. Active nests were identified from their persistent activity in Carrington coordinates. For each nest, we monitored the evolution of their photospheric magnetic field, extreme ultraviolet emission, and associated solar flares. Their impact on the solar corona was explored through potential field source surface extrapolations and coronagraph observations. These results were then compared with in-situ measurements of solar wind speed and magnetic field polarity. During 156 days of global monitoring in 2024, nearly 80% of all solar flares originated from three active nests in the southern hemisphere. Two of the nests converged over several months, colliding with a burst of activity in October 2024. The third nest was linked to the emergence of NOAA active region AR13664 in May 2024. Magnetically complex NOAA active regions were more prevalent within active nests. During 2024, 28% of the 147 nested active regions were complex, compared to just 16% of the 181 regions elsewhere. In January 2025, two active nests emerged from the collision site, and a new active nest appeared in the north. This reinforced a tilted dipole topology throughout 2025. Active nests modulated global solar activity in 2024 and shaped the solar corona and wind in 2025. Nearly continuous tracking was used to isolate their flaring and flux emergence rates. Active nest migration was driven by the emergence of magnetic flux, implying a coherent sub-surface origin. By anchoring the coronal magnetic field, active nests created more favourable source regions for solar wind connectivity studies.

astro-ph.SR

Early Exploration of the Scientific Discovery Space for the Habitable Worlds Observatory

The Habitable Worlds Observatory (HWO) is a future NASA flagship mission concept identified by the Astro2020 Decadal Survey as the highest priority for large space missions. HWO should conduct "transformative astrophysics" and search for biosignatures in the atmospheres of approximately 25 potentially Earth-like planets. To further the early-stage development of HWO, NASA formed the Science, Technology, Architecture Review Team (START). In turn, START invited the scientific community to join working groups to explore the potential discovery space. In this paper, we present 70 science cases that resulted from this process. The cases address four scientific pillars: growth of galaxies (15 cases), evolution of the elements (13 cases), solar systems in context (32 cases), and living worlds (10 cases). Combined, they would address 27 of the 30 science questions and discovery areas identified by Astro2020. The 140 observing programs needed for the 70 investigations encompass a rich variety of spectroscopic (for 87% of science cases) and photometric (for 30%) observations extending from the UV to the NIR. Additionally, high-contrast and polarimetric capabilities would be needed for 34% and 27% of science cases, respectively. Access to UV wavelengths is critical: 83% of science cases need data at wavelengths <400 nm, and 26% extend to <100 nm. In the NIR, 26% of science cases need observations at wavelengths >=2000 nm. Pursuing the full portfolio of science would also necessitate precise astrometry for planet mass measurement, rapid response capabilities, a large instantaneous field of regard, non-sidereal tracking, saturation mitigation strategies, and high dynamic range.

astro-ph.IM

Electron transport in a radiation-dominated plasma Application to solar corona brightenings

Bremsstrahlung scattering of fast electrons on ions can be enhanced by the microwave radiation present in the solar corona. It can account for the electron diffusive transport along magnetic loops and high precipitation rates. This process can also dominate the transport of thermal electrons confined in such loops. The influence of stimulated Bremsstrahlung scattering on the electron transport is studied, with focus on the return current induced by the fast electron population trapped in magnetic loops. Overall, transport coefficients are reevaluated in the radiation-dominated plasma, characterized by the stimulated action of radiation on the Bremsstrahlung electron-ion collision, down to thermal velocities. We develop a theoretical framework for electron transport driven by a large bandwidth, bright low-frequency part of the photon spectrum and compute a set of radiation-enhanced transport coefficients. UV, XEUV and hard X-ray signals from flares, evidencing anomalous resistivity, thermal conduction inhibition and high precipitation rates of fast electrons, are reinterpreted. The anomalous resistivity due to stimulated Bremsstrahlung scattering is found to dominate the classical resistivity in flares. The runaway effect due to Coulomb collisions is suppressed. Thermal conduction is inhibited compared to the Spitzer conduction, in agreement with coronal seismology of slow-mode waves. Stimulated Bremsstrahlung scattering is found to be a key collisional process in flaring events. It can explain the above loop-top hard X-ray signal due to the fast electrons, and the measured electrical conductivity due to the thermal electrons. As a perspective, the corresponding transport coefficients can be used in radiation MHD codes. The radiation model could also be applied to stimulate large-angle electron scattering in the kinetic or hybrid models used to study the solar corona.

astro-ph.SR

Star Planet Interactions

Star-planet interactions (SPIs) describe the continuous exchange of energy, momentum, and mass between exoplanets and their host stars through radiative, tidal, magnetic, and particle-driven processes. Together, these interactions shape the structure, evolution, and observable properties of exoplanetary systems. In this review, we bring together current theoretical and observational understanding of SPIs, highlighting how stellar radiation, winds, and magnetic activity influence planetary atmospheres, interiors, and orbital evolution, while using the Solar System as a valuable reference for interpreting these processes. High-energy stellar radiation, particularly in the far- and extreme-ultraviolet and X-ray bands, drives atmospheric heating, photochemistry, ionisation, and escape. These effects are further influenced by stellar winds and magnetic interactions, which can either protect planetary atmospheres or accelerate their loss over time. Tidal interactions redistribute energy and angular momentum, producing internal heating and driving orbital migration and circularisation. Magnetic star-planet coupling provides additional pathways for energy transfer through reconnection and current systems, potentially enhancing atmospheric escape, heating planetary ionospheres and interiors, and generating observable signatures such as radio emission and enhanced stellar activity. We discuss how these processes work together, emphasising that their long-term impact depends on stellar evolution, planetary properties, atmospheric structure, and magnetic field strength. By presenting radiative, tidal, and magnetic interactions within a unified framework, this review highlights the physical mechanisms that shape planetary environments and identifies the key observational signatures that will complement future studies of exoplanet evolution and habitability.

astro-ph.SR

Stellar Forcing of (exo)Planetary Environments

The environments of exoplanets are fundamentally shaped by the magnetic activity of their host stars through radiative, plasma, and particle-driven processes. This article presents a comprehensive overview of the four principal forms of stellar forcing that regulate atmospheric structure, chemistry, escape, and long-term planetary evolution: high-energy radiation, magnetized stellar winds, coronal mass ejections, and energetic particles. Using the Sun as a physically resolved benchmark, the discussion extends to increasingly active cool stars to establish a broader picture of star--planet interactions across the main sequence. The article first examines stellar X-ray and extreme ultraviolet emission from chromospheres and coronae, together with variability introduced by flares and magnetic reconnection. Particular attention is given to spectroscopic diagnostics, activity scalings with stellar rotation and age, flare energetics, and the observational links between impulsive and gradual phases of magnetic energy release. The treatment then shifts to magnetized stellar winds, describing the mechanisms that drive them and the role of multidimensional magnetohydrodynamic modeling in determining wind structure, angular momentum loss, and planetary interaction regimes. Solar and stellar coronal mass ejections are explored through their diagnostics, flare associations, propagation, and possible suppression by strong stellar magnetic fields. Finally, galactic and stellar energetic particles are discussed together with methods for estimating particle environments and their consequences for atmospheric chemistry and climate. The article concludes by outlining future observational and numerical developments needed to connect these coupled stellar forcing processes within a unified exoplanetary framework.

astro-ph.SR

Energetics of star-planet magnetic interactions: Novel insights from 3D modelling

Star-planet magnetic interactions (SPMI) occurring in the sub-Alfvenic regime can, in principle, induce stellar chromospheric hotspots. Currently, estimates of the power generated by SPMI primarily rely on analytical scaling laws that relate stellar and planetary parameters to the interaction energetics. The existing scaling laws published in the literature so far do not agree with each other by at least an order of magnitude. Our aim is to quantify an absolute upper limit on the power that a planet can channel back to its host star during such interactions, which in turn lead to the formation of stellar hotspots. By performing a series of 3D MHD simulations with varied parameters known to influence the energetics of SPMI, we derive a numerically supported scaling law that can be used to reliably estimate the energy channeled from the planet back to the star. Our results suggest that existing analytical scaling laws may not fully capture the power transferred from the planet to the star through SPMI. The scaling law derived from our numerical simulations appears to provide a more comprehensive estimate, reflecting dependencies on common stellar and planetary parameters also considered in earlier models. Moreover, our findings indicate that power generation involves not only the planetary obstacle itself but also the extended magnetic structure of the Alfven wings interacting with the streaming stellar wind. This study suggests that care should be taken when applying analogies directly from jovian sub-Alfvenic interactions to SPMI, as the underlying physical conditions (specifically the value of the Alfvenic Mach number) may not be directly comparable. Our numerically derived scaling law offers a potentially improved approach for estimating SPMI power, capturing some of the interaction's complexities exclusive to SPMI.

astro-ph.SR

Avalanches in Magnetohydrodynamical simulations

Scale invariance is a hallmark of many natural systems, including solar flares, where energy release spans a vast range of scales. Recent computational advances, at the level of both algorithmics and hardware, have enabled high-resolution magnetohydrodynamical (MHD) simulations to span multiple scales, offering new insights into magnetic energy dissipation processes. Here, we study scale invariance of magnetic energy dissipation in two distinct MHD simulations. Current sheets are identified and analyzed over time. Results demonstrate that dissipative events exhibit scale invariance, with power-law distributions characterizing their energy dissipation and lifetimes. Remarkably, these distributions are consistent across the two simulations, despite differing numerical and physical setups, suggesting universality in the process of magnetic energy dissipation. Comparisons between the evolution of dissipation regions reveals distinct growth behaviors in high plasma-beta regions (convective zone) and low plasma-beta regions (atmosphere). The latter display spatiotemporal dynamics similar to those of avalanche models, suggesting self-organized criticality and a common universality class.

astro-ph.SR

High-resolution Ultraviolet-to-nearinfrared Characterization of Exoplanet Atmospheres

The Habitable Worlds Observatory (HWO) offers a unique opportunity to revolutionize our understanding of planetary formation and evolution. The goal of this Science Case Development Document (SCDD) is to investigate the physical and chemical processes that shape the composition and atmospheric mass loss in exoplanets. We review the key observables currently known as diagnostics of mass loss via transit observations, i.e., absorption lines of escaping hydrogen (Lyman-alpha), helium, and metals (Fe, Mg, C, O). We also explore the challenges to infer planetary formation processes based on atmospheric composition characterization. HWO could enable a broad, continuous coverage from far-ultraviolet to near-infrared spectroscopy (~100--1600 nm) at high resolution (R > 60, 000), which is essential to make these measurements, disentangle their planetary origin from stellar activity, and ultimately, contextualize the escape rates by simultaneously characterizing the composition, cloud predominance, and thermal structure of exoplanet atmospheres.

astro-ph.IM

Avalanching together: A model for sympathetic flaring

Avalanche models running in a self-organized critical regime have proven powerful in reproducing the power-law distributions and scale invariance that characterize the statistical properties of solar flares. They are often interpreted as representing an individual active region of the Sun. As a result, this class of models has rarely been applied to describe sympathetic flares $\unicode{x2014}$ solar eruptions that occur in close spatial and temporal proximity, seemingly driven by their mutual interaction. In this study, we investigate the phenomenon of sympathetic flaring using avalanche models and compare their statistical properties with observations of sympathetic flares on the Sun. We developed a novel avalanche model featuring two connected lattices, each representing a distinct active region. This connectivity allows the transfer of nodal variable between the lattices, simulating the non-local effects expected to occur during sympathetic flares. Our results show that under strong connectivity, the lattices exhibit temporal synchronization, with correlations between their avalanche energies. Furthermore, increasing the connectivity between the lattices results in an excess of avalanches at short waiting times. A quantitative comparison with observational data suggests that only a weak connectivity allows our model to replicate the observed solar waiting time distributions. Consequently, we propose that if magnetic connectivity between distinct active regions drives sympathetic flaring on the Sun, it must remain relatively weak.

astro-ph.SR

A prolific solar flare factory. Nearly continuous monitoring of an active region nest with Solar Orbiter

The properties of active region nests, which are locations on the Sun with recurring flux emergence, are poorly constrained by observations from Earth alone. ESA's Solar Orbiter now monitors the far side of the Sun for extended periods of time. This facilitates observations of the entire Sun. We combined observations from near-Earth satellites and Solar Orbiter to evaluate the contribution of a long-lived active region nest to the global flaring activity of the Sun. We identified a location in Carrington coordinates with episodic bursts of flux emergence throughout 2022. The combined observations allowed a nearly continuous monitoring of this region from April to October, that is, during its most active period. GOES and Solar Orbiter/STIX were used to compare its flaring activity to that of the entire Sun. The region morphology was extracted from SDO/AIA and Solar Orbiter/EUI extreme-ultraviolet images and was combined with magnetic field measurements from SDO/HMI and Solar Orbiter/PHI to assess its unsigned magnetic flux. The active region nest grew in complexity from January to May due to repeated flux emergence events. The peak unsigned magnetic flux was $5\times10^{22}$Mx. The region caused 40% of the observed solar flares in 2022, including five months in which it produced 50-70% of all flares over the entire Sun (in the nearly continuous monitoring window). Of the 17 complex flaring NOAA active regions in 2022, this region contained 10, but they occupied less than 20% of the area in the active latitudes. Active region nests can maintain a high rate of flaring activity for several solar rotations and are more likely to produce complex active regions that can trigger X-class solar flares. Improving the identification and monitoring of long-lived active region nests would benefit space weather forecasts in the short to medium term.

astro-ph.SR

Global Turbulent Solar Convection: a Numerical Path Investigating Key Force Balances in the context of the Convective Conundrum

Understanding solar turbulent convection and its influence on differential rotation has been a challenge over the past two decades. Current models often overestimate giant convection cells amplitude, leading to an effective Rossby number too large and a shift towards an anti-solar rotation regime. This Convective Conundrum, underscores the need for improved comprehension of solar convective dynamics. We propose a numerical experiment in the parameter space that controls $Ro$ while increasing the Reynolds number ($Re$) and maintaining solar parameters. By controlling the Nusselt number ($Nu$), we limit the energy transport by convection while reducing viscous dissipation. This approach enabled us to construct a Sun-like rotating model (SBR97n035) with strong turbulence ($Re \sim 800$) that exhibits prograde equatorial rotation and aligns with observational data from helioseismology. We compare this model with an anti-solar rotating counterpart, and provide an in-depth spectral analysis to investigate the changes in convective dynamics. We also find the appearance of vorticity rings near the poles, which existence on the Sun could be probed in the future. The Sun-like model shows reduced buoyancy over the spectrum, as well as an extended quasi-geostrophic equilibrium towards smaller scales. This promotes a Coriolis-Inertia (CI) balance rather than a Coriolis-Inertia-Archimedes (CIA) balance, in order to favor the establishment of a prograde equator. The presence of convective columns in the bulk of the convection zone, with limited surface manifestations, also hints at such structures potentially occurring in the Sun.

astro-ph.SR

Magnetized winds of M-type stars and star-planet magnetic interactions: uncertainties and modeling strategy

M-type stars are the most common stars in the universe. They are ideal hosts for the search of exoplanets in the habitable zone (HZ), as their small size and low temperature make the HZ much closer in than their solar twins. Harboring very deep convective layers, they also usually exhibit very intense magnetic fields. Understanding their environment, in particular their coronal and wind properties, is thus very important, as they might be very different from what is observed in the solar system. The mass loss rate of M-type stars is poorly known observationally, and recent attempts to estimate it for some of them (TRAPPIST-1, Proxima Cen) can vary by an order of magnitude. In this work, we revisit the stellar wind properties of M-dwarfs in the light of the latest estimates of $\dot{M}$ through Lyman-$α$ absorption at the astropause and slingshot prominences. We outline a modeling strategy to estimate the mass loss rate, radiative loss and wind speed, with uncertainties, based on an Alfvén wave driven stellar wind model. We find that it is very likely that several TRAPPIST-1 planets lie within the Alfvén surface, which imply that these planets experience star-planet magnetic interactions (SPMI). We also find that SPMI between Proxima Cen b and its host star could be the reason of recently observed radio emissions.

astro-ph.SR

Magnetic activity evolution of solar-like stars: II. $S_{\rm ph}$-Ro evolution of Kepler main-sequence targets

There is now a large sample of stars observed by the Kepler satellite with measured rotation periods and photometric activity index $S_{\rm ph}$. We use this data, in conjunction with stellar interiors models, to explore the interplay of magnetism, rotation, and convection. Stellar activity proxies other than $S_{\rm ph}$ are correlated with the Rossby number, $Ro$, or ratio of rotation period to convective overturn timescale. We compute the latter using the Yale Rotating Evolution Code stellar models. We observe different $S_{\rm ph}$-$Ro$ relationships for different stellar spectral types. Though the overall trend of decreasing magnetic activity versus $Ro$ is recovered, we find a localized dip in $S_{\rm ph}$ around $Ro/Ro_{\odot} \sim$\,0.3 for the G and K dwarfs. F dwarfs show little to no dependence of $S_{\rm ph}$ on $Ro$ due to their shallow convective zones; further accentuated as $T_{\rm eff}$ increases. The dip in activity for the G and K dwarfs corresponds to the intermediate rotation period gap, suggesting that the dip in $S_{\rm ph}$ could be associated with the redistribution of angular momentum between the core and convective envelope inside stars. For G-type stars, we observe enhanced magnetic activity above solar $Ro$. Compared to other Sun-like stars with similar effective temperature and metallicity, we find that the Sun's current level of magnetic activity is comparable to its peers and lies near the transition to increasing magnetic activity at high $Ro$. We confirm that metal-rich stars have a systematically larger $S_{\rm ph}$ level than metal-poor stars, which is likely a consequence of their deeper convective zones.

astro-ph.SR

Secondary ionisation in hot atmospheres and interactions between planetary and stellar winds

The loss of close-in planetary atmospheres is influenced by various physical processes, such as photoionisation, which could potentially affect the atmosphere survivability on a secular timescale. The amount of stellar radiation converted into heat depends on the energy of the primary electrons produced by photoionisation and the local ionisation fraction. The Lyman-alpha line is an excellent probe for atmospheric escape. We study the interaction between the planetary and the stellar wind, the difference of the predicted mass-loss rates between 1D and 2D models, the signal of Ly-a and the impact of stellar flares. Using the PLUTO code, we perform 2D hydrodynamics simulations for four different planets. We consider planets in the size range from Neptune to Jupiter. We produce synthetic Ly-a profiles to comprehend the origin of the signal, and in particular its high velocity Doppler shift. Our results indicate a trend similar to the 1D models, with a decrease in the planetary mass-loss rate for all systems when secondary ionisation is taken into account. The mass-loss rates are found to decrease by 48% for the least massive planet when secondary ionisation is accounted for. We find nevertheless a decrease that is less pronounced in 2D than in 1D. We observe differences in the Ly-a profile between the different cases and significant asymmetries in all of them, especially for the lower mass planets. Finally, we observe that stellar flares do not affect the mass-loss rate because they act, in general, on a timescale that is too short. We find velocities in the escaping atmosphere up to 100 km/s, with the gas moving away from the star, which could be the result of the interaction with the stellar wind. Furthermore, we find that stellar flares generally occur on a timescale that is too short to have a visible impact on the mass-loss rate of the atmosphere.

astro-ph.EP

On stellar hotspots due to star-planet magnetic interactions: How much power can actually be transmitted to the chromosphere?

Star-Planet Magnetic Interactions (SPMI) have been proposed as a mechanism for generating stellar hot-spots with energy outputs on the order of $10^{19-21}$ watts. This interaction is primarily believed to be mediated by Alfvén waves propagating towards the star. The stellar atmosphere dictates where and how much of this incoming energy can actually be deposited as heat. The stellar transition region separating the chromosphere from the corona of cool stars gives rise to a significant variation of the Alfvén speed over a short distance, and therefore a reflection of the Alfvén waves at the transition region is naturally expected. We aim to characterize the efficiency of energy transfer due to SPMI by quantifying a frequency dependent reflection of the wave energy at the stellar transition region and its transmission to the stellar chromosphere. Magnetohydrodynamic simulations are employed to model the frequency-dependent propagation of Alfvén waves through a realistic background stellar wind profile. The transmission efficiency as a function of the wave frequency is quantified, and further analysis is conducted to characterize the overall energy transfer efficiency of SPMI in several candidate systems where chromospheric hotspots have been tentatively detected. Low-frequency waves experience greater reflection compared to high-frequency waves, resulting in reduced energy transfer efficiency for lower frequencies. Conversely, the parametric decay instability of Alfvén waves substantially diminishes the energy transfer efficiency at higher frequencies. As a result, there exists a frequency range where energy transfer is most efficient. A significant fraction of the Alfvén wave energy is reflected at the stellar transition region, and in most realistic scenarios, the transmission efficiency to the chromosphere is found to be approximately 10 percent.

astro-ph.SR

Flaring together: A preferred angular separation between sympathetic flares on the Sun

Sympathetic solar flares are eruptions that occur nearby in space and time, driven by an apparent interaction between the active regions in which they are triggered. Their statistical existence on the Sun has yet to be firmly established. The main goal of this paper is to identify a statistical signature of sympathetic flares, characterize their properties and determine a potential mechanism driving their interaction. We perform a statistical analysis of a large number of flares observed by the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO), the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) and the Spectrometer Telescope for Imaging X-rays (STIX) on Solar Orbiter during solar cycle 24 and 25. We examine the spatiotemporal distribution of consecutive flare pairs across solar cycle phases and hemispheres, along with the propagation velocity of potential causal interactions and the relationship between flare magnitudes. We observe an excess of hemispheric flares separated by about 30 degrees of longitude and triggered in less than 1.5 hours from each other. This peak in angular separation varies with the solar cycle phase and hemisphere. Moreover, we identify a deficit of transequatorial events separated by 25-30 degrees in latitude and less than 5 degrees in longitude, a phenomenon we term unsympathetic flares. We provide strong statistical evidence for the existence of sympathetic flares on the Sun, demonstrating that their occurrence rate reaches approximately 5% across the three instruments used in this study. Additionally, we propose an interpretation of the observed angular scale of the sympathetic phenomenon, based on the separation between magnetic field line footpoints derived from potential field source surface extrapolations.

astro-ph.SR

A multi-dimensional, robust, and cell-centered finite-volume scheme for the ideal MHD equations

We present a new multi-dimensional, robust, and cell-centered finite-volume scheme for the ideal MHD equations. This scheme relies on relaxation and splitting techniques and can be easily used at high order. A fully conservative version is not entropy satisfying but is observed experimentally to be more robust than standard constrained transport schemes at low plasma beta. At very low plasma beta and high Alfvén number, we have designed an entropy-satisfying version that is not conservative for the magnetic field but preserves admissible states and we switch locally a-priori between the two versions depending on the regime of plasma beta and Alfvén number. This strategy is robust in a wide range of standard MHD test cases, all performed at second order with a classic MUSCL-Hancock scheme.

physics.comp-ph

Impact of far-side structures observed by Solar Orbiter on coronal and heliospheric wind simulations

Solar Orbiter provides unique capabilities to understand the heliosphere. In particular, it has made observations of the far-side of the Sun and provides unique information to improve space weather monitoring. We aim to quantify how far-side data will affect simulations of the corona and the interplanetary medium, especially in the context of space weather forecasting. We focused on a time period with a single sunspot emerging on the far-side in February 2021. We used two different input magnetic maps: one with the far-side active region and one without. We used three different coronal models: a semi-empirical model (potential field source surface or PFSS) and two different magnetohydrodynamic models (Wind Predict and Wind Predict-AW). We compared all the models with both remote sensing and in situ observations. We find that the inclusion of the far-side active region in the various models has a small local impact due to the limited amount of flux of the sunspot (at most 8% of the total map flux), which leads to coronal hole changes of around 7% for all models. Interestingly, there is a more global impact on the magnetic structure seen in the current sheet, with clear changes in the coronal hole boundaries visible in extreme ultra-violet (EUV) on the western limb. For the Wind Predict-AW model, we demonstrate that the inclusion of the far-side data improves both the structure of the streamers and the connectivity to the spacecraft. In conclusion, the inclusion of a single far-side active region may have a small local effect with respect to the total magnetic flux, but it has global effects on the magnetic structure, and thus it must be taken into account to accurately describe the Sun-Earth connection. The flattening of the heliospheric current sheet for all models reveals an increase of the source surface height, which affects the open and closed magnetic field line distributions.

astro-ph.SR