pith. sign in

arxiv: 2304.06764 · v1 · submitted 2023-04-13 · 🌌 astro-ph.GA

What Does the Virial Coefficient of the Hb Broad-Line Region Depend On?

Lizvette Villafa\~na , Peter R. Williams , Tommaso Treu , Brendon J. Brewer , Aaron J. Barth , Vivian U , Vardha N. Bennert , Hengxiao Guo
show 32 more authors
Misty C. Bentz Gabriela Canalizo Alexei V. Filippenko Elinor Gates Michael D. Joner Matthew A. Malkan Jong-Hak Woo Bela Abolfathi Thomas Bohn K. Azalee Bostroem Andrew Brandel Thomas G. Brink Sanyum Channa Maren Cosens Edward Donohue Goni Halevi Carol E. Hood J. Chuck Horst Maxime de Kouchkovsky Benjamin Kuhn Douglas C. Leonard Raul Michel Melanie Kae B. Olaes Daeseong Park Jordan N. Runco Remington O. Sexton Isaac Shivvers Chance L. Spencer Benjamin E. Stahl Samantha Stegman Jonelle L. Walsh Weikang Zheng
This is my paper

Pith reviewed 2026-05-24 09:38 UTC · model grok-4.3

classification 🌌 astro-ph.GA
keywords virial coefficientreverberation mappingbroad-line regionblack hole massactive galactic nucleidynamical modelingscale factor f
0
0 comments X

The pith

The virial coefficient for the broad-line region correlates with black hole mass when calibrated against dynamical masses.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper recovers individual virial coefficients f by dividing dynamical modeling black hole masses by the virial products measured from reverberation mapping time lags and line widths. It then tests for correlations between these f values and AGN or BLR parameters such as mass, disk thickness, inclination, and line profile shape. The central result is evidence that log10(f_mean,σ) increases with black hole mass. Marginal evidence is reported for additional relations involving the rms-based coefficient, disk thickness, and inclination. These findings indicate that f is not a universal constant but varies systematically with source properties.

Core claim

By anchoring reverberation mapping data to independent dynamical black hole masses, the authors recover per-object scale factors and report evidence that the mean virial coefficient log10(f_mean,σ) correlates with black hole mass, along with marginal evidence that disk thickness and inclination anti-correlate with certain f measures and that line-profile shape correlates with log10(f_rms,σ).

What carries the argument

The virial coefficient f recovered as dynamical mass divided by the reverberation-mapping virial product (R ΔV²/G), allowing individual calibration instead of a single average value.

If this is right

  • Reverberation mapping masses estimated with a single average f carry mass-dependent systematic errors.
  • The broad-line region geometry or kinematics change with black hole mass.
  • The line dispersion measured in the mean spectrum produces a more mass-sensitive f than FWHM-based versions.
  • Inclination and disk thickness affect the observed line widths and therefore the recovered f values.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A mass-dependent f relation could be applied to improve single-epoch mass estimates for large quasar samples where dynamical modeling is impossible.
  • The correlation might reflect changes in accretion rate or radiation pressure that alter BLR structure at higher masses.
  • Orientation effects captured by the inclination anti-correlations could help explain part of the scatter in traditional f distributions.

Load-bearing premise

Black hole masses obtained from dynamical modeling are accurate and share no hidden assumptions with the reverberation mapping measurements used to compute the virial products.

What would settle it

No correlation between log10(f_mean,σ) and black hole mass appears in an independent sample of AGN that have both dynamical modeling masses and reverberation mapping data.

Figures

Figures reproduced from arXiv: 2304.06764 by Aaron J. Barth, Alexei V. Filippenko, Andrew Brandel, Bela Abolfathi, Benjamin E. Stahl, Benjamin Kuhn, Brendon J. Brewer, Carol E. Hood, Chance L. Spencer, Daeseong Park, Douglas C. Leonard, Edward Donohue, Elinor Gates, Gabriela Canalizo, Goni Halevi, Hengxiao Guo, Isaac Shivvers, J. Chuck Horst, Jonelle L. Walsh, Jong-Hak Woo, Jordan N. Runco, K. Azalee Bostroem, Lizvette Villafa\~na, Maren Cosens, Matthew A. Malkan, Maxime de Kouchkovsky, Melanie Kae B. Olaes, Michael D. Joner, Misty C. Bentz, Peter R. Williams, Raul Michel, Remington O. Sexton, Samantha Stegman, Sanyum Channa, Thomas Bohn, Thomas G. Brink, Tommaso Treu, Vardha N. Bennert, Vivian U, Weikang Zheng.

Figure 1
Figure 1. Figure 1: To propagate uncertainties, we assume Gaussian errors on the cross-correlation time lag (left) and line-width (right) measurements given by U et al. (2022). This allows us to create distribution functions that we can utilize with our caramel MBH posterior distribution function to determine the distribution of the scale factor of an individual source, from which we use the 68% confidence interval for 1σ unc… view at source ↗
Figure 3
Figure 3. Figure 3: Correlations between the scale factor log10 frms,σ (top) and log10 frms,FWHM (bottom) with select AGNs and model parameters. From left to right: MBH, optical luminosity, Eddington ratio, Hβ -emitting BLR opening angle (disk thickness), Hβ -emitting BLR inclination angle, and our “inflow-outflow” parameter. The colored dots and contours show the median and 68% confidence regions of the 2D posterior PDFs for… view at source ↗
Figure 4
Figure 4. Figure 4: Correlations between the scale factor log10 fmean,σ (top) and log10 fmean,FWHM (bottom) with select AGNs and model parameters. From left to right: MBH, optical luminosity, Eddington ratio, Hβ -emitting BLR opening angle (disk thickness), Hβ -emitting BLR inclination angle, and our “inflow-outflow” parameter. The colored dots and contours show the median and 68% confidence regions of the 2D posterior PDFs f… view at source ↗
Figure 5
Figure 5. Figure 5: Correlations between rms line-profile shape and scale factor determined using line dispersion (left) and FWHM (right). The dashed black lines and gray shaded regions give the median and 68% confidence intervals of the linear regression. Dotted lines are offset above and below the dashed line by the median value of the intrinsic scatter. Purple points are for the AGNs from V22, red points are from W18, gree… view at source ↗
Figure 6
Figure 6. Figure 6: Correlations between mean line-profile shape and scale factor determined using line dispersion (left) and FWHM (right). The dashed black lines and gray shaded regions give the median and 68% confidence intervals of the linear regression. Dotted lines are offset above and below the dashed line by the median value of the intrinsic scatter. Purple points are for the AGNs from V22, red points are from W18, gre… view at source ↗
Figure 8
Figure 8. Figure 8: ). The spherical BLR disk represented by 0.05 0.10 0.15 0.20 0.25 F 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 log10(FWHM/σ) Disk Thickness Effect on FWHM/σ θo = 5◦ θo = 15◦ θo = 25◦ θo = 45◦ [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: We investigate the role of BLR radius in line pro￾file shape using caramel models by varying the parameters µ and F, and holding all other model parameters constant. As described in the text, the parameter µ defines the mean BLR radius and the parameter F defines the minimum ra￾dius in units of µ. Different mean BLR radii, µ, are depicted in different colors: 1 light-day is shown in blue, 5 light-days is s… view at source ↗
Figure 9
Figure 9. Figure 9: Inflow (left) and Outflow (right) effects on line profile shape. Two different BLR disk thickness/opening angles, θo are used. In both plots, a thick disk with θo = 15◦ is shown in blue and a spherical structure with θo = 45◦ is shown in orange. The x -axis, fellip, represents the fraction of particles on elliptical orbits. Thus an increasing value of fellip represents a greater percentage of particles on … view at source ↗
Figure 10
Figure 10. Figure 10: We investigate the role of turbulent motion in line profile shape using caramel toy models by varying the parameter σturb, and holding all other model parameters con￾stant. Since macroturbulent velocities depend on both σturb and |vcirc| ∝ log10(MBH/M ), we test the effects of turbu￾lent motion using two different black hole masses. The blue points correspond to log10(MBH/M ) = 7.0 and the orange points c… view at source ↗
read the original abstract

We combine our dynamical modeling black hole mass measurements from the Lick AGN Monitoring Project 2016 sample with measured cross-correlation time lags and line widths to recover individual scale factors, f, used in traditional reverberation mapping analyses. We extend our sample by including prior results from Code for AGN Reverberation and Modeling of Emission Lines (caramel) studies that have utilized our methods. Aiming to improve the precision of black hole mass estimates, as well as uncover any regularities in the behavior of the broad-line region (BLR), we search for correlations between f and other AGN/BLR parameters. We find (i) evidence for a correlation between the virial coefficient log10(fmean,{\sigma}) and black hole mass, (ii) marginal evidence for a similar correlation between log10(frms,{\sigma}) and black hole mass, (iii) marginal evidence for an anti-correlation of BLR disk thickness with log10(fmean,FWHM)and log10(frms,FWHM), and (iv) marginal evidence for an anti-correlation of inclination angle with log10(fmean,FWHM), log10(frms,{\sigma}), and log10(fmean,{\sigma}). Lastly, we find marginal evidence for a correlation between line-profile shape, when using the root-meansquare spectrum, log10(FWHM/{\sigma})rms, and the virial coefficient, log10(frms,{\sigma}), and investigate how BLR properties might be related to line-profile shape using caramel models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper recovers individual virial coefficients f by dividing dynamical modeling black hole masses (from caramel on LAMP 2016 and prior samples) by reverberation-mapping virial products derived from the same light curves and spectra. It then searches for correlations between these f values (mean and rms, using σ and FWHM) and AGN/BLR parameters, reporting (i) evidence for a correlation between log10(f_mean,σ) and black hole mass, (ii) marginal evidence for a similar correlation with log10(f_rms,σ), and (iii-iv) marginal evidence for anti-correlations involving BLR disk thickness, inclination, and line-profile shape.

Significance. A robust, non-artifactual correlation between the virial coefficient and black hole mass would be significant for refining single-epoch or RM mass estimates and for constraining BLR geometry. The approach of calibrating f directly from dynamical models is a methodological strength when the two mass estimates are demonstrably independent.

major comments (2)
  1. [Abstract and f-recovery section] Abstract and the section deriving individual f values: f is recovered directly as M_dyn / VP with VP = c τ ΔV²/G from the identical LAMP 2016 data used for caramel dynamical modeling. The reported correlation between log10(f_mean,σ) and black hole mass (M_BH ≡ M_dyn) is therefore equivalent to a correlation between M_dyn and VP; without explicit mock-data tests or error-budget decomposition to exclude correlated uncertainties in τ or line width arising from shared sampling, S/N, or model assumptions, it is unclear whether the correlation is physical. This directly affects the central claim (i).
  2. [Results section] Results section on correlations: the abstract states specific findings but provides no error budgets, sample-selection criteria, or tests against modeling systematics. If these are also absent from the full analysis, small-number statistics or post-hoc choices could affect the significance of the reported f-M_BH correlation and the marginal results.
minor comments (2)
  1. [Notation] Notation for f_mean vs. f_rms and σ vs. FWHM should be defined once and used consistently in all figures and tables.
  2. [Figures] Correlation plots should include individual error bars on both axes and report the Spearman or Pearson coefficient with its p-value.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments, which have helped us improve the clarity and robustness of our analysis. We address each major comment below and indicate revisions made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract and f-recovery section] Abstract and the section deriving individual f values: f is recovered directly as M_dyn / VP with VP = c τ ΔV²/G from the identical LAMP 2016 data used for caramel dynamical modeling. The reported correlation between log10(f_mean,σ) and black hole mass (M_BH ≡ M_dyn) is therefore equivalent to a correlation between M_dyn and VP; without explicit mock-data tests or error-budget decomposition to exclude correlated uncertainties in τ or line width arising from shared sampling, S/N, or model assumptions, it is unclear whether the correlation is physical. This directly affects the central claim (i).

    Authors: We agree that f = M_dyn / VP by construction and that this implies a relationship between the reported log f–M_BH correlation and the underlying M_dyn–VP plane. However, M_dyn is not equivalent to a simple virial product: caramel performs a global fit to the full velocity-resolved reverberation dataset, constraining a physically motivated BLR geometry, kinematics, and inclination that are independent of the scalar cross-correlation lag and single line-width measure used for VP. The modeling therefore incorporates information (e.g., differential lags across the line profile) that is not present in VP. We acknowledge that shared data could introduce correlated uncertainties and that the original manuscript did not present explicit mock-data tests. In the revised version we have added (i) a dedicated error-budget decomposition separating statistical uncertainties in τ and ΔV from modeling systematics in M_dyn, and (ii) mock-data simulations that inject known BLR geometries and recover f values to demonstrate that the observed correlation is not reproduced by sampling or S/N effects alone. revision: yes

  2. Referee: [Results section] Results section on correlations: the abstract states specific findings but provides no error budgets, sample-selection criteria, or tests against modeling systematics. If these are also absent from the full analysis, small-number statistics or post-hoc choices could affect the significance of the reported f-M_BH correlation and the marginal results.

    Authors: The full manuscript already contains error budgets for the recovered f values (propagated from both M_dyn and VP uncertainties) and describes the sample-selection criteria (LAMP 2016 objects with successful caramel models plus prior caramel samples meeting the same quality cuts). We also tested robustness against modeling systematics by repeating the correlation analysis under alternate caramel priors and line-width definitions. Nevertheless, we accept that these elements were not sufficiently highlighted or cross-referenced in the results section. In revision we have (i) added explicit cross-references to the error budgets and selection criteria, (ii) included bootstrap and jackknife significance tests that account for the modest sample size, and (iii) clarified that the reported marginal correlations were pre-specified rather than post-hoc. These changes strengthen the presentation without altering the conclusions. revision: partial

Circularity Check

1 steps flagged

f recovered as M_dyn/VP; reported f-M_BH correlation reduces to M_dyn-VP relation by construction

specific steps
  1. self definitional [Abstract]
    "We combine our dynamical modeling black hole mass measurements from the Lick AGN Monitoring Project 2016 sample with measured cross-correlation time lags and line widths to recover individual scale factors, f, used in traditional reverberation mapping analyses. ... We find (i) evidence for a correlation between the virial coefficient log10(fmean,σ) and black hole mass"

    f is obtained exactly as f = M_dyn / VP. Correlating the resulting log f with log M_dyn is therefore identical to correlating log(M_dyn / VP) with log M_dyn, which is fixed once the M_dyn–VP relation is measured; the reported dependence is not an independent prediction but a restatement of how the two mass estimates differ.

full rationale

The paper recovers individual f values directly from dynamical masses divided by the reverberation-mapping virial product (VP = c τ ΔV²/G) using the same LAMP 2016 data. It then reports a correlation between log f and black hole mass (M_dyn). Because log f ≡ log M_dyn − log VP by definition, any such correlation is mathematically equivalent to a (negative) correlation between log M_dyn and log VP; no independent physical content about BLR structure is added. The abstract states the recovery method and the correlation finding without mock tests separating shared modeling assumptions or data. This matches the self-definitional pattern at score 7; other claims (e.g., inclination or thickness trends) are marginal and not load-bearing for the central result.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the accuracy of previously published dynamical masses and on the caramel modeling framework; no new free parameters are introduced in the abstract, but the recovery of f itself is data-driven.

free parameters (1)
  • individual f values
    Recovered per object from the ratio of dynamical mass to RM virial product; each is effectively fitted to the data for that AGN.
axioms (1)
  • domain assumption Dynamical modeling yields unbiased black hole masses independent of reverberation-mapping geometry assumptions
    Invoked when using those masses to solve for f; location implied in the description of combining the two techniques.

pith-pipeline@v0.9.0 · 6007 in / 1421 out tokens · 44733 ms · 2026-05-24T09:38:27.585148+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. OzDES Reverberation Mapping of Active Galactic Nuclei: Final Data Release, Black-Hole Mass Results, & Scaling Relations

    astro-ph.GA 2025-12 unverdicted novelty 6.0

    OzDES final release delivers 62 new reverberation-mapped black hole masses and tighter lag-luminosity relations for Hβ, MgII, and CIV in high-redshift AGN after correcting for survey-length selection effects.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · cited by 1 Pith paper

  1. [1]

    2012, ApJ, 752, 92

    Roca-Sogorb, M. 2012, ApJ, 752, 92

    work page 2012

  2. [2]

    J., Bennert, V

    Barth, A. J., Bennert, V. N., Canalizo, G., et al. 2015, ApJS, 217, 26

    work page 2015

  3. [3]

    C., Raimundo, S

    Batiste, M., Bentz, M. C., Raimundo, S. I., Vestergaard, M., & Onken, C. A. 2017, ApJL, 838, L10

    work page 2017

  4. [4]

    N., Treu, T., Woo, J.-H., et al

    Bennert, V. N., Treu, T., Woo, J.-H., et al. 2010, ApJ, 708, 1507

    work page 2010

  5. [5]

    C., Williams, P

    Bentz, M. C., Williams, P. R., & Treu, T. 2022, ApJ, 934, 168

    work page 2022

  6. [6]

    C., Denney, K

    Bentz, M. C., Denney, K. D., Cackett, E. M., et al. 2006, ApJ, 651, 775

    work page 2006

  7. [7]

    D., & McKee, C

    Blandford, R. D., & McKee, C. F. 1982, ApJ, 255, 419

    work page 1982

  8. [8]

    M., Bentz, M

    Cackett, E. M., Bentz, M. C., & Kara, E. 2021, iScience, 24, 102557

    work page 2021

  9. [9]

    D., Peterson, B

    Denney, K. D., Peterson, B. M., Pogge, R. W., et al. 2010, ApJ, 721, 715

    work page 2010

  10. [10]

    2020, The Astrophysical Journal, 888, 37

    Ding, X., Silverman, J., Treu, T., et al. 2020, The Astrophysical Journal, 888, 37

    work page 2020

  11. [11]

    T., Blandford, R

    Emmering, R. T., Blandford, R. D., & Shlosman, I. 1992, ApJ, 385, 460 What Does the Geometry of the Hβ BLR Depend On? 17

    work page 1992

  12. [12]

    2005, SSRv, 116, 523

    Ferrarese, L., & Ford, H. 2005, SSRv, 116, 523

    work page 2005

  13. [13]

    2000, ApJL, 539, L9

    Ferrarese, L., & Merritt, D. 2000, ApJL, 539, L9

    work page 2000

  14. [14]

    2000, ApJL, 539, L13

    Gebhardt, K., Bender, R., Bower, G., et al. 2000, ApJL, 539, L13

    work page 2000

  15. [15]

    R., Korista, K

    Goad, M. R., Korista, K. T., & Ruff, A. J. 2012, MNRAS, 426, 3086

    work page 2012

  16. [16]

    2011, MNRAS, 412, 2211 Gravity Collaboration, Sturm, E., Dexter, J., et al

    Combes, F. 2011, MNRAS, 412, 2211 Gravity Collaboration, Sturm, E., Dexter, J., et al. 2018, Nature, 563, 657 Gravity Collaboration, Amorim, A., Baub¨ ock, M., et al. 2020, A&A, 643, A154

    work page 2011

  17. [17]

    J., Pancoast, A., Barth, A

    Grier, C. J., Pancoast, A., Barth, A. J., et al. 2017, ApJ, 849, 146

    work page 2017

  18. [18]

    J., Peterson, B

    Grier, C. J., Peterson, B. M., Pogge, R. W., et al. 2012, ApJ, 755, 60

    work page 2012

  19. [19]

    J., Martini, P., Watson, L

    Grier, C. J., Martini, P., Watson, L. C., et al. 2013, ApJ, 773, 90

    work page 2013

  20. [20]

    G., Marscher, A

    Jorstad, S. G., Marscher, A. P., Lister, M. L., et al. 2005, AJ, 130, 1418

    work page 2005

  21. [21]

    F., & K¨ onigl, A

    Kartje, J. F., & K¨ onigl, A. 1996, Vistas in Astronomy, 40, 133

    work page 1996

  22. [22]

    Kelly, B. C. 2007, ApJ, 665, 1489

    work page 2007

  23. [23]

    F., Antonucci, R., et al

    Kishimoto, M., H¨ onig, S. F., Antonucci, R., et al. 2011, A&A, 536, A78

    work page 2011

  24. [24]

    2011, Baltic Astronomy, 20, 400 —

    Kollatschny, W., & Zetzl, M. 2011, Baltic Astronomy, 20, 400 —. 2013, A&A, 549, A100

    work page 2011

  25. [25]

    Konigl, A., & Kartje, J. F. 1994, ApJ, 434, 446

    work page 1994

  26. [26]

    Kormendy, J., & Ho, L. C. 2013, ARA&A, 51, 511

    work page 2013

  27. [27]

    1995, ARA&A, 33, 581

    Kormendy, J., & Richstone, D. 1995, ARA&A, 33, 581

    work page 1995

  28. [28]

    2015, in TORUS2015: The AGN Unification Scheme After 30 Years, ed

    Koshida, S. 2015, in TORUS2015: The AGN Unification Scheme After 30 Years, ed. P. Gandhi & S. F. Hoenig

    work page 2015

  29. [29]

    1998, AJ, 115, 2285

    Magorrian, J., Tremaine, S., Richstone, D., et al. 1998, AJ, 115, 2285

    work page 1998

  30. [30]

    A., Ferrarese, L., Merritt, D., et al

    Onken, C. A., Ferrarese, L., Merritt, D., et al. 2004, ApJ, 615, 645

    work page 2004

  31. [31]

    J., & Treu, T

    Pancoast, A., Brewer, B. J., & Treu, T. 2011, ApJ, 730, 139

    work page 2011

  32. [32]

    J., Treu, T., et al

    Pancoast, A., Brewer, B. J., Treu, T., et al. 2014, MNRAS, 445, 3073

    work page 2014

  33. [33]

    M., Barth, A

    Pei, L., Fausnaugh, M. M., Barth, A. J., et al. 2017, ApJ, 837, 131

    work page 2017

  34. [34]

    Peterson, B. M. 1993, PASP, 105, 247

    work page 1993

  35. [35]

    M., Ferrarese, L., Gilbert, K

    Peterson, B. M., Ferrarese, L., Gilbert, K. M., et al. 2004, ApJ, 613, 682

    work page 2004

  36. [36]

    Pringle, J. E. 1981, ARA&A, 19, 137

    work page 1981

  37. [37]

    A., Gebhardt, K., & Bonning, E

    Salviander, S., Shields, G. A., Gebhardt, K., & Bonning, E. W. 2007, The Astrophysical Journal, 662, 131

    work page 2007

  38. [38]

    2011, A&A, 535, A87

    Schulze, A., & Wisotzki, L. 2011, A&A, 535, A87

    work page 2011

  39. [39]

    2013, Bulletin of the Astronomical Society of India, 41, 61

    Shen, Y. 2013, Bulletin of the Astronomical Society of India, 41, 61

    work page 2013

  40. [40]

    A., & Salviander, S

    Shields, G. A., & Salviander, S. 2009, in Astronomical Society of the Pacific Conference Series, Vol. 419, Galaxy Evolution: Emerging Insights and Future Challenges, ed. S. Jogee, I. Marinova, L. Hao, & G. A. Blanc, 392

    work page 2009

  41. [41]

    A., Dunlop, J

    Targett, T. A., Dunlop, J. S., & McLure, R. J. 2012, Monthly Notices of the Royal Astronomical Society, 420, 3621

    work page 2012

  42. [42]

    A., & Blandford, R

    Treu, T., Malkan, M. A., & Blandford, R. D. 2004, ApJL, 615, L97 U, V., Barth, A. J., Vogler, H. A., et al. 2022, ApJ, 925, 52 Villafa˜ na, L., Williams, P. R., Treu, T., et al. 2022, The Astrophysical Journal, 930, 52

    work page 2004

  43. [43]

    R., & Treu, T

    Williams, P. R., & Treu, T. 2022, ApJ, 935, 128

    work page 2022

  44. [44]

    R., Pancoast, A., Treu, T., et al

    Williams, P. R., Pancoast, A., Treu, T., et al. 2018, ApJ, 866, 75 —. 2020, ApJ, 902, 74

    work page 2018

  45. [45]

    2013, ApJ, 772, 49

    Woo, J.-H., Schulze, A., Park, D., et al. 2013, ApJ, 772, 49

    work page 2013

  46. [46]

    A., & Blandford, R

    Woo, J.-H., Treu, T., Malkan, M. A., & Blandford, R. D. 2006, ApJ, 645, 900 —. 2008, ApJ, 681, 925

    work page 2006

  47. [47]

    Woo, J.-H., Yoon, Y., Park, S., Park, D., & Kim, S. C. 2015, ApJ, 801, 38

    work page 2015

  48. [48]

    J., et al

    Woo, J.-H., Treu, T., Barth, A. J., et al. 2010, ApJ, 716, 269

    work page 2010