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Christopher S. Kochanek

Publications and source records attributed to Christopher S. Kochanek.

At least 73 records · Page 4Linked to original sources

Discovery of a Candidate Black Hole - Giant Star Binary System in the Galactic Field

We report the discovery of the first likely black hole in a non-interacting binary system with a field red giant. By combining radial velocity measurements from the Apache Point Observatory Galactic Evolution Experiment (APOGEE) with photometric variability data from the All-Sky Automated Survey for Supernovae (ASAS-SN), we identified the bright rapidly-rotating giant 2MASS J05215658+4359220 as a binary system with a massive unseen companion. Subsequent radial velocity measurements reveal a system with an orbital period of 83 days and near-zero eccentricity. The photometric variability period of the giant is consistent with the orbital period, indicative of star spots and tidal synchronization. Constraints on the giant's mass and radius from its luminosity, surface gravity, and temperature imply an unseen companion with mass of $3.3^{+2.8}_{-0.7}$ M$_\odot$, indicating a low-mass black hole or an exceedingly massive neutron star. Measurement of the astrometric binary motion by {\it Gaia} will further characterize the system. This discovery demonstrates the potential of massive spectroscopic surveys like APOGEE and all-sky, high-cadence photometric surveys like ASAS-SN to revolutionize our understanding of the compact object mass function, and to test theories of binary star evolution and the supernova mechanism.

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Accretion Disk Size Measurement and Time Delays in the Lensed Quasar WFI 2033-4723

We present 13 seasons of $R$-band photometry of the quadruply-lensed quasar WFI 2033-4723 from the 1.3m SMARTS telescope at CTIO and the 1.2m Euler Swiss Telescope at La Silla, in which we detect microlensing variability of $\sim0.2$ mags on a timescale of $\sim$6 years. Using a Bayesian Monte Carlo technique, we analyze the microlensing signal to obtain a measurement of the size of this system's accretion disk of $\log (r_s/{\rm cm}) = 15.86^{+0.25}_{-0.27}$ at $λ_{rest} = 2481{\rm Å}$, assuming a $60^\circ$ inclination angle. We confirm previous measurements of the BC and AB time delays, and we obtain a tentative measurement of the delay between the closely spaced A1 and A2 images of $Δt_{A1A2} = t_{A1} - t_{A2} = -3.9^{+3.4}_{-2.2}$ days. We conclude with an update to the Quasar Accretion Disk Size - Black Hole Mass Relation, in which we confirm that the accretion disk size predictions from simple thin disk theory are too small.

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Energetic eruptions leading to a peculiar hydrogen-rich explosion of a massive star

Every supernova hitherto observed has been considered to be the terminal explosion of a star. Moreover, all supernovae with absorption lines in their spectra show those lines decreasing in velocity over time, as the ejecta expand and thin, revealing slower moving material that was previously hidden. In addition, every supernova that exhibits the absorption lines of hydrogen has one main light-curve peak, or a plateau in luminosity, lasting approximately 100 days before declining. Here we report observations of iPTF14hls, an event that has spectra identical to a hydrogen-rich core-collapse supernova, but characteristics that differ extensively from those of known supernovae. The light curve has at least five peaks and remains bright for more than 600 days; the absorption lines show little to no decrease in velocity; and the radius of the line-forming region is more than an order of magnitude bigger than the radius of the photosphere derived from the continuum emission. These characteristics are consistent with a shell of several tens of solar masses ejected by the star at supernova-level energies a few hundred days before a terminal explosion. Another possible eruption was recorded at the same position in 1954. Multiple energetic pre-supernova eruptions are expected to occur in stars of 95-130 solar masses, which experience the pulsational pair instability. That model, however, does not account for the continued presence of hydrogen, or the energetics observed here. Another mechanism for the violent ejection of mass in massive stars may be required.

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A Nova Outburst Powered by Shocks

Classical novae are runaway thermonuclear burning events on the surfaces of accreting white dwarfs in close binary star systems, sometimes appearing as new naked-eye sources in the night sky. The standard model of novae predicts that their optical luminosity derives from energy released near the hot white dwarf which is reprocessed through the ejected material. Recent studies with the Fermi Large Area Telescope have shown that many classical novae are accompanied by gigaelectronvolt gamma-ray emission. This emission likely originates from strong shocks, providing new insights into the properties of nova outflows and allowing them to be used as laboratories to study the unknown efficiency of particle acceleration in shocks. Here we report gamma-ray and optical observations of the Milky Way nova ASASSN-16ma, which is among the brightest novae ever detected in gamma-rays. The gamma-ray and optical light curves show a remarkable correlation, implying that the majority of the optical light comes from reprocessed emission from shocks rather than the white dwarf. The ratio of gamma-ray to optical flux in ASASSN-16ma directly constrains the acceleration efficiency of non-thermal particles to be ~0.005, favouring hadronic models for the gamma-ray emission. The need to accelerate particles up to energies exceeding 100 gigaelectronvolts provides compelling evidence for magnetic field amplification in the shocks.

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The BOSS Emission-Line Lens Survey V. Morphology and Substructure of Lensed Lyman-$α$ Emitters at redshift $z\approx2.5$ in the BELLS GALLERY

We present a morphological study of the 17 lensed Lyman-$α$ emitter (LAE) galaxies of the Baryon Oscillation Spectroscopic Survey Emission-Line Lens Survey (BELLS) for the GALaxy-Ly$α$ EmitteR sYstems (BELLS GALLERY) sample. This analysis combines the magnification effect of strong galaxy-galaxy lensing with the high resolution of the Hubble Space Telescope ($HST$) to achieve a physical resolution of $\sim$80 pc for this $2<z<3$ LAE sample, allowing a detailed characterization of the LAE rest-frame ultraviolet continuum surface brightness profiles and substructure. We use lens-model reconstructions of the LAEs to identify and model individual clumps, which we subsequently use to constrain the parameters of a generative statistical model of the LAE population. Since the BELLS GALLERY sample is selected primarily on the basis of Lyman-$α$ emission, the LAEs that we study here are likely to be directly comparable to those selected in wide-field narrow-band LAE surveys, in contrast with the lensed LAEs identified in cluster lensing fields. We find an LAE clumpiness fraction of approximately 88%, significantly higher than found in previous (non-lensing) studies. We find a well-resolved characteristic clump half-light radii of $\sim$350 pc, a scale comparable to the largest H II regions seen in the local universe. This statistical characterization of LAE surface-brightness profiles will be incorporated into future lensing analyses using the BELLS GALLERY sample to constrain the incidence of dark-matter substructure in the foreground lensing galaxies.

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The X-ray and Mid-Infrared luminosities in Luminous Type 1 Quasars

Several recent studies have reported different intrinsic correlations between the AGN mid-IR luminosity ($L_{MIR}$) and the rest-frame 2-10 keV luminosity ($L_{X}$) for luminous quasars. To understand the origin of the difference in the observed $L_{X}-L_{MIR}$ relations, we study a sample of 3,247 spectroscopically confirmed type 1 AGNs collected from Boötes, XMM-COSMOS, XMM-XXL-North, and the SDSS quasars in the Swift/XRT footprint spanning over four orders of magnitude in luminosity. We carefully examine how different observational constraints impact the observed $L_{X}-L_{MIR}$ relations, including the inclusion of X-ray non-detected objects, possible X-ray absorption in type 1 AGNs, X-ray flux limits, and star formation contamination. We find that the primary factor driving the different $L_{X}-L_{MIR}$ relations reported in the literature is the X-ray flux limits for different studies. When taking these effects into account, we find that the X-ray luminosity and mid-IR luminosity (measured at rest-frame $6μm$, or $L_{6μm}$) of our sample of type 1 AGNs follow a bilinear relation in the log-log plane: $\log L_X =(0.84\pm0.03)\times\log L_{6μm}/10^{45}{\rm erg\;s^{-1}} + (44.60\pm0.01)$ for $L_{6μm} < 10^{44.79}{\rm erg\;s^{-1}} $, and $\log L_X = (0.40\pm0.03)\times\log L_{6μm}/10^{45}{\rm erg\;s^{-1}} +(44.51\pm0.01)$ for $L_{6μm} \geq 10^{44.79}{\rm erg\;s^{-1}} $. This suggests that the luminous type 1 quasars have a shallower $L_{X}-L_{MIR}$ correlation than the approximately linear relations found in local Seyfert galaxies. This result is consistent with previous studies reporting a luminosity-dependent $L_{X}-L_{MIR}$ relation, and implies that assuming a linear $L_{X}-L_{MIR}$ relation to infer the neutral gas column density for X-ray absorption might overestimate the column densities in luminous quasars.

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Survey of Period Variations of Superhumps in SU UMa-Type Dwarf Novae. IX: The Ninth Year (2016-2017)

Continuing the project described by Kato et al. (2009, arXiv:0905.1757), we collected times of superhump maxima for 127 SU UMa-type dwarf novae observed mainly during the 2016--2017 season and characterized these objects. We provide updated statistics of relation between the orbital period and the variation of superhumps, the relation between period variations and the rebrightening type in WZ Sge-type objects. We obtained the period minimum of 0.05290(2)d and confirmed the presence of the period gap above the orbital period ~0.09d. We note that four objects (NY Her, 1RXS J161659.5+620014, CRTS J033349.8-282244 and SDSS J153015.04+094946.3) have supercycles shorter than 100d but show infrequent normal outbursts. We consider that these objects are similar to V503 Cyg, whose normal outbursts are likely suppressed by a disk tilt. These four objects are excellent candidates to search for negative superhumps. DDE 48 appears to be a member of ER UMa-type dwarf novae. We identified a new eclipsing SU UMa-type object MASTER OT J220559.40-341434.9. We observed 21 WZ Sge-type dwarf novae during this interval and reported 18 out of them in this paper. Among them, ASASSN-16js is a good candidate for a period bouncer. ASASSN-16ia showed a precursor outburst for the first time in a WZ Sge-type superoutburst. ASASSN-16kg, CRTS J000130.5+050624 and SDSS J113551.09+532246.2 are located in the period gap. We have newly obtained 15 orbital periods, including periods from early superhumps.

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Discovery of a Very Bright and Intrinsically Very Luminous, Strongly Lensed Lyα Emitting Galaxy at z = 2.82 in the BOSS Emission-Line Lens Survey

We report the discovery of a very bright (r = 20.16), highly magnified, and yet intrinsically very luminous Lyα emitter (LAE) at z = 2.82. This system comprises four images in the observer plane with a maximum separation of ~ 6" and it is lensed by a z = 0.55 massive early-type galaxy. It was initially identified in the Baryon Oscillation Spectroscopic Survey (BOSS) Emission-Line Lens Survey for GALaxy-Lyα EmitteR sYstems (BELLS GALLERY) survey, and follow-up imaging and spectroscopic observations using the Gran Telescopio Canarias (GTC) and William Herschel Telescope (WHT) confirmed the lensing nature of this system. A lens model using a singular isothermal ellipsoid in an external shear field reproduces quite well the main features of the system, yielding an Einstein radius of 2.95" +/- 0.10", and a total magnification factor for the LAE of 8.8 +/- 0.4. This LAE is one of the brightest and most luminous galaxy-galaxy strong lenses known. We present initial imaging and spectroscopy showing the basic physical and morphological properties of this lensed system.

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The Mysterious Dimmings of the T Tauri Star V1334 Tau

We present the discovery of two extended $\sim$0.12 mag dimming events of the weak-lined T-Tauri star V1334. The start of the first event was missed but came to an end in late 2003, and the second began in February 2009, and continues as of November 2016. Since the egress of the current event has not yet been observed, it suggests a period of $>$13 years if this event is periodic. Spectroscopic observations suggest the presence of a small inner disk, although the spectral energy distribution shows no infrared excess. We explore the possibility that the dimming events are caused by an orbiting body (e.g. a disk warp or dust trap), enhanced disk winds, hydrodynamical fluctuations of the inner disk, or a significant increase in the magnetic field flux at the surface of the star. We also find a $\sim$0.32 day periodic photometric signal that persists throughout the 2009 dimming which appears to not be due to ellipsoidal variations from a close stellar companion. High precision photometric observations of V1334 Tau during K2 campaign 13, combined with simultaneous photometric and spectroscopic observations from the ground, will provide crucial information about the photometric variability and its origin.

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Extended X-ray Monitoring of Gravitational Lenses with Chandra and Joint Constraints on X-ray Emission Regions

We present an X-ray photometric analysis of six gravitationally lensed quasars, with observation campaigns spanning from 5 to 14 years, measuring the total (0.83 - 21.8 keV restframe), soft (0.83 - 3.6 keV), and hard (3.6 - 21.8 keV) band image flux ratios or each epoch. Using the ratios of the model-predicted macro-magnifications as baselines, we build differential microlensing light curves and obtain joint likelihood functions for the average X-ray emission region sizes. Our analysis yields a Probability Distribution Function for the average half-light radius of the X-Ray emission region in the sample that peaks slightly above 1 gravitational radius and with nearly indistinguishable 68% confidence (one-sided) upper limits of 17.8 and 18.9 gravitational radii for the soft and hard X-ray emitting regions, assuming a mean stellar mass of 0.3 solar masses. We see hints of energy dependent microlensing between the soft and hard bands in two of the objects. In a separate analysis on the root-mean-square of the microlensing variability, we find significant differences between the soft and hard bands but the sign of the difference is not consistent across the sample. This suggests the existence of some kind of spatial structure to the X-ray emission in an otherwise extremely compact source. We also discover a correlation between the RMS microlensing variability and the average microlensing amplitude.

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The BOSS Emission-Line Lens Survey. IV. : Smooth Lens Models for the BELLS GALLERY Sample

We present \textsl{Hubble Space Telescope} (\textsl{HST}) F606W-band imaging observations of 21 galaxy-Ly$α$ emitter lens candidates in the Baryon Oscillation Spectroscopic Survey (BOSS) Emission-Line Lens Survey (BELLS) for GALaxy-Ly$α$ EmitteR sYstems (BELLS GALLERY) survey. 17 systems are confirmed to be definite lenses with unambiguous evidence of multiple imaging. The lenses are primarily massive early-type galaxies (ETGs) at redshifts of approximately $0.55$, while the lensed sources are Ly$α$ emitters (LAEs) at redshifts from 2 to 3. Although the \textsl{HST} imaging data are well fit by smooth lens models consisting of singular isothermal ellipsoids in an external shear field, a thorough exploration of dark substructures in the lens galaxies is required. The Einstein radii of the BELLS GALLERY lenses are on average $60\%$ larger than those of the BELLS lenses because of the much higher source redshifts which will allow a detailed investigation of the radius evolution of the mass profile in ETGs. With the aid of the average $\sim 13 \times$ lensing magnification, the LAEs are resolved to comprise individual star-forming knots of a wide range of properties with characteristic sizes from less than 100 pc to several kpc, rest-frame far UV apparent AB magnitudes from 29.6 to 24.2, and typical projected separations of 500 pc to 2 kpc.

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The TDE ASASSN-14li and its host resolved at parsec scales with the EVN

We report European Very Long Baseline Interferometry Network (EVN) radio continuum observations of ASASSN-14li, one of the best studied tidal disruption events (TDEs) to date. At 1.7 GHz with ~12x6mas resolution, the emission is unresolved. At 5.0 GHz with ~3x2mas resolution, the radio emission shows an extended structure that can be modeled with two components: a core-like component and a fainter, possibly elongated source 4.3mas (~2pc) away. Our observations are not conclusive as to the nature of the components, but three scenarios are discussed. One possibility is a core-jet/outflow morphology, thus making of ASASSN-14li the first TDE jet/outflow directly imaged. For this case, the projected separation between the two components can only be explained by superluminal motion, rather than the lower velocities inferred from low-resolution radio observations. However, typical fast moving jets have brightness temperatures ~5 orders of magnitude higher than we find, thus making this scenario less likely. The second possibility is that we are imaging a non-relativistic jet from past AGN/TDE activity. In this case a past TDE is preferred given that the spatial extension and radio luminosity of the elongated component are consistent with the theoretical predictions for a TDE outflow. Alternatively, the two sources could indicate the presence of a binary black hole, which would then naturally explain the enhanced TDE rates of post-starburst galaxies. Future EVN observations will help us to distinguish between these scenarios.

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The Eruption of the Candidate Young Star ASASSN-15qi

Outbursts on young stars are usually interpreted as accretion bursts caused by instabilities in the disk or the star-disk connection. However, some protostellar outbursts may not fit into this framework. In this paper, we analyze optical and near-infrared spectra and photometry to characterize the 2015 outburst of the probable young star ASASSN-15qi. The $\sim 3.5$ mag brightening in the $V$ band was sudden, with an unresolved rise time of less than one day. The outburst decayed exponentially by 1 mag for 6 days and then gradually back to the pre-outburst level after 200 days. The outburst is dominated by emission from $\sim10,000$ K gas. An explosive release of energy accelerated matter from the star in all directions, seen in a spectacular cool, spherical wind with a maximum velocity of 1000 km/s. The wind and hot gas both disappeared as the outburst faded and the source returned to its quiescent F-star spectrum. Nebulosity near the star brightened with a delay of 10-20 days. Fluorescent excitation of H$_2$ is detected in emission from vibrational levels as high as $v=11$, also with a possible time delay in flux increase. The mid-infrared spectral energy distribution does not indicate the presence of warm dust emission, although the optical photospheric absorption and CO overtone emission could be related to a gaseous disk. Archival photometry reveals a prior outburst in 1976. Although we speculate about possible causes for this outburst, none of the explanations are compelling.

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The BOSS Emission-Line Lens Survey. III. : Strong Lensing of Ly$α$ Emitters by Individual Galaxies

We introduce the Baryon Oscillation Spectroscopic Survey (BOSS) Emission-Line Lens Survey (BELLS) for GALaxy-Ly$α$ EmitteR sYstems (BELLS GALLERY) Survey, which is a Hubble Space Telescope program to image a sample of galaxy-scale strong gravitational lens candidate systems with high-redshift Ly$α$ emitters (LAEs) as the background sources. The goal of the BELLS GALLERY Survey is to illuminate dark substructures in galaxy-scale halos by exploiting the small-scale clumpiness of rest-frame far-UV emission in lensed LAEs, and to thereby constrain the slope and normalization of the substructure-mass function. In this paper, we describe in detail the spectroscopic strong-lens selection technique, which is based on methods adopted in the previous Sloan Lens ACS (SLACS) Survey, BELLS, and SLACS for the Masses Survey. We present the BELLS GALLERY sample of the 21 highest-quality galaxy--LAE candidates selected from $\approx 1.4 \times 10^6$ galaxy spectra in the BOSS of the Sloan Digital Sky Survey III. These systems consist of massive galaxies at redshifts of approximately 0.5 strongly lensing LAEs at redshifts from 2--3. The compact nature of LAEs makes them an ideal probe of dark substructures, with a substructure-mass sensitivity that is unprecedented in other optical strong-lens samples. The magnification effect from lensing will also reveal the structure of LAEs below 100 pc scales, providing a detailed look at the sites of the most concentrated unobscured star formation in the universe. The source code used for candidate selection is available for download as a part of this release.

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DM Ori: A Young Star Occulted by a Disturbance in its Protoplanetary Disk

In some planet formation theories, protoplanets grow gravitationally within a young star's protoplanetary disk, a signature of which may be a localized disturbance in the disk's radial and/or vertical structure. Using time-series photometric observations by the Kilodegree Extremely Little Telescope South (KELT-South) project and the All-Sky Automated Survey for SuperNovae (ASAS-SN), combined with archival observations, we present the discovery of two extended dimming events of the young star, DM Ori. This young system faded by $\sim$1.5 mag from 2000 March to 2002 August and then again in 2013 January until 2014 September (depth $\sim$1.7 mag). We constrain the duration of the 2000-2002 dimming to be $<$ 860 days, and the event in 2013-2014 to be $<$ 585 days, separated by $\sim$12.5 years. A model of the spectral energy distribution (SED) indicates a large infrared excess consistent with an extensive circumstellar disk. Using basic kinematic arguments, we propose that DM Ori is likely being periodically occulted by a feature (possibly a warp or perturbation) in its circumstellar disk. In this scenario, the occulting feature is located $>$6 AU from the host star, moving at $\sim$14.6 km/s, and is $\sim$4.9 AU in width. This localized structure may indicate a disturbance such as may be caused by a protoplanet early in its formation.

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Survey of Period Variations of Superhumps in SU UMa-Type Dwarf Novae. VIII: The Eighth Year (2015-2016)

Continuing the project described by Kato et al. (2009, arXiv:0905.1757), we collected times of superhump maxima for 128 SU UMa-type dwarf novae observed mainly during the 2015-2016 season and characterized these objects. The data have improved the distribution of orbital periods, the relation between the orbital period and the variation of superhumps, the relation between period variations and the rebrightening type in WZ Sge-type objects. Coupled with new measurements of mass ratios using growing stages of superhumps, we now have a clearer and statistically greatly improved evolutionary path near the terminal stage of evolution of cataclysmic variables. Three objects (V452 Cas, KK Tel, ASASSN-15cl) appear to have slowly growing superhumps, which is proposed to reflect the slow growth of the 3:1 resonance near the stability border. ASASSN-15sl, ASASSN-15ux, SDSS J074859.55+312512.6 and CRTS J200331.3-284941 are newly identified eclipsing SU UMa-type (or WZ Sge-type) dwarf novae. ASASSN-15cy has a short (~0.050 d) superhump period and appears to belong to EI Psc-type objects with compact secondaries having an evolved core. ASASSN-15gn, ASASSN-15hn, ASASSN-15kh and ASASSN-16bu are candidate period bouncers with superhump periods longer than 0.06 d. We have newly obtained superhump periods for 79 objects and 13 orbital periods, including periods from early superhumps. In order that the future observations will be more astrophysically beneficial and rewarding to observers, we propose guidelines how to organize observations of various superoutbursts.

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An Extreme Analogue of $ε$ Aurigae: An M-giant Eclipsed Every 69 Years by a Large Opaque Disk Surrounding a Small Hot Source

We present TYC 2505-672-1 as a newly discovered and remarkable eclipsing system comprising an M-type red giant that undergoes a ~3.45 year long, near-total eclipse (depth of ~4.5 mag) with a very long period of ~69.1 yr. TYC 2505-672-1 is now the longest-period eclipsing binary system yet discovered, more than twice as long as that of the currently longest-period system, $ε$ Aurigae. We show from analysis of the light curve including both our own data and historical data spanning more than 120 yr and from modeling of the spectral energy distribution, both before and during eclipse, that the red giant primary is orbited by a moderately hot source (T$_{eff}$~8000 K) that is itself surrounded by an extended, opaque circumstellar disk. From the measured ratio of luminosities, the radius of the hot companion must be in the range 0.1-0.5 Rsun (depending on the assumed radius of the red giant primary), which is an order of magnitude smaller than that for a main sequence A star and 1-2 orders of magnitude larger than that for a white dwarf. The companion is therefore most likely a "stripped red giant" subdwarf-B type star destined to become a He white dwarf. It is however somewhat cooler than most sdB stars, implying a very low mass for this "pre-He-WD" star. The opaque disk surrounding this hot source may be a remnant of the stripping of its former hydrogen envelope. However, it is puzzling how this object became stripped, given that it is at present so distant (orbital semi-major axis of ~24 AU) from the current red giant primary star. Extrapolating from our calculated ephemeris, the next eclipse should begin in early UT 2080 April and end in mid UT 2083 September (eclipse center UT 2081 December 24). This system is poised to become an exemplar of a very rare class of systems, even more extreme in several respects than the well studied archetype $ε$ Aurigae.

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Quasar Variability in the Mid-Infrared

The Decadal IRAC Bootes Survey is a mid-IR variability survey of the ~9 sq. deg. of the NDWFS Bootes Field and extends the time baseline of its predecessor, the Spitzer Deep, Wide-Field Survey (SDWFS), from 4 to 10 years. The Spitzer Space Telescope visited the field five times between 2004 and 2014 at 3.6 and 4.5 microns. We provide the difference image analysis photometry for a half a million mostly extragalactic sources. In mid-IR color-color plane, sources with quasar colors constitute the largest variability class (75%), 16% of the variable objects have stellar colors and the remaining 9% have the colors of galaxies. Adding the fifth epoch doubles the number of variable active galactic nuclei (AGNs) for the same false positive rates as in SDWFS, or increases the number of sources by 20% while decreasing the false positive rates by factors of 2-3 for the same variability amplitude. We quantify the ensemble mid-IR variability of ~1500 spectroscopically confirmed AGNs using single power-law structure functions (SFs), which we find to be steeper (index $γ=0.45$) than in the optical ($γ=0.3$), leading to much lower amplitudes at short time-lags. This provides evidence for large emission regions, smoothing out any fast UV/optical variations, as the origin of infrared quasar variability. The mid-IR AGN SF slope $γ$ seems to be uncorrelated with both the luminosity and rest-frame wavelength, while the amplitude shows an anti-correlation with the luminosity and a correlation with the rest-frame wavelength.

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