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F. Ménard

Publications and source records attributed to F. Ménard.

At least 91 records · Page 5Linked to original sources

Non-azimuthal linear polarization in protoplanetary disks

Several studies discussing imaging polarimetry observations of protoplanetary disks use the so-called radial Stokes parameters Q_phi and U_phi to discuss the results. This approach has the advantage of providing a direct measure of the noise in the polarized images under the assumption that the polarization is azimuthal only, i.e., perpendicular to the direction towards the illuminating source. However, a detailed study of the validity of this assumption is currently missing. We aim to test whether departures from azimuthal polarization can naturally be produced by scattering processes in optically thick protoplanetary disks at near infrared wavelengths. We use the radiative transfer code MCFOST to create a generic model of a transition disk using different grain size distributions and dust masses. From these models we generate synthetic polarized images at 2.2\mum. We find that even for moderate inclinations (e.g., i = 40degr), multiple scattering alone can produce significant (up to ~4.5% of the Q_phi image) non-azimuthal polarization reflected in the U_phi images. We also find that different grain populations can naturally produce radial polarization (negative values in the Q_phi images). Our results suggest that caution is recommended when interpreting polarized images by only analyzing the Q_phi and U_phi images. We find that there can be astrophysical signal in the U_phi images and negative values in the Q_phi images, which indicate departures from azimuthal polarization. If significant signal is detected in the U_phi images, we recommend to check the standard Q and U images to look for departures from azimuthal polarization. On the positive side, signal in the U_phi images once all instrumental and data-reduction artifacts have been corrected for means that there is more information to be extracted regarding the dust population and particle density.

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The inner environment of Z~CMa: High-Contrast Imaging Polarimetry with NaCo

Context. Z\,CMa is a binary composed of an embedded Herbig Be and an FU Ori class star separated by $\sim100$ au. Observational evidence indicate a complex environment in which each star has a circumstellar disk and drives a jet, and the whole system is embedded in a large dusty envelope. Aims. We aim to probe the circumbinary environment of Z\,CMa in the inner 400 au in scattered light. Methods. We use high contrast imaging polarimetry with VLT/NaCo at $H$ and $K_s$ bands. Results. The central binary is resolved in both bands. The polarized images show three bright and complex structures: a common dust envelope, a sharp extended feature previously reported in direct light, and an intriguing bright clump located $0\farcs3$ south of the binary, which appears spatially connected to the sharp extended feature. Conclusions.We detect orbital motion when compared to previous observations, and report a new outburst driven by the Herbig star. Our observations reveal the complex inner environment of Z\,CMa with unprecedented detail and contrast.

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Gas inside the 97 au cavity around the transition disk Sz\,91

We present ALMA (Cycle 0) band-6 and band-3 observations of the transition disk Sz\,91. The disk inclination and position angle are determined to be $i=49.5\degr\pm3.5\degr$ and $\mathrm{PA}=18.2\degr\pm3.5\degr$ and the dusty and gaseous disk are detected up to $\sim220$ au and $\sim400$ au from the star, respectively. Most importantly, our continuum observations indicate that the cavity size in the mm-sized dust distribution must be $\sim97$ au in radius, the largest cavity observed around a T Tauri star. Our data clearly confirms the presence of \co(2-1) well inside the dust cavity. Based on these observational constrains we developed a disk model that simultaneously accounts for the \co and continuum observations (i.e., gaseous and dusty disk). According to our model, most of the millimeter emission comes from a ring located between 97 and 140 au. We also find that the dust cavity is divided into an innermost region largely depleted of dust particles ranging from the dust sublimation radius up to 85 au, and a second, moderately dust-depleted region, extending from 85 to 97 au. The extremely large size of the dust cavity, the presence of gas and small dust particles within the cavity and the accretion rate of Sz\,91 are consistent with the formation of multiple (giant) planets.

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CO gas inside the protoplanetary disk cavity in HD 142527: disk structure from ALMA

Inner cavities and annular gaps in circumstellar disks are possible signposts of giant planet formation. The young star HD 142527 hosts a massive protoplanetary disk with a large cavity that extends up to 140 au from the central star, as seen in continuum images at infrared and millimeter wavelengths. Estimates of the survival of gas inside disk cavities are needed to discriminate between clearing scenarios. We present a spatially and spectrally resolved carbon monoxide isotopologue observations of the gas-rich disk HD 142527, in the J=2-1 line of 12CO, 13CO and C18O, obtained with the Atacama Large Millimeter Array (ALMA). We detect emission coming from inside the dust-depleted cavity in all three isotopologues. Based on our analysis of the gas in the dust cavity, the 12CO emission is optically thick, while 13CO and C18O emission are both optically thin. The total mass of residual gas inside the cavity is about 1.5-2 Jupiter masses. We model the gas with an axisymmetric disk model. Our best fit model shows that the cavity radius is much smaller in CO than it is in millimeter continuum and scattered light observations, with a gas cavity that does not extend beyond 105 au (at 3-sigma). The gap wall at its outer edge is diffuse and smooth in the gas distribution, while in dust continuum it is manifestly sharper. The inclination angle, as estimated from the high velocity channel maps, is 28+/-0.5 degrees, higher than in previous estimates, assuming a fix central star mass of 2.2 Solar masses.

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Signatures of warm carbon monoxide in protoplanetary discs observed with Herschel SPIRE

Molecular gas constitutes the dominant mass component of protoplanetary discs. To date, these sources have not been studied comprehensively at the longest far-infrared and shortest submillimetre wavelengths. This paper presents Herschel SPIRE FTS spectroscopic observations toward 18 protoplanetary discs, covering the entire 450-1540 GHz (666-195 $μ$m) range at R~400-1300. The spectra reveal clear detections of the dust continuum and, in six targets, a significant amount of spectral line emission primarily attributable to $^{12}$CO rotational lines. Other targets exhibit little to no detectable spectral lines. Low signal-to-noise detections also include signatures from $^{13}$CO, [CI] and HCN. For completeness, we present upper limits of non-detected lines in all targets, including low-energy transitions of H2O and CH$^+$ molecules. The ten $^{12}$CO lines that fall within the SPIRE FTS bands trace energy levels of ~50-500 K. Combined with lower and higher energy lines from the literature, we compare the CO rotational line energy distribution with detailed physical-chemical models, for sources where these are available and published. Our 13CO line detections in the disc around Herbig Be star HD 100546 exceed, by factors of ~10-30, the values predicted by a model that matches a wealth of other observational constraints, including the SPIRE $^{12}$CO ladder. To explain the observed $^{12}$CO/$^{13}$CO ratio, it may be necessary to consider the combined effects of optical depth and isotope selective (photo)chemical processes. Considering the full sample of 18 objects, we find that the strongest line emission is observed in discs around Herbig Ae/Be stars, although not all show line emission. In addition, two of the six T Tauri objects exhibit detectable $^{12}$CO lines in the SPIRE range.

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The Taurus Boundary of Stellar/Substellar (TBOSS) Survey I: far-IR disk emission measured with Herschel

With Herschel/PACS 134 low mass members of the Taurus star-forming region spanning the M4-L0 spectral type range and covering the transition from low mass stars to brown dwarfs were observed. Combining the new Herschel results with other programs, a total of 150 of the 154 M4-L0 Taurus members members have observations with Herschel. Among the 150 targets, 70um flux densities were measured for 7 of the 7 ClassI objects, 48 of the 67 ClassII members, and 3 of the 76 ClassIII targets. For the detected ClassII objects, the median 70um flux density level declines with spectral type, however, the distribution of excess relative to central object flux density does not change across the stellar/substellar boundary in the M4-L0 range. Connecting the 70um TBOSS values with the results from K0-M3 ClassII members results in the first comprehensive census of far-IR emission across the full mass spectrum of the stellar and substellar population of a star-forming region, and the median flux density declines with spectral type in a trend analogous to the flux density decline expected for the central objects. SEDs were constructed for all TBOSS targets covering the optical to far-IR range and extending to the submm/mm for a subset of sources. Based on an initial exploration of the impact of different physical parameters; inclination, scale height and flaring have the largest influence on the PACS flux densities. From the 24um to 70um spectral index of the SEDs, 5 new candidate transition disks were identified. The steep 24um to 70um slope for a subset of 8 TBOSS targets may be an indication of truncated disks in these systems.Two examples of mixed pair systems that include secondaries with disks were measured. Finally, comparing the TBOSS results with a Herschel study of Ophiuchus brown dwarfs reveals a lower fraction of disks around the Taurus substellar population.

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A CO survey in planet-forming disks: characterizing the gas content in the epoch of planet formation

We carried out a 12CO(3-2) survey of 52 southern stars with a wide range of IR excesses (LIR/L*) using the single dish telescopes APEX and ASTE. The main aims were (1) to characterize the evolution of molecular gas in circumstellar disks using LIR/L* values as a proxy of disk dust evolution, and (2) to identify new gas-rich disk systems suitable for detailed study with ALMA. About 60% of the sample (31 systems) have LIR/L* > 0.01 typical of T-Tauri or Herbig AeBe stars, and the rest (21 systems) have LIR/L* < 0.01 typical of debris disks. We detect CO(3-2) emission from 20 systems, and 18 (90%) of these have LIR/L* > 0.01. However, the spectra of only four of the newly detected systems appear free of contamination from background or foreground emission from molecular clouds. These include the early-type stars HD 104237 (A4/5V, 116 pc) and HD 98922 (A2 III, 507 pc, as determined in this work), where our observations reveal the presence of CO-rich circumstellar disks for the first time. Of the other detected sources, many could harbor gaseous circumstellar disks, but our data are inconclusive. For these two newly discovered gas-rich disks, we present radiative transfer models that simultaneously reproduce their spectral energy distributions and the 12CO(3-2) line profiles. For both of these systems, the data are fit well by geometrically flat disks, placing them in the small class of non-flaring disks with significant molecular gas reservoirs.

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Constraining the structure of the transition disk HD 135344B (SAO 206462) by simultaneous modeling of multiwavelength gas and dust observations

HD 135344B is an accreting (pre-) transition disk that displays the emission of warm CO extending tens of AU inside its 30 AU dust cavity. We used the dust radiative transfer code MCFOST and the thermochemical code ProDiMo to derive the disk structure from the simultaneous modeling of the spectral energy distribution (SED), VLT/CRIRES CO P(10) 4.75 micron, Herschel/PACS [O I] 63 micron, Spitzer-IRS, and JCMT 12CO J=3-2 spectra, VLTI/PIONIER H-band visibilities, and constraints from (sub-)mm continuum interferometry and near-IR imaging. We found a disk model able to describe the current observations simultaneously. This disk has the following structure. (1) To reproduce the SED, the near-IR interferometry data, and the CO ro-vibrational emission, refractory grains (we suggest carbon) are present inside the silicate sublimation radius (0.08 100 to account for the 870 micron continuum upper limit and the CO P(10) line flux. (5) The gas/dust ratio at 30<R<200 AU is < 10 to simultaneously describe the [O I] 63 micron line flux and the CO P(10) line profile. (6) In the outer disk, most of the mass should be located in the midplane, and a significant fraction of the dust is in large grains. Conclusions: Simultaneous modeling of the gas and dust is required to break the model degeneracies and constrain the disk structure. An increasing gas surface density with radius in the inner dust cavity echoes the effect of a migrating Jovian planet. The low gas mass (a few MJupiter) in the HD 135344B's disk suggests that it is an evolved disk that has already lost a large portion of its mass.

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On the structure of the transition disk around TW Hya

For over a decade, the structure of the inner cavity in the transition disk of TW Hydrae has been a subject of debate. Modeling the disk with data obtained at different wavelengths has led to a variety of proposed disk structures. Rather than being inconsistent, the individual models might point to the different faces of physical processes going on in disks, such as dust growth and planet formation. Our aim is to investigate the structure of the transition disk again and to find to what extent we can reconcile apparent model differences. A large set of high-angular-resolution data was collected from near-infrared to centimeter wavelengths. We investigated the existing disk models and established a new self-consistent radiative-transfer model. A genetic fitting algorithm was used to automatize the parameter fitting. Simple disk models with a vertical inner rim and a radially homogeneous dust composition from small to large grains cannot reproduce the combined data set. Two modifications are applied to this simple disk model: (1) the inner rim is smoothed by exponentially decreasing the surface density in the inner ~3 AU, and (2) the largest grains (>100 um) are concentrated towards the inner disk region. Both properties can be linked to fundamental processes that determine the evolution of protoplanetary disks: the shaping by a possible companion and the different regimes of dust-grain growth, respectively. The full interferometric data set from near-infrared to centimeter wavelengths requires a revision of existing models for the TW Hya disk. We present a new model that incorporates the characteristic structures of previous models but deviates in two key aspects: it does not have a sharp edge at 4 AU, and the surface density of large grains differs from that of smaller grains. This is the first successful radiative-transfer-based model for a full set of interferometric data.

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Evolution from protoplanetary to debris discs: The transition disc around HD 166191

HD 166191 has been identified by several studies as hosting a rare and extremely bright warm debris disc with an additional outer cool disc component. However, an alternative interpretation is that the star hosts a disc that is currently in transition between a full gas disc and a largely gas-free debris disc. With the help of new optical to mid-IR spectra and Herschel imaging, we argue that the latter interpretation is supported in several ways: i) we show that HD 166191 is co-moving with the ~4 Myr-old Herbig Ae star HD 163296, suggesting that the two have the same age, ii) the disc spectrum of HD 166191 is well matched by a standard radiative transfer model of a gaseous protoplanetary disc with an inner hole, and iii) the HD 166191 mid-IR silicate feature is more consistent with similarly primordial objects. We note some potential issues with the debris disc interpretation that should be considered for such extreme objects, whose lifetime at the current brightness is mush shorter than the stellar age, or in the case of the outer component requires a mass comparable to the solid component of the Solar nebula. These aspects individually and collectively argue that HD 166191 is a 4-5 Myr old star that hosts a gaseous transition disc. Though it does not argue in favour of either scenario, we find strong evidence for 3-5 um disc variability. We place HD 166191 in context with discs at different evolutionary stages, showing that it is a potentially important object for understanding the protoplanetary to debris disc transition.

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Unveiling the gas and dust disk structure in HD 163296 using ALMA observations

Aims: The aim of this work is to study the structure of the protoplanetary disk surrounding the Herbig Ae star HD 163296. Methods: We have used high-resolution and high-sensitivity ALMA observations of the CO(3-2) emission line and the continuum at 850 microns, as well as the 3- dimensional radiative transfer code MCFOST to model the data presented in this work. Results: The CO(3-2) emission unveils for the first time at sub-millimeter frequencies the vertical structure details of a gaseous disk in Keplerian rotation, showing the back- and the front-side of a flared disk. Continuum emission at 850 microns reveals a compact dust disk with a 240 AU outer radius and a surface brightness profile that shows a very steep decline at radius larger than 125 AU. The gaseous disk is more than two times larger than the dust disk, with a similar critical radius but with a shallower radial profile. Radiative transfer models of the continuum data confirms the need for a sharp outer edge to the dust disk. The models for the CO(3-2) channel map require the disk to be slightly more geometrically thick than previous models suggested, and that the temperature at which CO gas becomes depleted (frozen-out) from the outer regions of the disk midplane is T < 20 K, in agreement with previous studies.

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Nature of the gas and dust around 51 Oph

Circumstellar disc evolution is paramount for the understanding of planet formation. The GASPS program aims at determining the circumstellar gas and solid mass around ~250 pre-main-sequence Herbig Ae and TTauri stars. We aim to understand the origin and nature of the circumstellar matter orbiting 51 Oph, a young (<1 Myr) luminous B9.5 star. We obtained continuum and line observations with the PACS instrument on board the Herschel Space Observatory and continuum data at 1.2 mm with the IRAM 30m telescope. The SED and line fluxes were modelled using the physico-chemo radiative transfer code ProDiMo. We detected a strong emission by OI at 63 microns using the Herschel Space Observatory. The [OI] emission at 145 microns, the [CII] emission at 158 microns, the high-J CO emissions, and the warm water emissions were not detected. Continuum emission was detected at 1.2 mm. The continuum from the near- to the far-infrared and the [OI] emission are well explained by the emission from a compact hydrostatic disc model with a gas mass of 5E-6 MSun, 100 times that of the solid mass. However, this model fails to match the continuum millimeter flux, which hints at a cold outer disc with a mass in solids of 1E-6 MSun or free-free emission from a photoevaporative disc wind. This outer disc can either be devoid of gas and/or is to cold to emit in the [OI] line. A very flat extended disc model (Rout=400 AU) with a fixed vertical structure and dust settling matches all photometric points and most of the [O I] flux. The observations can be explained by an extended flat disc where dust grains have settled. However, a flat gas disc cannot be reproduced by hydrostatic disc models. The low mass of the 51 Oph inner disc in gas and dust may be explained either by the fast dissipation of an initial massive disc or by a very small initial disc mass.

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Planet gaps in the dust layer of 3D protoplanetary disks. II. Observability with ALMA

[Abridged] Aims: We provide predictions for ALMA observations of planet gaps that account for the specific spatial distribution of dust that results from consistent gas+dust dynamics. Methods: In a previous work, we ran full 3D, two-fluid Smoothed Particle Hydrodynamics (SPH) simulations of a planet embedded in a gas+dust T Tauri disk for different planet masses and grain sizes. In this work, the resulting dust distributions are passed to the Monte Carlo radiative transfer code MCFOST to construct synthetic images in the ALMA wavebands. We then use the ALMA simulator to produce images that include thermal and phase noise for a range of angular resolutions, wavelengths, and integration times, as well as for different inclinations, declinations and distances. We also produce images which assume that gas and dust are well mixed with a gas-to-dust ratio of 100 to compare with previous ALMA predictions, all made under this hypothesis. Results: Our findings clearly demonstrate the importance of correctly incorporating the dust dynamics. We show that the gap carved by a 1 M_J planet orbiting at 40 AU is visible with a much higher contrast than the well-mixed assumption would predict. In the case of a 5 M_J planet, we clearly see a deficit in dust emission in the inner disk, and point out the risk of interpreting the resulting image as that of a transition disk with an inner hole if observed in unfavorable conditions. Planet signatures are fainter in more distant disks but declination or inclination to the line-of-sight have little effect on ALMA's ability to resolve the gaps. Conclusions: ALMA has the potential to see signposts of planets in disks of nearby star-forming regions. We present optimized observing parameters to detect them in the case of 1 and 5 M_J planets on 40 AU orbits.

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The dynamically disrupted gap in HD 142527

The vestiges of planet formation have been observed in debris disks harboring young and massive gaseous giants. The process of giant planet formation is terminated by the dissipation of gas in the protoplanetary disk. The gas-rich disk around HD142527 features a small inner disk, a large gap from \sim10 to \sim140AU, and a massive outer disk extending out to \sim300AU. The gap could have been carved-out by a giant planet. We have imaged the outer regions of this gap using the adaptive-optics camera NICI on Gemini South. Our images reveal that the disk is dynamically perturbed. The outer boundary of the roughly elliptical gap appears to be composed of several segments of spiral arms. The stellar position is offset by 0.17+-0.02" from the centroid of the cavity, consistent with earlier imaging at coarser resolutions. These transient morphological features are expected in the context of disk evolution in the presence of a perturbing body located inside the cavity. We perform hydro-dynamical simulations of the dynamical clearing of a gap in a disk. A 10Mjup body in a circular orbit at r = 90AU, perturbs the whole disks, even after thousands of orbits. By then the model disk has an eccentric and irregular cavity, flanked by tightly wound spiral arms, but it is still evolving far from steady state. A particular transient configuration that is a qualitative match to HD142527 is seen at 1.7Myr.

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Herschel PACS Observations and Modeling of Debris Disks in the Tucana-Horologium Association

We present Herschel PACS photometry of seventeen B- to M-type stars in the 30 Myr-old Tucana-Horologium Association. This work is part of the Herschel Open Time Key Programme "Gas in Protoplanetary Systems" (GASPS). Six of the seventeen targets were found to have infrared excesses significantly greater than the expected stellar IR fluxes, including a previously unknown disk around HD30051. These six debris disks were fitted with single-temperature blackbody models to estimate the temperatures and abundances of the dust in the systems. For the five stars that show excess emission in the Herschel PACS photometry and also have Spitzer IRS spectra, we fit the data with models of optically thin debris disks with realistic grain properties in order to better estimate the disk parameters. The model is determined by a set of six parameters: surface density index, grain size distribution index, minimum and maximum grain sizes, and the inner and outer radii of the disk. The best fitting parameters give us constraints on the geometry of the dust in these systems, as well as lower limits to the total dust masses. The HD105 disk was further constrained by fitting marginally resolved PACS 70 micron imaging.

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Detection of warm water vapour in Taurus protoplanetary discs by Herschel

Line spectra of 68 Taurus T Tauri stars were obtained with the Herschel-PACS (Photodetector Array Camera & Spectrometer) instrument as part of the GASPS (Gas Evolution in Protoplanetary Systems) survey of protoplanetary discs. A careful examination of the line scans centred on the [OI] 63.18 microns fine-structure line unveiled a line at 63.32 micron in some of these spectra. We identify this line with a transition of ortho-water. It is detected confidently (i.e., >3 sigma) in eight sources, i.e., 24% of the sub-sample with gas-rich discs. Several statistical tests were used to search for correlations with other disc and stellar parameters such as line fluxes of [OI] 6300 Armstrong and 63.18 microns; X-ray luminosity and continuum levels at 63 microns and 850 microns. Correlations are found between the water line fluxes and the [OI] 63.18 microns line luminosity, the dust continuum, and possibly with the stellar X-ray luminosity. This is the first time that this line of warm water vapour has been detected in protoplanetary discs. We discuss its origins, in particular whether it comes from the inner disc and/or disc surface or from shocks in outflows and jets. Our analysis favours a disc origin, with the observed water vapour line produced within 2-3AU from the central stars, where the gas temperature is of the order of 500-600 K.

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An icy Kuiper-Belt around the young solar-type star HD 181327

HD 181327 is a young F5/F6V star belonging to the Beta Pictoris moving group (12 Myr). It harbors an optically thin belt of circumstellar material at 90 AU. We aim to study the dust properties in the belt in details, and to constrain the gas-to-dust ratio. We obtained far-IR observations with the Herschel/PACS instrument, and 3.2 mm observations with the ATCA array. The geometry of the belt is constrained with newly reduced HST/NICMOS images that break the degeneracy between the disk geometry and the dust properties. We use the radiative transfer code GRaTer to compute a large grid of models, and we identify the grain models that best reproduce the Spectral Energy Distribution through a Bayesian analysis. We attempt to detect the [OI] and [CII] lines with PACS spectroscopy, providing observables to our photochemical code ProDiMo. The HST observations confirm that the dust is confined in a narrow belt. The continuum is detected in the far-IR with PACS and the disk is resolved with both PACS and ATCA. A medium integration of the gas spectral lines only provides upper limits on the line fluxes. We show that the HD 181327 dust disk consists of micron-sized grains of porous amorphous silicates and carbonaceous material surrounded by an important layer of ice, for a total dust mass of 0.05 M\oplus (up to 1 mm). We discuss evidences that the grains are fluffy aggregates. The upper limits on the gas lines do not provide unambiguous constraints: only if the PAH abundance is high, the gas mass must be lower than 17 M\oplus. Despite the weak constraints on the gas disk, the age of HD 181327 and the properties of the dust disk suggest that it has passed the stage of gaseous planets formation. The dust reveals a population of icy planetesimals, similar to the primitive Kuiper Belt, that may be a source for the future delivery of water and volatiles onto forming terrestrial planets.

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Gas modelling in the disc of HD 163296

We present detailed model fits to observations of the disc around the Herbig Ae star HD 163296. This well-studied object has an age of ~ 4 Myr, with evidence of a circumstellar disc extending out to ~ 540AU. We use the radiation thermo-chemical disc code ProDiMo to model the gas and dust in the circumstellar disc of HD 163296, and attempt to determine the disc properties by fitting to observational line and continuum data. These include new Herschel/PACS observations obtained as part of the open-time key program GASPS (Gas in Protoplanetary Systems), consisting of a detection of the [OI]63mic line and upper limits for several other far infrared lines. We complement this with continuum data and ground-based observations of the 12CO 3-2, 2-1 and 13CO J=1-0 line transitions, as well as the H2 S(1) transition. We explore the effects of stellar ultraviolet variability and dust settling on the line emission, and on the derived disc properties. Our fitting efforts lead to derived gas/dust ratios in the range 9-100, depending on the assumptions made. We note that the line fluxes are sensitive in general to the degree of dust settling in the disc, with an increase in line flux for settled models. This is most pronounced in lines which are formed in the warm gas in the inner disc, but the low excitation molecular lines are also affected. This has serious implications for attempts to derive the disc gas mass from line observations. We derive fractional PAH abundances between 0.007 and 0.04 relative to ISM levels. Using a stellar and UV excess input spectrum based on a detailed analysis of observations, we find that the all observations are consistent with the previously assumed disc geometry.

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