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Christine H. Chen

Publications and source records attributed to Christine H. Chen.

At least 37 records · Page 2Linked to original sources

Vertical Structure of Gas and Dust in Four Debris Disks

We present high-spectral resolution M-band spectra from iSHELL on NASA's Infrared Telescope Facility (IRTF) along the line of sight to the debris disk host star HD 32297. We also present a Gemini Planet Imager (GPI) H-band polarimetric image of the HD 131488 debris disk. We search for fundamental CO absorption lines in the iSHELL spectra of HD 32297 but do not detect any. We place an upper limit on the CO column density of $\sim$6$\times10^{15}$ cm$^{-2}$. By combining the column density upper limit, the CO mass measured with ALMA, and the geometrical properties of the disk, we estimate the scale height of the CO to be $\lesssim$ 2 au across the radial extent of the disk ($\sim$80-120 au). We use the same method to estimate the CO scale height of three other edge-on, CO-rich debris disks that all have CO observed in absorption with HST as well as in emission with ALMA: $β$ Pictoris, HD 110058, and HD 131488. We compare our estimated CO scale heights of these four systems to the millimeter dust scale heights and find that, under the assumption of hydrostatic equilibrium, there is a potential correlation between the CO and millimeter dust scale heights. There are multiple factors that affect the gas vertical structure such as turbulence, photodissociation with weak vertical mixing, as well as where the gas originates. One possible explanation for the potential correlation could be that the gas and dust are of a similar secondary origin in these four systems.

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A Uniform Analysis of Debris Disks with the Gemini Planet Imager I: An Empirical Search for Perturbations from Planetary Companions in Polarized Light Images

The Gemini Planet Imager (GPI) has excelled in imaging debris disks in the near-infrared. The GPI Exoplanet Survey (GPIES) imaged twenty-four debris disks in polarized $H$-band light, while other programs observed half of these disks in polarized $J$- and/or $K1$-bands. Using these data, we present a uniform analysis of the morphology of each disk to find asymmetries suggestive of perturbations, particularly those due to planet-disk interactions. The multi-wavelength surface brightness, the disk color and geometry permit identification of any asymmetries such as warps or disk offsets from the central star. We find that nineteen of the disks in this sample exhibit asymmetries in surface brightness, disk color, disk geometry, or a combination of the three, suggesting that for this sample, perturbations, as seen in scattered light, are common. The relationship between these perturbations and potential planets in the system are discussed. We also explore correlations among stellar temperatures, ages, disk properties, and observed perturbations. We find significant trends between the vertical aspect ratio and the stellar temperature, disk radial extent, and the dust grain size distribution power-law, $q$. We also confirm a trend between the disk color and stellar effective temperature, where the disk becomes increasingly red/neutral with increasing temperature. Such results have important implications on the evolution of debris disk systems around stars of various spectral types.

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The first scattered light images of HD 112810, a faint debris disk in the Sco-Cen association

Context: Circumstellar debris disks provide insight into the formation and early evolution of planetary systems. Resolved belts in particular help to locate planetesimals in exosystems, and can hint at the presence of disk-sculpting exoplanets. Aims: We study the circumstellar environment of HD 112810 (HIP 63439), a mid-F type star in the Sco-Cen association with a significant infrared excess indicating the presence of a circumstellar debris disk. Methods: We collected five high-contrast observations of HD 112810 with VLT/SPHERE. We identified a debris disk in scattered light, and found that the debris signature is robust over a number of epochs and a variety of reduction techniques. We modelled the disk, accounting for self-subtraction and assuming that it is optically thin. Results: We find a single-belt debris disk, with a radius of 118$\pm$9au and an inclination angle of ${75.7}^{+1.1}_{-1.3}$$°$. This is in good agreement with the constraints from SED modelling and from a partially-resolved ALMA image of the system. No planets are detected, though planets below the detection limit ($\sim$2.6M$_\textrm{J}$ at a projected separation of 118au) could be present and could have contributed to sculpting the ring of debris. Conclusions: HD 112810 adds to the growing inventory of debris disks imaged in scattered light. The disk is faint, but the radius and the inclination of the disk are promising for follow-up studies of the dust properties.

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The JWST Early Release Science Program for Direct Observations of Exoplanetary Systems I: High Contrast Imaging of the Exoplanet HIP 65426 b from 2-16 $μ$m

We present JWST Early Release Science (ERS) coronagraphic observations of the super-Jupiter exoplanet, HIP 65426 b, with the Near-Infrared Camera (NIRCam) from 2-5 $μ$m, and with the Mid-Infrared Instrument (MIRI) from 11-16 $μ$m. At a separation of $\sim$0.82" (86$^{+116}_{-31}$ au), HIP 65426 b is clearly detected in all seven of our observational filters, representing the first images of an exoplanet to be obtained by JWST, and the first ever direct detection of an exoplanet beyond 5 $μ$m. These observations demonstrate that JWST is exceeding its nominal predicted performance by up to a factor of 10, depending on separation and subtraction method, with measured 5$σ$ contrast limits of $\sim$1$\times10^{-5}$ and $\sim$2$\times10^{-4}$ at 1" for NIRCam at 4.4 $μ$m and MIRI at 11.3 $μ$m, respectively. These contrast limits provide sensitivity to sub-Jupiter companions with masses as low as 0.3$M_\mathrm{Jup}$ beyond separations of $\sim$100 au. Together with existing ground-based near-infrared data, the JWST photometry are well fit by a BT-SETTL atmospheric model from 1-16 $μ$m, and span $\sim$97% of HIP 65426 b's luminous range. Independent of the choice of model atmosphere we measure an empirical bolometric luminosity that is tightly constrained between $\mathrm{log}\!\left(L_\mathrm{bol}/L_{\odot}\right)$=-4.31 to $-$4.14, which in turn provides a robust mass constraint of 7.1$\pm$1.2 $M_\mathrm{Jup}$. In totality, these observations confirm that JWST presents a powerful and exciting opportunity to characterise the population of exoplanets amenable to high-contrast imaging in greater detail.

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Testing the Interaction Between a Substellar Companion and a Debris Disk in the HR 2562 System

The HR 2562 system is a rare case where a brown dwarf companion resides in a cleared inner hole of a debris disk, offering invaluable opportunities to study the dynamical interaction between a substellar companion and a dusty disk. We present the first ALMA observation of the system as well as the continued GPI monitoring of the companion's orbit with 6 new epochs from 2016 to 2018. We update the orbital fit and, in combination with absolute astrometry from GAIA, place a 3$σ$ upper limit of 18.5 $M_J$ on the companion's mass. To interpret the ALMA observations, we used radiative transfer modeling to determine the disk properties. We find that the disk is well resolved and nearly edge on. While the misalignment angle between the disk and the orbit is weakly constrained due to the short orbital arc available, the data strongly support a (near) coplanar geometry for the system. Furthermore, we find that the models that describe the ALMA data best have an inner radius that is close to the companion's semi-major axis. Including a posteriori knowledge of the system's SED further narrows the constraints on the disk's inner radius and place it at a location that is in reasonable agreement with, possibly interior to, predictions from existing dynamical models of disk truncation by an interior substellar companion. HR\,2562 has the potential over the next few years to become a new testbed for dynamical interaction between a debris disk and a substellar companion.

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Debris Disk Color with the Hubble Space Telescope

Multi-wavelength scattered light imaging of debris disks may inform dust properties including typical size and mineral composition. Existing studies have investigated a small set of individual systems across a variety of imaging instruments and filters, calling for uniform comparison studies to systematically investigate dust properties. We obtain the surface brightness of dust particles in debris disks by post-processing coronagraphic imaging observations, and compare the multi-wavelength reflectance of dust. For a sample of resolved debris disks, we perform a systematic analysis on the reflectance properties of their birth rings. We reduced the visible and near-infrared images of 23 debris disk systems hosted by A through M stars using two coronagraphs onboard the Hubble Space Telescope: the STIS instrument observations centering at 0.58 $μ$m, and the NICMOS instrument at 1.12 $μ$m or 1.60 $μ$m. For proper recovery of debris disks, we used classical reference differential imaging for STIS, and adopted non-negative matrix factorization with forward modeling for NICMOS. By dividing disk signals by stellar signals to take into account of intrinsic stellar color effects, we systematically obtained and compared the reflectance of debris birth rings at ~90 deg scattering angle. Debris birth rings typically exhibit a blue color at ~90 deg scattering angle. As the stellar luminosity increases, the color tends to be more neutral. A likely L-shaped color-albedo distribution indicates a clustering of scatterer properties. The observed color trend correlates with the expected blow-out size of dust particles. The color-albedo clustering likely suggests different populations of dust in these systems. More detailed radiative transfer models with realistic dust morphology will contribute to explaining the observed color and color-albedo distribution of debris systems.

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Trends in Silicates in the $β$ Pictoris Disk

While beta Pic is known to host silicates in ring-like structures, whether the properties of these silicate dust vary with stellocentric distance remains an open question. We re-analyze the beta Pictoris debris disk spectrum from the Spitzer Infrared Spectrograph (IRS) and a new IRTF/SpeX spectrum to investigate trends in Fe/Mg ratio, shape, and crystallinity in grains as a function of wavelength, a proxy for stellocentric distance. By analyzing a re-calibrated and re-extracted spectrum, we identify a new 18 micron forsterite emission feature and recover a 23 micron forsterite emission feature with a substantially larger line-to-continuum ratio than previously reported. We find that these prominent spectral features are primarily produced by small submicron-sized grains, which are continuously generated and replenished from planetesimal collisions in the disk and can elucidate their parent bodies' composition. We discover three trends about these small grains: as stellocentric distance increases, (1) small silicate grains become more crystalline (less amorphous), (2) they become more irregular in shape, and (3) for crystalline silicate grains, the Fe/Mg ratio decreases. Applying these trends to beta Pic's planetary architecture, we find that the dust population exterior to the orbits of beta Pic b and c differs substantially in crystallinity and shape. We also find a tentative 3-5 micron dust excess due to spatially unresolved hot dust emission close to the star. From our findings, we infer that the surfaces of large planetesimals are more Fe-rich and collisionally-processed closer to the star but more Fe-poor and primordial farther from the star.

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A Layered Debris Disk around M Star TWA 7 in Scattered Light

We have obtained Hubble Space Telescope (HST) coronagraphic observations of the circumstellar disk around M star TWA 7 using the STIS instrument in visible light. Together with archival observations including HST/NICMOS using the F160W filter and Very Large Telescope/SPHERE at $H$-band in polarized light, we investigate the system in scattered light. By studying this nearly face-on system using geometric disk models and Henyey--Greenstein phase functions, we report new discovery of a tertiary ring and a clump. We identify a layered architecture: three rings, a spiral, and an ${\approx}150$ au$^2$ elliptical clump. The most extended ring peaks at $28$ au, and the other components are on its outskirts. Our point source detection limit calculations demonstrate the necessity of disk modeling in imaging fainter planets. Morphologically, we witness a clockwise spiral motion, and the motion pattern is consistent with both solid body and local Keplerian; we also observe underdensity regions for the secondary ring that might result from mean motion resonance or moving shadows: both call for re-observations to determine their nature. Comparing multi-instrument observations, we obtain blue STIS-NICMOS color, STIS-SPHERE radial distribution peak difference for the tertiary ring, and high SPHERE-NICMOS polarization fraction; these aspects indicate that TWA 7 could retain small dust particles. By viewing the debris disk around M star TWA 7 at a nearly face-on vantage point, our study allows for the understanding of such disks in scattered light in both system architecture and dust property.

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A Deep Polarimetric Study of the Asymmetrical Debris Disk HD 106906

HD 106906 is a young, binary stellar system, located in the Lower Centaurus Crux (LCC) group. This system is unique among discovered systems in that it contains an asymmetrical debris disk, as well as an 11 M$_{Jup}$ planet companion, at a separation of $\sim$735 AU. Only a handful of other systems are known to contain both a disk and directly imaged planet, where HD 106906 is the only one in which the planet has apparently been scattered. The debris disk is nearly edge on, and extends roughly to $>$500 AU, where previous studies with HST have shown the outer regions to have high asymmetry. To better understand the structure and composition of the disk, we have performed a deep polarimetric study of HD 106906's asymmetrical debris disk using newly obtained $H$-, $J$-, and $K1$-band polarimetric data from the Gemini Planet Imager (GPI). An empirical analysis of our data supports a disk that is asymmetrical in surface brightness and structure, where fitting an inclined ring model to the disk spine suggests that the disk may be highly eccentric ($e\gtrsim0.16$). A comparison of the disk flux with the stellar flux in each band suggests a blue color that also does not significantly vary across the disk. We discuss these results in terms of possible sources of asymmetry, where we find that dynamical interaction with the planet companion, HD 106906b, is a likely candidate.

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Multiband GPI Imaging of the HR 4796A Debris Disk

We have obtained Gemini Planet Imager (GPI) J-, H-, K1-, and K2-Spec observations of the iconic debris ring around the young, main-sequence star HR 4796A. We applied several point-spread function (PSF) subtraction techniques to the observations (Mask-and-Interpolate, RDI-NMF, RDI-KLIP, and ADI-KLIP) to measure the geometric parameters and the scattering phase function for the disk. To understand the systematic errors associated with PSF subtraction, we also forward-modeled the observations using a Markov Chain Monte Carlo framework and a simple model for the disk. We found that measurements of the disk geometric parameters were robust, with all of our analyses yielding consistent results; however, measurements of the scattering phase function were challenging to reconstruct from PSF-subtracted images, despite extensive testing. As a result, we estimated the scattering phase function using disk modeling. We searched for a dependence of the scattering phase function with respect to the GPI filters but found none. We compared the H-band scattering phase function with that measured by Hubble Space Telescope STIS at visual wavelengths and discovered a blue color at small scattering angles and a red color at large scattering angles, consistent with predictions and laboratory measurements of large grains. Finally, we successfully modeled the SPHERE H2 HR 4796A scattered phase function using a distribution of hollow spheres composed of silicates, carbon, and metallic iron.

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Debris Disk Results from the Gemini Planet Imager Exoplanet Survey's Polarimetric Imaging Campaign

We report the results of a ${\sim}4$-year direct imaging survey of 104 stars to resolve and characterize circumstellar debris disks in scattered light as part of the Gemini Planet Imager Exoplanet Survey. We targeted nearby (${\lesssim}150$ pc), young (${\lesssim}500$ Myr) stars with high infrared excesses ($L_{\mathrm{IR}} / L_\star > 10^{-5}$), including 38 with previously resolved disks. Observations were made using the Gemini Planet Imager high-contrast integral field spectrograph in $H$-band (1.6 $μ$m) coronagraphic polarimetry mode to measure both polarized and total intensities. We resolved 26 debris disks and three protoplanetary/transitional disks. Seven debris disks were resolved in scattered light for the first time, including newly presented HD 117214 and HD 156623, and we quantified basic morphologies of five of them using radiative transfer models. All of our detected debris disks but HD 156623 have dust-poor inner holes, and their scattered-light radii are generally larger than corresponding radii measured from resolved thermal emission and those inferred from spectral energy distributions. To assess sensitivity, we report contrasts and consider causes of non-detections. Detections were strongly correlated with high IR excess and high inclination, although polarimetry outperformed total intensity angular differential imaging for detecting low inclination disks (${\lesssim} 70 °$). Based on post-survey statistics, we improved upon our pre-survey target prioritization metric predicting polarimetric disk detectability. We also examined scattered-light disks in the contexts of gas, far-IR, and millimeter detections. Comparing $H$-band and ALMA fluxes for two disks revealed tentative evidence for differing grain properties. Finally, we found no preference for debris disks to be detected in scattered light if wide-separation substellar companions were present.

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Multiband Polarimetric Imaging of HR 4796A with the Gemini Planet Imager

HR4796A hosts a well-studied debris disk with a long history due to its high fractional luminosity and favorable inclination lending itself well to both unresolved and resolved observations. We present new J- and K1-band images of the resolved debris disk HR4796A taken in the polarimetric mode of the Gemini Planet Imager (GPI). The polarized intensity features a strongly forward scattered brightness distribution and is undetected at the far side of the disk. The total intensity is detected at all scattering angles and also exhibits a strong forward scattering peak. We use a forward modelled geometric disk in order to extract geometric parameters, polarized fraction and total intensity scattering phase functions for these data as well as H-band data previously taken by GPI. We find the polarized phase function becomes increasingly more forward scattering as wavelength increases. We fit Mie and distribution of hollow spheres grain (DHS) models to the extracted functions. We find that while it is possible to describe generate a satisfactory model for the total intensity using a DHS model, but not with a Mie model. We find that no single grain population of DHS or Mie grains of arbitrary composition can simultaneously reproduce the polarized fraction and total intensity scattering phase functions, indicating the need for more sophisticated grain models.

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The Gemini Planet Imager view of the HD 32297 debris disk

We present new $H$-band scattered light images of the HD 32297 edge-on debris disk obtained with the Gemini Planet Imager (GPI). The disk is detected in total and polarized intensity down to a projected angular separation of 0.15", or 20au. On the other hand, the large scale swept-back halo remains undetected, likely a consequence of its markedly blue color relative to the parent body belt. We analyze the curvature of the disk spine and estimate a radius of $\approx$100au for the parent body belt, smaller than past scattered light studies but consistent with thermal emission maps of the system. We employ three different flux-preserving post-processing methods to suppress the residual starlight and evaluate the surface brightness and polarization profile along the disk spine. Unlike past studies of the system, our high fidelity images reveal the disk to be highly symmetric and devoid of morphological and surface brightness perturbations. We find the dust scattering properties of the system to be consistent with those observed in other debris disks, with the exception of HR 4796. Finally, we find no direct evidence for the presence of a planetary-mass object in the system.

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First Resolved Scattered-Light Images of Four Debris Disks in Scorpius-Centaurus with the Gemini Planet Imager

We present the first spatially resolved scattered-light images of four debris disks around members of the Scorpius-Centaurus (Sco-Cen) OB Association with high-contrast imaging and polarimetry using the Gemini Planet Imager (GPI). All four disks are resolved for the first time in polarized light and one disk is also detected in total intensity. The three disks imaged around HD 111161, HD 143675, and HD 145560 are symmetric in both morphology and brightness distribution. The three systems span a range of inclinations and radial extents. The disk imaged around HD 98363 shows indications of asymmetries in morphology and brightness distribution, with some structural similarities to the HD 106906 planet-disk system. Uniquely, HD 98363 has a wide co-moving stellar companion Wray 15-788 with a recently resolved disk with very different morphological properties. HD 98363 A/B is the first binary debris disk system with two spatially resolved disks. All four targets have been observed with ALMA, and their continuum fluxes range from one non-detection to one of the brightest disks in the region. With the new results, a total of 15 A/F-stars in Sco-Cen have resolved scattered light debris disks, and approximately half of these systems exhibit some form of asymmetry. Combining the GPI disk structure results with information from the literature on millimeter fluxes and imaged planets reveals a diversity of disk properties in this young population. Overall, the four newly resolved disks contribute to the census of disk structures measured around A/F-stars at this important stage in the development of planetary systems.

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Direct Imaging of the HD 35841 Debris Disk: A Polarized Dust Ring from Gemini Planet Imager and an Outer Halo from HST/STIS

We present new high resolution imaging of a light-scattering dust ring and halo around the young star HD 35841. Using spectroscopic and polarimetric data from the Gemini Planet Imager in H-band (1.6 microns), we detect the highly inclined (i=85 deg) ring of debris down to a projected separation of ~12 au (~0.12") for the first time. Optical imaging from HST/STIS shows a smooth dust halo extending outward from the ring to >140 au (>1.4"). We measure the ring's scattering phase function and polarization fraction over scattering angles of 22-125 deg, showing a preference for forward scattering and a polarization fraction that peaks at ~30% near the ansae. Modeling of the scattered-light disk indicates that the ring spans radii of ~60-220 au, has a vertical thickness similar to that of other resolved dust rings, and contains grains as small as 1.5 microns in diameter. These models also suggest the grains have a low porosity, are more likely to consist of carbon than astrosilicates, and contain significant water ice. The halo has a surface brightness profile consistent with that expected from grains pushed by radiation pressure from the main ring onto highly eccentric but still bound orbits. We also briefly investigate arrangements of a possible inner disk component implied by our spectral energy distribution models, and speculate about the limitations of Mie theory for doing detailed analyses of debris disk dust populations.

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Science Impacts of the SPHEREx All-Sky Optical to Near-Infrared Spectral Survey II: Report of a Community Workshop on the Scientific Synergies Between the SPHEREx Survey and Other Astronomy Observatories

SPHEREx is a proposed NASA MIDEX mission selected for Phase A study. SPHEREx would carry out the first all-sky spectral survey in the near infrared. At the end of its two-year mission, SPHEREx would obtain 0.75-to-5$μ$m spectra of every 6.2 arcsec pixel on the sky, with spectral resolution R>35 and a 5-$σ$ sensitivity AB$>$19 per spectral/spatial resolution element. More details concerning SPHEREx are available at http://spherex.caltech.edu. The SPHEREx team has proposed three specific science investigations to be carried out with this unique data set: cosmic inflation, interstellar and circumstellar ices, and the extra-galactic background light. Though these three themes are undoubtedly compelling, they are far from exhausting the scientific output of SPHEREx. Indeed, SPHEREx would create a unique all-sky spectral database including spectra of very large numbers of astronomical and solar system targets, including both extended and diffuse sources. These spectra would enable a wide variety of investigations, and the SPHEREx team is dedicated to making the data available to the community to enable these investigations, which we refer to as Legacy Science. To that end, we have sponsored two workshops for the general scientific community to identify the most interesting Legacy Science themes and to ensure that the SPHEREx data products are responsive to their needs. In February of 2016, some 50 scientists from all fields met in Pasadena to develop these themes and to understand their implications for the SPHEREx mission. The 2016 workshop highlighted many synergies between SPHEREx and other contemporaneous astronomical missions, facilities, and databases. Consequently, in January 2018 we convened a second workshop at the Center for Astrophysics in Cambridge to focus specifically on these synergies. This white paper reports on the results of the 2018 SPHEREx workshop.

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Space-Based Coronagraphic Imaging Polarimetry of the TW Hydrae Disk: Shedding New Light on Self-Shadowing Effects

We present Hubble Space Telescope Near-Infrared Camera and Multi-Object Spectrometer coronagraphic imaging polarimetry of the TW Hydrae protoplanetary disk. These observations simultaneously measure the total and polarized intensity, allowing direct measurement of the polarization fraction across the disk. In accord with the self-shadowing hypothesis recently proposed by Debes et al., we find that the total and polarized intensity of the disk exhibits strong azimuthal asymmetries at projected distances consistent with the previously reported bright and dark ring-shaped structures (~45-99 au). The sinusoidal-like variations possess a maximum brightness at position angles near ~268-300 degrees and are up to ~28% stronger in total intensity. Furthermore, significant radial and azimuthal variations are also detected in the polarization fraction of the disk. In particular, we find that regions of lower polarization fraction are associated with annuli of increased surface brightness, suggesting that the relative proportion of multiple-to-single scattering is greater along the ring and gap structures. Moreover, we find strong (~20%) azimuthal variation in the polarization fraction along the shadowed region of the disk. Further investigation reveals that the azimuthal variation is not the result of disk flaring effects, but instead from a decrease in the relative contribution of multiple-to-single scattering within the shadowed region. Employing a two-layer scattering surface, we hypothesize that the diminished contribution in multiple scattering may result from shadowing by an inclined inner disk, which prevents direct stellar light from reaching the optically thick underlying surface component.

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Infrared Spectroscopy of HR 4796A's Bright Outer Cometary Ring + Tenuous Inner Hot Dust Cloud

We have obtained new NASA IRTF SpeX spectra of the HR 4796A debris ring system. We find a unique red excess flux that extends out to ~9 um in Spitzer IRS spectra, where thermal emission from cold, ~100K dust from the system's ring at ~75 AU takes over. Matching imaging ring photometry, we find the excess consists of NIR reflectance from the ring which is as red as that of old, processed comet nuclei, plus a tenuous thermal emission component from close-in, T ~ 850 K circumstellar material evincing an organic plus silicate emission feature complex at 7 - 13 um. Unusual, emission-like features due to atomic Si, S, Ca, and Sr were found at 0.96 - 1.07 um, likely sourced by rocky dust evaporating in the 850 K component. An empirical cometary dust phase function can reproduce the scattered light excess and 1:5 balance of scattered vs. thermal energy for the ring with optical depth Tau > 0.10 in an 8 AU wide belt of 4 AU vertical height and Mdust > 0.1-0.7 M_Mars. Our results are consistent with HR 4796A consisting of a narrow sheparded ring of devolatilized cometary material associated with multiple rocky planetesimal subcores, and a small steady stream of dust inflowing from this belt to a rock sublimation zone at approximately 1 AU from the primary. These subcores were built from comets that have been actively emitting large, reddish dust for > 0.4 Myr at 100K, the temperature at which cometary activity onset is seen in our Solar System.

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