arXiv subjects

H. Klahr

Publications and source records attributed to H. Klahr.

At least 19 recordsLinked to original sources

arXiv2022

Hydro-, Magnetohydro-, and Dust-Gas Dynamics of Protoplanetary Disks

The building of planetary systems is controlled by the gas and dust dynamics of protoplanetary disks. While the gas is simultaneously accreted onto the central star and dissipated away by winds, dust grains aggregate and collapse to form planetesimals and eventually planets. This dust and gas dynamics involves instabilities, turbulence and complex non-linear interactions which ultimately control the observational appearance and the secular evolution of these disks. This chapter is dedicated to the most recent developments in our understanding of the dynamics of gaseous and dusty disks, covering hydrodynamic and magnetohydrodynamic turbulence, gas-dust instabilities, dust clumping and disk winds. We show how these physical processes have been tested from observations and highlight standing questions that should be addressed in the future.

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arXiv2021

The CARMENES search for exoplanets around M dwarfs -- Planet occurrence rates from a subsample of 71 stars

The CARMENES exoplanet survey of M dwarfs has obtained more than 18 000 spectra of 329 nearby M dwarfs over the past five years as part of its guaranteed time observations (GTO) program. We determine planet occurrence rates with the 71 stars from the GTO program for which we have more than 50 observations. We use injection-and-retrieval experiments on the radial-velocity (RV) time series to measure detection probabilities. We include 27 planets in 21 planetary systems in our analysis. We find 0.06+0.04-0.03 giant planets (100 M_Earth < M_pl sin i < 1000 M_Earth) per star in periods of up to 1000 d, but due to a selection bias this number could be up to a factor of five lower in the whole 329-star sample. The upper limit for hot Jupiters (orbital period of less than 10 d) is 0.03 planets per star, while the occurrence rate of planets with intermediate masses (10 M_Earth < M_pl sin i < 100 M_Earth) is 0.18+0.07-0.05 planets per star. Less massive planets with 1 M_Earth < M_pl sin i < 10 M_Earth are very abundant, with an estimated rate of 1.32+0.33-0.31 planets per star for periods of up to 100 d. When considering only late M dwarfs with masses M_star < 0.34 M_sol, planets more massive than 10 M_Earth become rare. Instead, low-mass planets with periods shorter than 10 d are significantly overabundant. For orbital periods shorter than 100 d, our results confirm the known stellar mass dependences from the Kepler survey: M dwarfs host fewer giant planets and at least two times more planets with M_pl sin i < 10 M_Earth than G-type stars. In contrast to previous results, planets around our sample of very low-mass stars have a higher occurrence rate in short-period orbits of less than 10 d. Our results demonstrate the need to take into account host star masses in planet formation models.

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arXiv2021

A multiwavelength analysis of the spiral arms in the protoplanetary disk around WaOph 6

[Full abstract in the paper] In recent years, protoplanetary disks with spiral structures have been detected in scattered light, millimeter continuum, and CO gas emission. The mechanisms causing these structures are still under debate. A popular scenario to drive the spiral arms is the one of a planet perturbing the material in the disk. However, if the disk is massive, gravitational instability is usually the favored explanation. Multiwavelength studies could be helpful to distinguish between the two scenarios. So far, only a handful of disks with spiral arms have been observed in both scattered light and millimeter continuum. We aim to perform an in-depth characterization of the protoplanetary disk morphology around WaOph 6 analyzing data obtained at different wavelengths, as well as to investigate the origin of the spiral features in the disk. We present the first near-infrared polarimetric observations of WaOph 6 obtained with SPHERE at the VLT and compare them to archival millimeter continuum ALMA observations. We traced the spiral features in both data sets and estimated the respective pitch angles. We discuss the different scenarios that can give rise to the spiral arms in WaOph 6. We tested the planetary perturber hypothesis by performing hydrodynamical and radiative transfer simulations to compare them with scattered light and millimeter continuum observations.

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arXiv2021

The New Generation Planetary Population Synthesis (NGPPS). V. Predetermination of planet types in global core accretion models

State-of-the-art planet formation models are now capable of accounting for the full spectrum of known planet types. This comes at the cost of increasing complexity of the models, which calls into question whether established links between their initial conditions and the calculated planetary observables are preserved. In this paper, we take a data-driven approach to investigate the relations between clusters of synthetic planets with similar properties and their formation history. We trained a Gaussian Mixture Model on typical exoplanet observables computed by a global model of planet formation to identify clusters of similar planets. We then traced back the formation histories of the planets associated with them. Using cluster affiliation as labels, we trained a Random Forest classifier to predict planet species from properties of the originating protoplanetary disk. Without presupposing any planet types, we identified four distinct classes in our synthetic population. They roughly correspond to the observed populations of (sub-)Neptunes, giant planets, and (super-)Earths, plus an additional unobserved class we denote as "icy cores". These groups emerge already within the first 0.1 Myr of the formation phase and are predicted from disk properties with an overall accuracy of >90%. The most reliable predictors are the initial orbital distance of planetary nuclei and the total planetesimal mass available. Giant planets form only in a particular region of this parameter space that is in agreement with purely analytical predictions. Including N-body interactions between the planets decreases the predictability, especially for sub-Neptunes that frequently undergo giant collisions and turn into super-Earths. The impact of gravitational interactions highlights the need for N-body integrators for realistic predictions of systems of low-mass planets. (abridged)

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arXiv2020

The New Generation Planetary Population Synthesis (NGPPS). III. Warm super-Earths and cold Jupiters: A weak occurrence correlation, but with a strong architecture-composition link

Recent observational findings have suggested a positive correlation between the occurrence rates of inner super-Earths and outer giant planets. These results raise the question of whether this trend can be reproduced and explained by planet formation theory. Here, we investigate the properties of inner super-Earths and outer giant planets that form according to a core accretion scenario. We study the mutual relations between these planet species in synthetic planetary systems and compare them to the observed exoplanet population. We invoked the Generation 3 Bern model of planet formation and evolution to simulate 1000 multi-planet systems. We then confronted these synthetic systems with the observed sample, taking into account the detection bias that distorts the observed demographics. The formation of warm super-Earths and cold Jupiters in the same system is enhanced compared to the individual appearances, although it is weaker than what has been proposed through observations. We attribute the discrepancy to warm and dynamically active giant planets that frequently disrupt the inner systems, particularly in high-metallicity environments. In general, a joint occurrence of the two planet types requires intermediate solid reservoirs in the originating protoplanetary disk. Furthermore, we find differences in the volatile content of planets in different system architectures and predict that high-density super-Earths are more likely to host an outer giant. This correlation can be tested observationally.

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arXiv2020

The CARMENES search for exoplanets around M dwarfs -- LP 714-47b (TOI 442.01): Populating the Neptune desert

We report the discovery of a Neptune-like planet (LP 714-47 b, P = 4.05204 d, m_b = 30.8 +/- 1.5 M_earth , R_b = 4.7 +/- 0.3 R_earth ) located in the 'hot Neptune desert'. Confirmation of the TESS Object of Interest (TOI 442.01) was achieved with radial-velocity follow-up using CARMENES, ESPRESSO, HIRES, iSHELL, and PFS, as well as from photometric data using TESS, Spitzer, and ground-based photometry from MuSCAT2, TRAPPIST- South, MONET-South, the George Mason University telescope, the Las Cumbres Observatory Global Telescope network, the El Sauce telescope, the TUBITAK National Observatory, the University of Louisville Manner Telescope, and WASP-South. We also present high-spatial resolution adaptive optics imaging with the Gemini Near-Infrared Imager. The low uncertainties in the mass and radius determination place LP 714-47 b among physically well-characterised planets, allowing for a meaningful comparison with planet structure models. The host star LP 714-47 is a slowly rotating early M dwarf (T_eff = 3950 +/- 51 K) with a mass of 0.59 +/- 0.02 M_sun and a radius of 0.58 +/- 0.02 R_sun. From long-term photometric monitoring and spectroscopic activity indicators, we determine a stellar rotation period of about 33 d. The stellar activity is also manifested as correlated noise in the radial-velocity data. In the power spectrum of the radial-velocity data, we detect a second signal with a period of 16 days in addition to the four-day signal of the planet. This could be shown to be a harmonic of the stellar rotation period or the signal of a second planet. It may be possible to tell the difference once more TESS data and radial-velocity data are obtained.

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arXiv2020

VLT/SPHERE survey for exoplanets around young, early-type stars including systems with multi-belt architectures

Dusty debris disks around pre- and main-sequence stars are potential signposts for the existence of planetesimals and exoplanets. Giant planet formation is therefore expected to play a key role in the evolution of the disk. This is indirectly confirmed by extant sub-millimeter near-infrared images of young protoplanetary and cool dusty debris disks around main sequence stars usually showing substantial spatial structures. A majority of recent discoveries of imaged giant planets have been obtained around young, early-type stars hosting a circumstellar disk. In this context, we have carried out a direct imaging program designed to maximize our chances of giant planet discovery and targeting twenty-two young, early-type stars. About half of them show indication of multi-belt architectures. Using the IRDIS dual-band imager and the IFS integral field spectrograph of SPHERE to acquire high-constrast coronagraphic differential near-infrared images, we have conducted a systematic search in the close environment of these young, dusty and early-type stars. We confirmed that companions detected around HIP 34276, HIP 101800 and HIP 117452 are stationary background sources and binary companions. The companion candidates around HIP 8832, HIP 16095 and HIP 95619 are determined as background contamination. For stars for which we infer the presence of debris belts, a theoretical minimum mass for planets required to clear the debris gaps can be calculated . The dynamical mass limit is at least $0.1 M_J$ and can exceed $1 M_J$. Direct imaging data is typically sensitive to planets down to $\sim 3.6 M_J$ at 1 $''$, and $1.7 M_J$ in the best case. These two limits tightly constrain the possible planetary systems present around each target. These systems will be probably detectable with the next generation of planet imagers.

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arXiv2019

A giant exoplanet orbiting a very low-mass star challenges planet formation models

Statistical analyses from exoplanet surveys around low-mass stars indicate that super-Earth and Neptune-mass planets are more frequent than gas giants around such stars, in agreement with core accretion theory of planet formation. Using precise radial velocities derived from visual and near-infrared spectra, we report the discovery of a giant planet with a minimum mass of 0.46 Jupiter masses in an eccentric 204-day orbit around the very low-mass star GJ 3512. Dynamical models show that the high eccentricity of the orbit is most likely explained from planet-planet interactions. The reported planetary system challenges current formation theories and puts stringent constraints on the accretion and migration rates of planet formation and evolution models, indicating that disc instability may be more efficient in forming planets than previously thought.

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arXiv2019

The CARMENES search for exoplanets around M dwarfs. Two temperate Earth-mass planet candidates around Teegarden's Star

Context. Teegarden's Star is the brightest and one of the nearest ultra-cool dwarfs in the solar neighbourhood. For its late spectral type (M7.0V), the star shows relatively little activity and is a prime target for near-infrared radial velocity surveys such as CARMENES. Aims. As part of the CARMENES search for exoplanets around M dwarfs, we obtained more than 200 radial-velocity measurements of Teegarden's Star and analysed them for planetary signals. Methods. We find periodic variability in the radial velocities of Teegarden's Star. We also studied photometric measurements to rule out stellar brightness variations mimicking planetary signals. Results. We find evidence for two planet candidates, each with $1.1M_\oplus$ minimum mass, orbiting at periods of 4.91 and 11.4 d, respectively. No evidence for planetary transits could be found in archival and follow-up photometry. Small photometric variability is suggestive of slow rotation and old age. Conclusions. The two planets are among the lowest-mass planets discovered so far, and they are the first Earth-mass planets around an ultra-cool dwarf for which the masses have been determined using radial velocities.

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arXiv2019

Planetary system around the nearby M dwarf GJ 357 including a transiting, hot, Earth-sized planet optimal for atmospheric characterization

We report the detection of a transiting Earth-size planet around GJ 357, a nearby M2.5V star, using data from the Transiting Exoplanet Survey Satellite (TESS). GJ 357 b (TOI-562.01) is a transiting, hot, Earth-sized planet (Teq=525+-11 K) with a radius of Rb=1.217+-0.084 Re and an orbital period of Pb=3.93 d. Precise stellar radial velocities from CARMENES and PFS, as well as archival data from HIRES, UVES, and HARPS also display a 3.93-day periodicity, confirming the planetary nature and leading to a planetary mass of Mb=1.84+-0.31 Me. In addition to the radial velocity signal for GJ 357 b, more periodicities are present in the data indicating the presence of two further planets in the system: GJ 357 c, with a minimum mass of Mc=3.40+-0.46 Me in a 9.12 d orbit, and GJ 357 d, with a minimum mass of Md=6.1+-1.0 Me in a 55.7 d orbit inside the habitable zone. The host is relatively inactive and exhibits a photometric rotation period of Prot=78+-2 d. GJ 357 b is to date the second closest transiting planet to the Sun, making it a prime target for further investigations such as transmission spectroscopy. Therefore, GJ 357 b represents one of the best terrestrial planets suitable for atmospheric characterization with the upcoming JWST and ground-based ELTs.

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arXiv2019

Highly structured disk around the planet host PDS 70 revealed by high-angular resolution observations with ALMA

PDS 70b is the most robustly detected young planet imaged in the gap of a transition disk so far, found at a separation of ~195 mas (~22 au) from its host star and at a position angle of about 155 deg. This system is therefore a unique laboratory to characterize the properties of young planetary systems at the stage of their formation. We aim to trace direct and indirect imprints of PDS 70b on the gas and dust emission of the circumstellar disk in order to study the properties of this ~5 Myr young planetary system. We obtained ALMA band 7 observations of PDS 70 in dust continuum and $^{12}CO$ (3-2) and combined them with archival data resulting in an unprecedented angular resolution of about 70 mas (~8 au). We derive an upper limit on circumplanetary material at the location of PDS 70b of ~0.01 $M_{\oplus}$ and find a highly structured circumstellar disk in both dust and gas. The outer dust ring peaks at ~0.65" (74 au) and reveals a possible second unresolved peak at ~0.53" (60 au). The integrated intensity of CO also shows evidence of a depletion of emission at ~0.2" (23 au) with a width of ~0.1" (11 au). The gas kinematics show evidence of a deviation from Keplerian rotation inside $\lesssim$0.8" (91 au). This implies a pressure gradient that can account for the location of the dust ring well beyond the location of PDS 70b. Further in, we detect an inner disk which appears to be connected to the outer disk by a possible bridge feature in the North-West region in both gas and dust. We compare the observations to hydrodynamical simulations including a planet with different masses covering the estimated mass range previously derived from NIR photometry (~5-9 M$_{Jup}$). We find that even a planet with a mass of 10 M$_{Jup}$ may not be sufficient to explain the extent of the wide gap and an additional low-mass companion may be needed to account for the observed disk morphology.

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arXiv2018

VLT/SPHERE astrometric confirmation and orbital analysis of the brown dwarf companion HR 2562 B

Context. A low-mass brown dwarf has been recently imaged around HR 2562 (HD 50571), a star hosting a debris disk resolved in the far infrared. Interestingly, the companion location is compatible with an orbit coplanar with the disk and interior to the debris belt. This feature makes the system a valuable laboratory to analyze the formation of substellar companions in a circumstellar disk and potential disk-companion dynamical interactions. Aims. We aim to further characterize the orbital motion of HR 2562 B and its interactions with the host star debris disk. Methods. We performed a monitoring of the system over ~10 months in 2016 and 2017 with the VLT/SPHERE exoplanet imager. Results. We confirm that the companion is comoving with the star and detect for the first time an orbital motion at high significance, with a current orbital motion projected in the plane of the sky of 25 mas (~0.85 au) per year. No orbital curvature is seen in the measurements. An orbital fit of the SPHERE and literature astrometry of the companion without priors on the orbital plane clearly indicates that its orbit is (quasi-)coplanar with the disk. To further constrain the other orbital parameters, we used empirical laws for a companion chaotic zone validated by N-body simulations to test the orbital solutions that are compatible with the estimated disk cavity size. Non-zero eccentricities (>0.15) are allowed for orbital periods shorter than 100 yr, while only moderate eccentricities up to ~0.3 for orbital periods longer than 200 yr are compatible with the disk observations. A comparison of synthetic Herschel images to the real data does not allow us to constrain the upper eccentricity of the companion.

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arXiv2018

The GJ 504 system revisited. Combining interferometric, radial velocity, and high contrast imaging data

The G-type star GJ504A is known to host a 3 to 35 MJup companion whose temperature, mass, and projected separation all contribute to make it a test case for the planet formation theories and for atmospheric models of giant planets and light brown dwarfs. We collected data from the CHARA interferometer, SOPHIE spectrograph, and VLT/SPHERE high contrast imager to revisit the properties of the system. We measure a radius of 1.35+/- 0.04Rsun for GJ504A which yields isochronal ages of 21+/-2Myr or 4.0+/-1.8Gyr for the system and line-of-sight stellar rotation axis inclination of $162.4_{-4.3}^{+3.8}$ degrees or $18.6_{-3.8}^{+4.3}$ degrees. We re-detect the companion in the Y2, Y3, J3, H2, and K1 dual band SPHERE images. The complete 1-4 $μ$m SED shape of GJ504b is best reproduced by T8-T9.5 objects with intermediate ages ($\leq1.5$Gyr), and/or unusual dusty atmospheres and/or super-solar metallicities. All six atmospheric models used yield $\mathrm{T_{eff}=550 \pm 50}$K for GJ504b and point toward a low surface gravity (3.5-4.0 dex). The accuracy on the metallicity value is limited by model-to-model systematics. It is not degenerate with the C/O ratio. We derive $\mathrm{log\:L/L_{\odot}=-6.15\pm0.15}$ dex for the companion compatible with masses of $\mathrm{M=1.3^{+0.6}_{-0.3}M_{Jup}}$ and $\mathrm{M=23^{+10}_{-9} M_{Jup}}$ for the young and old age ranges, respectively. The semi-major axis (sma) is above 27.8 au and the eccentricity lower than 0.55. The posterior on GJ~504b's orbital inclination suggests a misalignment with GJ~504A rotation axis. We combine the radial velocity and multi-epoch imaging data to exclude additional objects (90\% prob.) more massive than 2.5 and 30 $\mathrm{M_{Jup}}$ with sma in the range 0.01-80 au for the young and old system ages, respectively. The companion is in the envelope of the population of planets synthetized with our core-accretion model.

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arXiv2018

Discovery of a brown dwarf companion to the star HIP 64892

We report the discovery of a bright, brown dwarf companion to the star HIP 64892, imaged with VLT/SPHERE during the SHINE exoplanet survey. The host is a B9.5V member of the Lower-Centaurus-Crux subgroup of the Scorpius Centaurus OB association. The measured angular separation of the companion ($1.2705\pm0.0023$") corresponds to a projected distance of $159\pm12$ AU. We observed the target with the dual-band imaging and long-slit spectroscopy modes of the IRDIS imager to obtain its SED and astrometry. In addition, we reprocessed archival NACO L-band data, from which we also recover the companion. Its SED is consistent with a young (<30 Myr), low surface gravity object with a spectral type of M9$_γ\pm1$. From comparison with the BT-Settl atmospheric models we estimate an effective temperature of $T_{\textrm{eff}}=2600 \pm 100$ K, and comparison of the companion photometry to the COND evolutionary models yields a mass of $\sim29-37$ M$_{\text{J}}$ at the estimated age of $16^{+15}_{-7}$ Myr for the system. HIP 64892 is a rare example of an extreme-mass ratio system ($q\sim0.01$) and will be useful for testing models relating to the formation and evolution of such low-mass objects.

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arXiv2018

First scattered light detection of a nearly edge-on transition disk around the T Tauri star RY Lup

Transition disks are considered sites of ongoing planet formation, and their dust and gas distributions could be signposts of embedded planets. The transition disk around the T Tauri star RY Lup has an inner dust cavity and displays a strong silicate emission feature. Using high-resolution imaging we study the disk geometry, including non-axisymmetric features, and its surface dust grain, to gain a better understanding of the disk evolutionary process. Moreover, we search for companion candidates, possibly connected to the disk. We obtained high-contrast and high angular resolution data in the near-infrared with the VLT/SPHERE extreme adaptive optics instrument whose goal is to study the planet formation by detecting and characterizing these planets and their formation environments through direct imaging. We performed polarimetric imaging of the RY~Lup disk with IRDIS (at 1.6 microns), and obtained intensity images with the IRDIS dual-band imaging camera simultaneously with the IFS spectro-imager (0.9-1.3 microns). We resolved for the first time the scattered light from the nearly edge-on circumstellar disk around RY~Lup, at projected separations in the 100 \,au range. The shape of the disk and its sharp features are clearly detectable at wavelengths ranging from 0.9 to 1.6 microns. We show that the observed morphology can be interpreted as spiral arms in the disk. This interpretation is supported by in-depth numerical simulations. We also demonstrate that these features can be produced by one planet interacting with the disk. We also detect several point sources which are classified as probable background objects.

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arXiv2017

Efficiency of thermal relaxation by radiative processes in protoplanetary discs: constraints on hydrodynamic turbulence

Hydrodynamic, non-magnetic instabilities can provide turbulent stress in the regions of protoplanetary discs, where the MRI can not develop. The induced motions influence the grain growth, from which formation of planetesimals begins. Thermal relaxation of the gas constrains origins of the identified hydrodynamic sources of turbulence in discs. We estimate the radiative relaxation timescale of temperature perturbations and study the dependence of this timescale on the perturbation wavelength, the location within the disc, the disc mass, and the dust-to-gas mass ratio. We then apply thermal relaxation criteria to localise modes of the convective overstability, the vertical shear instability, and the zombie vortex instability. Our calculations employed the latest tabulated dust and gas mean opacities and we account for the collisional coupling to the emitting species. The relaxation criterion defines the bulk of a typical T Tauri disc as unstable to the development of linear hydrodynamic instabilities. The midplane is unstable to the convective overstability from at most $2\mbox{ au}$ and up to $40\mbox{ au}$, as well as beyond $140\mbox{ au}$. The vertical shear instability can develop between $15\mbox{ au}$ and $180\mbox{ au}$. The successive generation of (zombie) vortices from a seeded noise can work within the inner $0{.}8\mbox{ au}$. Dynamic disc modelling with the evolution of dust and gas opacities is required to clearly localise the hydrodynamic turbulence, and especially its non-linear phase.

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arXiv2016

Gaps, rings, and non-axisymmetric structures in protoplanetary disks - Emission from large grains

Dust grains with sizes around (sub)mm are expected to couple only weakly to the gas motion in regions beyond 10 au of circumstellar disks. In this work, we investigate the influence of the spatial distribution of such grains on the (sub)mm appearance of magnetized protoplanetary disks. We perform non-ideal global 3D magneto-hydrodynamic stratified disk simulations including particles of different sizes (50 micron to 1 cm), using a Lagrangian particle solver. We calculate the spatial dust temperature distribution, including the dynamically coupled submicron-sized dust grains, and derive ideal continuum re-emission maps of the disk through radiative transfer simulations. Finally, we investigate the feasibility to observe specific structures in the thermal re-emission maps with the Atacama Large Millimeter/submillimeter Array (ALMA). The pressure bump close to the outer edge of the dead-zone leads to particle trapping in ring structures. More specifically, vortices in the disk concentrate the dust and create an inhomogeneous distribution of the solid material in the azimuthal direction. The large-scale disk perturbations are preserved in the (sub)mm re-emission maps. The observable structures are very similar to those expected to result from planet-disk interaction. The larger dust particles increase the brightness contrast between gap and ring structures. We find that rings, gaps and the dust accumulation in the vortex could be traced with ALMA down to a scale of a few astronomical units in circumstellar disks located in nearby star-forming regions. Finally, we present a brief comparison of these structures with those recently found with ALMA in the young circumstellar disks of HL Tau and Oph IRS 48.

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arXiv2015

Vortex Formation and Evolution in Planet Harboring Disks under Thermal Relaxation

We study the evolution of planet-induced vortices in radially stratified disks, with initial conditions allowing for radial buoyancy. For this purpose we run global two dimensional hydrodynamical simulations, using the PLUTO code. Planet-induced vortices are a product of the Rossby wave instability (RWI) triggered in the edges of a planetary gap. In this work we assess the influence of radial buoyancy for the development of the vortices. We found that radial buoyancy leads to smoother planetary gaps, which generates weaker vortices. This effect is less pronounced for locally isothermal and quasi-isothermal (very small cooling rate) disks. We observed the formation of two generations of vortices. The first generation of vortices is formed in the outer wall of the planetary gap. The merged primary vortex induces accretion, depleting the mass on its orbit. This process creates a surface density enhancement beyond the primary vortex position. The second generation of vortices arise in this surface density enhancement, indicating that the bump in this region is sufficient to trigger the RWI. The merged secondary vortex is a promising explanation for the location of the vortex in the Oph IRS48 system. Finally, we observed a nonmonotonic behavior for the vortex lifetimes as a function of the thermal relaxation timescale, agreeing with previous studies. The birth times of the secondary vortices also display a nonmonotonic behavior, which is correlated with the growth time of the primary vortex and its induced accretion.

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