1

A dust-parallax distance of 19 megaparsecs to the supermassive in NGC 4151

Sebastian F. Hönig1,2, Darach Watson1, Makoto Kishimoto3, Jens Hjorth1

The active NGC 4151 has a crucial role as one the order of 0.1 pc, with a square-root dependence on of only two active galactic nuclei for which black luminosity12, 15. hole mass measurements based on emission line In parallel, IR interferometry taken at the same reverberation mapping can be calibrated against wavelength measures a corresponding angular size, ρ, 1–3 other dynamical methods . Unfortunately, effect- of the same emission region. The angular and physical ive calibration requires an accurate distance to NGC sizes are trigonometrically related by sin ρ = Rτ/DA, 4 4151, which is currently not available . Recently where DA is the angular-diameter distance to the object. reported distances range from 4 to 29 megaparsecs For small angles sin ρ ≈ ρ and accounting for 5–7 (Mpc) . Strong peculiar motions make a - cosmological time dilation, we obtain DA(Mpc) = 0.173 × based distance very uncertain, and the geometry of τ(days) / (ρ(mas)·(1+z)),which forms the basis of the the galaxy and its nucleus prohibit accurate distance measurement presented here (see Fig. 1). A measurements using other techniques. Here we geometric technique was first proposed for broad 8 report a dust-parallax distance to NGC 4151 of DA= emission lines . Unfortunately, the typical angular size

Mpc. The measurement is based on an of the broad-emission line region for bright AGN is of adaptation of a geometric method proposed the order of 0.001 − 0.01 milliarcseconds, which is too previously using the emission line regions of active small to be spatially resolved with current optical long- galaxies8. Since this region is too small for current baseline interferometers. The dust continuum emission, imaging capabilities, we use instead the ratio of the on the other hand, is larger by a factor of ~4 and IR physical-to-angular sizes of the more extended hot interferometers have now managed to resolve about a 9 10 11,12,14,16 dust emission as determined from time-delays dozen AGN . Moreover, using dust emission and infrared interferometry11–14. This new distance requires only photometric reverberation mapping leads to an approximately 1.4-fold increase in the instead of spectrally resolving emission lines. Finally, dynamical black hole mass, implying a dust physics is arguably easier to model than gas line corresponding correction to emission line emission. reverberation masses of black holes if they are To determine the distance to the supermassive black calibrated against the two objects with additional hole in NGC 4151, we make use of interferometry dynamical masses. obtained with the two Keck telescopes and monitoring The central black hole in AGN is surrounded by a data from literature. V- (wavelength 0.55 μm) and K- putative that emits pre-dominantly at band (2.2 μm) photometric monitoring from 2001 to 10 ultraviolet and optical wavelengths. At large distances 2006 trace the UV/optical and hot dust emission, from this central emission source, the gas is cool enough respectively. Since long-term brightness changes can for dust to survive (temperature ≤1500 K). This “dusty cause τ and ρ to increase or decrease, monitoring and torus” absorbs the UV/optical radiation and thermally interferometry data should be taken more or less reemits the energy in the infrared (IR). Thus, any contemporaneously. We use six Keck interferometry variability in the UV/optical emission will be detected measurements taken between 2003 and 201011–14. in the dust emission with some time delay. Near-IR These overlap with the monitoring data, and inspection reverberation mapping measures the time lag, τ, of long-term brightness trends showed that fluctuations between the UV/optical variability and the were moderate between 2000 and 201016. Indeed, no corresponding changes in emission of the hot dust, significant change in size has been detected for this set 16,18 which is located about the sublimation radius Rsub = R(T of interferometry and variability data . ~ 1,500 K). The time lag can be converted into a When comparing angular and physical sizes, it is physical size via Rτ = τ · c, where c is the speed of light. important to make sure they refer to the same physical Typically, time lags are found in the range of several 10s region. First, observations of the dust emission are to 100s of days, which corresponds to physical sizes of preferably made at the same waveband (here: K-band).

1Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen Ø, Denmark. 2School of Physics & Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom. 3Department of Physics, Faculty of Science, Kyoto Sangyo University, Kamigamo-motoyama, Kita-ku, Kyoto 603-8555, Japan

2

Figure 1 | Impact of the brightness distribution on the observed sizes and time lags. The bottom row (b) illustrates effects of varying the brightness distribution on the time lag signal (blue: optical; red: near-IR), while the upper row (a) outlines the corresponding (interferometric) brightness distribution in the near-IR. Compact distributions lead to shorter time lags and smaller interferometric radii than shallow profiles. This information is encoded in the shape (width, amplitude) of the light curve. With a simple power law parametrization, this smearing effect can be accounted for to determine the time lag and angular size of the innermost radius of the brightness distribution. The simultaneous modeling of both light curves and interferometry results in a very precise angular distance measurement.

Second, the spatial distribution of the dust around the ρin ≈ rin/DA of the inner boundary of the brightness AGN impacts the observed sizes (see Fig. 1): Dust that is distribution, respectively. Further parameters that may homogeneously distributed will result in a larger influence the observationally inferred physical and apparent size than a compact dusty region, since the angular size of rin are the disk geometry of the emission former involves a larger region that contributes to the region and the dust properties. For the inclination and emission at a given wavelength. As a consequence, the disk orientation, we use observational constraints variability signal of the K-band will show some degree based on the dynamics of the emission line region in of smoothening with respect to the V-band signal, 23–25 NGC 4151 and polarimetry . We do not consider the depending on the distribution. This also involves a shift radio jet in this study26, since the available data does not of the peak time lag. At the same time the size measured allow for reliably establishing both position angles and by interferometry will appear correspondingly smaller inclinations simultaneously. The absorption efficiency or larger. As shown in literature, this distribution effect of the dust is implicitly included in our parametrization in both types of data can be effectively modeled by of the brightness distribution. Moreover, the means of a disk model17,18, assuming that the dust is sublimation temperature does not affect the distance heated by AGN radiation and the projected brightness α determination since it scales in the same way for the distribution is represented by a power law S(r) ∝ r (see reference angular sizes and the reference time lags. Methods for details). This geometry is in line with Since the light curves are sampled with finite and theoretical expectations and observational evidence of varying gaps between each observation, we simulated the hot-dust region19,20. Indeed, the model has been 1,250 random, continuous representations of data using successfully applied to reproduce multi-wavelength, the AGN variability pattern derived from the structure multi-baseline interferometry of several AGN (including 14,21 function. τin, α, and ρin were calculated simultaneously NGC 4151) as well as the light curve of NGC 18,22 given the observationally constrained inclination and 4151 . disk orientation as priors. This resulted in 1,250 The common reference size in such a model is the inner estimates of DA, which are shown in Fig. 2. An important radius rin of the brightness distribution. For feature of this process is that although determining the reverberation mapping and interferometry, this corres- reference time lag or the reference angular size ponds to a reference time lag τin = rin/c and angular size individually is quite uncertain, both parameters are 3

14 Mpc 16 Mpc 18 Mpc 20 Mpc

+2.4 DA = 19.0-2.6 Mpc ) 22 Mpc

s 0.5 (68% confidence interval)

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a 0.00 m 15 20 25 distance (Mpc) 0.1 10 20 30 40 50 60 time lag of the inner boundary tin (days)

Figure 2 | Relating time lags and angular sizes to measure the absolute distance to NGC 4151. The colored circles show the modeled reference time lags τin and associated angular sizes ρin (1,250 random realizations of the V-band light curve; 68% confidence interval shown as error bars). The blue-dashed lines mark τin/ρin ratios corresponding to distance DA = 14 − 24 Mpc. The distribution median and 68% confidence interval are marked by the red-dashed line and orange-shaded area. Brightness distribution power law indices α are color coded from α = 2 (red; shallow) to α = −15 (blue; steep). The small inset shows the probability distribution function of the inferred distance, with mean and 68% confidence interval indicated by blue-dashed and -dotted lines, respectively. strongly correlated with the dust brightness velocities. Attempts to estimate the distance this way distribution. Thus, the ratio, i.e. DA , can be constrained resulted in a wide range of values between ∼4 and 20 with much higher precision than the reference time lag Mpc5, 6. More recently, a luminosity distance of 29.2±0.5 or reference angular size individually, if τin and ρin are Mpc was suggested based on near-IR reverberation calculated simultaneously given the inferred α. mapping only and a model for absorption and re- We obtain an distance to NGC 4151 of emission of dust, which cannot be reconciled with our 7 DA = Mpc (see probability distribution in Fig. 2). new result or the other estimates . The error bars include statistical uncertainties from the The new precise distance to NGC 4151 is important reverberation and interferometric observations, as well since this galaxy is a cornerstone in calibrating black as the systematic uncertainties introduced by the hole masses inferred from different methods2; aside geometry, brightness distribution, and the uncertainty from NGC 3227, it is the only galaxy with a suitable in host and putative accretion disk contribution to the mass estimated from reverberation mapping, stellar K-band interferometry. These uncertainties have been and gas dynamics. Of both , NGC 4151 has much accounted for in the Monte Carlo simulations when better mass constraints2, so that any systematic offset in sampling the data (see Methods). The new distance distance will almost equally affect the calibration of clarifies the uncertain situation for NGC 4151. The black hole masses. Dynamical mass estimates relate the galaxy is in the vicinity of the Virgo cluster (angular rotational velocity field of or gas surrounding the o separation to the Virgo cluster center <30 ), resulting in black hole to their distances from the AGN. In the strong peculiar motion with respect to the Hubble process, observed angular distances have to be flow6,27. Therefore, any recession velocity-dependent converted into physical distances for which an absolute distance has to be considered uncertain. Geometric distance to the galaxy is required. The most recent mass 4,28 megamaser distances require that the nuclear region is estimates assume DA = 13.2 Mpc . A stellar velocity- SD 7 seen very close to edge-on, which is generally not the based mass was reported as M BH = (3.76 ± 1.15) × 10 case for unobscured AGN. Moreover, a direct distance M 4. Our new measurement implies that this mass is estimate based on the Tully-Fisher relation is difficult ⊙ underestimated by a factor of ~1.4, leading to a revised because of the face-on view of the galactic disk, which SD 7 mass of M = (5.4 ± 1.8) × 10 M . Similarly, the makes it difficult to determine the required rotational BH ⊙ 4 correction in distance increases the gas-dynamical 18. Hönig, S, F., & Kishimoto, M. Constraining properties of 28 GD 7 GD dusty environments by infrared variability. Astron. mass from M BH = × 10 M⊙ to M BH = 7 Astrophys. 534, A121 (2011) × 10 M . ⊙ 19. Kawaguchi, T., & Mori, M. Near-infrared Reverberation by Dusty Clumpy Tori in Active Galactic Nuclei. Astrophys. J. The new distance and corrected MBH values also affect the correction factor f that has to be invoked when 737, 105 (2011) converting reverberation time lags and velocities into 20. Landt, H., et al. The near-infrared broad emission line region black hole masses. The most recent canonical value has of active galactic nuclei - II. The 1-μm continuum. Mon. Not. R. Astron. Soc. 414, 218-240 (2011) been found as f = 4.31 ± 1.05 inferred from comparing 21. Hönig, S. F., et al. Dust in the Polar Region as a Major reverberation mapping masses to black hole masses 29 Contributor to the Infrared Emission of Active Galactic based on the MBH-σ∗ relation . Using our corrected Nuclei. Astrophys. J. 771, 87 (2013) values for the dynamical black hole masses on the 22. Schnülle, et al. Dust physics in the nucleus of NGC 4151. reverberation data1, we find a range of f = 5.2 − 6.5 Astron. Astrophys. 557, L13 (2013) (reflecting the difference between gas and stellar 23. Müller-Sanchez, F., et al. Outflows from Active Galactic dynamical masses), implying a systematic shift to larger Nuclei: Kinematics of the Narrow-line and Coronal-line masses. Such larger f-values may be generally Regions in Seyfert Galaxies. Astrophys. J. 739, 69 (2011) applicable, as also suggested by complex modeling of 24. Das, V., et al. Mapping the Kinematics of the Narrow-Line 30 Region in the NGC 4151. Astron. J. 130, 945- velocity-resolved reverberation mapping data . 956 (2005) 25. Martel, A. R. New Hα Spectropolarimetry of NGC 4151: The 1. Bentz, M. C., et al. A reverberation-based mass for the central Broad-Line Region-Host Connection. Astrophys. J. 508, 657- black hole in NGC 4151. Astrophys. J. 651, 775-781 (2006) 663 (1998) 2. Peterson, B.M. Measuring the mass of supermassive black 26. Mundell, C. G., et al. The Nuclear Regions of the Seyfert holes. Space Sci. Rev. 183,253-275 (2014) Galaxy NGC 4151: -Scale H I Absorption and a 3. Woo, J.-H., et al. Do quiescent and active galaxies have Remarkable Radio Jet. Astrophys. J. 583, 192-204 (2003) different MBH −σ∗ relations? Astrophys. J. 772, 49 (2013) 27. Mould, J. R, et al. The Key Project 4. Onken, C. A., et al. The black hole mass of NGC 4151. II. Stellar on the Extragalactic Distance Scale. XXVIII. Combining the dynamical measurement from near-infrared integral field Constraints on the Hubble Constant. Astrophys. J. 529, 786- spectroscopy. Astrophys. J. 791, 37 (2014) 794 (2000) 5. Theureau, G., et al. Kinematics of the Local Universe. XIII. 21- 28. Hicks, E. K. S., & Malkan, M. A. Circumnuclear Gas in Seyfert cm line measurements of 452 galaxies with the Nancay 1 Galaxies: Morphology, Kinematics, and Direct radiotelescope, JHK Tully-Fisher relation, and preliminary Measurement of Black Hole Masses. Astrophys. J. Suppl. maps of the peculiar velocity field. Astron. Astrophys. 465, Series 174, 31-73 (2008) 71-85 (2007) 29. Grier, C. J., et al. Stellar Velocity Dispersion Measurement in 6. NASA/IPAC Extragalactic Database, High-Luminosity Quasar Hosts and Implications for the AGN http://http://ned.ipac.caltech.edu Black Hole Mass Scale. Astrophys. J. 773, 90 (2013) 7. Yoshii, Y., et al. A New Method for Measuring Extragalactic 30. Pancoast, A., et al. The Lick AGN Monitoring Project 2011: Distances. Astrophys. J. Letters 784, 11 (2014) Dynamical Modeling of the Broad-line Region in Mrk 50. 8. Elvis, M., & Karovska, M. Quasar Parallax: A Method for Astrophys. J. 754, 49 (2012) Determining Direct Geometrical Distances to Quasars. Astrophys. J. Letters 581, 67-70 (2002) Acknowledgements We thank Radek Wojtak for helpful 9. Hönig, S. F. Dust Reverberation Mapping in the Era of Big discussions on peculiar velocities near the Virgo cluster. S.F.H. Optical Surveys and its Cosmological Application. Astrophys. acknowledges support from a Marie Curie International Incoming J. Letters 784, 4 (2014) Fellowship within the 7th European Community Framework 10. Koshida, S., et al. Variation of Inner Radius of Dust Torus in Programme (PIIF-GA-2013-623804). The Dark Cosmology Centre NGC4151. Astrophys. J. Letters 700, 109-113 (2009) is funded by the Danish National Research Foundation. This 11. Swain, M., et al. Interferometer Observations of Subparsec- research has made use of the NASA/IPAC Extragalactic Database Scale Infrared Emission in the Nucleus of NGC 4151. (NED), which is operated by JPL, Caltech, under contract with the Astrophys. J. Letters 596, 163-166 (2003) National Aeronautics and Space Administration. 12. Kishimoto,M., et al. Exploring he Inner Region of Type 1 AGNs with the Keck Interferometer. Astron. Astrophys. 507, Author Contributions S. F. H. and D. W. conceived the project. S. L57-L60 (2009) F. H. collected the data, developed the model, and wrote up the 13. Pott, J.-U., et al. Luminosity-variation Independent Location paper. D. W. assisted in interpreting the results and helped of the Circumnuclear Hot Dust in NGC 4151. Astrophys. J. writing the manuscript. M. K. contributed the interferometry 715, 736-742 (2010) data and help with the data analysis. J. H. contributed to the 14. Kishimoto, M., et al. The Innermost Dusty Structure in Active modeling and interpretation. All authors engaged in discussion Galactic Nuclei as Probed by the Keck Interferometer. and provided comments on the manuscript. Astron. Astrophys. 527, A121 (2011) 15. Suganuma, M., et al. Reverberation Measurements of the Author Information Reprints and permissions information is Inner Radius of the Dust Torus in Nearby Seyfert 1 Galaxies. available at www.nature.com/reprints. The authors declare no Astrophys. J. 639, 46-63 (2006) competing financial interests. Correspondence and requests for 16. Kishimoto, M., et al. Evidence for a Receding Dust materials should be addressed to S.F.H. ([email protected]). Sublimation Region around a Supermassive Black Hole. Astrophys. J. Letters 775, 36 (2013) 17. Kishimoto, M., et al. Mapping the radial structure of AGN tori. Astron. Astrophys. 536, A78 (2011) 5 Methods

Principle of the measurement The present study makes use of the fact that the To measure the angular-diameter distance DA to the distance to an object DA is the ratio between its physical AGN, we have to constrain a time lag and an angular size size and the observed angular size. The dust emission in that are representative of the brightness distribution AGN, however, is not a fixed-size object but has with power-law index α. Conveniently, we choose the emission from a range of radii contributing to any given scaling time lag τin and ρin that correspond to the inner observed band. Here we focus on the K-band emission size of the emission region. The power-law index α may for the hottest dust around an AGN. As we previously be constrained from both interferometry and showed using IR interferometric data17,21,31, the radius- reverberation measurements. In practice, we do not dependent brightness distribution F(r) in any infrared cover enough baseline lengths with the available band can be approximated by a power law interferometry, so that we use the α-constraint from fitting the light curves to determine ρ (see below for in more details). ( ) ∫ ( ( )) ( ) The absolute values of both scaling sizes τin and ρin

depend on the choice of T . However, according to sub equations (2) and (3), both T-dependencies are exactly where B (T(r)) is the frequency-dependent Planck ν the same. Therefore, the ratio of τ and ρ is function for temperature T(r) at radius r and Q is the in in abs;ν independent of T. Nevertheless, we sample T from a absorption efficiency of the dust. The frequency- sub normal distribution with mean of 1,400 K and a integration can be approximated by a power law as standard deviation of 100 K. This choice is based on our Q B (T)dν ∝ T4+γ 32. The power law index γ describes abs;ν ν previous work, where we were simultaneously fitting the change of absorption efficiency in the IR with the near-IR SED and interferometry of NGC 4151 by respect to the optical. For black-body radiation, Q = 1 17 abs;ν means of a power-law brightness model . We also and γ = 0, while typical astronomical dust has γ ≈ 1.6 − tested and confirmed that the distance measurement is, 33 1.8 . Therefore, using the equation of local thermal indeed, not depending on temperature. Extended Data equilibrium, the emission from radius r can be Figure 1 shows the DA-Tsub plane of our 1,250 samples. 18 expressed by a power law The Spearman correlation coefficient of 0.01±0.03 is

( ) ( ) ( ) ( ) ( ) (1)

)

K 1,600 (

b

u s where F(r ) is the emission from the innermost radius T

in e

r u

rin = r(Tsub), where the dust temperature reaches the t

a r sublimation temperature T . Equation (1) is used to e

sub p

m 1,400

model the observed light curves and interferometry e

t

n

(see below). As such, the dust opacity of the emitting o

i t

medium is implicitly accounted for since γ is absorbed a

m i

l

in α = α’ − 2/(4 + γ), which is directly constrained by b

u s observations. Therefore, we do not need to assume any

d 1,200

particular dust composition but solve self-consistently e m

u

for the effect of dust absorption efficiencies. s s

a The rest-frame light-travel time τ from the AGN to radius r can be expressed as τ = r/c, so that we can 1,000 rewrite equation (1) as 10 15 20 25 distance DA (Mpc)

( ) ( ) ( ) . (2) Extended Data Figure 1 | Dependence of the

measured angular diameter distance D on the dust A sublimation temperature Tsub . The red circles Equivalently, the observed angular size ρ is related to r represent the distribution of sublimation temperatures via ρ = r/DA (for small ρ), so that equation (1) can be in the 1,250 Monte Carlo runs to model the light curve. written as Overplotted are dashed contours at 5%, 15%, 33%, and 50% peak density. The distribution of DA is consistent

with being independent of Tsub, illustrating that the ( ) ( ) ( ) . (3) distance determination is insensitive of the detailed dust properties.

6 very close to zero, implying that the distribution is has been inferred from kinematic modeling of emission consistent with being drawn from an uncorrelated lines and optical polarimetry (see Extended Data Table parent distribution. 1 for a compilation of data). The various constraints originate from different spatial scales: Optical Data selection and AGN variability polarization in this type 1 AGN is a result of scattering of the putative accretion disk emission by gas clouds on NGC 4151 has been extensively monitored in the V- and the scale of the broad-line region, i.e. about a factor of 4 K-band using the MAGNUM telescope in the period 10 smaller than the hot dust emission. These data indicate between 2001 and 2006 . In 2003, NGC 4151 became o 25,34 an orientation approximately toward PA 90 . The the first AGN to be spatially resolved with IR 11 outflow kinematics of the narrow emission line region interferometry using the Keck interferometer . Further on arcsecond scales point ap- proximately toward PA Keck interferometry were taken in 2009, 2010, and o 23 o 24 12,14 35 or 60 , with the two different values presumably 2011 . originating from a different modeling approach. While Changes in the luminosity of the putative accretion disk the smallest spatial resolutions obtained by polarimetry might influence the size of the hot dust region via may arguably best trace the orientation of the AGN, we o increased or suppressed sublimation. It has been chose a prior on the position angle of 75 and sample demonstrated, however, that the V-to-K-band time lag from a normal distribution with a standard deviation of o in NGC 4151 between 2001 and 2006 did not vary 15 to reflect the range of inferred directions. It is significantly despite a peak-to-valley change in important to note that with future multi-baseline IR luminosity of a factor of ~20 in the V-band, which could interferometry, both inclination and position angle will have potentially changed the radius by a factor of 4 − 18 16 be self-consistently inferred from the data without the 5 . Evidence has been found that any change in the need to invoke extra information. size of the hot dust-emitting region in NGC 4151 Priors for the inclination of the disk come from the requires much stronger variability in terms of kinematic modeling of the emission lines in the outflow luminosity and has to be sustained over some time. 23,24 Furthermore, structural changes appear to be delayed cone . Despite the differences in position angle from the different modeling approaches, the inclination of the on time scales ∼40 times longer than the time lag (∼5 o 16 system has been consistently found as 45 , with varying years) . Based on separate modeling of the light curve degrees of uncertainty. Accordingly, we sample and interferometry, respectively, we previously inclinations from a normal distribution with a mean concluded that the inner radius remained unchanged o o within measurement uncertainties between 2003 and value of 45 and a standard deviation of ±10 . 201016,18. Therefore, the six independent K-band interferometry data sets from 2003, 2009, and 2010 can Modeling of the data be considered contemporaneous with the photometric V- and K-band light curves: In a first step, the V-band monitoring and we include them for our analysis. light curve was resampled to obtain a uniform coverage without seasonal gaps. The gaps were filled using a Inclination and orientation of the dusty disk standard stochastic interpolation technique that makes use of the AGN’s known variability pattern via the The model that is used to recover the time lag τ and in structure function15,36. The resampled V-band light angular size ρ of the inner radius of the K-band in curve was then used as the AGN variability pattern and brightness distribution accounts for the geometry of the put into a simplified radiative transfer model of a dust 18 emitting region in terms of a (projected) disk disk to predict a model K-band light curve given τ , configuration. We showed previously that it is a in simplified, yet accurate, representation of the direct the power law index α of the projected dust (unobscured) hot dust emission of more complex (brightness) distribution, the energy conversion efficiency w , the sublimation temperature T of the clumpy or smooth torus models31. eff sub dust, and the inclination i of the disk. Here, w The position angle of the AGN system/polar axis eff (=direction of the minor axis of the projected ellipse) represents the fraction of K-band light that reacts to the

Extended Data Table 1 | Constraints on the inclination and position angle of the AGN structure from various observations. The inferred distances for each individual observation do not include any uncertainties on the position angles and assume an inclination of 45o± 10o.

method inclination position angle approx. scale reference distance optical continuum . . . 91◦ < 0. 01 continuum near Hα line25 20.4 ± 1.6Mpc polarimetry . . . 92◦ ± 2◦ < 0. 01 UB V RI polarimetry34 20.5 ± 1.6Mpc kinematic modeling 45◦ ± 5◦ 35◦ > 1 bicone23 15.7 ± 2.4Mpc of emission lines 45◦ 60◦ < 2 bicone24 18.0 ± 1.8Mpc

◦ ◦ ◦ ◦ +2.4 combined inference 45 ± 10 75 ± 15 ...... 19.0− 2.6 Mpc 7 18 variability of the V -band . In the framework of this . Similarly, we found an efficiency factor of w = study, it can be considered as a scaling factor of the These values are consistent with our amplitudes that does not have any influence on the time previously published results18. delay or shape of the variability pattern. As an important step in the process, we base our modeling on K-band interferometry: The light curve resampling and the relative light curve g (t) = f (t)/f (t) − 1 (i = V, K) i i i modeling provides us with 1,250 sets of parameters rather than the light curve f (t) to remove any i (τin;j , αj , Tsub;j , ij ). As shown in Fig. 2, the key element in dependencies on the dust covering factor. Here, fi(t) is obtaining a precise angular distance is to determine the mean flux of the respective band over the entire both time lag and IR interferometry size using the same monitoring period. set of parameters that define the brightness distribution To obtain the best representation of the resampled light of the object, in particular α. This parameter is curve, we implemented a fitting scheme based on the idl correlated with both τin and ρin. While the α-dependence 37 mpfit fitting package . For that, the K-band light curve contributes a high uncertainty for constraints on τin and model fluxes have been extracted at the observed ρin individually, it cancels as a source of uncertainty for epochs. This model K-band fluxes have been compared the absolute distance, i.e. the ratio between both to the observed K-band fluxes, and the squared parameters, when determining τin and ρin self- residuals have been minimized by varying τ , α, and w . in eff consistently given the inferred α. The fitting scheme has been repeated for 1,250 From α , T , and i we simulated model images of the randomly resampled V-band light curves to properly j sub;j j account for the uncertainties due to the finite sampling. K-band emission and fit the six observed squared visibilities with an angular size ρin of the inner In this process, Tsub and i have been treated as fixed parameter rather than actual model parameters since brightness distribution. This con- version from visibility the light curves do not constrain these parameters. to size depends on the orientation of the emitting disk Instead we randomly draw a value from the prior since two-telescope long-baseline interferometry distributions of T and i for each resampled light curve provides spatial information only along a specific sub position angle. Therefore, we randomly pick a disk that is fit. This accounts for the systematic uncertainty orientation for each individual parameter set based on in these parameters. the prior distribution as constrained by observations Extended Data Figure 2 shows the 2-dimensional (see previous section). This accounts for the systematic distribution of fitted α and τin. The fully marginalized uncertainty in the orientation of the disk. We also mean values and 68% confidence levels of these correct for the contributions of the host galaxy and 12,14 distributions are determined as τin = days and α = putative accretion disk to the K-band visibilities and consider the respective uncertainties in our simulations. This additional Monte Carlo sampling has been

repeated 200 times for each of the 1,250 initial sets of a (α , T , i ) samples and each of the six independent

x j sub;j j

e d

n interferometric measurements.

i 0

w

a

l

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e Error budget and potential sources of systematic

w o

p uncertainty

n o

i The geometric distance measurement is based on a

t u

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i power-law parametrization for the brightness

r

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i distribution in the K-band. Therefore, the uncertainties

d

s

s reflect the probability distribution given this model of

e n

t 1,250 Monte Carlo samples to recover τin and α from the

h

g i r light curve and additional sampling to recover ρin from b -10 the interferometric data. It accounts for uncertainties in the geometric parameters (disk inclination and position 10 20 30 40 50 60 angle) as well as the statistical errors of the six time lag of inner boundary tin (days) interferometric observations and the uncertainties in their correction for host and putative accretion disk Extended Data Figure 2 | Dependence of the time lag contributions as far as they are constrained by these data. The observations do not allow to distinguish τin of the inner boundary on the brightness distribution power-law index α. The black circles between different parametrizations of the brightness represent the fitted τin and α for each of the 1,250 Monte distribution, which is an unknown uncertainty at this Carlo representations of the V-band light curve given the time. However, this parametrization was successful in inclination and sublimation temperature distributions simultaneously repro- ducing near- and mid-IR 17 (68% fitting error levels mostly smaller than symbol photometry and resolved interferometry of NGC 4151 size). and, therefore, seems a justified choice.

8 The error on the distance results from a combination of correlated and uncorrelated uncertainties in

determining the reference time lag τin, reference

a

angular size ρin, and the brightness distribution power- x

e d

law index α. In Extended Data Figure 2, we show the n

i 0

distribution of τ and α values obtained from fitting the w in a

l

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1,250 random representations of the finite-sampled V- e w band light curve given inclination and T from the o sub p

n o

distributions discussed previously. The total scatter is i

t u

b -5

dominated by the uncertainty in defining the “true” light i

r

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curve from the limited number of epochs, with only a i

d

small contribution from the sampled range of s

s

e n

inclinations. The relative contribution of this fitting to t

h g

the total distance uncertainty can be estimated from the i r b -10 relative width of the distribution in τin given α and is approximately 6-7% (68% confidence level). Extended Data Figure 3 shows the distribution of ρ for the 1,250 in 0.1 0.2 0.3 0.4 0.5 0.6 fitted α values, inclinations, and sublimation angular size of inner boundary rin (mas) temperatures given the six interferometric observations. The gray error bars represent the Extended Data Figure 3 | Dependence of the angular additional error on ρ from the uncertainty in position in size ρin of the inner boundary on the brightness angle and in removing the host and putative accretion distribution power-law index α. The black circles disk contributions, based on additional sampling of the represent the distribution of ρin determined from the six interferometric data (see precious section). Similarly as interferometric data points for given α, sublimation before, we can estimate the relative contribution from temperature, and inclination for each of the 1,250 Monte

ρin from a combination of the scatter of the data points Carlo representations of the V-band light curve. The and the individual uncertainties as approximately 12% green error bars represent the additional uncertainties (68% confidence level). This combines to a total error in (68% confidence levels) from the combined statistical the distance of about 13.5% as obtained from the errors of the observations, the position angle of the emitting disk, as well as the corrections for host and probability distribution function of DA shown in Fig. 2. The major sources of systematic uncertainty are the putative accretion disk contributions. object’s inclination and orientation. The prior distributions of i and the disk symmetry axis orientation are the dominant factors that contribute to the width of sublimation zone, which is unconstrained as of now. the τ-ρ correlation. The reason for this is that while the However, our assumption of a small covering factor is disk orientation influences the conversion of justified by the observed K-band covering factor of interferometric visibility to de-projected size, it does cK∼0.2 as estimated from the mean fluxes of the V- and not have any influence on the time-delay signal. The K-band light curves. inclination, on the other hand, affects both interferometry and reverberation signal. In 31. Hönig, S. F., & Kishimoto, M., The dusty heart of nearby interferometry, it causes a projection effect that active galaxies. II. From clumpy torus models to physical decreases the observed size by approximately a factor properties of dust around AGN. Astron. Astrophys. 523, A27 of cos i · sin θ as a function of the position angle offset θ (2010) from the disk symmetry axis. For reverberation 32. Barvainis, R. Hot dust and the near-infrared bump in the mapping, however, it only broadens/smoothens the continuum spectra of quasars and active galactic nuclei. time lag signal symmetrically around the mean without Astrophys. J. 320, 537-544 (1987) a significant shift in τ . 33. Mor, R., & Netzer, H. Hot graphite dust and the infrared Although the dust covering factor cancels out when spectral energy distribution of active galactic nuclei. Mon. modeling relative light curves18, we implicitly assume Not. R. Astron. Soc. 420, 526-541 (2012) 34. Merkulova, N. I., & Shakhovskoy, N. M. UBVRI Linear that the covering factor in the K-band is small. In the Polarimetry of the Nucleus of the Seyfert Galaxy NGC 4151 present model, the brightness distribution is a projected at Maximum and Minimum Light. ASP Conf. Ser., 360, 65-68 disk and, thus, geometrically thin. This simplifies the (2006) convolution of the light curves with the transfer 35. Pedlar, A., et al. The Radio Nucleus of NGC 4151 at 5-GHZ function to a 2-dimensional problem. If there were and 8-GHZ. Mon. Not. R. Astron. Soc. 263, 471-480 (1993) significant amount of emission coming from material 36. Peterson, B. M., et al On Uncertainties in Cross-Correlation highly elevated above the disk, then we would obtain Lags and the Reality of Wavelength-dependent Continuum additional contributions to the transfer from the 3- Lags in Active Galactic Nuclei. Publ. Astron. Soc. Pac. 110, dimensional space. Whether the presence of such 660-670 (1998) emission would bias the results toward longer or 37. Markwardt, C. B. Non-Linear Least Squares Fitting in IDL shorter lags or a more compact or shallow brightness with MPFIT, ASP Conf. Series 411, 251-254 (2009) distribution, depends on the exact shape of the inner