arXiv subjects

E. Gendron

Publications and source records attributed to E. Gendron.

At least 19 recordsLinked to original sources

arXiv2026

The MICADO first light imager for the ELT: first on-bench performance tests of the SCAO hybrid LQG+Integrator controller

The Extremely Large Telecope (ELT) is expected to begin scientific operations in 2030, with MICADO as its first-light imager. The Single Conjugate Adaptive Optics (SCAO) system for MICADO is currently being assembled and tested at the Observatoire de Paris. MICADO's baseline AO control strategy uses a hybrid approach: a Linear Quadratic Gaussian (LQG) regulator for low-order modes, optimised to adapt to the disturbance caused by vibrations and ELT windshake, and an integral action controller for higher-oder modes. Despite the non linearity of the Pyramid Wavefront Sensor (PyWFS), the so-called "optical gains", LQG control has proven to be highly effective in simulation. It has been shown that the LQG regulator is resilient to optical gain variations in the loop or to inaccuracies in their estimation. Increasing the number of modes controlled by the LQG regulator would therefore make the controller less sensitive to the optical gains, which are notoriously difficult to estimate during operation. A first hybrid controller, with only tip-tilt controlled with LQG control, has been validated using COMPASS, the end-to-end, GPU-accelerated, AO simulation platform. We will present the first experimental validation of this hybrid controller on the MICADO-SCAO bench. We will also present results obtained by increasing the number of modes controlled with LQG control and the work carried out to go towards an autonomous LQG regulator.

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arXiv2026

The GRAVITY young stellar object survey -- XV. The star-disk interaction region of the T Tauri star DO Tau

Protoplanetary disks around young Sun-like stars are the cradles of the vast majority of detected exoplanets. Probing these disks at multiple spatial scales is key to uncovering how planets form. We aim to spatially and spectrally resolve the inner disk and star-disk interaction region of the M0.3 T Tauri star DO Tau by combining two complementary techniques. We used high-resolution near-infrared spectra from CFHT/SPIRou to constrain the magnetospheric star-disk interaction process and optical long-baseline interferometry with ESO VLTI/GRAVITY to determine the sizes of the K-band continuum and Br$\gamma$ line emitting regions. From the SPIRou spectra, we confirmed that this ~0.5 M$_\odot$ star is a strong accretor. The HI and HeI lines exhibit strong variability on a daily timescale, consistent with the burster classification of DO Tau derived from its K2 light curve. We derived an upper limit of 0.35 on the ratio between the magnetospheric truncation radius and the disk corotation radius, indicative of an ordered unstable accretion regime. The size of the Br$\gamma$ line emitting region obtained from GRAVITY is much smaller than the K-band continuum emitting region. This compact Br$\gamma$ emission region ($R_{Br\gamma} \sim$ 0.011 au) suggests that most of the line flux originates from the magnetospheric accretion region and/or from an inner wind close to the magnetosphere-disk interface. The inclination we derived for the inner disk (45-55{\deg}) differs from that of the outer disk inferred from the ALMA continuum (30{\deg}). This points toward a misalignment or warp of the outer disk that may originate from the suspected past encounter with the neighboring HV Tau system.

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arXiv2026

Non-common path aberration compensation and a dark hole loop with a pyramid adaptive optics system: Application to SAXO+

In ground-based high-contrast instruments, non-common path aberrations (NCPAs) limit detection performance, as they are unseen by the adaptive optics (AO) wavefront sensor but impact the astrophysical image, creating quasi-static speckles. SAXO+, the upgrade of the SAXO (SPHERE AO system) includes a second loop of AO downstream of the SAXO loop that is equipped with a near-infrared pyramid wavefront sensor whose nonlinearities, usually described with modal optical gains, might be challenging for removing quasi-static speckles. We investigated two methods of quasi-static speckle removal : NCPA compensation and a dark hole loop, behind a pyramid AO system, measuring the interest of compensating for the pyramid optical gains. We performed end-to-end numerical simulations under various astrophysical conditions. We offset the pyramid wavefront sensor operating point to apply both the speckle suppression methods, with or without optical gain calibration. We evaluated the performance by measuring the residual starlight in the coronagraph image. A by-product of our study is an on-sky calibration method of measuring the pyramid optical gains. NCPA compensation reduces the residual starlight in the coronagraph image by a factor of 20 for seeing between 0.7" and 1" for a bright star and a factor of 2 at 0.7" for a faint star. Optical gains compensation enhances the performance at poor seeing and small pyramid modulation radius with a bright star, but shows a useless or even negative impact due to estimation inaccuracies at faint targets. On the other hand, the dark hole loop reduces the residual starlight by a factor of 200. The optical gain calibration enhances the dark hole performance behind a single pyramid AO system but is useless behind the SAXO+ system. Our parametric study gives baseline values for the efficient control of the dark hole loop for the SAXO+ system.

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arXiv2025

Exomoon search with VLTI/GRAVITY around the substellar companion HD 206893 B

Direct astrometric detection of exomoons remains unexplored. This study presents the first application of high-precision astrometry to search for exomoons around substellar companions. We investigate whether the orbital motion of the companion HD 206893 B exhibits astrometric residuals consistent with the gravitational influence of an exomoon or binary planet. Using the VLTI/GRAVITY instrument, we monitored the astrometric positions of HD 206893 B and c across both short (days to months) and long (yearly) timescales. This enabled us to isolate potential residual wobbles in the motion of component B attributable to an orbiting moon. Our analysis reveals tentative astrometric residuals in the HD 206893 B orbit. If interpreted as an exomoon signature, these residuals correspond to a candidate (HD 206893 B I) with an orbital period of approximately 0.76 years and a mass of $\sim$0.4 Jupiter masses. However, the origin of these residuals remains ambiguous and could be due to systematics. Complementing the astrometry, our analysis of GRAVITY $R=4000$ spectroscopy for HD 206893 B confirms a clear detection of water, but no CO is found using cross-correlation. We also find that AF Lep b, and $\beta$ Pic b are the best short-term candidates to look for moons with GRAVITY+. Our observations demonstrate the transformative potential of high-precision astrometry in the search for exomoons, and proves the feasibility of the technique to detect moons with masses lower than Jupiter and potentially down to less than Neptune in optimistic cases. Crucially, further high-precision astrometric observations with VLTI/GRAVITY are essential to verify the reality and nature of this signal and attempt this technique on a variety of planetary systems.

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arXiv2025

The ExoGRAVITY survey: A K-band spectral library of giant exoplanet and brown dwarf companions

Direct observations of exoplanet and brown dwarf companions with near-infrared interferometry, first enabled by the dual-field mode of VLTI/GRAVITY, provide unique measurements of the objects' orbital motions and atmospheric compositions. Here, we compile a homogeneous library of all exoplanet and brown dwarf K-band spectra observed by GRAVITY thus far. We re-reduced all the available GRAVITY dual-field high-contrast data and, where companions are detected, extract their ~2.0-2.4 $\mu$m K-band contrast spectra. We then derived stellar model atmospheres for all employed flux references, which we used to convert the companion contrast into companion flux spectra. Solely from the resulting GRAVITY spectra, we extracted spectral types, spectral indices, and bulk physical properties for all companions. Finally, and with the help of age constraints from the literature, we also derived isochronal masses for most companions using evolutionary models. The resulting library contains R ~ 500 GRAVITY spectra of 39 substellar companions from late M to late T spectral types, including the entire L-T transition. Throughout this transition, a shift from CO-dominated late M- and L-type dwarfs to CH4-dominated T-type dwarfs can be observed in the K-band. The GRAVITY spectra alone constrain the objects' bolometric luminosity to typically within $\pm$0.15 dex. The derived isochronal masses agree with dynamical masses from the literature where available, except for HD 4113 c for which we confirm its previously reported potential underluminosity. Medium-resolution spectroscopy of substellar companions with GRAVITY provides insight into the carbon chemistry and the cloudiness of these objects' atmospheres. It also constrains these objects' bolometric luminosities which can yield measurements of their formation entropy if combined with dynamical masses, for instance from Gaia and GRAVITY astrometry.

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arXiv2025

Constraints on the Orbit of the Young Substellar Companion GQ Lup B from High-Resolution Spectroscopy and VLTI/GRAVITY Astrometry

Understanding the orbits of giant planets is critical for testing planet formation models, particularly at wide separations greater than 10 au where traditional core accretion becomes inefficient. However, constraining orbits at these separations has been challenging because of sparse orbital coverage and degeneracies in the orbital parameters. We use existing high-resolution spectroscopic measurements from CRIRES+ (R ~ 100000), astrometric data from SPHERE, NACO, and ALMA, and new high-precision GRAVITY astrometry to refine the orbit of GQ Lup B, a ~30 M_J companion at ~100 au, in a system that also hosts a circumstellar disk and a wide companion, GQ Lup C. Including radial velocity data significantly improves orbital constraints by breaking the degeneracy between inclination and eccentricity that affects astrometry-only fits for long-period companions. This work is among the first to combine high-precision astrometry with the companion's relative radial velocity to achieve improved orbital constraints. The eccentricity is refined from e = 0.47 (+0.14, -0.16) with GRAVITY alone to e = 0.35 (+0.10, -0.09) when RVs and GRAVITY data are combined. The orbit is misaligned by 63 (+6, -14) deg relative to the circumstellar disk and 52 (+19, -24) deg relative to the host star spin axis, and is more consistent (34 (+6, -13) deg) with the inclination of the wide tertiary companion GQ Lup C disk. These results support a formation scenario for GQ Lup B consistent with cloud fragmentation and highlight the power of combining companion RV constraints with interferometric astrometry to probe the dynamics and formation of wide-orbit substellar companions.

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arXiv2025

Direct imaging discovery of a young giant planet orbiting on Solar System scales

HD 135344 AB is a young visual binary system that is best known for the protoplanetary disk around the secondary star. The circumstellar environment of the A0-type primary star, on the other hand, is already depleted. HD 135344 A is therefore an ideal target for the exploration of recently formed giant planets because it is not obscured by dust. We searched for and characterized substellar companions to HD 135344 A down to separations of about 10 au. We observed HD 135344 A with VLT/SPHERE in the $H23$ and $K12$ bands and obtained $YJ$ and $YJH$ spectroscopy. In addition, we carried out VLTI/GRAVITY observations for the further astrometric and spectroscopic confirmation of a detected companion. We discovered a close-in young giant planet, HD 135344 Ab, with a mass of about 10 $M_\mathrm{J}$. The multi-epoch astrometry confirms the bound nature based on common parallax and common proper motion. This firmly rules out the scenario of a non-stationary background star. The semi-major axis of the planetary orbit is approximately 15-20 au, and the photometry is consistent with that of a mid L-type object. The inferred atmospheric and bulk parameters further confirm the young and planetary nature of the companion. HD 135344 Ab is one of the youngest directly imaged planets that has fully formed and orbits on Solar System scales. It is a valuable target for studying the early evolution and atmosphere of a giant planet that could have formed in the vicinity of the snowline.

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arXiv2025

Orbit and atmosphere of HIP 99770 b through the eyes of VLTI/GRAVITY

Context: Inferring the likely formation channel of giant exoplanets and brown dwarf companions from orbital and atmospheric observables remains a formidable challenge. Further and more precise directly measured dynamical masses of these companions are required to inform and gauge formation, evolutionary, and atmospheric models. We present an updated study of HIP 99770 b based on observations conducted with VLTI/GRAVITY. Aims: Combining the new data with previous observations from the literature, we characterise HIP 99770 b to better constrain its orbit, dynamical mass, and atmospheric properties, as well as to shed light on its likely formation channel. Methods: We ran a renewed orbit fit to further constrain the dynamical mass of the companion and the orbit solution. We also analysed the GRAVITY K-band spectrum, placing it into context with literature data, and extracting magnitude, age, spectral type, bulk properties and atmospheric characteristics of HIP 99770 b. Results: We detected the companion at a radial separation of $417\,\mathrm{mas}$ from its host. The new orbit fit yields a dynamical mass of $17_{-5}^{+6}\,\mathrm{M}_\mathrm{Jup}$ and an eccentricity of $0.31_{-0.12}^{+0.06}$. We also find that additional relative astrometry epochs in the future will not enable further constraints on the dynamical mass due to the dominating relative uncertainty on the Hipparcos-Gaia proper motion anomaly. The publication of Gaia DR4 will likely ease this predicament. We find that the companion is consistent with spectral type L8 and exhibits a potential metal enrichment in its atmosphere. Conclusions: These results do not yet allow for a definite inference of the companion's formation channel. Nevertheless, the new constraints on its bulk properties and the additional GRAVITY spectrum presented here will aid future efforts to determine the formation history of HIP 99770 b.

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arXiv2025

Exploring the presence of a fifth force at the Galactic Center

Aims: The presence of a Yukawa-like correction to Newtonian gravity is investigated at the Galactic Center, leading to a new upper limit for the intensity of such a correction. Methods: We perform a Markov Chain Monte Carlo analysis using the astrometric and spectroscopic data of star S$2$ collected at the Very Large Telescope by GRAVITY, NACO and SINFONI instruments, covering the period from $1992$ to $2022$. Results: The precision of the GRAVITY instrument allows us to derive the most stringent upper limit at the Galactic Center for the intensity of the Yukawa contribution ($\propto \, \alpha e^{- \lambda r}$) to be $|\alpha| < 0.003$ for a scale length $\lambda = 3 \cdot 10^{13}\, \rm m\, (\sim 200 \, \rm AU)$. This improves by roughly one order of magnitude all estimates obtained in previous works.

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arXiv2024

The GRAVITY young stellar object survey XIV : Investigating the magnetospheric accretion-ejection processes in S CrA N

The dust- and gas-rich protoplanetary disks around young stellar systems play a key role in star and planet formation. While considerable progress has recently been made in probing these disks on large scales of a few tens of astronomical units (au), the central au needs to be more investigated. We aim at unveiling the physical processes at play in the innermost regions of the strongly accreting T Tauri Star S CrA N by means of near-infrared interferometric observations. The K-band continuum emission is well reproduced with an azimuthally-modulated dusty ring. As the star alone cannot explain the size of this sublimation front, we propose that magnetospheric accretion is an important dust-heating mechanism leading to this continuum emission. The differential analysis of the Hydrogen Br$\gamma$ line is in agreement with radiative transfer models combining magnetospheric accretion and disk winds. Our observations support an origin of the Br$\gamma$ line from a combination of (variable) accretion-ejection processes in the inner disk region.

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arXiv2024

The MICADO first light imager for the ELT: off-axis performance of PSF reconstruction

The highest scientific return, for adaptive optics (AO) observations, is achieved with a reliable reconstruction of the PSF. This is especially true for MICADO@ELT. In this presentation, we will focus on extending the MICADO PSF reconstruction (PSF-R) method to the off-axis case. Specifically, a novel approach based on temporal-based tomography of AO telemetry data has been recently implemented. Results from the PSF-R of both simulated and real data show that, at half isoplanatic angle distances, a precision of about 10-15% is achievable in both Strehl ratio and full-width at half maximum, paving the way to extend the MICADO PSF-R tool also to the multi-conjugated AO case.

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arXiv2024

High contrast at short separation with VLTI/GRAVITY: Bringing Gaia companions to light

Since 2019, GRAVITY has provided direct observations of giant planets and brown dwarfs at separations of down to 95 mas from the host star. Some of these observations have provided the first direct confirmation of companions previously detected by indirect techniques (astrometry and radial velocities). We want to improve the observing strategy and data reduction in order to lower the inner working angle of GRAVITY in dual-field on-axis mode. We also want to determine the current limitations of the instrument when observing faint companions with separations in the 30-150 mas range. To improve the inner working angle, we propose a fiber off-pointing strategy during the observations to maximize the ratio of companion-light-to-star-light coupling in the science fiber. We also tested a lower-order model for speckles to decouple the companion light from the star light. We then evaluated the detection limits of GRAVITY using planet injection and retrieval in representative archival data. We compare our results to theoretical expectations. We validate our observing and data-reduction strategy with on-sky observations; first in the context of brown dwarf follow-up on the auxiliary telescopes with HD 984 B, and second with the first confirmation of a substellar candidate around the star Gaia DR3 2728129004119806464. With synthetic companion injection, we demonstrate that the instrument can detect companions down to a contrast of $8\times 10^{-4}$ ($\Delta \mathrm{K}= 7.7$ mag) at a separation of 35 mas, and a contrast of $3\times 10^{-5}$ ($\Delta \mathrm{K}= 11$ mag) at 100 mas from a bright primary (K<6.5), for 30 min exposure time. With its inner working angle and astrometric precision, GRAVITY has a unique reach in direct observation parameter space. This study demonstrates the promising synergies between GRAVITY and Gaia for the confirmation and characterization of substellar companions.

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arXiv2024

Astrometric detection of a Neptune-mass candidate planet in the nearest M-dwarf binary system GJ65 with VLTI/GRAVITY

The detection of low-mass planets orbiting the nearest stars is a central stake of exoplanetary science, as they can be directly characterized much more easily than their distant counterparts. Here, we present the results of our long-term astrometric observations of the nearest binary M-dwarf Gliese 65 AB (GJ65), located at a distance of only 2.67 pc. We monitored the relative astrometry of the two components from 2016 to 2023 with the VLTI/GRAVITY interferometric instrument. We derived highly accurate orbital parameters for the stellar system, along with the dynamical masses of the two red dwarfs. The GRAVITY measurements exhibit a mean accuracy per epoch of 50-60 microarcseconds in 1.5h of observing time using the 1.8m Auxiliary Telescopes. The residuals of the two-body orbital fit enable us to search for the presence of companions orbiting one of the two stars (S-type orbit) through the reflex motion they imprint on the differential A-B astrometry. We detected a Neptune-mass candidate companion with an orbital period of p = 156 +/- 1 d and a mass of m = 36 +/- 7 Mearth. The best-fit orbit is within the dynamical stability region of the stellar pair. It has a low eccentricity, e = 0.1 - 0.3, and the planetary orbit plane has a moderate-to-high inclination of i > 30{\deg} with respect to the stellar pair, with further observations required to confirm these values. These observations demonstrate the capability of interferometric astrometry to reach microarcsecond accuracy in the narrow-angle regime for planet detection by reflex motion from the ground. This capability offers new perspectives and potential synergies with Gaia in the pursuit of low-mass exoplanets in the solar neighborhood.

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arXiv2024

Four-of-a-kind? Comprehensive atmospheric characterisation of the HR 8799 planets with VLTI/GRAVITY

With four companions at separations from 16 to 71 au, HR 8799 is a unique target for direct imaging, presenting an opportunity for the comparative study of exoplanets with a shared formation history. Combining new VLTI/GRAVITY observations obtained within the ExoGRAVITY program with archival data, we perform a systematic atmospheric characterisation of all four planets. We explore different levels of model flexibility to understand the temperature structure, chemistry and clouds of each planet using both petitRADTRANS atmospheric retrievals and fits to self-consistent radiative-convective equilibrium models. Using Bayesian Model Averaging to combine multiple retrievals, we find that the HR 8799 planets are highly enriched in metals, with [M/H] $\gtrsim$1, and have stellar to super-stellar C/O ratios. The C/O ratio increases with increasing separation from $0.55^{+0.12}_{-0.10}$ for d to $0.78^{+0.03}_{-0.04}$ for b, with the exception of the innermost planet which has a C/O ratio of $0.87\pm0.03$. By retrieving a quench pressure and using a disequilibrium chemistry model we derive vertical mixing strengths compatible with predictions for high-metallicity, self-luminous atmospheres. Bayesian evidence comparisons strongly favour the presence of HCN in HR 8799 c and e, as well as CH$_{4}$ in HR 8799 c, with detections at $>5\sigma$ confidence. All of the planets are cloudy, with no evidence for patchiness. The clouds of c, d and e are best fit by silicate clouds lying above a deep iron cloud layer, while the clouds of the cooler HR 8799 b are more likely composed of Na$_{2}$S. With well defined atmospheric properties, future exploration of this system is well positioned to unveil further detail in these planets, extending our understanding of the composition, structure, and formation history of these siblings.

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arXiv2024

Combining Gaia and GRAVITY: Characterising five new Directly Detected Substellar Companions

Precise mass constraints are vital for the characterisation of brown dwarfs and exoplanets. Here we present how the combination of data obtained by Gaia and GRAVITY can help enlarge the sample of substellar companions with measured dynamical masses. We show how the Non-Single-Star (NSS) two-body orbit catalogue contained in Gaia DR3 can be used to inform high-angular-resolution follow-up observations with GRAVITY. Applying the method presented in this work to eight Gaia candidate systems, we detect all eight predicted companions, seven of which were previously unknown and five are of a substellar nature. Among the sample is Gaia DR3 2728129004119806464 B, which - detected at an angular separation of (34.01 $\pm$ 0.15) mas from the host - is the closest substellar companion ever imaged. This translates to a semi-major axis of (0.938 $\pm$ 0.023) AU. WT 766 B, detected at a greater angular separation, was confirmed to be on an orbit exhibiting an even smaller semi-major axis of (0.676 $\pm$ 0.008) AU. The GRAVITY data were then used to break the host-companion mass degeneracy inherent to the Gaia NSS orbit solutions as well as to constrain the orbital solutions of the respective target systems. Knowledge of the companion masses enabled us to further characterise them in terms of their ages, effective temperatures, and radii via the application of evolutionary models. The inferred ages exhibit a distinct bias towards values younger than what is to be expected based on the literature. The results serve as an independent validation of the orbital solutions published in the NSS two-body orbit catalogue and show that the combination of astrometric survey missions and high-angular-resolution direct imaging holds great promise for efficiently increasing the sample of directly imaged companions in the future, especially in the light of Gaia's upcoming DR4 and the advent of GRAVITY+.

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arXiv2024

A catalogue of dual-field interferometric binary calibrators

Dual-field interferometric observations with VLTI/GRAVITY sometimes require the use of a "binary calibrator", a binary star whose individual components remain unresolved by the interferometer, with a separation between 400 and 2000 mas for observations with the Units Telescopes (UTs), or 1200 to 3000 mas for the Auxiliary Telescopes (ATs). The separation vector also needs to be predictable to within 10 mas for proper pointing of the instrument. Up until now, no list of properly vetted calibrators was available for dual-field observations with VLTI/GRAVITY on the UTs. Our objective is to compile such a list, and make it available to the community. We identify a list of candidates from the Washington Double Star (WDS) catalogue, all with appropriate separations and brightness, scattered over the Southern sky. We observe them as part of a dedicated calibration programme, and determine whether these objects are true binaries (excluding higher multiplicities resolved interferometrically but unseen by imaging), and extract measurements of the separation vectors. We combine these new measurements with those available in the WDS to determine updated orbital parameters for all our vetted calibrators. We compile a list of 13 vetted binary calibrators for observations with VLTI/GRAVITY on the UTs, and provide orbital estimates and astrometric predictions for each of them. We show that our list guarantees that there are always at least two binary calibrators at airmass < 2 in the sky over the Paranal observatory, at any point in time. Any Principal Investigator wishing to use the dual-field mode of VLTI/GRAVITY with the UTs can now refer to this list to select an appropriate calibrator. We encourage the use of "whereistheplanet" to predict the astrometry of these calibrators, which seamlessly integrates with "p2Gravity" for VLTI/GRAVITY dual-field observing material preparation.

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arXiv2024

The GRAVITY young stellar object survey XIII. Tracing the time-variable asymmetric disk structure in the inner AU of the Herbig star HD98922

Temporal variability in the photometric and spectroscopic properties of protoplanetary disks is common in YSO. However, evidence pointing toward changes in their morphology over short timescales has only been found for a few sources, mainly due to a lack of high cadence observations at mas resolution. We combine GRAVITY multi-epoch observations of HD98922 at mas resolution with PIONIER archival data covering a total time span of 11 years. We interpret the interferometric visibilities and spectral energy distribution with geometrical models and through radiative transfer techniques. We investigated high-spectral-resolution quantities to obtain information on the properties of the HI BrG-line-emitting region. The observations are best fitted by a model of a crescent-like asymmetric dust feature located at 1 au and accounting for 70% of the NIR emission. The feature has an almost constant magnitude and orbits the central star with a possible sub-Keplerian period of 12 months, although a 9 month period is another, albeit less probable, solution. The radiative transfer models show that the emission originates from a small amount of carbon-rich (25%) silicates, or quantum-heated particles located in a low-density region. Among different possible scenarios, we favor hydrodynamical instabilities in the inner disk that can create a large vortex. The high spectral resolution differential phases in the BrG-line show that the hot-gas component is offset from the star and in some cases is located between the star and the crescent feature. The scale of the emission does not favor magnetospheric accretion as a driving mechanism. The scenario of an asymmetric disk wind or a massive accreting substellar or planetary companion is discussed. With this unique observational data set for HD98922, we reveal morphological variability in the innermost 2 au of its disk region.

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arXiv2024

A dynamical measure of the black hole mass in a quasar 11 billion years ago

Tight relationships exist in the local universe between the central stellar properties of galaxies and the mass of their supermassive black hole. These suggest galaxies and black holes co-evolve, with the main regulation mechanism being energetic feedback from accretion onto the black hole during its quasar phase. A crucial question is how the relationship between black holes and galaxies evolves with time; a key epoch to probe this relationship is at the peaks of star formation and black hole growth 8-12 billion years ago (redshifts 1-3). Here we report a dynamical measurement of the mass of the black hole in a luminous quasar at a redshift of 2, with a look back time of 11 billion years, by spatially resolving the broad line region. We detect a 40 micro-arcsecond (0.31 pc) spatial offset between the red and blue photocenters of the H$\alpha$ line that traces the velocity gradient of a rotating broad line region. The flux and differential phase spectra are well reproduced by a thick, moderately inclined disk of gas clouds within the sphere of influence of a central black hole with a mass of 3.2x10$^{8}$ solar masses. Molecular gas data reveal a dynamical mass for the host galaxy of 6x10$^{11}$ solar masses, which indicates an under-massive black hole accreting at a super-Eddington rate. This suggests a host galaxy that grew faster than the supermassive black hole, indicating a delay between galaxy and black hole formation for some systems.

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