arXiv subjects

M. Flock

Publications and source records attributed to M. Flock.

At least 19 recordsLinked to original sources

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

Gas chemistry in the dust depleted inner regions of protoplanetary disks. I. Near-IR spectra and overtones

The molecular composition inside the dust sublimation zones of protoplanetary disks is mostly unknown but important to understanding terrestrial planet formation. A few molecules have been observed from this region, specifically CO, H2O, OH and SiO. The small surface area makes observing this region difficult, hence modeling is required to disentangle the innermost disk from regions further out. We model a protoplanetary disk around a Herbig-type star including the dust depleted inner region (approx. 0.1-0.3 au) and aim to investigate the chemistry of this region and explain existing and future observations. Methods. We post-process the dust and gas distribution of a magnetohydrostatic model with the radiation thermochemical code ProDiMo to study the chemistry and to produce observables. We find that the dust free inner disk is a molecular rich environment, where besides CO we also find H2, H2O and SiO. The gas temperature profile is complex and fluctuates between 700 and 2000 K, which is warm enough to produce CO overtone line emission. Next to the CO overtone lines we also find strong high J-level fundamental CO lines between 4.3 and 4.6 micron. The elemental enrichment of Si due to dust sublimation leads to 2 orders of magnitude more SiO abundance. The SiO gas has average temperatures of approx. 1000 K resulting in strong SiO overtone emission in the spectral range between 4 and 4.3 micron. We predict that the gas density in the dust depleted inner disk is high enough to allow for H2 formation, resulting in an molecular rich environment. For our representative Herbig model, the dust-depleted inner disk is responsible for at least 90% of the line emission for CO and H2O between 1 and 28 micron. Next to CO overtone lines, SiO overtone lines are expected to be an important tracer of a dust free inner disk.

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arXiv2024

Forbidden Emission Line spectro-imaging of the RU Lupi jet and Low Velocity Component

The first images of the jet and low velocity component (LVC) from the strongly accreting classical T Tauri star RU Lupi are presented. Adaptive optics assisted spectro-imaging of forbidden emission lines was used. The main aim of the observations was to test the conclusion from a recent spectro-astrometric study that the narrow component of the LVC is tracing an MHD disk wind, and to estimate the mass loss rate in the wind. The structure and morphology of the component supports a wind origin for the NC. An upper limit to the launch radius and semi-opening angle of the wind in [O I]{\lambda}6300 emission are estimated to be 2 au and 19{\deg} in agreement with MHD wind models for high accretors. The height of the [O I]{\lambda}6300 wind emitting region, a key parameter for the derivation of the mass loss rate, is estimated for the first time at approximately 35 au giving M_out = 2.6 x 10^-11 M_sun/yr. When compared to the derived mass accretion rate of M_acc = 1.6 x 10^-7 M_sun/yr, the efficiency in the wind is too low for the wind to be significantly contributing to angular momentum removal.

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arXiv2023

The GRAVITY young stellar object survey: XI. Imaging the hot gas emission around the Herbig Ae star HD 58647

We aim to investigate the origin of the HI Br$\gamma$ emission in young stars by using GRAVITY to image the innermost region of circumstellar disks, where important physical processes such as accretion and winds occur. With high spectral and angular resolution, we focus on studying the continuum and the HI Br$\gamma$-emitting area of the Herbig star HD58647. Using VLTI-GRAVITY, we conducted observations of HD58647 with both high spectral and high angular resolution. Thanks to the extensive $uv$ coverage, we were able to obtain detailed images of the circumstellar environment at a sub-au scale, specifically capturing the continuum and the Br$\gamma$-emitting region. Through the analysis of velocity-dispersed images and photocentre shifts, we were able to investigate the kinematics of the HI Br$\gamma$-emitting region. The recovered continuum images show extended emission where the disk major axis is oriented along a position angle of 14\degr. The size of the continuum emission at 5-sigma levels is $\sim$ 1.5 times more extended than the sizes reported from geometrical fitting (3.69 mas $\pm$ 0.02 mas). This result supports the existence of dust particles close to the stellar surface, screened from the stellar radiation by an optically thick gaseous disk. Moreover, for the first time with GRAVITY, the hot gas component of HD58647 traced by the Br$\gamma$ ,has been imaged. This allowed us to constrain the size of the Br$\gamma$-emitting region and study the kinematics of the hot gas; we find its velocity field to be roughly consistent with gas that obeys Keplerian motion. The velocity-dispersed images show that the size of the hot gas emission is from a more compact region than the continuum (2.3 mas $\pm$ 0.2 mas). Finally, the line phases show that the emission is not entirely consistent with Keplerian rotation, hinting at a more complex structure in the hot gaseous disk.

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arXiv2023

A Collection of German Science Interests in the Next Generation Very Large Array

The Next Generation Very Large Array (ngVLA) is a planned radio interferometer providing unprecedented sensitivity at wavelengths between 21 cm and 3 mm. Its 263 antenna element array will be spatially distributed across North America to enable both superb low surface brightness recovery and sub-milliarcsecond angular resolution imaging. The project was developed by the international astronomy community under the lead of the National Radio Astronomy Observatory (NRAO), and is anticipated to be built between 2027 and 2037. Two workshops have been held in 2022 and 2023 with the goal to discuss and consolidate the scientific interests in the ngVLA within the German astronomical community. This community paper constitutes a collection of 48 science ideas which the German community aims to pursue with the ngVLA in the 2030s. This is not a complete list and the ideas are not developed at the level of a "Science Book", such that the present document is mainly meant provide a basis for further discussion within the community. As such, additional contributions are welcome, and will be considered for inclusion in future revisions.

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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

Observable scattered light features from inclined and non-inclined planets embedded in protoplanetary discs

Over the last few years instruments such as VLT/SPHERE and Subaru/HiCIAO have been able to take detailed scattered light images of protoplanetary discs. Many of the features observed in these discs are generally suspected to be caused by an embedded planet, and understanding the cause of these features requires detailed theoretical models. In this work we investigate disc-planet interactions using the PLUTO code to run 2D and 3D hydrodynamic simulations of protoplanetary discs with embedded 30 M$_{\oplus}$ and 300 M$_{\oplus}$ planets on both an inclined ($i = 2.86^{\circ}$) and non-inclined orbit, using an $\alpha$-viscosity of $4 \times 10^{-3}$. We produce synthetic scattered-light images of these discs at \emph{H-band} wavelengths using the radiative transfer code RADMC3D. We find that while the surface density evolution in 2D and 3D simulations of inclined and non-inclined planets remain fairly similar, their observational appearance is remarkably different. Most of the features seen in the synthetic \emph{H-band} images are connected to density variations of the disc at around 3.3 scale heights above and below the midplane, which emphasizes the need for 3D simulations. Planets on sustained orbital inclinations disrupt the disc's upper-atmosphere and produce radically different observable features and intensity profiles, including shadowing effects and intensity variation in the order of 10-20 times the surrounding background. The vertical optical depth to the disc midplane for \emph{H-band} wavelengths is $\tau \approx 20$ in the disc gap created by the high-mass planet. We conclude that direct imaging of planets embedded in the disc remains difficult to observe, even for massive planets in the gap.

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arXiv2021

Disk Evolution Study Through Imaging of Nearby Young Stars (DESTINYS): Late infall causing disk misalignment and dynamic structures in SU Aur

Gas-rich circumstellar disks are the cradles of planet formation. As such, their evolution will strongly influence the resulting planet population. In the ESO DESTINYS large program, we study these disks within the first 10 Myr of their development with near-infrared scattered light imaging. Here we present VLT/SPHERE polarimetric observations of the nearby class II system SU Aur in which we resolve the disk down to scales of ~7 au. In addition to the new SPHERE observations, we utilize VLT/NACO, HST/STIS and ALMA archival data. The new SPHERE data show the disk around SU Aur and extended dust structures in unprecedented detail. We resolve several dust tails connected to the Keplerian disk. By comparison with ALMA data, we show that these dust tails represent material falling onto the disk. The disk itself shows an intricate spiral structure and a shadow lane, cast by an inner, misaligned disk component. Our observations suggest that SU Aur is undergoing late infall of material, which can explain the observed disk structures. SU Aur is the clearest observational example of this mechanism at work and demonstrates that late accretion events can still occur in the class II phase, thereby significantly affecting the evolution of circumstellar disks. Constraining the frequency of such events with additional observations will help determine whether this process is responsible for the spin-orbit misalignment in evolved exoplanet systems.

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arXiv2020

ALMA and VLA Observations of EX Lupi in its Quiescent State

Extreme outbursts in young stars may be a common stage of pre-main-sequence stellar evolution. These outbursts, caused by enhanced accretion and accompanied by increased luminosity, can also strongly impact the evolution of the circumstellar environment. We present ALMA and VLA observations of EX Lupi, a prototypical outburst system, at 100 GHz, 45 GHz, and 15 GHz. We use these data, along with archival ALMA 232 GHz data, to fit radiative transfer models to EX Lupi's circumstellar disk in its quiescent state following the extreme outburst in 2008. The best fit models show a compact disk with a characteristic dust radius of 45 au and a total mass of 0.01 M$_{\odot}$. Our modeling suggests grain growth to sizes of at least 3 mm in the disk, possibly spurred by the recent outburst, and an ice line that has migrated inward to $0.2-0.3$ au post-outburst. At 15 GHz, we detected significant emission over the expected thermal disk emission which we attribute primarily to stellar (gyro)synchrotron and free-free disk emission. Altogether, these results highlight what may be a common impact of outbursts on the circumstellar dust.

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arXiv2020

Population synthesis of exocometary gas around A stars

The presence of CO gas around 10-50 Myr old A stars with debris discs has sparked debate on whether the gas is primordial or secondary. Since secondary gas released from planetesimals is poor in H$_2$, it was thought that CO would quickly photodissociate never reaching the high levels observed around the majority of A stars with bright debris discs. Kral et al. 2019 showed that neutral carbon produced by CO photodissociation can effectively shield CO and potentially explain the high CO masses around 9 A stars with bright debris discs. Here we present a new model that simulates the gas viscous evolution, accounting for carbon shielding and how the gas release rate decreases with time as the planetesimal disc loses mass. We find that the present gas mass in a system is highly dependant on its evolutionary path. Since gas is lost on long timescales, it can retain a memory of the initial disc mass. Moreover, we find that gas levels can be out of equilibrium and quickly evolving from a shielded onto an unshielded state. With this model, we build the first population synthesis of gas around A stars, which we use to constrain the disc viscosity. We find a good match with a high viscosity ($\alpha\sim0.1$), indicating that gas is lost on timescales $\sim1-10$ Myr. Moreover, our model also shows that high CO masses are not expected around FGK stars since their planetesimal discs are born with lower masses, explaining why shielded discs are only found around A stars. Finally, we hypothesise that the observed carbon cavities could be due to radiation pressure or accreting planets.

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arXiv2019

A Particle Module for the PLUTO Code: III -- Dust

The implementation of a new particle module describing the physics of dust grains coupled to the gas via drag forces is the subject of this work. The proposed particle-gas hybrid scheme has been designed to work in Cartesian as well as in cylindrical and spherical geometries. The numerical method relies on a Godunov-type second-order scheme for the fluid and an exponential midpoint rule for dust particles which overcomes the stiffness introduced by the linear coupling term. Besides being time-reversible and globally second-order accurate in time, the exponential integrator provides energy errors which are always bounded and it remains stable in the limit of arbitrarily small particle stopping times yielding the correct asymptotic solution. Such properties make this method preferable to the more widely used semi-implicit or fully implicit schemes at a very modest increase in computational cost. Coupling between particles and grid quantities is achieved through particle deposition and field-weighting techniques borrowed from Particle-In-Cell simulation methods. In this respect, we derive new weight factors in curvilinear coordinates that are more accurate than traditional volume- or area-weighting. A comprehensive suite of numerical benchmarks is presented to assess the accuracy and robustness of the algorithm in Cartesian, cylindrical and spherical coordinates. Particular attention is devoted to the streaming instability which is analyzed in both local and global disk models. The module is part of the PLUTO code for astrophysical gas-dynamics and it is mainly intended for the numerical modeling of protoplanetary disks in which solid and gas interact via aerodynamic drag.

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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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arXiv2019

Spatial segregation of dust grains in transition disks. SPHERE observations of 2MASS J16083070-3828268 and RXJ1852.3-3700

Context. The mechanisms governing the opening of cavities in transition disks are not fully understood. Several processes have been proposed but their occurrence rate is still unknown. Aims. We present spatially resolved observations of two transition disks and aim at constraining their vertical and radial structure using multiwavelength observations. Methods. We have obtained near-IR scattered light observations with VLT/SPHERE of the transition disks J1608 and J1852. We complement our datasets with ALMA observations and with unresolved photometric observations covering a wide range of wavelengths. We performed radiative transfer modeling to analyze the morphology of the disks and compare the results with a sample of 20 other transition disks observed with both SPHERE and ALMA. Results. The scattered light image of J1608 reveals a very inclined disk, with two bright lobes and a large cavity. J1852 shows an inner ring extending beyond the coronagraphic radius up to 15au, a gap and a second ring at 42au. Our radiative transfer model of J1608 indicates that the millimeter-sized grains are less extended vertically and radially than the micron-sized grains, indicating advanced settling and radial drift. We find good agreement with the observations of J1852 with a similar model, but due to the low inclination of the system, the model remains partly degenerate. The analysis of 22 transition disks shows that, in general, the cavities observed in scattered light are smaller than the ones detected at millimeter wavelengths. Conclusions. The analysis of a sample of transition disks indicates that the small grains can flow inward of the region where millimeter grains are trapped. While 15 out of the 22 cavities in our sample could be explained by a planet of less than 13 Jupiter masses, the others either require the presence of a more massive companion or of several low-mass planets.

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arXiv2019

Imaging the Distribution of Solids in Planet-forming Disks undergoing Hydrodynamical Instabilities with the Next Generation Very Large Array

We present simulations of the capabilities of the Next Generation Very Large Array to image at high angular resolution substructures in the dust emission of protoplanetary disks. The main goal of this study is to investigate the kinds of substructures that are expected by state-of-the-art 3D simulations of disks and that an instrument like the ngVLA, with its current design, can detect. The disk simulations adopted in this investigation consist of global 3D radiation-hydrodynamics models with embedded particles, the latter representing dust grains. Our work shows that the ngVLA can detect and spatially resolve, down to sub-astronomical unit scales in disks in nearby star forming regions, the dust continuum emission at 3mm from azimuthal asymmetric structures, as well as from weak rings and gaps produced in these models as a consequence of the vertical shear instability (VSI). This hydrodynamical instability has been proposed to generate turbulence in regions of weak coupling between the disk gas and magnetic field, as well as to form vortices which may be preferred locations of planetesimal formation.

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arXiv2018

Discovery of a planetary-mass companion within the gap of the transition disk around PDS 70

Young circumstellar disks are of prime interest to understand the physical and chemical conditions under which planet formation takes place. Only very few detections of planet candidates within these disks exist, and most of them are currently suspected to be disk features. In this context, the transition disk around the young star PDS 70 is of particular interest, due to its large gap identified in previous observations, indicative of ongoing planet formation. We aim to search for the presence of planets and search for disk structures indicative for disk-planet interactions and other evolutionary processes. We analyse new and archival near-infrared (NIR) images of the transition disk PDS 70 obtained with the VLT/SPHERE, VLT/NaCo and Gemini/NICI instruments in polarimetric differential imaging (PDI) and angular differential imaging (ADI) modes. We detect a point source within the gap of the disk at about 195 mas (about 22 au) projected separation. The detection is confirmed at five different epochs, in three filter bands and using different instruments. The astrometry results in an object of bound nature, with high significance. The comparison of the measured magnitudes and colours to evolutionary tracks suggests that the detection is a companion of planetary mass. We confirm the detection of a large gap of about 54 au in size within the disk in our scattered light images, and detect a signal from an inner disk component. We find that its spatial extent is very likely smaller than about 17 au in radius. The images of the outer disk show evidence of a complex azimuthal brightness distribution which may in part be explained by Rayleigh scattering from very small grains. Future observations of this system at different wavelengths and continuing astrometry will allow us to test theoretical predictions regarding planet-disk interactions, planetary atmospheres and evolutionary models.

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arXiv2018

First direct detection of a polarized companion outside of a resolved circumbinary disk around CS Cha

In the present study we aim to investigate the circumstellar environment of the spectroscopic binary T Tauri star CS Cha. From unresolved mid- to far-infrared photometry it is predicted that CS Cha hosts a disk with a large cavity. In addition, SED modeling suggests significant dust settling, pointing towards an evolved disk that may show signs of ongoing or completed planet formation. We observed CS Cha with the high contrast imager VLT/SPHERE in polarimetric differential imaging mode to resolve the circumbinary disk in near infrared scattered light. These observations were followed-up by VLT/NACO L-band observations and complemented by archival VLT/NACO K-band and HST/WFPC2 I-band data. We resolve the compact circumbinary disk around CS Cha for the first time in scattered light. We find a smooth, low inclination disk with an outer radius of $\sim$55 au (at 165 pc). We do not detect the inner cavity but find an upper limit for the cavity size of $\sim$15 au. Furthermore, we find a faint co-moving companion with a projected separation of 210 au from the central binary outside of the circumbinary disk. The companion is detected in polarized light and shows an extreme degree of polarization (13.7$\pm$0.4 \% in J-band). The companion's J- and H-band magnitudes are compatible with masses of a few M$_\mathrm{Jup}$. However, K-, L- and I-band data draw this conclusion into question. We explore with radiative transfer modeling whether an unresolved circum-companion disk can be responsible for the high polarization and complex photometry. We find that the set of observations is best explained by a heavily extincted low mass ($\sim 20 \mathrm{M}_\mathrm{Jup}$) brown dwarf or high mass planet with an unresolved disk and dust envelope.

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arXiv2017

Impact of convection and resistivity on angular momentum transport in dwarf novae

The eruptive cycles of dwarf novae (DN) are thought to be due to a thermal-viscous instability in the accretion disk surrounding the white dwarf (WD). This model has long been known to imply a stress to pressure ratio \alpha ~0.1 in outburst compared to \alpha ~ 0.01 in quiescence. Such an enhancement in $\alpha$ has recently been observed in simulations of turbulent transport driven by the magneto-rotational instability (MRI) when convection is present, without requiring a net magnetic flux. We independently recover this result by carrying out PLUTO MHD simulations of vertically stratified, radiative, shearing boxes with the thermodynamics and opacities appropriate to DN. The results are robust against the choice of vertical boundary conditions. The thermal equilibrium solutions found by the simulations trace the well-known S-curve in the density-temperature plane. We confirm that the high values of \alpha ~ 0.1 occur near the tip of the hot branch of the S-curve, where convection is active. However, we also present thermally-stable simulations at lower temperatures that have standard values of \alpha ~ 0.03 despite the presence of vigorous convection. We find no simple relationship between \alpha and the strength of the convection, as measured by the ratio of convective to radiative flux. The cold branch is only very weakly ionized so, in the second part of this work, we studied the impact of non-ideal MHD effects on transport. We include resistivity in the simulations and find that the MRI-driven transport is quenched (\alpha ~ 0) below the critical density at which the magnetic Reynolds number R_m \leq 10^4. This is problematic as X-ray emission observed in quiescent systems requires ongoing accretion onto the WD. We verify that these X-rays cannot self-sustain MRI-driven turbulence by photo-ionizing the disk and discuss possible solutions to the issue of accretion in quiescence.

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arXiv2017

Magnetic fields in circumstellar disks: The potential of Zeeman observations

Context. Recent high angular resolution polarimetric continuum observations of circumstellar disks provide new insights into their magnetic field. However, direct constraints are limited to the plane of sky component of the magnetic field. Observations of Zeeman split spectral lines are a potential approach to enhance these insights by providing complementary information. Aims. We investigate which constraints for magnetic fields in circumstellar disks can be obtained from Zeeman observations of the $113~\mathrm{GHz}$ CN lines. Furthermore, we analyze the requirements to perform these observations and their dependence on selected quantities. Methods. We simulate the Zeeman splitting with the radiative transfer (RT) code POLARIS (Reissl et al. 2016) extended by our Zeeman splitting RT extension ZRAD (Brauer et al. 2017), which is based on the line RT code Mol3D (Ober et al. 2015). Results. We find that Zeeman observations of the $113~\mathrm{GHz}$ CN lines provide significant insights into the magnetic field of circumstellar disks. However, with the capabilities of recent and upcoming instrument/observatories, even spatially unresolved observations would be challenging. Nevertheless, these observations are feasible for the most massive disks with a strong magnetic field and high abundance of CN/H. The most restrictive quantity is the magnetic field strength, which should be at least in the order of $\sim1~\mathrm{mG}$. In addition, the inclination of the disk should be around $60\deg$ to preserve the ability to derive the line-of-sight (LOS) magnetic field strength and to obtain a sufficiently high circularly polarized flux.

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