The Most Powerful Lenses in the Universe: Quasar Micro-lensing Dave Pooley (Trinity University) Paul Schechter (MIT), Joachim Wambsganss (Heidelberg), Saul Rappaport (MIT), Jeffrey Blackburne (Arete), Jordan Koeller, Brian Guerrero (Trinity), Naim Barnett (Trinity), Jackson Braley (Trinity) Subject: Chandra data Date: December 7, 1999 at 7:56:02 PM EST This Saturday will be my To: [email protected], [email protected], [email protected]

th From [email protected] Tue Dec 7 17:58:28 1999 20 anniversary of Message-Id: <[email protected]> X-Mailer: exmh version 2.0.2 2/24/98 To: LEWIN (Walter Lewin) analyzing Chandra data. cc: [email protected] (Pepi Fabbiano), [email protected], [email protected] (Arcops Account), [email protected] (Joy S. Nichols), [email protected] (Roger J. Brissenden) Subject: Your Chandra Observation 500059 Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Date: Tue, 07 Dec 1999 17:57:27 -0500 SN 1999em, ObsID 763 From: "Tomas P. Girnius" Status: RO

Dear Dr. Lewin,

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The .contents file contains a listing of the .tar file. The vv file contains the V&V report. Fundamental astronomy has a fundamental problem. Nobody ever measures the stellar mass. That is not a measurable thing; it’s an inferred quantity. You measure light, OK? You can measure light in many bands, but you infer stellar mass. Everybody seems to agree on certain assumptions that are completely unproven. — Carlos Frenk, 2017 May 15

The Astrophysical Journal, 845:157 (18pp), 2017 August 20 https://doi.org/10.3847/1538-4357/aa816d © 2017. The American Astronomical Society. All rights reserved.

The Initial Mass Function in the Nearest Strong Lenses from SNELLS: Assessing the Consistency of Lensing, Dynamical, and Spectroscopic Constraints

Andrew B. Newman1 , Russell J. Smith2 , Charlie Conroy3 , Alexa Villaume4 , and Pieter van Dokkum5 1 The Observatories of the Carnegie Institution for Science, Pasadena, CA, USA; [email protected] 2 Centre for Extragalactic Astronomy, University of Durham, South Road, Durham, UK 3 Department of Astronomy, Harvard University, Cambridge, MA, USA 4 Department ofMain Astronomy and sources Astrophysics, University of ofuncertainty California, Santa Cruz, CA, USA 5 Department of Astrophysical Sciences, Yale University, New Haven, CT, USA Received 2016 December 16;lensing: revised 2017 July 5;contribution accepted 2017 July 19; published of 2017 dark August 22 matter stellar dynamics:Abstractcontribution of dark matter We present new observations of the three nearest early-type galaxy (ETG) strong lenses discovered in the SINFONI Nearby Elliptical Lens Locator Survey (SNELLS). Based on their lensing masses, these ETGs were inferredstellar to have a stellar pop. initial masssynthesis: function (IMF) consistentlow-mass with that of the cutoff Milky Way, of not theIMF bottom-heavy IMF that has been reported as typical for high-σ ETGs based on lensing, dynamical, and stellar population synthesis techniques. We use these unique systems to test the consistency of IMF estimates derived from different The Astrophysical Journal, 837:166 (8pp), 2017 Marchfi 10 https://doi.org/10.3847/1538-4357/aa6190fi methods. We rst estimate the stellar M*/L using lensing and stellar dynamics. We then t high-quality“To illustrate optical the sensitivity of © 2017. The American Astronomical Society. All rightsspectra reserved. of the lenses using an updated version of the stellar population synthesis models developed by Conroy & van Dokkum. When examined individually, we find good agreement among these methods for onethe galaxy. total The mass to the cutoff, for a The Stellar Initialother Mass two Functiongalaxies show in 2– Early-type3σ tension with Galaxies lensing estimates, from dependingAbsorption on the Line dark matter contribution,single whenpower law with α=2.7, considering IMFs that extend to 0.08 Me. Allowing a variable low-mass cutoff or a nonparametric form of the IMF Spectroscopy.reduces IV. the A tension Super-Salpeter among the IMF IMF estimates in the to < Center2σ. There of is moderate NGC 1407 evidence for a reduced numberthe mass-to-light of low- ratio is 70% mass stars infrom the SNELLS Non-parametric spectra, but no Modelssuch evidence in a composite spectrum of matched-σ ETGs drawn from the SDSS. Such variation in the form of the IMF at low stellar masses (m 0.3 M ), if present, couldhigher reconcile if the cutoff is 0.05M 1 2 3  e ⊙ lensingCharlie/dynamical Conroy , Pieter and spectroscopic G. van Dokkum IMF, and estimates Alexa Villaume for the SNELLS lenses and account for their lighter M /L 1 Department of Astronomy, Harvard University, Cambridge, MA, 02138, USA compared* to 0.08M .” relative2 Department to the of mean Astronomy, matched- Yale University,σ ETG. New We Haven, provide CT, 06511, the spectra USA used in this study to facilitate future comparisons. ⊙ 3 DepartmentKey of words: Astronomygalaxies: and Astrophysics, elliptical University and lenticular, of California, cD Santa– galaxies: Cruz, CA 95064, stellar USA content – gravitational lensing: strong Received 2016 November 30; revised 2017 February 8; accepted 2017 February 16; published 2017 March 15 Supporting material: machine-readable table Abstract It is now well-established that the stellar initial mass function (IMF) can be determined from the absorption line spectra of old stellar systems, and this1. has Introduction been used to measure the IMF and its variationanomalies, across which the early-type Schechter et al. (2014) have demonstrated using ten multiply imaged quasars. galaxy population.The Previous stellar initial work mass focused function on measuring(IMF) is the both slope fundamental of the IMF over one or more stellar mass The second method is based on a detailed analysis of absorption intervals, implicitlyas a basic assuming outcome that this of the is a star good formation description process of the IMF and anda critical that the IMF has a universal low-mass cutoff. In this work we consider more flexible IMFs, including two-component powerline laws spectra. with a variable Recent low- stellar population synthesis (SPS) models ingredient for interpreting numerous extragalactic observations. mass cutoff and a general non-parametric model. We demonstrate with mock spectra thathave the detailedopened upshape studies of the of the chemical abundance patterns and Although the IMF is consistent with being independent of the−1 IMF can be accurately recovered as long as the data quality is high (S/N  300 Å ) andthe cover IMF a wide in unresolved wavelength old populations at a remarkable level of range (0.4–1.0environmentμm). We apply within these theflexible Milky IMF Way models(e.g., Bastian to a high et S al./N2010 spectrum), ofdetail the center(Conroy of the & massive van Dokkum 2012a, 2012b;vanDokkum& elliptical galaxythere NGC is little 1407. theoretical Fitting the reason spectrum for with this non-parametric to hold across IMFs, all star- we find thatConroy the IMF2012;LaBarberaetal. in the center 2013;Spinielloetal.2014).While shows a continuousforming rise regions extending where toward the physical the hydrogen-burning conditions can limit, vary with widely a behaviorvarying that is well-approximated the age and metallicity imparts large and well-known by a power law(e.g., with Hennebelle an index of − &2.7. Chabrier These results2008; provide Krumholz strong et evidence al. 2011 for; the existencechanges of to extreme integrated(super- galaxy spectra, variations in the IMF also Salpeter) IMFs in the cores of massive galaxies. Hopkins 2012, 2013; Chabrier et al. 2014). Since only the affect surface gravity-sensitive absorption lines at levels that are Key words: galaxies:brightest evolution stars beyond– galaxies: the local stellar volume content – canstars: be luminosity individually function,subtle mass function but detectable (;1%) with high-quality data. detected, constraints on the IMF in external systems are When these two methods are applied to ETGs, both are challenging and must rely on integrated light observations. consistent with a trend of “heavier” IMFs in higher-σ (Cappellari 1.Two Introduction such approaches have been applied tofor early-type radial variation galaxies in the IMF (McConnell et al. 2016; Zieleniewski et al. 2017), thoughet al. 2013a it is not; La clear Barbera if the different et al. 2015), more metal-rich (Martín- The initial mass function(ETGs).The(IMFfi) rstconnects is to model a variety the total of mass distribution using conclusions are due to smallNavarro sample et al.sizes2015c or the), or analysis more α-enhanced (Conroy & van astrophysical phenomenagravitational including lensing, star formation, stellar kinematics, stellar or both in combination techniques. The existence ofDokkum strong IMF2012b gradients) galaxies. implies(Since that these quantities are mutually feedback, the relative number of stellar remnants, and heavy (Auger et al. 2010b;Treuetal.2010;Spinielloetal.direct comparison2011 between; differentcorrelated, techniques the primary requires driver careful of the correlation is more difficult element production, andThomas is a critical et al. ingredient2011;Barnabèetal. in modeling2013 the ;Cappellarietal.consideration2013a of aperture; effects.to establish.) The “heaviness” of the IMF can be expressed via integrated light from distantConroy galaxies. et al. 2013 Precise;Newmanetal. knowledge of2013a the , 2013b).Sincetheradial When inferring the IMF fromthe mass the depth factor ofα stellar= (M absorption*/L)/(M*/L)MW, where (M*/L)MW is IMF is essential fordistributions understanding and the shapes dark of matter the dark content matter andlines, stars all are analyses thought to to date have focused on parameterized IMFs, fi within galaxies (e.g., Cappellari et al. 2006) and measuring inferred from SPS models assuming a ducial IMF and thereby be distinct, it is possible to separate the componentse.g., a single with powerthe aid law ornormalizes a broken power differences law, with in a fi agexed or metallicity. Lensing and supermassive black hole masses (e.g., Gebhardt & Tho- low-mass cutoff. Recently, Spiniello et al. (2015) and mas 2009; McConnellof et a al. few2012 assumptions.). Unfortunately, The the principal IMF is assumptions are that the dynamical methods infer higher α in high-σ ETGs, but cannot fi stellar mass density follows the luminosity densityLyubenova and et that al. the(2016) distinguishhave combined between stellar an population increased number of low-mass stars very dif cult to measure when individual stars cannot be and dynamical constraints for individual galaxies in order to dark matter profile follows a parameterized form, such as a power versus stellar remnants, while absorption line studies of such resolved, owing to the intrinsic faintness of lower main place stronger constraints on the shape of the IMF. law or an NFW halo (Navarro-Frenk-White, Navarro et al. 1996). sequence stars relative to the evolved giants (e.g., Conroy & Here we go one step furthergalaxies and directly attempt point to constrain to a larger the number of low-mass (< 1 Me) van Dokkum 2012a). This leads to an integral constraint on thegeneral IMF, i.e., formM* of/L.A the IMFstars, directly i.e., from a “bottom-heavy the absorption” IMF. line Nonetheless, in recentpromising years there alternative have been route many to reportedM*/L is to measurespectra of the old“graini- stellar systems.This We apparent consider convergence several models, of entirely independent techni- measurements of theness IMF” inof distant the gravitational galaxies by potential a variety via of microlensing-inducedincluding a two-partflux power lawques with has a variable lent credence low-mass to cutoff claims of a non-universal IMF. techniques, including kinematic analysis, strong lensing, stellar and a general non-parametric IMF (see also Dries et al. 2016, population synthesis, and microlensing (e.g., Cenarro who explored the ability1 to reconstruct the IMF from mock et al. 2003; Treu et al. 2010; van Dokkum & Conroy 2010; data). Our goals are (1) to assess what is measurable from Cappellari et al. 2012; Conroy & van Dokkum 2012b; Spiniello absorption line spectra in the idealized limit of perfect models, et al. 2012; Conroy et al. 2013; Ferreras et al. 2013; La Barbera and (2) to reconstruct from real data the IMF under different et al. 2013; McDermid et al. 2014; Schechter et al. 2014; assumptions and assess how the derived mass-to-light ratio Posacki et al. 2015; Lyubenova et al. 2016). The emerging depends on these assumptions. consensus is that the IMF appears to vary systematically, from a Milky Way-like IMF (Kroupa 2001; Chabrier 2003) at lower galaxy masses, to a Salpeter (1955) IMF, or even more bottom- 2. Models and Spectral Fitting heavy IMF, for the most massive galaxies. This consensus is not without challenges (e.g., Smith 2014; Smith et al. 2015; 2.1. Overview Newman et al. 2016). Our approach to constructing stellar population models At the same time, evidence has emerged that the IMF may follows (Conroy & van Dokkum 2012a, CvD12a), with several vary strongly within galaxies, at least at high masses (Martín- important updates. First and foremost, the models now span a Navarro et al. 2015; La Barbera et al. 2016; van Dokkum et al. wider range of ages and metallicities, with ages extending from 2016; Davis & McDermid 2017). Not all authors find evidence 1–13.5 Gyr and metallicities from [Z/H]=−1.5 to [Z/

1 Chandra enabled the use of micro-lensing as powerful tool.

Magellan Lensing Model Chandra X-ray Observatory Flux ratio anomalies in lensed images of quasar quasars were known in the optical for decades, but the explanation was unclear Lensing galaxy (millilensing vs. microlensing). should be brightest image anomalously dim in X-rays RX J1131−1231 RX J1131−1231 RX J1131−1231 Blackburne, DP, & Rappaport 2006 DM subhalos✘ stars ✔

Chandra observations established that stronger X-ray anomalies are nearly universal, indicating microlensing is the cause of the anomalies. DP et al. 2007

This allowed microlensing to be used as a tool to study both the sources (quasars) and the lenses (elliptical galaxies) on scales of micro-arseconds.

1/2 Einstein radius of a star in m Gpc typical lensing galaxy θEin ≈ 3 μas ( M⊙ ) ( DL ) The probability of strong micro-lensing effects is a non-monotonic function of stellar density.

1% stars 10% stars 100% stars 0.05 0.10

0.15 0.04 0.08

0.03 0.06 0.10

0.02 0.04

Relative Frequency 0.05 Relative Frequency Relative Frequency 0.01 0.02

0.00 0.00 0.00 0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00 0.01 0.10 1.00 10.00 100.00 Magnification (relative to average) Magnification (relative to average) Magnification (relative to average) LensingThe Astrophysical galaxy Journal ,744:111(13pp),2012January10contents: Pooley et al. The ensemble average stellar fraction is ~7% at typical impact

parameterThe Astrophysical Journal of,744:111(13pp),2012January10 7 kpc. Pooley et al. Matter in Relative Probability Stars 1.5% 0 2.2% 0.003 3.2% 0.034 The Astrophysical Journal,744:111(13pp),2012January10 Pooley et al. × 4.6% 0.186

The Astrophysical Journal,744:111(13pp),2012January10 6.8% Pooley et al. 0.356 10% 0.237 = 15% 0.08 22% 0.028 The Astrophysical Journal,744:111(13pp),2012January10 Pooley et al. × 32% 0.006 ensemble of 13 systems 46% 0.005 68% 0.01 100% 0.056

DP et al. 2012

Figure 7. Probability distributions for the stellar fraction, Sj,atthecharacteristicradialdistanceRc from the center of the lensing galaxy for 14 quadruply lensed quasar systems. Those labeled in italics do not have a measured lens redshift zl.

which can be seen in the bottom panels of Figure 6 using values as the internal motions of the microlensing stars, can be thought fromFigure ObsID 7. Probability 363. distributions for the stellar fraction, Sj,atthecharacteristicradialdistanceof as an effectiveRc from the motion center of of the the lensing source galaxy through for 14 quadruply the field lensed of quasarAll systems. of the above Those labeled has been in italics workeddo not out have for a a measured single observation lens redshift zl. the microlensing map (Wyithe et al. 2000). The more time of a system, but several systems have been observed multiple between observations, the higher the chance that the source times with Chandra. We combine these multiple observations is in a different enough region of the map to be considered usingwhich conditional can be seen probability: in the bottom panels of Figure 6 using values anas the independent internal motions sampling of the of it. microlensing We therefore stars, include can be a thought term in from ObsID 363. Pof(obs ask an)proportionaltohowisolatedintimetheobservationis, effective motion of the source through the field of All of the above has been worked out for a single observation the microlensing map (Wyithe et al. 2000). The more time P (Sj ) P (Sj obsk)P (obsk), (15) defined as the sum of the intervals between the observation and = | of a system, but several systemsk have been observed multiple allbetween other observations. observations, the higher the chance that the source ! times with Chandra. We combine these multiple observations isThe in a second different ingredient enough region in P (obs ofk the)isbasedonthequality map to be considered whereusing conditional we take P probability:(obsk)asaweightingfactor(normalizedto ofan the independent information sampling that the of observation it. We therefore provides. include Observations a term in unity) that combines two measures of the effectiveness of the whichP (obsk yield)proportionaltohowisolatedintimetheobservationis, tight constraints on the individual fractions and observationFigure 7. Probability toP provide(S distributionsj ) uniqueP for( andS thej obs stellarusefulk)P fraction,(obs information.k),S ,atthecharacteristicradialdistance(15) thedefined total as fluxR thefrom consequently sum the of center the intervals of give the lensing much between galaxy better thefor defined 14 observation quadruply probability lensed and = | j c quasarThe systems. first ingredient Those labeled in inkPitalics(obsdok)concernstheuniquenessof not have a measured lens redshift zl. functionsall other observations. for the stellar fraction (we point out specific examples ! the information from the observation. Over time, the proper below).The second We use ingredient the measured in P (obs uncertaintiesk)isbasedonthequality (Table 1)onthe where we take P (obs )asaweightingfactor(normalizedto Figure 7. Probability distributions for the stellarmotions fraction, S of,atthecharacteristicradialdistance the lensingk galaxy and backgroundR from the quasar, center of as the well lensing galaxyfractionsof the for 14information quadruply(symmetrized) lensed that the and observation the total flux provides. to calculate Observations this. Our unity) thatj combines two measures of thec effectiveness of the quasar systems. Those labeled in italics do not havewhich a measured can be lens seen redshift in thezl. bottom panels of Figure 6 using values whichas the internal yield tight motions constraints of the microlensing on the individual stars, can fractions be thought and observationfrom ObsID to 363. provide unique and useful information. theof as total an flux effective consequently motion give of the much source better through defined the probability field of 9 TheAll of first the ingredient above has been in P (obs workedk)concernstheuniquenessof out for a single observation functionsthe microlensing for the stellar map fraction (Wyithe (we et al.point2000 out). specific The more examples time which can be seen in the bottom panels ofofthe Figure a information system,6 using but values from several the systems observation.as the have internal been Over motions observed time, of the themultiple proper microlensingbelow).between stars, canWe observations, be use thought the measured the higher uncertainties the chance (Table that the1)onthe source from ObsID 363. timesmotions with ofChandra the lensing. We galaxy combineof and as thesebackgroundan effective multiple quasar, motion observations as of well the sourcefractionsis through in a different (symmetrized) the field enough of and region the total of the flux map to calculate to be considered this. Our All of the above has been worked out forusing a single conditional observation probability:the microlensing map (Wyithe et al. 2000an independent). The more sampling time of it. We therefore include a term in of a system, but several systems have been observed multiple between observations, the higher the9 chanceP (obsk)proportionaltohowisolatedintimetheobservationis, that the source times with Chandra. We combine theseFigure multiple 7. Probability observationsP (S distributionsj ) P foris(S the inj obs stellar a differentk)P fraction,(obsk) enough,S ,atthecharacteristicradialdistance region(15) of thedefined map to as beR the consideredfrom sum the of center the intervals of the lensing between galaxy thefor 14 observation quadruply lensed and = | j c using conditional probability: quasar systems. Those labeled ink italicsando independent not have a measured sampling lens ofredshift it. Wezl. thereforeall other include observations. a term in ! P (obsk)proportionaltohowisolatedintimetheobservationis,The second ingredient in P (obsk)isbasedonthequality where we take P (obs )asaweightingfactor(normalizedto P (Sj ) P (Sj obsk)P (obsk), (15)k defined as the sum of the intervals betweenof the the information observation that and the observation provides. Observations = | unity) that combines two measures of the effectiveness of the k which can be seen in the bottomall otherpanels observations. of Figure 6 using values whichas the internal yield tight motions constraints of the microlensing on the individual stars, can fractions be thought and ! observationfrom ObsID to 363. provide unique andThe useful second information. ingredient in P (obsk)isbasedonthequalitytheof as total an flux effective consequently motion give of the much source better through defined the probability field of where we take P (obsk)asaweightingfactor(normalizedtoTheAll of first the ingredient above has been in P (obs workedof thek)concernstheuniquenessof information out for a single that observation the observationfunctionsthe provides. microlensing forObservations the stellar map fraction (Wyithe (we et al.point2000 out). specific The more examples time unity) that combines two measures of theofthe a effectiveness information system, but from ofseveral the the systems observation.which have yield been Over tight observed time, constraints the multiple proper on the individualbelow).between We observations, fractions use the and measured the higher uncertainties the chance (Table that the1)onthe source observation to provide unique and usefultimesmotions information. with ofChandra the lensing. We galaxy combinethe and total thesebackground flux multiple consequently quasar, observations as give well much betterfractionsis in defined a different (symmetrized) probability enough and region the total of the flux map to calculate to be considered this. Our The first ingredient in P (obsk)concernstheuniquenessofusing conditional probability:functions for the stellar fraction (we pointan out independent specific examples sampling of it. We therefore include a term in the information from the observation. Over time, the proper below). We use the measured uncertainties9 P (obs (Tablek)proportionaltohowisolatedintimetheobservationis,1)onthe motions of the lensing galaxy and background quasar,P as(Sj well) Pfractions(Sj obsk) (symmetrized)P (obsk), and the(15) total fluxdefined to calculate as the sumthis. ofOur the intervals between the observation and = | k all other observations. ! 9 The second ingredient in P (obsk)isbasedonthequality where we take P (obsk)asaweightingfactor(normalizedto of the information that the observation provides. Observations unity) that combines two measures of the effectiveness of the which yield tight constraints on the individual fractions and observation to provide unique and useful information. the total flux consequently give much better defined probability The first ingredient in P (obsk)concernstheuniquenessof functions for the stellar fraction (we point out specific examples the information from the observation. Over time, the proper below). We use the measured uncertainties (Table 1)onthe motions of the lensing galaxy and background quasar, as well fractions (symmetrized) and the total flux to calculate this. Our

9 Lensing galaxy contents: Chandra data determine stellar M/L via the Fundamental Plane. • Overall mass density of lensing galaxy is known from macro-lensing. • Chandra gives level of micro-lensing → mass in individual stars, including stellar remnants, brown dwarfs, and red dwarfs too faint to produce photometric or spectroscopic signatures. • We assess stellar M/L via a calibration factor ℱ that multiplies the stellar mass fundamental plane. The Astrophysical Journal,793:96(18pp),2014October1 Schechter et al.

0.77 < ℱ < 2.10 Uncertainty dominated by small sample size.

Figure 2. Likelihood of the calibration factor applied to the stellar mass fundamental plane to computeSchechter the probability et al. distribution 2014 of micro-lensing fluctuations. The dashed line shows the median likelihood value,F 1.23. F = the constant c in the stellar mass fundamental plane down by data, it would seem best to use an Einstein ring radius likewise 2 1.453 afactor1/f . But, since Σe re− , the net factor by which uncorrected for a possible mass sheet. the predicted surface densities∼ are smaller is 1/f 0.547. There is, however, some reason to believe (Holder & Schechter 2003)thatquadruplyimagedquasarsexperience stronger shear than the SLACS and SL2S lensing galaxies. 7.6. Effects of Mass Sheet Degeneracy Koopmans et al. (2006)placealimitontheexternalshearfor asubsetoftheSLACSlensesof0.035.Bycontrast,weseein The well known mass sheet degeneracy permits the addition Table 6 that the typical shear for our lensed quasars is 0.1. On of a uniform surface density mass sheet to a lens model that, with the hypothesis that the external shear is the result of a larger corresponding adjustment of the model parameters, produces the isothermal cluster, the additional convergence would be larger same image positions and relative magnifications. To the extent for the present sample than for the SLACS sample. that lensing galaxies lie in clusters of galaxies, the cluster dark The effect would not be large except for the case of RX matter along the line of sight to the lens acts as such a mass J0911+0551, for which the lensing galaxy is clearly part of a sheet. cluster of galaxies. However, the shear, with γ 0.294, does In one of our lenses, PG 1115+080, we have taken this not seem to be directed to the center of the cluster= (Kneib et al. into account explicitly by modeling the associated group of 2000). galaxies as an isothermal sphere, contributing an additional convergence at the position of the lens galaxy of ∆κ 0.105. = The convergence for an isothermal sphere is equal to the shear, 7.7. Systematic Errors in Lens Model and QSO Magnifications so to gauge the possible effect of the mass sheet degeneracy, we might add a convergence, ∆κ,comparabletotheexternalshear In modeling the expected fluctuations, one must specify that we measure. a convergence, κ,andashear,γ at the positions of the Such an additional smooth contribution to the convergence quasar images. These depend upon the particular model for does not change the magnification histogram. It does, however, the gravitational lens potential. Our adopted SIE+X model is a increase the observed size of the Einstein ring, by a factor singular isothermal ellipsoid with ellipticity and position angle (1 κ)/(1 κ ∆κ). Our proxy velocity dispersion is taken taken from optical observations, with an external source of tidal to be− a property− − of the lensing galaxy as opposed to the galaxy shear providing as much if not more quadrupole than the SIE. plus mass sheet system. In the presence of a mass sheet we then Acommonlyadoptedalternative(andtheoneexplicitly overestimate this as well. adopted by the SLACS and SL2S groups) is to attribute all Our convergence values cluster around κ 1/2, as expected of the quadrupole to an SIE. While there are several systems for for an isothermal sphere. The observed Einstein≈ ring radii which the SIE is manifestly inferior (there are obvious sources of therefore overestimate the velocity dispersion in the galaxy by tides), we have constructed SIE models for our systems and use afactor (1 2∆κ). We see from Table 1 that for the same them to gauge how large a systematic error we might be making measured≈ effective− radius we will get a smaller predicted surface in adopting our SIE+X models. For the SIE models we find a mass density, by a factor (1 2∆κ)1.748.Thesewouldleadto median likelihood value for the factor by which the Salpeter larger values of the calibration≈ − factor . stellar mass must be multiplied which is lower by roughly 17% While it might seem to be more appropriateF to use the smaller than for the SIE+X models. Einstein ring radii, we note that both the SLACS and SL2S radii We have also produced models with a steeper than isothermal γ were derived assuming no mass sheet. Since we seek to apply mass density profile, ρ(r) ρ0(r/r0)− ′ ,withexponentγ ′ acalibratingfactortoafundamentalplanederivedfromthese 2.1. The resulting calibration= factor is higher by roughly 32%.=

11 Stellar Masses from Micro-lensing 5

to reconcile those discrepancies by adopting a toy model in which 50% of the flux was pointlike and 50% of the light was very extended and not subject to micro-lensing. Subsequent work by Pooley et al (2007), Morgan et al (2010) andBlackburneet al (2011) showed that the continuum emitting regions of bright lensed quasars were factors of 3 - 30 larger than predicted by the venerated Shakura and Sunyaev (1973) model. Schechter et al (2014) used X-ray flux ratios in their estimateofthefactorby which Salpteter mass surface densities needed to be multiplied to allay concerns about the size of the continuum emitting region. Jiménez-Vicente et al (2015) took a different tack, carrying out a joint analysis of stellar mass fraction and emitting region size. The two approaches yield consistent results, albeit with large uncertainties. The size of the X-ray sample will continue to grow as long as theChandraX-ray Lensing galaxyObservatory contents: continues to operate. Unfortunatly none of the currently planned X-ray missions will be able to make such measurements as they lack Chandra’s resolution. Chandra dataAs constrain discussed above, newly fraction discovered of quadruply dark lensed halo quasars in are likely to be less luminous than those first discovered. It is reasonable toexpecttheircontinuum MaCHOs emittingto ≲10%. regions to be correspondingly smaller, mitigatingtheeffect of their partial • Previousresolution. assumption: dark halo component is smooth → halo particles of at most planetary mass (depends on X-ray size of quasar) • Instead,5. Limitstake onstellar MaCHOs, surface Including mass Primordial density Black to Holesbe known and let the factor ℱ representIn our calculations, the fraction we implicitly of the assume dark that halo the dark in haloMassivecomponent Compact of a lens is Halo smoothly distributed. This translates to halo particles of at most planetary mass, de- Objectspending (MaCHOs), upon the poorly which known sizesincludes of quasar ~20M X-ray emitting⊙ blackregions. holes.

See also Mediavilla et al. (2009)

Figure 3. Likelihoods for a range of fractional contributioSchechterns of MaCHOs2018 to the dark matter surface density in ten lensed quasars. Note the finite likeli- hood for a negative fraction, which would result if a SalpeterIMFoveresti- mates the surface mass density. We can invert our assumptions, and take the stellar surface mass density to be known (adopting in our case a Salpeter IMF) and instead let thefactorF represent the 4 A. J. Shajib et al.

Table 1. Observation information and references for the lens systems.

Total exposure time System name Observation date (seconds) Reference F160W F814W F475X 4 A. J. ShajibPS et J0147+4630 al. 2017 Sept 13 2196.9 1348.0 1332.0 Berghea et al. (2017) SDSS J0248+1913 2017 Sept 5 2196.9 1428.0 994.0 Ostrovski et al. 2018b (in preparation) ATLASJ0259-1635 2017Sept7 2196.9 1428.0 994.0 Schechter et al. (2018) Table 1. Observation informationDES J0405-3308 and references 2017 for Sept the 6 lens systems. 2196.9 1428.0 1042.0 Anguita et al. (2018) DES J0408-5354 2018 Jan 17 2196.9 1428.0 1348.0 Lin et al. (2017); Diehl et al. (2017); Agnello et al. (2017b) DES J0420-4037 2017Total Nov exposure 23 time 2196.9 1428.0 1158.0 Ostrovski et al. 2018b (in preparation) System namePS Observation J0630-1201 date 2017 Oct(seconds) 5 2196.9Reference 1428.0 980.0 Ostrovski et al. (2018); Lemon et al. (2018) SDSS J1251+2935F160W 2018 Apr F814W 26 F475X 2196.9 1428.0 1010.0 Kayo et al. (2007) 4 A. J. ShajibPS et J0147+4630 al. SDSS 2017 Sept J1433+6007 13 2196.9 2018 May 1348.0 4 1332.0 2196.9Berghea 1428.0 et al. 1504.0(2017) Agnello et al. (2018a) SDSS J0248+1913PS 2017 J1606-2333 Sept 5 2196.9 2017 Sept 1428.0 1 994.0 2196.9 Ostrovski 1428.0 et al. 994.0 2018bLemon (in preparation) et al. (2018) ATLASJ0259-1635DES 2017Sept7 J2038-4008 2196.9 2017 Aug 1428.0 29 994.02196.9Schechter 1428.0 et 1158.0 al. (2018)Agnello et al. (2017a) Table 1. Observation informationDES J0405-3308 and referencesWISE 2017 for Sept J2344-3056 the 6 lens systems. 2196.9 2017 Sept 1428.0 9 1042.0 2196.9Anguita 1428.0 et al. 1042.0(2018) Schechter et al. (2017) DES J0408-5354 2018 Jan 17 2196.9 1428.0 1348.0 Lin et al. (2017); Diehl et al. (2017); Agnello et al. (2017b) DES J0420-4037 2017Total Nov exposure 23 time 2196.9 1428.0 1158.0 Ostrovski et al. 2018b (in preparation) System namePS Observation J0630-1201 date 2017 Oct(seconds) 5 2196.9Reference 1428.0 980.0 Ostrovski et al. (2018); Lemon et al. (2018) SDSS J1251+2935F160W 2018 Apr F814W 26 F475X 2196.9 1428.0 1010.0 Kayo et al. (2007) PS J0147+46304SDSS 2017A. Sept J1433+6007 J. 13 Shajib 2196.9 2018 et al. May 1348.0 4 1332.0 2196.9Berghea 1428.0 et al. 1504.0(2017) Agnello et al. (2018a) SDSS J0248+1913PS 2017 J1606-2333 Sept 5 2196.9 2017 Sept 1428.0 1 994.0 2196.9 Ostrovski 1428.0 et al. 994.0 2018bLemon (in preparation) et al. (2018) ATLASJ0259-1635DES 2017Sept7 J2038-4008 2196.9 2017 Aug 1428.0 29 994.0 2196.9Schechter 1428.0 et 1158.0 al. (2018)Agnello et al. (2017a) DES J0405-3308TableWISE 2017 Sept 1. J2344-3056Observation 6 2196.9 information 2017 Sept 1428.0 9 and references 1042.0 2196.9 forAnguita the 1428.0 lens et systems. al. 1042.0(2018) Schechter et al. (2017) DES J0408-5354 2018 Jan 17 2196.9 1428.0 1348.0 Lin et al. (2017); Diehl et al. (2017); Agnello et al. (2017b) DES J0420-4037 2017 Nov 23 2196.9 1428.0 1158.0Total Ostrovski exposure et time al. 2018b (in preparation) PS J0630-1201 2017System Oct name 5 2196.9 Observation 1428.0 date 980.0 Ostrovski(seconds) et al. (2018);ReferenceLemon et al. (2018) SDSS J1251+2935 2018 Apr 26 2196.9 1428.0 1010.0F160WKayo F814W et al. (2007 F475X) 4SDSSA. J1433+6007 J. Shajib 2018PS et J0147+4630 al. May 4 2196.9 2017 Sept 1428.0 13 1504.0 2196.9Agnello 1348.0 et al. 1332.0(2018a) Berghea et al. (2017) PS J1606-2333 2017SDSS Sept J0248+1913 1 2196.9 2017 Sept 1428.0 5 994.0 2196.9Lemon 1428.0 et al. (2018 994.0) Ostrovski et al. 2018b (in preparation) DES J2038-4008 2017ATLASJ0259-1635 Aug 29 2196.9 2017Sept7 1428.0 1158.0 2196.9Agnello 1428.0 et al. ( 994.02017a) Schechter et al. (2018) TableWISE J2344-30561. Observation information 2017DES Sept J0405-3308 9 and references 2196.9 2017 for Sept the 1428.0 6 lens systems. 1042.0 2196.9Schechter 1428.0 et al. 1042.0(2017) Anguita et al. (2018) DES J0408-5354 2018 Jan 17 2196.9 1428.0 1348.0 Lin et al. (2017); Diehl et al. (2017); Agnello et al. (2017b) DES J0420-4037 2017Total Nov exposure 23 time 2196.9 1428.0 1158.0 Ostrovski et al. 2018b (in preparation) System namePS Observation J0630-1201 date 2017 Oct(seconds) 5 2196.9Reference 1428.0 980.0 Ostrovski et al. (2018); Lemon et al. (2018) SDSS J1251+2935F160W 2018 Apr F814W 26 F475X 2196.9 1428.0 1010.0 Kayo et al. (2007) 4 A. J. ShajibPS et J0147+4630 al. SDSS 2017 Sept J1433+6007 13 2196.9 2018 May 1348.0 4 1332.0 2196.9Berghea 1428.0 et al. 1504.0(2017) Agnello et al. (2018a) SDSS J0248+1913PSFurther 2017 J1606-2333 Sept 5progress: 2196.9 2017 Sept 1428.0 1 2196.9994.0 Ostrovski 1428.0 et al. 994.0 2018bLemon (in preparation) et al. (2018) ATLASJ0259-1635DESChandra 2017Sept7 J2038-4008 observations 2196.9 2017 Aug 1428.0 29 of newly 2196.9 994.0 Schechter discovered 1428.0 et 1158.0 al. (2018 quads)Agnello etwill al. (2017a improve) Table 1. Observation informationDES J0405-3308 and referencesWISE 2017 for Sept J2344-3056 the 6 lens systems. 2196.9 2017 Sept 1428.0 9 1042.0 2196.9Anguita 1428.0 et al. 1042.0(2018) Schechter et al. (2017) DES J0408-5354the 2018 Janensemble 17 2196.9 results. 1428.0 1348.0 Lin et al. (2017); Diehl et al. (2017); Agnello et al. (2017b) DES J0420-4037 2017Total Nov exposure 23 time 2196.9 1428.0 1158.0 Ostrovski et al. 2018b (in preparation) System name Observation date (seconds) Reference PS J0630-1201 2017 Oct 5 2196.9 1428.0 980.0 Ostrovski et al. (2018); Lemon et al.New(2018 surveys) are SDSS J1251+2935F160W 2018 Apr F814W 26 F475X 2196.9 1428.0 1010.0 Kayo et al. (2007) 10 ks 1.5 ks 1.5 ks PS J0147+4630SDSS 2017 Sept J1433+6007 13 2196.9 2018 May 1348.0 4 1332.0 2196.9Berghea 1428.0 et al. 1504.0(2017) Agnello et al. (2018a) finding many SDSS J0248+1913PS 2017 J1606-2333 Sept 5 2196.9 2017 Sept 1428.0 1 994.0 2196.9 Ostrovski 1428.0 et al. 994.0 2018bLemon (in preparation) et al. (2018) more quads. ATLASJ0259-1635DES 2017Sept7 J2038-4008 2196.9 2017 Aug 1428.0 29 994.02196.9Schechter 1428.0 et 1158.0 al. (2018)Agnello et al. (2017a) DES J0405-3308WISE 2017 Sept J2344-3056 6 2196.9 2017 Sept 1428.0 9 1042.0 2196.9Anguita 1428.0 et al. 1042.0(2018) Schechter et al. (2017) DES J0408-5354 2018 Jan 17 2196.9 1428.0 1348.0 Lin et al. (2017); Diehl et al. (2017); Agnello et al. (2017b) DES J0420-4037 2017 Nov 23 2196.9 1428.0 1158.0 Ostrovski et al. 2018b (in preparation) PS J0630-1201 2017 Oct 5 2196.9 1428.0 980.0 Ostrovski et al. (2018); Lemon et al. (2018) We are obtaining SDSS J1251+2935 2018 Apr 26 2196.9 1428.0 1010.0 Kayo et al. (2007) 1.5 ks 6 ks 25 ks Chandra SDSS J1433+6007 2018 May 4 2196.9 1428.0 1504.0 Agnello et al. (2018a) observations of PS J1606-2333 2017 Sept 1 2196.9 1428.0Figure 994.0 1.LemonComparison et al. (2018 between) the observed (first, third and fifth columns) and reconstructed (second, fourth and sixth columns) DES J2038-4008 2017 Aug 29 2196.9 1428.0strong-lens 1158.0 Agnello systems. et The al. (2017a three )HST bands: F160W, F814W, and F475Xall publicly are used in the red, green, and blue channels, respectively, to WISE J2344-3056 2017 Sept 9 2196.9 1428.0create 1042.0 theSchechter red-green-blue et al. (RGB)(2017) images. Horizontal white lines forannounced each system new are rulers showing 1 arcsec. The relative intensities of the bands have been adjusted for each lens system for clear visualisationquads. of the features in the system.

25time-delays ks between25 ks the quasar images:25 ks t = 112 1.52. 1ks near the quasar images that are possibly multiple images of AB ± days, t = 155.5 12.8 days, and t = 42.4 17.6 days extra components in the source plane. Figure 1. Comparison betweenAD the observed± (first, third andBD fifth columns)± and reconstructed (second, fourth and sixth columns) strong-lens systems. The(Courbin three HST et al.bands:2018). F160W, F814W, and F475X are used in the red, green, and blue channels, respectively, to create the red-green-blue (RGB) images. Horizontal white lines for each system are rulers showing 1 arcsec. The relative intensities of the bands have been adjusted for each lens system for clear visualisation of the features in the2.2.7 system. PS J0630-1201 2.2.6 DES J0420-4037 Shajib et al. 2018 This system is a five-image lensed quasar system (Ostrovski time-delays betweenThe the discovery quasar images: of this quadt = is112 reported2.1 by Ostrovskinear the quasar et al. imageset al. that2018 are). possibly The discovery multiple was images the result of of a lens search AB ± days, t = 155.5 (2018b,12.8 days, in preparation). and t = 42 Several.4 17.6 smalldays knotsextra are noticeable components infrom the sourceGaia plane.data from a selection of lens candidates from Figure 1. Comparison betweenAD the observed± (first, third andBD fifth columns)± and reconstructed (second, fourth and sixth columns) strong-lens systems. The(Courbin three HST et al.bands:2018). F160W, F814W, and F475X are used in the red, green, and blue channels, respectively, to MNRAS 000, 1–18 (2018) create the red-green-blue (RGB) images. Horizontal white lines for each system are rulers showing 1 arcsec. The relative intensities of the bands have been adjusted for each lens system for clear visualisation of the features in the2.2.7 system. PS J0630-1201 2.2.6 DES J0420-4037 This system is a five-image lensed quasar system (Ostrovski time-delays betweenThe the discovery quasar images: of thist quad= is112 reported2.1 by Ostrovskinear the quasar et al. imageset al. that2018 are). possibly The discovery multiple was images the resultof of a lens search AB ± days, t = 155.5 (2018b,12.8 days, in preparation).and t = 42 Several.4 17.6 smalldays knotsextra are noticeable components infrom the sourceGaia plane.data from a selection of lens candidates from AD ± Figure 1. ComparisonBD between± the observed (first, third and fifth columns) and reconstructed (second, fourth and sixth columns) (Courbin et al. 2018). strong-lens systems. The three HST bands: F160W, F814W, and F475X are used in the red, green, and blueMNRAS channels,000 respectively,, 1–18 (2018) to create the red-green-blue (RGB) images. Horizontal white lines for each system are rulers showing 1 arcsec. The relative intensities of the bands have been adjusted for each lens system2.2.7 for clear PS J0630-1201 visualisation of the features in the system. 2.2.6 DES J0420-4037 This system is a five-image lensed quasar system (Ostrovski The discovery of thistime-delays quad is reported between by the Ostrovski quasar et images: al. t et= al. 2018112 ).2 The.1 discoverynear the was quasar the result images of that a lens are search possibly multiple images of AB ± (2018b, in preparation).days, Severalt = small155. knots5 12.8 aredays, noticeable and t = from42.4 Gaia17.6 daysdata fromextra a selection components of lens in the candidates source plane. from Figure 1. Comparison betweenAD the observed± (first, third andBD fifth columns)± and reconstructed (second, fourth and sixth columns) strong-lens systems. The(Courbin three HST et al.bands:2018). F160W, F814W, and F475X are used in the red, green, and blue channels, respectively, to MNRAS 000, 1–18 (2018) create the red-green-blue (RGB) images. Horizontal white lines for each system are rulers showing 1 arcsec. The relative intensities of the bands have been adjusted for each lens system for clear visualisation of the features in the2.2.7 system. PS J0630-1201 2.2.6 DES J0420-4037 This system is a five-image lensed quasar system (Ostrovski time-delays betweenThe the discovery quasar images: of this quadt = is112 reported2.1 by Ostrovskinear the etquasar al. imageset al. that2018 are). possibly The discovery multiple was images the result of of a lens search AB ± days, t = 155.5 (2018b,12.8 days, in preparation). and t = 42 Several.4 17. small6 days knotsextra are noticeable components infrom the sourceGaia data plane. from a selection of lens candidates from Figure 1. Comparison betweenAD the observed± (first, third andBD fifth columns)± and reconstructed (second, fourth and sixth columns) strong-lens systems. The(Courbin three HST et al.bands:2018). F160W, F814W, and F475X are used in the red, green, and blue channels, respectively, to MNRAS 000, 1–18 (2018) create the red-green-blue (RGB) images. Horizontal white lines for each system are rulers showing 1 arcsec. The relative intensities of the bands have been adjusted for each lens system for clear visualisation of the features in the2.2.7 system. PS J0630-1201 2.2.6 DES J0420-4037 This system is a five-image lensed quasar system (Ostrovski time-delays betweenThe the discovery quasar images: of this quadt = is112 reported2.1 by Ostrovskinear the quasar et al. imageset al. that2018 are). possibly The discovery multiple was images the result of of a lens search AB ± days, t = 155.5 (2018b,12.8 days, in preparation). and t = 42 Several.4 17.6 smalldays knotsextra are noticeable components infrom the sourceGaia plane.data from a selection of lens candidates from AD ± BD ± (Courbin et al. 2018). MNRAS 000, 1–18 (2018) 2.2.7 PS J0630-1201 2.2.6 DES J0420-4037 This system is a five-image lensed quasar system (Ostrovski The discovery of this quad is reported by Ostrovski et al. et al. 2018). The discovery was the result of a lens search (2018b, in preparation). Several small knots are noticeable from Gaia data from a selection of lens candidates from

MNRAS 000, 1–18 (2018) Further progress: Chandra observations of newly discovered quads will improve the ensemble results. Matter in Matter in Relative Probability Relative Probability Stars Stars 1.5% 0 1.5% 0 2.2% 0.003 2.2% 0 3.2% 0.034 3.2% 0.051 4.6% 0.186 4.6% 0.649 6.8% 0.356 6.8% 0.27 10% 0.237 10% 0.029 15% 0.08 15% 0.001 22% 0.028 22% 0 possible realization 32% 0.006 ensemble of 13 systems 32% 0 of 30 systems 46% 0.005 46% 0 68% 0.01 68% 0 100% 0.056 100% 0

DP et al. 2012 Further progress: Future X-ray observations will unlock the full potential of future LSST discoveries. • LSST will discover thousands of quads, but there are fundamental degeneracies between the assumptions one makes about the optical emitting region(s) and what you can determine about the lensing galaxy contents. • The full power of those discoveries will be unlocked with Lynx, which could study several hundred with a modest observing program. • The snapshot ensemble analysis could be done as a function of redshift.

artist’s representation of Lynx

Oguri & Marshall 2010 Main takeaways:

• Quasar micro-lensing is the only way to determine stellar M/L beyond the solar neighborhood. • Chandra established that quasars are micro-lensed. • Sub-arcsecond X-ray imaging is the best way to determine micro-lensing effects. • I gave a micro-lensing talk without showing a single micro-lensing magnification map. • I can’t leave out my three year old.

Further progress: Long term Chandra (and then Lynx) follow-up of quads will determine stellar fraction and M/L for individual galaxies.

Additional Microlensing Additional Microlensing Additional Microlensing Magnifcation: ×2 Magnifcation: ×5 Magnifcation: ×0.3 Observations spaced years to decades apart will provide independent samples of the lensing galaxy’s stellar population.

HE 0230-2130 MG 0414+0534 HE 0435-1223 RX 0911+0551 Current state of SDSS 0924+0219 HE 1113-0641 observations: PG 1115+080 SDSS 1138+0314 H 1413+117 B 1422+231 WFI 2026-4536 WFI 2033-4723

1999 2002 2005 2008 2011 2014 2017 2020 Micro-lensing is not a single star phenomenon.

Observed Quasar Each of the four Images stars “macro-images” Zoom in comprises dozens of ×105 “micro-images.”

Lensing microimages of Galaxy quasar

6’’ 60 μas

Additional Microlensing Additional Microlensing Additional Microlensing Additional Microlensing Additional Microlensing Magnifcation: ×1.2 Magnifcation: ×7 Magnifcation: ×2 Magnifcation: ×5 Magnifcation: ×0.3 Quasar structure: Micro-lensing has ruled out certain corona geometries and established that the corona must be compact.

The Astrophysical Journal,769:53(8pp),2013May20 Corona Fe line AccretionMosquera Disk et al. This research was supported by NASA/SAO grants GO6- 7093X, GO0-11121A/B/C, and GO1-12139A/B/C, and NSF grants AST-0708082 and AST-1009756. Further support for this work was provided by the National Aeronautics and SpaceNo! Administration through Chandra Award Number 11121✘ issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. A.M.M. also acknowledges the support optical continuum size of Generalitat Valenciana, grant APOSTD/2010/030.✘ No! REFERENCES

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7 The Astrophysical Journal,789:125(11pp),2014July10 Blackburne et al.

0.40 0.40 Optical 0.35 log Prior UV 0.35 Quasar structure: 0.30 Soft X-ray The Astrophysical Journal,789:125(11pp),2014July10 0.30 Blackburne et al. 0.25 Hard X-ray The optical disk is 3–300.40 times larger than standard00..4025 thin-disk 0.20 Optical 0.35 log Prior theory predicts. 0.15 UV 00..3520 0.30 Soft X-ray 0.10 00..3015 0.25 Individual systemsHard X-ray Population The Astrophysical0.05 Journal,789:125(11pp),2014July10 The AstrophysicalRelative Probability Journal,729:34(20pp),2011March1 Blackburne et al. Blackburne et al. No. 1, 2007 EnsembleFLUX average RATIO ANOMALIES IN QUADRUPLY LENSED QUASARS.0 I..25 25 0.20HE 0435−1223 Blackburne et al. 2014 0.10 Blackburne et al. 2011 00..4000 0.40 and to Chandra X-ray flux ratios, both from the literature and 0.15 Optical 00..2005 newly reported here, we see chromatic variations, which we

00..3535 logLinear Prior Prior 1115 UV 0.35 attribute to the microlensing of a multicolor accretion disk. The 0.10 1017 2033 Relative Probability 0.15 standard thin accretion disk model predicts a temperature profile 0.30 0.00 0230 0.30 Soft X-ray −1 2026 0 1 0.05 UV Optical Relative Probability 0.30 10 10 10 that falls as the β 3/4powerofradius,implyingthatthe

0435 = 4/3 00..2525 Hard X-ray 0.10 Mean Mass M/M half-light radius of the disk is proportional to λ , with an 1138

0.00 cm overall normalization that depends on the black hole mass and / Figure 7.0.Posterior25 probability distribution for the mean mass M of the 0924 0.20 0414 0.20 1016 microlens0 stars..05 All other parameters are marginalized. This result is⟨ taken⟩ (weakly) from on the Eddington fraction and accretion efficiency. 2500 ,

0 35 0911 . Linear Prior 2 / the optical-only simulation to avoid biases arising from the lack of convergenceThis chromatic dependence combines with the dependence of 1 0 15 1131 . r 0 20 0.15 at small X-ray. and UV sizes; see the text for details. The solid line uses a Relative Probability 0.30 0.00 microlensing magnification on source size to produce higher- 0.10 UV Optical flat logarithmic prior,1113 while10−1 the dashed line uses100 the result of Poindexter101 amplitude & microlensing variations at blue wavelengths than at 0.10 Kochanek0. (152010b)asaprior. 0.25 15 Mean Mass M/M red wavelengths. For single-epoch observations like ours, this

10 Relative Probability 00..0505 usually means that the observed flux ratios at blue wavelengths Figure 7.0Posterior10 probability distribution for the mean mass M of the 0.20 soft X-ray.. These distributions use a logarithmic prior⟨ only,⟩ are as a more anomalous than at red wavelengths, compared to the 00..0000 microlens stars. All other parameters are marginalized. This result is taken from 0.15 14.014.515.015.516.016.517.0 thelinear optical-only prior does simulation not make to avoid sense biases forarising such from ratios. the lack We of convergence also markratios predicted by smooth lens models. at7 small X-ray0.05 and UV8 sizes; see the text9 for details. The solid10 line uses a 0.35 Scale Radius log(rs/cm) 10the expected value10 of the UV to10 optical ratio for10 the thin diskIn addition to formal statistical uncertainties, we have esti- Linear Prior flat logarithmic prior, while the dashed line uses the result of Poindexter & 0 10 MBH/M mated the systematic errors in the flux ratios due to spectral con- Figure 5. Posterior. probability distributions for the scale radius of the source case where ∆ log A 8/3∆ log λ. This value is consistent with Relative Probability Kochanek0 (.2010b00 )asaprior.= 0The.30 AstrophysicalQ 2237+0305 Journal,769:53(8pp),2013May20 Mosquera et al. 2013 Figure 10.ourHalf-light probability radii at a rest distribution, wavelength−1 ofMosquera which 2500 Å, et is plotted al. not0 surprisingagainst black given1 tamination the by broad emission lines, quasar variability combined at a variety0.05 of wavelengths. TheUV top (bottom)Optical panel uses a logarithmic (linear)Standard SS Disk10 10 10 prior. The arrows mark the observed-frame UV and R-band scale radii expectedhole mass. Black hole masses estimated using the virial method are plotted with lensing time delays, and microlensing variability combined 0.25 width of the UV area distribution.Mean Mass AlthoughM/M it is not relevant from thin disk theory; the flux estimates are smaller by 0.38 dex. The verticalas red squares;soft X-ray. those estimated These distributions using bolometric use luminosity a logarithmic are plotted prior as only, aswith a delays between measurement epochs. We have described DP et al. 2007 0.00 1132 blue diamonds.for the The thin solid linediskMORGAN is model, the ETprediction AL. theMorgan ratioof the thinof et hard disk al. model2010 X-ray (with area toVol. soft 712 Fig. 3.—Comparison of X-ray (dark blue crosses) and opticalline on ( thered left circles marks14.014 the) ratios estimated.515 to. gravitational015 lens. model516 radius.016 ratios of the.517 black for. select hole.0 The imagelinearFigure pairs prior 7. forPosterior doeseach probabilitynot lensed make quasar. distribution sense for The forsuch theleft ratios. mean panel mass We alsoM markofaBayesianmethodfordeterminingtheprobabilitydistribution the 0.20 fEdd 0.25X-ray and η area0.15), is interesting while the dotted because line is the of bestthe fitpossibility to these data. of⟨ structure⟩ shows the ratio of the highly magnified saddle point ( HS) to themean highly mass M magnifiedis fixed at 0.3 minimumScaleM , and Radius all other ( log( HM parametersr ),cm) while are marginalized. the middle and= microlens right panels= stars. All show other parameters the ratio are of marginalized. each of This these, result is taken fromof the half-light radius of the source in each filter. For quasars ⟨ ⟩ ⊙ s/ The microlensingthein optical-only the expected radii X-ray of simulation Morgan value corona. of et toal. Wethe avoid (2010 UV do biases)areplottedinthinblacklines not to optical findarising evidence from ratio the for lack of the of a convergence difference thin disk Diminished micro- 0.15 for comparison. They have been corrected to 2500 Å (assuming that r λ4/3) respectively, to the less magnified minimum ( LM). The ratios for the X-ray are based on the observation with the highest signal-to-noisecaseatin small sizes: where X-ray the∆ and distributionlog UV ratio,A sizes;8 and/3 seepeaks∆ log tho theλ justtextse. This for for at unitydetails.the value optical (though Theis consistent solid it line is usesbroad withwithout a X-ray measurements, we are able to place upper limits Figure 5. Posterior0.20 probability distributions for the scale radius of the sourceand converted to half-light radii by multiplying by 2.44 (see Equation (2∝)). The ourflat logarithmic probability prior, distribution, while= the dashed which line is uses not the surprising result of Poindexter givenon the & the half-light radii, by virtue of the departures of the flux are based on the observation closest in time to the chosen X-rayat a data. variety0 The. of10 wavelengths. light blue The crosses top (bottom) showUV/Optical panel the uses variation a logarithmic in (linear) theinclination X-rayenough correction ratios that for has been quadssignificant removed. observed differences Their error mu barsltiple in are the larger source times because areaby they at different lensing in optical prior. The arrows mark the observed-frame UV and R-band scale radii expected widthKochanek of (2010b the UV)asaprior. area distribution. Although it is not relevantratios from the model predictions. When X-ray ratios are avail- Chandra. Soft X-ray/Optical take severalX-ray sources energies of systematic are error not into ruled account. out). The This black runs hole masses a bit counterhave to the from thin0 disk.05 theory; the flux estimates are smaller by 0.38 dex. The verticalunplottedfor uncertaintiesresults the ofthin of Chen about disk et 0.4 model, al. dex. (2011 the), ratio who of measure hard X-ray X-ray area light to curves softable, we assume that they originate from a very compact region, line on the left marks the estimated gravitational radius of the black hole. The and we are able to place both upper and lower limits on the half- Hard X-ray/Optical (A color versionX-rayindicating of area this figure energy-dependentis interesting is available becausein the sizeonline of differences journal.) the possibility in Q of2237+0305. structure mean mass00..00M15 is fixed at 0.3 M , and all other parameters are marginalized. soft X-ray. These distributions use a logarithmic prior only,light as a radius, reasoning that the differences between the X-ray ⟨ ⟩ ⊙ Hard X-ray/Soft X-ray inMore the denselyX-ray corona. sampled We X-ray do not light find curves evidence would of be a difference invaluable 14.014.515.015.516.016.517.0 linear prior does not make sense for such ratios. We also markand optical ratios arise because of the appreciable optical radius. the X-ray ratio is less than the optical. The middle and right0.20 panels 5. SIZES OF QUASARthat theinin effective sizes: addressing EMISSION the temperature distribution this question. is peaksREGIONS a steeply just falling at unity function (though of it is broad Scale RadiusThin log(r Disks/cm) the expected value of the UV to optical ratio for the thin diskThe resulting size estimates are larger overall than is predicted UV/Optical radius. Itenough is worth that noting significant that differencesnearly all the in the measured source slopesarea at different show whether the discrepancy with the model comesFigure from 5. Posterior the HSprobability distributions for the scale radius of the source case where ∆ log A 8/3∆ log λ. This value is consistent withby the standard thin disk model by nearly an order of magnitude, Estimate size of optical For the purpose of interpretingare consistentX-ray energies with ourν results, are0,4.5. not= i.e., ruled Mean we a source out).adopt Microlens This with the runs a Mass wavelength- work- a bit counter to the at a variety0 of.10 wavelengths. The top (bottom)Soft panel X-ray/Optical uses a logarithmic (linear)independentour probability size, though= distribution, the existence which of such is an not object surprising seems givenon the average. The scaling of the half-light radius with mass or the HM image (or a combination of the two). Inprior. general, The arrows mark the the observed-frame UV and R-band scale radii expected resultswidth of of the Chen UV et area al. ( distribution.2011), who measure Although X-ray it is notlight relevant curves ing hypothesisHard X-ray/Optical that thehighly anomalous unlikely,Figure from7 fluxshows both theoretical ratios the posterior presented (e.g., probability energy conservation) in distribution this foris the consistent with the expected slope, but the scaling with disk necessary to givefrom thin0 disk.15 theory; the flux estimates are smaller by 0.38 dex. The vertical indicatingfor the thin energy-dependent disk model, the size ratio differences of hard X-ray in Q 2237+0305. area to soft LM image is much less susceptible to microlensingline on than the left markseither the estimated gravitationalHard radius X-ray/Soft of the blackX-ray hole. Theand observationalmean mass (e.g., of the quasar stars variability)M . Like standpoints. the source area, this measure-wavelength is shallower than expected. Although the large paper are the result of microlensing.MoreX-ray densely area is Microlensing interesting sampled X-ray because light by of curves stars the possibility would in the be of invaluable structure

Relative Probability ⟨ ⟩ error bars on the wavelength slopes mean that a single slope the HS or HM image (Kochanek & Dalal 2004).mean mass M is fixed at 0.3 M , and all other parameters are marginalized. MG J0414,inment addressing is the affected only this quasar by question. the that resolution matches limits the thin of disk our sizemagnification ⟨ ⟩ ⊙ Thin Disk in the X-ray corona. We do not find evidence of a differencemeasurement would have little weight, the fact that all but one that less variability 0.05 lensing galaxyFigure can1. Inclination-corrected accountprediction accretion for diskpatterns size (and theR2500 thevs. observed (see black only hole Figure mass radio-loudMBH2. The). flux To solid quasar avoid line ratio through thisin the ouranomalies, data problem, shows sample), our best which power-law is also fit would to the data bias and the dot-dashed line shows the prediction from thin-disk theory (L/LE 1andη 0.1). The shaded band surrounding the best fit shows the expected variance due to The group statistics for the flux ratio anomalies presented0Figure20 4. Probability in distributions for the half-light radii of the V-band (solid), Figure 5. Probabilityin sizes: distributions the distribution= for the half-light= radii peaks of the justV-band at (solid) unity (though it is broadof the measured slopes are too shallow commands attention. . inclination. Disk sizes are correctedthe only to a restquasar wavelength of withλrest 2500 a slope Å and theν blackconsistent hole masses were with estimated 4 using/3. emission Although line widths. The filled points but only if the source is smallour compared mass results= toward4.5. to theMean smaller Einstein Microlens mean masses, Mass radii we of use the results soft X-ray (dash-dotted), and hard X-ray (dashed) emission.without The gray error bars area are R2500 andestimates full X-ray based on(dashed) the observed, emission. magnification-corrected The gray area correspondsI-band fluxes. to They predicted have typical values uncertainties of 0.1–0.2dex.Solidlinesatthe 0.10 UV/Optical MG J0414enough was that the significant only case where differences we9 used in the empirical source area mid- at differentSince the scaling of radius with wavelength is the reciprocal Figure 3 and Table 5 are summarized in Figure 4. The errorcorresponds bars to predicted values of rg, where the lower and upperbottom limits of the are plot for show theof innermostrg, where stable theof circular lower the orbit and simulation for upper a maximally limits rotating are that for KerrMBH is black constrained hole0.9 and10 a SchwarzschildM (C onlyiv; black by hole, representing the R-band a plausible light range of 9 9 9 = × ⊙ MBH 0.9 10 M (C iv; Morganthe et al. microlensing2010)andMBH radii2.4 for the10 inner stars.M edge of anMorgan FigureIR accretion flux et disk. al.X-ray ratios2010 3Figure shows)and energies inM7 placeBHshows dramatic2 are.4 of the10 modelnotM posterior ruled(H ratiosβ evidence; Assef out). probability et(asThis al. we2011 runs), forsuspected distributiona mic- bit counter the for toof the the scaling of temperature with radius, our results indicate (Hβ;Assefetal.= × 2011⊙), respectively. Soft X-ray/Optical= × ⊙ respectively. curve. Like Poindexter= × ⊙ & Kochanek (2010b), we partially break represent the rms spread in the logarithm of the flux ratios presencemeanresults of millilensing), mass of Chen of the et itstarsTable al. is 1unlikely(2011M .), Like who that the its measure source unique area, X-ray behavior this light measure- curvesthat if we assume a blackbody accretion disk with a power-law (A0. color00 version of this figure is availablerolensing inHard the online X-ray/Optical journal.) in the X-ray band(A color version forthe of at this degeneracyleast figure isMeasured available 7 of between and in theDerived the online Quantities 10velocity journal.) lensing and mean systems mass by virtue of

Æ Relative Probability indicating energy-dependent⟨ ⟩ size differences in Q 2237+0305.temperature profile the temperature slope is in general steeper (normalized by the smooth model values) between various0.15−6 im- −4 −2 0 24is due tomentour this dynamicchoice, is affected since magnification mid-IRby the resolutionradios patterns, for other limits but lensed as of we our quasars have magnification mentioned, Hard X-ray/Soft2 X-rayObject Line FWHM MBH log(RS /cm) λrest BLR Icorr log(RS /cm) in keeping solutions.Projected We keepin hardArea our X-ray Ratio study. fits∆ withlog(χA In/N) dof general,! 4 (including theMore optical PG densely 1115)9 emission match sampled the X-raymodels of these light very curves well same (Chiba would sys- et be al. invaluablethan the standard thin disk model predicts. However, other 0.05 2 Å(Observed)patterns (10 (seeM ) Figure (Microlensing)2). To (µm) avoid Contaminant this problem,(mag) which would (Thin Diskbias Flux) and then fit the soft X-ray band saving results with χ /N 3. the small⊙ random stellar velocities in the HE 0435 system re- age pairs for our quasar sample. The black outer bars result from QJ 0158dof !4325 Mg2005ii ;Minezakietal.40in addressing0.162009 this 14.9+0 question.)..3 0.306 Balmer, Fe ii UV, Mg ii 19.09 0.12 15.2 0explanations.1 for the relative insensitivity of radius to wavelength Figure 6. Posterior probability distributions for the ratios of projected− areas, our mass results toward0.3 smaller mean masses, we± use the results± Dropped solutions would contributetems,Thin to our while Disk Bayesian still probability being microlensed,duceH/V the power has of+0− .5 less this method. extreme Indeed, flux our ratio posterior velocity 2 2 HE 0435 1223 C iv 70 S/V 0.50 15.7 0.7 0.260 Balmer, Fe ii UV 20.76 0.25 14.9 0are.1 also possible, though none immediately presents itself. expressed as logarithmic differences. The arrow denotes the ratio− of UV area Two lenses,S/H HE 0230− and WFI J2033, have microlensing± ± including all 10 quasars; the thick blue bars result whenintegrals we as ex- exp[ (χ χmin)/2], so the dropped casesSDSS should 0924+0219 Mg ii 61of the simulation0.11 15.0+0 that.3 is0.277 constrained Balmer, Fe ii UV, only Mg ii by21.24 the0.25R-band 14.8 light0.1 − − distribution is indistinguishable0.4 from the prior,± indicating± that to opticalbe area exponentially predicted unimportantby the standardanomalies to the thin final disk results. theory; than compare in the it to X-ray thesize estimates band bythat a decrease factor+0−4.5..4 with of Mean wavelength.2(seeFig.4and Microlens This Mass result does There is growing evidence from microlensing studies that FBQ 0951+2635 Mg ii 70curve. Like0.89 Poindexter 16.1 0.4 &0.313 Kochanek Balmer, Fe ii (UV,2010b Mg ii ),17. we16 0 partially.11 15.6 break0.1 0.10 we have not strongly− broken the degeneracy.± Our mean± mass clude the systems Q2237+0305 and SDSS J0924+0219.dot–dashed Q2237+Once curve. the The initial mean set mass of trialsM hasis fixedbeen “filtered” at 0.3 MSDSS through, 1004+4112 and all other Mg ii 134 0.39 14.9+0.3 0.228 Balmer, Fe ii UV 20.97 0.44 14.9 0quasar.2 accretion disks are larger at optical wavelengths than not seem to arise from a gross0.3 failure of our analysis method, 0.00 ⊙ − ± ± the three bands, we use athe⟨ Bayesian⟩ discussion analysis to determine above). Sincethe probability degeneracy X-rays distribution are between+0.2 expected is velocity consistent to and be with mean emitted other mass estimates by virtue of the of parameters are marginalized. We use a logarithmic prior. HE 1104 1805 C iv 103 Figure2.377 shows 15.9 0. the3 posterior0.211 Balmer, probability Fe ii UV 18 distribution.17 0.31 15. for4 0simple.1 the accretion models predict. Our result is a confirmation −6 −4 −2 0 24− since the flux ratios themselves− become more anomalous± with ± 0305 is excluded because the uniquely small redshift of itsthe lensing parameters and their uncertainties. Figure 4 showsPG 1115+080 the Mg ii 127our dynamic1.23 magnification 16.6+0.3 0.257 patterns, Balmer, Fe butii UV as we 19.52 have0.27 mentioned, 15.1 0.1 meanmean mass mass of (Poindexter the stars0.4 M & Kochanek. Like the2010b source). area, Since± this our measure- posterior± Projectedvery Area Ratio near∆ thelog(A black) hole,wavelength, the condition which in most for+0− .2 microlensing cases implies a smaller is easy source to size of earlier, more qualitative work by Pooley et al. (2007). It also probability distribution for the scale radii obtainedRXJ 1131with1231 a Hβ 90 0.06 15.3 0.2 0.422 Balmer, Fe ii Optical 20.73 0.11 14.8 0.1 Relative Probability − the small random stellar⟨ velocities⟩ in the HE 0435± system± re- galaxy causes the projected microlens Einstein radiuswe tocalculate be bigger posterior probability distributions for the ratio of mentprobability is affected distribution by+0− .6 the isresolution independent limits of of that our of magnification Poindextercorroborates & the results of Morgan et al. (2010), who use several logarithmic prior on the sizes. We found that the X-ray-emittingSDSS 1138+0314 C(seeiv Table25 5). It0.04 cannotbe 14.9 due0.6 to0.203 microlensing Balmer, Fe ii variability,UV 21.97 0 since.19 14.6 0.1 Figure 6. Posterior probability distributionsmeet—the for the ratios source of projected should areas, indeed be quite− small compared to the± ± areas. Figure0regions.05 6 areshows significantly distributions smaller than for the the optical, logarithm andSBS found 1520+530 of no the UV C iv 75duceKochanek the0.88 power (2010b 15. of7+0),.2 this we0.245 method. multiply Balmer, Indeed, the Fe ii twoUV our distributions 18. posterior92 0.13 15 velocity together.3 0.1 patterns (see Figure0.2 2). To avoid this problem, which± would± biaslensed quasars to study the mass dependence of the accretion than any region of the source, while SDSS J0924+0219expressed as logarithmic might differences. The arrow denotes the ratio of UV areathe optical/IR measurementsa +0− .3 are coeval. But it should not be significant difference between soft and hard bands,Q2237+030 although Civ 48distribution0.9 is indistinguishable15.6 0.3 0.208 fromBalmer the prior,17.90 0 indicating.44 15.5 0 that.2 toarea optical to areaoptical predicted area by ratio, the standard alongEinstein with thin disk similar radius theory; distributions compare of the it to microlensing theforconcludedourand that mass plot the stars. resultsthem temperature as By toward− a dashed contrast, smaller profile curve mean of the in these Figure masses,markedly quasars7.Thisisequivalent we± is use in the results± disk radius, and find it difficult to reconcile their radii with thin the uncertainties are large. Between the different bands we Full X-ray/V-band also be excluded because the source size is thought to be so small Notes. RS from microlensing is the accretionweto using havedisk size at theirnotλrest, thestrongly distribution rest-frame wavelength broken as corresponding a the prior. degeneracy. to the center of the monitoring Our mean filter used massfor that dot–dashedsoft X-rayfound curve. to half-light optical, The mean radius hard massratios X-ray ofM log( toisR fixedoptical,/R at) 0 and.3 M0 hard.,65 and+0.47 X-ray all, otherfact to inverted,of the as simulation the errors that do is not constrained rule out positive only by slopes. the R-band It lightdisk theory. Our results for the observer-frame r′-band size of lower⟨ ⟩ 1/2 degree,soft V ofquasar’s⊙ microlensing light0.55 curve. Use half-light radii (R1 in/2 2 the.44RS ) to optical compare these size band measurements implies to other disk models. that Significant the sources of unmicrolensed parameters are marginalized. We use a logarithmic prior.= − probability= distribution is consistent with other estimates of the +0.47 flux from− the QSO broad-line region (BLR)curve. that fall Like into or overlapPoindexter with the observing & Kochanek passband are indicated. (2010b Balmer), continuum we partially (λ 3650 Å), FebreakiiMGUV J0414 and u′-band size of SDSS J0924 are consistent with that even its broad-line region is partially microlensed (Keetonlog(R1/2 et,hard al./RV ) 0.51 0.57,andlog(R1/2,soft/R1/2,hard) is possible that the sources lie on an unusual region of the! 0.00 +0.60 = − size− of the opticalcontinuum (λ=! emission3100 Å), Fe ii optical10 continuummeanthe region degeneracy (4240 mass Å ! (Poindexterinλ ! 5400 many between Å), or Mg ii &(λ of2798 velocity Kochanek Å). theseIcorr is the and corrected2010b sources mean (unmagnified)). Since massI-band is our by magnitude. posteriorvirtue Typicalthe of upper limits of Bate et al. (2008)andFloydetal.(2009), 0.14 0.72.WealsoranasequenceconsideringonlytheOGLEI-band measurement errorscaustic are 0.1mag,butthelargererrorson pattern whereI thecorr come smoothing from uncertainties caused in the lens magnification. by a largerRS calculated source using corrected I-band − −6− −4 −2 0 24! 2006). we calculateV band posterior and the full probability X-ray band, since distributions the X-ray uncertainties formagnitude the and ratio thin-disk ofcauses theory is alsomore unscaled;probabilityour dynamic itanomalous is the disk distributionsize magnification at the flux rest-frame ratios. wavelength is independent patterns, These corresponding lenses but to the of as center that underscore we of the haveofHST Poindexter I-band mentioned, filter (F814W).respectively. & Likewise, our estimates of the radius of RX J1131 are smaller whenProjected consideringroughly Area the complete Ratio ∆ comparable energylog(ABoth range.) disk sizes We assume to an average the inclination size angle ofi 60 the◦. stellar microlens Einstein areas. Figure 6 shows distributions for the logarithma of the UV Kochanek= (2010b), we multiply the two distributions togetherat around 4000 Å in the rest frame, and of HE 0435 at around It can be seen from the blue bars in Figure 4 that the ratios of 2 The C iv emission linethe width fromimportance Yeethe & De small Robertis of (1991 randomhaving) depends a strongly stellar large on the sample velocities fit to several of blended objects in C theiv absorption when HE features, 0435 using so systemwe report MBH re-at saved trials with full X-ray band fits with χ /Ndoflower! precision.5and areaFigure to 6. opticalPosterior area probability ratio, distributions alongradius.+0 with.45 for similar the ratios distributions of projected areas, forsingle-epochandduce plot measurements.the them power as of a dashed this method. curve Indeed,in Figure our7.Thisisequivalent posterior velocity6500 Å in the observed frame, are consistent with those of Dai found log(RX,full/RV ) 0.52 0.54 (Figure 5). Thus the X-ray the HS to HM images deviate more (from their expectedsoftexpressed X-ray as values) logarithmic to optical, differences. hard= X-ray− The− to arrow optical, denotes and the hard ratio of X-ray UV area to to using their distribution as a prior. half-light radius is on scales of 101.7 0.5 r ,whiletheoptical distribution is indistinguishable from the prior, indicatinget that al. (2010)andMosqueraetal.(2010), respectively. to optical area predicted by the standardMany thin± diskg theory; authorsestimates compare are have appropriate. it to the studied While the formal the uncertainties effect in MBH ofThis source implies either size that 0.33 on dex of the random uncertainty for half-light radius is on scales of 10∼2.2 0.2 r for M from10 the9.1 lineM width. relations are only 0.1dex,thesystematic each individual black hole mass is too large or that we haveThe picture is less clear regarding the temperature profile in the X-ray band than in the optical band by a factordot–dashed of 2.4. curve. The The mean mass M is± fixedg atBH 0.3 M , and all other we∼ have8. SUMMARY not strongly AND broken CONCLUSIONS the degeneracy. Our mean mass We also examined the distributions of the projecteduncertainties= area ratios⊙ are generallyFigure believed 6. Top panel: to be probabilitycloser to 0 distributions.33 dex foroverestimated the logarithm the of uncertainties the half- in the size measurements. parameters are marginalized. We usemicrolensing a⟨ logarithmic⟩ prior. of⊙ quasars10 byprobability intervening distribution galaxies. is consistent Typically with other the estimates ofslope the β.Usingtime-domainmeasurementsatseveralwave- discrepancy is somewhat smaller for the HS/LM ratios atbetween a factor the different wavelengths to see if the structure(McLure of & the Jarvis 2002light;Kollmeieretal.We radius have hard X-ray obtained2006 area;Vestergaard to optical flux area ratios ratio (solid),The in eightlow along efficiency withoptical similar estimate and is closely IR filters related to the argument results are presented&Peterson2006;Pengetal. asdistributions plots2006mean for of). soft If X-ray masswe microlensing fit to the optical(Poindexter relation (dashed) andby & hardPooley Kochanek light to etsoft al. X-ray ( curves20072010b (dash-) that the). sizes (e.g., Since estimated our from posterior microlens-lengths, Poindexter et al. (2008)findβ 0.6forthequasarHE X-ray-emitting region could be better defined (seeassuming Figure that6). this is andotted).spanning additional Bottom random panel: a factor errorsimilar in distribution of each six black forin the wavelength, fullingX-ray are larger area thanto optical for expected aarea sample from thin-disk of theory, 12 but the1104 com- 1805, consistent with the thin disk∼ slope of 3/4. They of 1.7. The HM/LM and LS/LM ratios are not as anomalousThese ratios in have values of log(A /A ) 1.01+0.73 , ratio. we calculate posterior probability distributionsX,full V hole for mass, the0 then.86 ratio we find of log (R2500/probabilitycm) (15.9 0.2) distribution + (1.0 parison is independent is not exact because of Pooley that et of al. ( Poindexter2007)usedblackhole & − +0.Wambsganss83 = − &+0− Paczyn.86 9 ´stronglyski 1991) lensed= quasars.rather±2 Comparing± than rms these fluctuations ratios to each inother point out that a flatter temperature slope (i.e., β < 3/4) would  log(A /A ) 1.21 ,log(A /A ) 0.3)1 log.01MBH/,10 M (Awith color a versiongoodness of thisof fit figureχ /N isdof available0.59. in the onlinemasses journal.) determined largely from estimates of the bolometric areas. FigureX,6softshowsV distributions1.07 forX,hard the logarithmV of1.12 the UV⊙ Kochanek (2010b= ), we multiply the two distributions together either band, but the X-ray ratios still have a wider range than do the= − − +0.95 = − − 10 area to opticaland log(A areaX,soft/A ratio,X,hard) alongthe0.21 with logarithm0.91 that similar are consistent distributions of! with the flux. for" Thereand plot themare no as a analytic dashed curve techniques in Figure 7.Thisisequivalent for = − − Part of the problem is that it is probably exponentially harder optical ratios. It is on the larger anomalies in the X-raysoft band X-raythe derived to for optical, the half-light hard radius X-ray ratios. to optical, and hard X-ray to to using their distribution as a prior. 19 Despite finding large numbersestimating of good fits to the rmsV-band fluctuations, data to fit temporally so longer one light must curves using simulate the Monte Carlo the ap- micro- HS/HM and HS/LM ratios, as compared to those for thealone, optical when we included the X-ray data we had difficulty ob- proach (see Poindexter & Kochanek 2010a). But another prob- taining large numbers of trialslensing that were good process. fits to the data.5 lem could10 be that we do not have fully aligned optical and band, that we base our quantitative analysis of the size of the op- X-ray data because we have no extension of the OGLE light 5 2 2 Best (χ /Ndof )hard 1.9, and best (χIdeally/Ndof )soft 1.3. we would runcurves point-source to the most recent X-ray simulations epochs. We tested thisfor hypothe- each of the tically emitting regions of the accretion disks in the next section. = = 40 images in our sample,5 taking into account the theoretical mag- nification (which in turn depends on two independent parameters, aconvergenceandashear)andthefractionofbaryonicmatter. Each simulation would produce a magnification map, which might then be convolved with sources of different sizes, producing mag- nification histograms. Such an effort lies beyond the scope of the present paper, but we can draw on such simulations that have been carried out. In particular we use the work of Mortonson et al. (2005), who studied in detail the effect of source size on minima and saddle points with magnifications of +6 and 6, respectively, assuming that the convergence (a dimensionlessÀ surface density) is due entirely to equal-mass stars and taking the convergence to be equal to the shear, as would be the case for an unperturbed iso- Fig. 4.—The rms of the flux ratio anomalies in the optical vs. X-ray (see Fig. 3 thermal sphere. The magnifications for our highly magnified and Table 5 for the flux ratios of individual sources). The black outer bars result from including all 10 quasars in our sample, the thick blue bars result from ex- cluding Q2237+0305 and SDSS J0924+0219, and the red bars result when we ex- 10 While the rms microlensing fluctuations in flux are formally divergent, rms clude only Q2237+0305. fluctuations in the logarithm of the flux are not (e.g., Witt et al. 1995). Quasar structure: Micro-lensing of Fe line strongly constrains inner disk. Fe line micro-lensing has been observed in several systems (e.g., Chartas et al. 2002, Dai et al. 2003, Ota et al. 2006, Chartas et al. 2007, Chen et al. 2012, Chartas et al. 2012, Chartas et al. 2017) 2 Chartas et. al.: Gravitational Lensing Size Scales for Quasars

1017 Fe Kα ) 0.1 − 1 Utilizing such observations requires RXJ1131 Q0158 Q0924 PG1115 Q0435 HE1104 Q2237 1016 keV − 1 advanced modeling of both strong-

(cm) 15 0.01 1/2 10 Chandra texp = 4.70 ks Image B field gravity and micro-lensing

X-ray R Date: Jan 1, 2007 1014 Rate (cnts s

2 Innermost Stable Circular Orbit, a = 1 2 Innermost Stable Circular Orbit,/c a = 0 features. /c BH 6GMBH GM 1013 2 107 108 109 1010

MBH(MÍ) ∆χ 0 −2 Fig. 1 X-ray half-light radii of quasars as determined • Heyrovský & Loeb (1997) Fe Kα from our microlensing analysis versus their black hole ) 0.1 − 1 • Popovic et al. (2001, 2003a, 2003b, masses. keV − 1 2006) 0.01 dius of the X-ray emission region to have an upper limit Chandra texp = 4.71 ks Image B of log(r1/2/cm) = 15.33 (95% confidence), a low inclination Date: Feb 13, 2007 • Jovanovic et al. (2009) Rate (cnts s angle is preferred statistically, the mean mass of the stars in < > • Neronov & Vovk (2016) the lensing galaxy ( M )rangesbetween0.1and0.4M⊙ 2 ξ and the slope of the size-wavelength relation r1/2 ∝ λ ,is

∆χ 0 +0.30 • Krawczynski & Chartas (2017) ξ =1.0−0.56.ThemajorityoftheobservedcontinuumX- −2 ray emission is found to originate within ∼ 30rg,assum- 8 0.5 1 25 • Ledvina et al. (2018) ing a black hole estimate of MBH =5.9× 10 M⊙ based Observed−Frame Energy (keV) on the width of the Hβ line (Assef et al. 2011). Based on this black hole mass estimate the gravitational radius of Chartas et al. 2012 Fig. 2 Evolution of the Fe Kα line possibly caused by the HE 1104−1805 is r =8.7× 1013 cm. g motion of a magnification caustic as it moves away from the In MacLeod et al. 2015 we analyze the light-curves of center of the black hole. RX J1131−1231: RISCO ≲ 9 Rg z =1.524quasarSDSS0924+0219 using static microlens- ing magnification patterns. SDSS J0924+0219 has been ob- served at a variety of wavelengths ranging from the near- al. 2009). Included in Figure 1 are the uncertainties of the infrared to X-ray. Our microlensing analysis in this system black-hole mass estimates and uncertainties in the size es- constrains the soft-X-ray, UV, and optical half-light radiito timates. The X-ray sizes of the quasars in our sample are +10 14 +24 14 +5. 15 found to be close to the sizes of their innermost stable circu- be 2.5−2 ×10 cm, 8−7 × 10 cm, and ∼ 5−2.5 × 10 cm, respectively. Assuming the MgII based black-hole es- lar orbits. Assuming that most of the X-ray emission in the 8 timate of MBH =2.8× 10 M⊙ the majority of the soft band detected originates from the hot X-ray corona, these X-ray emission of SDSS 0924+0219 originates within ∼ results indicate that the corona is very compact and not ex- 30 rg.ThegravitationalradiusofSDSS0924+0219 is rg = tended over a large portion of the accretion disk. 4.12 × 1013 cm. In Dai et al. 2010 and Chartas et al. 2009 we analyze the 3 Estimating the Inner Most Stable Circular light-curves of z =0.658quasarRXJ1131−1231. We find Orbit Using Microlensing the X-ray and optical half-light radii to be 2.3 × 1014 cm 15 and 1.3 × 10 cm, respectively. These sizes correspond to RX J1131−1231 has been monitored 38 times over a pe- ∼ 26rg and ∼ 147rg,respectively. riod of 10 years with the Chandra X-ray Observatory.As An important result found in all microlensing studies is reported in Chartas el al. 2012, redshifted and blueshifted that optical sizes of quasar accretion disks as inferred from Fe Ka lines have been detected in the spectra of the lensed the microlensing analysis are significantly larger than those images. predicted by thin-disk theory. Specifically, measurements of In Figure 2 we show the evolution of the red and blue the radius of the accretion disk at 2500A˚ rest-frame indicate components of the Fe Kα line possibly caused by the mo- that the sizes obtained from microlensing measurements are tion of a magnification caustic as it moves away from the 2–3 times larger than the values predicted by thin-disk the- center of the black hole. We interpret the shift of the Fe Kα ory (e.g., Morgan et al. 2010, Mosquera et al. 2013). line as resulting from general relativistic and special rela- In Figure 1 we present the X-ray half-light radii of tivistic Doppler effects. As shown in Figure 3, the two red- quasars from recent microlensing studies of lensed systems shifted iron lines in the Jan 1, 2007 observation are each observed as part of our monitoring program (MacLeod et al. detected at the ≥ 99% confidence and the iron lines in the 2015, Blackburne et al. 2014, 2015, Mosquera et al. 2013, Feb 13, 2007 observation are each marginally detected at Morgan et al. 2008, 2012, Dai et al. 2010, and Chartas et the ≥ 90% confidence level (Figure 3).

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