arXiv subjects

Luke T. Maud

Publications and source records attributed to Luke T. Maud.

17 recordsLinked to original sources

arXiv2024

Multiple chemical tracers finally unveil the intricate NGC\,1333 IRAS\,4A outflow system. FAUST XVI

The exploration of outflows in protobinary systems presents a challenging yet crucial endeavour, offering valuable insights into the dynamic interplay between protostars and their evolution. In this study, we examine the morphology and dynamics of jets and outflows within the IRAS\,4A protobinary system. This analysis is based on ALMA observations of SiO(5--4), H$_2$CO(3$_{0,3}$--2$_{0,3}$), and HDCO(4$_{1,4}$--3$_{1,3}$) with a spatial resolution of $\sim$150\,au. Leveraging an astrochemical approach involving the use of diverse tracers beyond traditional ones has enabled the identification of novel features and a comprehensive understanding of the broader outflow dynamics. Our analysis reveals the presence of two jets in the redshifted emission, emanating from IRAS\,4A1 and IRAS\,4A2, respectively. Furthermore, we identify four distinct outflows in the region for the first time, with each protostar, 4A1 and 4A2, contributing to two of them. We characterise the morphology and orientation of each outflow, challenging previous suggestions of bends in their trajectories. The outflow cavities of IRAS\,4A1 exhibit extensions of 10$''$ and 13$''$ with position angles (PA) of 0$^{\circ}$ and -12$^{\circ}$, respectively, while those of IRAS\,4A2 are more extended, spanning 18$''$ and 25$''$ with PAs of 29$^{\circ}$ and 26$^{\circ}$. We propose that the misalignment of the cavities is due to a jet precession in each protostar, a notion supported by the observation that the more extended cavities of the same source exhibit lower velocities, indicating they may stem from older ejection events.

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arXiv2024

Into the thick of it: ALMA 0.45 mm observations of HL Tau at 2 au resolution

Aims. To comprehend the efficiency of dust evolution within protoplanetary disks, it is crucial to conduct studies of these disks using high-resolution observations at multiple wavelengths with the Atacama Large Millimeter/submillimeter Array (ALMA). Methods. In this work, we present high-frequency ALMA observations of the HL Tau disk using its Band 9 centered at a wavelength of 0.45 mm. These observations achieve the highest angular resolution in a protoplanetary disk to date, 12 milliarcseconds (mas), allowing the study of the dust emission at scales of 2 au. We use these data to extend the previously published multi-wavelength analysis of the HL Tau disk. Results. Our new 0.45 mm data traces mainly optically thick emission, providing a tight constraint to the dust temperature profile. We derive maximum particle sizes of $\sim$1 cm from the inner disk to $\sim$60 au. Beyond this radius, we find particles between 300 $μ$m and 1 mm. Moreover, an intriguing asymmetry is observed at 32 au in the northeast inner part of the HL Tau disk at 0.45 mm. We propose that this asymmetry is the outcome of a combination of factors including the optically thick nature of the emission, the orientation of the disk, and a relatively large dust scale height of the grains. To validate this, we conducted a series of radiative transfer models using the RADMC-3D software. If this scenario is correct, our measured dust mass within 32 au would suggest a dust scale height H/R> 0.08 for the inner disk. Finally, the unprecedented resolution allowed us to probe for the first time the dust emission down to a few au scales. We observed an increase in brightness temperature inside the estimated water snowline and speculate whether it could indicate the presence of a traffic jam effect in the inner disk. Abridge

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arXiv2023

ALMA High-frequency Long Baseline Campaign in 2021: Highest Angular Resolution Submillimeter Wave Images for the Carbon-rich Star R Lep

The Atacama Large Millimeter/submillimeter Array (ALMA) was used in 2021 to image the carbon-rich evolved star R Lep in Bands 8-10 (397-908 GHz) with baselines up to 16 km. The goal was to validate the calibration, using band-to-band (B2B) phase referencing with a close phase calibrator J0504-1512, 1.2 deg from R Lep in this case, and the imaging procedures required to obtain the maximum angular resolution achievable with ALMA. Images of the continuum emission and the hydrogen cyanide (HCN) maser line at 890.8 GHz, from the J=10-9 transition between the (1110) and (0400) vibrationally excited states, achieved angular resolutions of 13, 6, and 5 mas in Bands 8-10, respectively. Self-calibration (self-cal) was used to produce ideal images as to compare with the B2B phase referencing technique. The continuum emission was resolved in Bands 9 and 10, leaving too little flux for self-cal of the longest baselines, so these comparisons are made at coarser resolution. Comparisons showed that B2B phase referencing provided phase corrections sufficient to recover 92%, 83%, and 77% of the ideal image continuum flux densities. The HCN maser was sufficiently compact to obtain self-cal solutions in Band 10 for all baselines (up to 16 km). In Band 10, B2B phase referencing as compared to the ideal images recovered 61% and 70% of the flux density for the HCN maser and continuum, respectively.

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arXiv2023

The ALMA Interferometric Pipeline Heuristics

We describe the calibration and imaging heuristics developed and deployed in the ALMA interferometric data processing pipeline, as of ALMA Cycle 9. The pipeline software framework is written in Python, with each data reduction stage layered on top of tasks and toolkit functions provided by the Common Astronomy Software Applications package. This framework supports a variety of tasks for observatory operations, including science data quality assurance, observing mode commissioning, and user reprocessing. It supports ALMA and VLA interferometric data along with ALMA and NRO45m single dish data, via different stages and heuristics. In addition to producing calibration tables, calibrated measurement sets, and cleaned images, the pipeline creates a WebLog which serves as the primary interface for verifying the data quality assurance by the observatory and for examining the contents of the data by the user. Following the adoption of the pipeline by ALMA Operations in 2014, the heuristics have been refined through annual development cycles, culminating in a new pipeline release aligned with the start of each ALMA Cycle of observations. Initial development focused on basic calibration and flagging heuristics (Cycles 2-3), followed by imaging heuristics (Cycles 4-5), refinement of the flagging and imaging heuristics with parallel processing (Cycles 6-7), addition of the moment difference analysis to improve continuum channel identification (2020 release), addition of a spectral renormalization stage (Cycle 8), and improvement in low SNR calibration heuristics (Cycle 9). In the two most recent Cycles, 97% of ALMA datasets were calibrated and imaged with the pipeline, ensuring long-term automated reproducibility. We conclude with a brief description of plans for future additions, including self-calibration, multi-configuration imaging, and calibration and imaging of full polarization data.

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arXiv2023

Updates to ALMA Site Properties: using the ESO-Allegro Phase RMS database -- ALMA Memo 624

We present a long-term overview of the atmospheric phase stability at the Atacama Large Millimeter/submillimeter Array (ALMA) site, using >5 years of data, that acts as the successor to the studies summarized two decades ago by Evans et al 2003. Importantly, we explore the atmospheric variations, the `phase RMS', and associated metadata of over 17000 accrued ALMA observations taken since Cycle 3 (2015) by using the Bandpass calibrator source scans. We indicate the temporal phase RMS trends for average baseline lengths of 500, 1000, 5000, and 10000m, in contrast to the old stability studies that used a single 300m baseline phase monitor system. At the ALMA site, on the Chajnantor plateau, we report the diurnal variations and monthly changes in the phase RMS on ALMA relevant baselines lengths, measured directly from data, and we reaffirm such trends in atmospheric transmission (via Precipitable Water Vapour - PWV). We confirm that day observations have respectively higher phase RMS and PWV in contrast to night, while the monthly variations show Chilean winter (June - August) providing the best, high-frequency and long-baseline observing conditions - low (stable) phase RMS and low PWV. Yet, not all good phase stability condition occur when the PWV is low. Measurements of the phase RMS as a function of short timescales, 30 to 240s, that tie with typical target source scan times, and as a function of baseline length indicate that phase variations are smaller for short timescales and baselines and larger for longer timescales and baselines. We illustrate that fast-switching phase-referencing techniques, that allow short target scan times, could work well in reducing the phase RMS to suitable levels specifically for high-frequencies (Band 8, 9 and 10), long-baselines, and the two combined.

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arXiv2022

FAUST III. Misaligned rotations of the envelope, outflow, and disks in the multiple protostellar system of VLA 1623$-$2417

We report a study of the low-mass Class-0 multiple system VLA 1623AB in the Ophiuchus star-forming region, using H$^{13}$CO$^+$ ($J=3-2$), CS ($J=5-4$), and CCH ($N=3-2$) lines as part of the ALMA Large Program FAUST. The analysis of the velocity fields revealed the rotation motion in the envelope and the velocity gradients in the outflows (about 2000 au down to 50 au). We further investigated the rotation of the circum-binary VLA 1623A disk as well as the VLA 1623B disk. We found that the minor axis of the circum-binary disk of VLA 1623A is misaligned by about 12 degrees with respect to the large-scale outflow and the rotation axis of the envelope. In contrast, the minor axis of the circum-binary disk is parallel to the large-scale magnetic field according to previous dust polarization observations, suggesting that the misalignment may be caused by the different directions of the envelope rotation and the magnetic field. If the velocity gradient of the outflow is caused by rotation, the outflow has a constant angular momentum and the launching radius is estimated to be $5-16$ au, although it cannot be ruled out that the velocity gradient is driven by entrainments of the two high-velocity outflows. Furthermore, we detected for the first time a velocity gradient associated with rotation toward the VLA 16293B disk. The velocity gradient is opposite to the one from the large-scale envelope, outflow, and circum-binary disk. The origin of its opposite gradient is also discussed.

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arXiv2021

FAUST II. Discovery of a Secondary Outflow in IRAS 15398-3359: Variability in Outflow Direction during the Earliest Stage of Star Formation?

We have observed the very low-mass Class 0 protostar IRAS 15398-3359 at scales ranging from 50 au to 1800 au, as part of the ALMA Large Program FAUST. We uncover a linear feature, visible in H2CO, SO, and C18O line emission, which extends from the source along a direction almost perpendicular to the known active outflow. Molecular line emission from H2CO, SO, SiO, and CH3OH further reveals an arc-like structure connected to the outer end of the linear feature and separated from the protostar, IRAS 15398-3359, by 1200 au. The arc-like structure is blue-shifted with respect to the systemic velocity. A velocity gradient of 1.2 km/s over 1200 au along the linear feature seen in the H2CO emission connects the protostar and the arc-like structure kinematically. SO, SiO, and CH3OH are known to trace shocks, and we interpret the arc-like structure as a relic shock region produced by an outflow previously launched by IRAS 15398-3359. The velocity gradient along the linear structure can be explained as relic outflow motion. The origins of the newly observed arc-like structure and extended linear feature are discussed in relation to turbulent motions within the protostellar core and episodic accretion events during the earliest stage of protostellar evolution.

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arXiv2020

ALMA High-frequency Long-baseline Campaign in 2017: A Comparison of the Band-to-band and In-band Phase Calibration Techniques and Phase-calibrator Separation Angles

The Atacama Large millimeter/submillimeter Array (ALMA) obtains spatial resolutions of 15 to 5 milli-arcsecond (mas) at 275-950GHz (0.87-0.32mm) with 16km baselines. Calibration at higher-frequencies is challenging as ALMA sensitivity and quasar density decrease. The Band-to-Band (B2B) technique observes a detectable quasar at lower frequency that is closer to the target, compared to one at the target high-frequency. Calibration involves a nearly constant instrumental phase offset between the frequencies and the conversion of the temporal phases to the target frequency. The instrumental offsets are solved with a differential-gain-calibration (DGC) sequence, consisting of alternating low and high frequency scans of strong quasar. Here we compare B2B and in-band phase referencing for high-frequencies ($>$289GHz) using 2-15km baselines and calibrator separation angles between $\sim$0.68 and $\sim$11.65$^{\circ}$. The analysis shows that: (1) DGC for B2B produces a coherence loss $<$7% for DGC phase RMS residuals $<$30$^{\circ}$. (2) B2B images using close calibrators ( $<$1.67$^{\circ}$ ) are superior to in-band images using distant ones ( $>$2.42$^{\circ}$ ). (3) For more distant calibrators, B2B is preferred if it provides a calibrator $\sim$2$^{\circ}$ closer than the best in-band calibrator. (4) Decreasing image coherence and poorer image quality occur with increasing phase calibrator separation angle because of uncertainties in the antenna positions and sub-optimal phase referencing. (5) To achieve $>$70% coherence for long-baseline (16 km) band 7 (289GHz) observations, calibrators should be within $\sim$4$^{\circ}$ of the target.

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arXiv2020

ALMA Band-to-band Phase Referencing: Imaging Capabilities on Long Baselines and High Frequencies

High-frequency long-baseline experiments with the Atacama Large Millimeter/submillimeter Array were organized to test the high angular resolution imaging capabilities in the submillimeter wave regime using baselines up to 16 km. Four experiments were conducted, two Band 7 (289 GHz) and two Band 8 (405 GHz) observations. Phase correction using band-to-band (B2B) phase referencing was used with a phase calibrator only 0.7deg away observed in Band 3 (96 GHz) and Band 4 (135 GHz), respectively. In Band 8, we achieved the highest resolution of 14x11 mas. We compared the synthesis images of the target quasar using 20 and 60 s switching cycle times in the phase referencing. In Band 7, the atmosphere had good stability in phase rms (<0.5 rad over 2 minutes), and there was little difference in image coherence between the 20 and 60 s switching cycle times. One Band 8 experiment was conducted under a worse phase rms condition (>1 rad over 2 minutes), which led to a significantly reduced coherence when using the 60 s switching cycle time. One of our four experiments indicates that the residual phase rms error after phase referencing can be reduced to 0.16 rad at 289 GHz in using the 20 s switching cycle time. Such conditions would meet the phase correction requirement of image coherence of >70% in Band 10, assuming a similar phase calibrator separation angle, emphasizing the need for such B2B phase referencing observing at high frequencies.

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arXiv2020

ALMA High-frequency Long Baseline Campaign in 2017: Band-to-band Phase Referencing in Submillimeter Waves

In 2017, an Atacama Large Millimeter/submillimeter Array (ALMA) high-frequency long baseline campaign was organized to test image capabilities with baselines up to 16 km at submillimeter (submm) wavelengths. We investigated image qualities using ALMA receiver Bands 7, 8, 9, and 10 (285-875 GHz) by adopting band-to-band (B2B) phase referencing in which a phase calibrator is tracked at a lower frequency. For B2B phase referencing, it is expected that a closer phase calibrator to a target can be used, comparing to standard in-band phase referencing. In the first step, it is ensured that an instrumental phase offset difference between low- and high-frequency Bands can be removed using a differential gain calibration in which a phase calibrator is certainly detected while frequency switching. In the next step, comparative experiments are arranged to investigate the image quality between B2B and in-band phase referencing with phase calibrators at various separation angles. In the final step, we conducted long baseline imaging tests for a quasar at 289 GHz in Band 7 and 405 GHz in Band 8 and complex structure sources of HL Tau and VY CMa at ~670 GHz in Band 9. The B2B phase referencing was successfully applied, allowing us to achieve an angular resolution of 14x11 and 10x8 mas for HL Tau and VY CMa, respectively. There is a high probability of finding a low-frequency calibrator within 5.4 deg in B2B phase referencing, bright enough to use an 8 s scan length combined with a 7.5 GHz bandwidth.

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arXiv2018

Evidence for the start of planet formation in a young circumstellar disk

The growth of dust grains in protoplanetary disks is a necessary first step towards planet formation. This growth has been inferred via observations of thermal dust emission towards mature protoplanetary systems (age >2 million years) with masses that are, on average, similar to Neptune3. In contrast, the majority of confirmed exoplanets are heavier than Neptune. Given that young protoplanetary disks are more massive than their mature counterparts, this suggests that planet formation starts early, but evidence for grain growth that is spatially and temporally coincident with a massive reservoir in young disks remains scarce. Here, we report observations on a lack of emission of carbon monoxide isotopologues within the inner ~15 au of a very young (age ~100,000 years) disk around the Solar-type protostar TMC1A. By using the absence of spatially resolved molecular line emission to infer the gas and dust content of the disk, we conclude that shielding by millimeter-size grains is responsible for the lack of emission. This suggests that grain growth and millimeter-size dust grains can be spatially and temporally coincident with a mass reservoir sufficient for giant planet formation. Hence, planet formation starts during the earliest, embedded phases in the life of young stars.

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arXiv2018

Radiative transfer modelling of W33A MM1: 3-D structure and dynamics of a complex massive star forming region

We present a composite model and radiative transfer simulations of the massive star forming core W33A MM1. The model was tailored to reproduce the complex features observed with ALMA at $\approx 0.2$ arcsec resolution in CH$_3$CN and dust emission. The MM1 core is fragmented into six compact sources coexisting within $\sim 1000$ au. In our models, three of these compact sources are better represented as disc-envelope systems around a central (proto)star, two as envelopes with a central object, and one as a pure envelope. The model of the most prominent object (Main) contains the most massive (proto)star ($M_\star\approx7~M_\odot$) and disc+envelope ($M_\mathrm{gas}\approx0.4~M_\odot$), and is the most luminous ($L_\mathrm{Main} \sim 10^4~L_\odot$). The model discs are small (a few hundred au) for all sources. The composite model shows that the elongated spiral-like feature converging to the MM1 core can be convincingly interpreted as a filamentary accretion flow that feeds the rising stellar system. The kinematics of this filament is reproduced by a parabolic trajectory with focus at the center of mass of the region. Radial collapse and fragmentation within this filament, as well as smaller filamentary flows between pairs of sources are proposed to exist. Our modelling supports an interpretation where what was once considered as a single massive star with a $\sim 10^3$ au disc and envelope, is instead a forming stellar association which appears to be virialized and to form several low-mass stars per high-mass object.

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arXiv2017

Sulphur monoxide exposes a potential molecular disk wind from the planet-hosting disk around HD100546

Sulphur-bearing volatiles are observed to be significantly depleted in interstellar and circumstellar regions. This missing sulphur is postulated to be mostly locked up in refractory form. With ALMA we have detected sulphur monoxide (SO), a known shock tracer, in the HD 100546 protoplanetary disk. Two rotational transitions: $J=7_{7}-6_{6}$ (301.286 GHz) and $J=7_8-6_7$ (304.078 GHz) are detected in their respective integrated intensity maps. The stacking of these transitions results in a clear 5$σ$ detection in the stacked line profile. The emission is compact but is spectrally resolved and the line profile has two components. One component peaks at the source velocity and the other is blue-shifted by 5 km s$^{-1}$. The kinematics and spatial distribution of the SO emission are not consistent with that expected from a purely Keplerian disk. We detect additional blue-shifted emission that we attribute to a disk wind. The disk component was simulated using LIME and a physical disk structure. The disk emission is asymmetric and best fit by a wedge of emission in the north east region of the disk coincident with a `hot-spot' observed in the CO $J=3-2$ line. The favoured hypothesis is that a possible inner disk warp (seen in CO emission) directly exposes the north-east side of the disk to heating by the central star, creating locally the conditions to launch a disk wind. Chemical models of a disk wind will help to elucidate why the wind is particularly highlighted in SO emission and whether a refractory source of sulphur is needed. An alternative explanation is that the SO is tracing an accretion shock from a circumplanetary disk associated with the proposed protoplanet embedded in the disk at 50 au. We also report a non-detection of SO in the protoplanetary disk around HD 97048.

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arXiv2017

ALMA Long Baseline Campaigns: Phase Characteristics of Atmosphere at Long Baselines in the Millimeter and Submillimeter Wavelengths

This paper presents the first detailed investigation of the characteristics of mm/submm phase fluctuation and phase correction methods obtained using ALMA with baseline lengths up to ~15 km. Most of the spatial structure functions (SSFs) show that the phase fluctuation increases as a function of baseline length, with a power-law slope of ~0.6. In many cases, we find that the slope becomes shallower (average of ~0.2-0.3) at baseline lengths longer than ~1 km, namely showing a turn-over in SSF. The phase correction method using water vapor radiometers (WVRs) works well, especially for the cases where PWV >1 mm, which reduces the degree of phase fluctuations by a factor of two in many cases. However, phase fluctuations still remain after the WVR phase correction, suggesting the existence of other turbulent constituent that cause the phase fluctuation. This is supported by occasional SSFs that do not exhibit any turn-over; these are only seen when the PWV is low or after WVR phase correction. This means that the phase fluctuation caused by this turbulent constituent is inherently smaller than that caused by water vapor. Since there is no turn-over in the SSF up to the maximum baseline length of ~15 km, this turbulent constituent must have scale height of 10 km or more, and thus cannot be water vapor, whose scale height is around 1 km. This large scale height turbulent constituent is likely to be water ice or a dry component. Excess path length fluctuation after the WVR phase correction at a baseline length of 10 km is large (>200 micron), which is significant for high frequency (>450 GHz or <700 micron) observations. These results suggest the need for an additional phase correction method, such as fast switching, in addition to the WVR phase correction. We simulated the fast switching, and the result suggests that it works well, with shorter cycle times linearly improving the coherence.

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arXiv2016

The Winds from HL Tau

Outflowing motions, whether a wind launched from the disk, a jet launched from the protostar, or the entrained molecular outflow, appear to be an ubiquitous feature of star formation. These outwards motions have a number of root causes, and how they manifest is intricately linked to their environment as well as the process of star formation itself. Using the ALMA Science Verification data of HL Tau, we investigate the high velocity molecular gas being removed from the system as a result of the star formation process. We aim to place these motions in context with the optically detected jet, and the disk. With these high resolution ($\sim 1"$) ALMA observations of CO (J=1-0), we quantify the outwards motions of the molecular gas. We find evidence for a bipolar outwards flow, with an opening angle, as measured in the red-shifted lobe, starting off at 90$^\circ$, and narrowing to 60$^\circ$ further from the disk, likely because of magnetic collimation. Its outwards velocity, corrected for inclination angle is of order 2.4 km s$^{-1}$.

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arXiv2016

Submillimeter Array Observations of NGC 2264-C: Molecular Outflows and Driving Sources

We present 1.3mm Submillimeter Array (SMA) observations at $\sim$3$^{\prime\prime}$ resolution towards the brightest section of the intermediate/massive star forming cluster NGC 2264-C. The millimetre continuum emission reveals ten 1.3mm continuum peaks, of which four are new detections. The observed frequency range includes the known molecular jet/outflow tracer SiO (5-4), thus providing the first high resolution observations of SiO towards NGC 2264-C. We also detect molecular lines of twelve additional species towards this region, including CH$_3$CN, CH$_3$OH, SO, H$_2$CO, DCN, HC$_3$N, and $^{12}$CO. The SiO (5-4) emission reveals the presence of two collimated, high velocity (up to 30kms$^{-1}$ with respect to the systemic velocity) bi-polar outflows in NGC 2264-C. In addition, the outflows are traced by emission from $^{12}$CO, SO, H$_2$CO, and CH$_3$OH. We find an evolutionary spread between cores residing in the same parent cloud. The two unambiguous outflows are driven by the brightest mm continuum cores, which are IR-dark, molecular line weak, and likely the youngest cores in the region. Furthermore, towards the RMS source AFGL 989-IRS1, the IR-bright and most evolved source in NGC 2264-C, we observe no molecular outflow emission. A molecular line rich ridge feature, with no obvious directly associated continuum source, lies on the edge of a low density cavity and may be formed from a wind driven by AFGL 989-IRS1. In addition, 229GHz class I maser emission is detected towards this feature.

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arXiv2013

Disk evolution in the solar neighborhood. I Disk frequencies from 1 to 100 Myr

We study the evolution of circumstellar disks in 22 young (1 to 100 Myr) nearby (within 500 pc) associations over the entire mass spectrum using photometry covering from the optical to the mid-infrared. We compiled a catalog of 2340 spectroscopically-confirmed members of these nearby associations. We analyzed their spectral energy distributions and searched for excess related to the presence of protoplanetary disks in a homogeneous way. Sensitivity limits and spatial completeness were also considered. We derive disk fractions as probed by mid-infrared excess in these regions. The unprecedented size of our sample allows us to confirm the timescale of disk decay reported in the literature and to find new trends. The fraction of excess sources increases systematically if measured at longer wavelengths. Disk percentages derived using different wavelength ranges should therefore be compared with caution. The dust probed at 22-24 um evolves slower than that probed at shorter wavelengths (3.4-12 um). Assuming an exponential decay, we derive a timescale tau=4.2-5.8 Myr at 22-24 um for primordial disks, compared to 2-3 Myr at shorter wavelength (3.4-12 um). Primordial disks disappear around 10 Myr, matching in time a brief increase of the number of 'evolved' disks. The increase in timescale of excess decay at longer wavelength is compatible with inside-out disk clearing scenarios. The increased timescale of decay and larger dispersion in the distribution of disk fractions at 22-24 um suggest that the inner and outer zones evolve differently, the latter potentially following a variety of evolutionary paths. The drop of primordial disks and the coincident rise of evolved disks at 10 Myr are compatible with planet formation theories suggesting that the disappearance of the gas is immediately followed by the dynamical stirring of the disk.

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