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arxiv: 1907.00315 · v1 · pith:GUXI5J6Pnew · submitted 2019-06-30 · 🌌 astro-ph.GA

Calibration of the virial factor f in supermassive black hole masses of reverberation-mapped AGNs

Li-Ming Yu , Wei-Hao Bian , Chan Wang , Bi-Xuan Zhao , Xue Ge (NJNU) This is my paper

Pith reviewed 2026-05-25 13:16 UTC · model grok-4.3

classification 🌌 astro-ph.GA
keywords supermassive black holesreverberation mappingvirial factoractive galactic nucleiblack hole massstellar velocity dispersionbroad line regionH beta line
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The pith

The virial factor f for AGN black hole masses is derived by forcing agreement with the M_BH-σ_* relation of quiescent galaxies.

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

The paper calculates the virial factor f for 34 low-redshift reverberation-mapped AGNs by assuming these objects obey exactly the same supermassive black hole mass to stellar velocity dispersion relation as quiescent galaxies. Four versions of f are obtained using Hβ full width at half maximum or line dispersion measured from either the mean or rms spectrum. FWHM-based factors show correlations with line width parameters that disappear when line dispersion is used instead, which the authors interpret as an inclination effect in broad-line region geometry. With the calibrated mean-spectrum FWHM version of f, the black hole masses for the sample scatter by 0.39 dex around the relation; a separate high-redshift sample of 30 SDSS reverberation-mapped AGNs shows larger scatter.

Core claim

By assuming that the 34 reverberation-mapped AGNs intrinsically obey the same M_BH-σ_* relation as quiescent galaxies, four kinds of virial factor f are derived using Hβ velocity tracers from mean and rms spectra. The FWHM-based f from the mean spectrum correlates significantly with observational parameters such as FWHM itself, while the line-dispersion versions show weaker or no correlations, implying inclination dependence in the broad-line region. The calibrated FWHM-based f yields a 0.39 dex scatter in SMBH masses for the low-redshift sample, whereas the high-redshift SDSS sample exhibits larger scatter, suggesting possible evolution of the M_BH-σ_* relation.

What carries the argument

The virial factor f, obtained by enforcing equality of the M_BH-σ_* relation between the reverberation-mapped AGNs and quiescent galaxies so that measured time lags and line widths convert directly into black hole mass.

If this is right

  • FWHM-based f values correlate with line width, while line-dispersion-based values do not, indicating that line dispersion is less sensitive to broad-line region inclination.
  • The variable nature of f means that single-epoch mass estimates that use a fixed f and luminosity plus FWHM will carry additional scatter.
  • A thick-disk model of the broad-line region reproduces the observed dependence of FWHM on inclination while leaving line dispersion largely unaffected.
  • The 0.39 dex scatter obtained with the calibrated mean-spectrum FWHM f holds for the 34 low-redshift objects but increases for the 30 high-redshift SDSS objects.
  • Larger scatter at high redshift is consistent with evolution in the underlying M_BH-σ_* relation between low- and high-redshift active galaxies.

Where Pith is reading between the lines

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

  • If the assumption holds, future single-epoch surveys should incorporate a luminosity- or width-dependent f rather than a universal constant.
  • The inclination interpretation could be tested by comparing f values against independent orientation indicators such as radio jet angles in the same objects.
  • The reported increase in scatter at higher redshift motivates repeating the calibration exercise on larger intermediate-redshift reverberation-mapped samples once stellar velocity dispersions become available.
  • Any systematic offset between active and quiescent M_BH-σ_* relations would propagate directly into the derived f values and all reported scatters.

Load-bearing premise

The 34 reverberation-mapped AGNs intrinsically obey exactly the same supermassive black hole mass to stellar velocity dispersion relation as quiescent galaxies.

What would settle it

An independent determination of black hole masses in these same 34 objects, for example via stellar dynamical modeling, that yields a different M_BH-σ_* slope or scatter than the one assumed to solve for f.

Figures

Figures reproduced from arXiv: 1907.00315 by Bi-Xuan Zhao, Chan Wang, Li-Ming Yu, Wei-Hao Bian, Xue Ge (NJNU).

Figure 1
Figure 1. Figure 1: The distributions of four kinds of the factor f, i.e., fF,mean, fF,rms, fσ,mean, fσ,rms from left to right and from top to bottom, respectively. The red line is the f distribution for our sample excluding pseudobulges. The black line and the blue dash-line are the f distributions for our total sample adopting different MBH − σ∗ relation for pseudobulges with α = −0.51 (PB unscaled f) and α = −1.09 (PB scal… view at source ↗
Figure 2
Figure 2. Figure 2: Left: fF,mean vs. FWHMmean . Black circles denote ellipticals. Red squares denote classical bulges. Gray triangles represent pseudobulges. The solid line is our best-fitting relation derived from BCES(Y|X), the dash lines represent the 1 σ scatter range. The red lines are that only for 17 classical bulges and ellipticals. Right: the SMBH mass calculated based on the fF,mean by Equation 6 (the left panel) v… view at source ↗
Figure 3
Figure 3. Figure 3: Left: fF,mean vs. DHβ . Right: the SMBH mass versus σ∗ . The symbols and the lines are the same as [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: fF,rms vs. FWHMrms . Right: the SMBH mass versus σ∗ . The symbols and the lines are the same as [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Cumulative fraction of values of four kind of f (top left: fF,mean; top right fF,rms; bottom left: fσ,mean, bottom right: fσ,rms), compared to the theoretical distributions (blue dot dash lines) for two thick-disk models of a=0.1 and a=0.3. The offset factor is the value by which the theoretical value has been divided to aid in comparison of the two distributions. The black line is for the total sample. Th… view at source ↗
Figure 6
Figure 6. Figure 6: fModel F,mean versus FWHMmean of Hβ , where fmodel F,mean is cal￾culated from the BLRs dynamical modelling of the continuum light curve and the Hβ line profiles, X-ray variability and resolved Paα emission re￾gion. The solid line is our calibration correlation of Equation 6 and the dash lines show the range of 1 σ uncertainty.The blue line is the relation from the method by the standard accretion disk mode… view at source ↗
read the original abstract

Using a compiled sample of 34 broad-line active galactic nuclei (AGNs) with measured H$\beta$ time lags from the reverberation mapping (RM) method and measured bulge stellar velocity dispersions $\sigma_*$, we calculate the virial factor $f$ by assuming that the RM AGNs intrinsically obey the same $M_{\rm BH}-\sigma_*$ relation as quiescent galaxies, where $M_{\rm BH}$ is the mass of the supermassive black hole (SMBH). Considering four tracers of the velocity of the broad-line regions (BLRs), i.e., the H$\beta$ line width or line dispersion from the mean or rms spectrum, there are four kinds of the factor $f$. Using the \hb Full-width at half-maximum (FWHM) to trace the BLRs velocity, we find significant correlations between the factor $f$ and some observational parameters, e.g., FWHM, the line dispersion. Using the line dispersion to trace the BLRs velocity, these relations disappear or become weaker. It implies the effect of inclination in BLRs geometry. It also suggests that the variable $f$ in $M_{\rm BH}$ estimated from luminosity and FWHM in a single-epoch spectrum is not negligible. Using a simple model of thick-disk BLRs, we also find that, as the tracer of the BLRs velocity, H$\beta$ FWHM has some dependence on the inclination, while the line dispersion $\sigma_{\rm H\beta }$ is insensitive to the inclination. Considering the calibrated FWHM-based factor $f$ from the mean spectrum, the scatter of the SMBH mass is 0.39 dex for our sample of 34 low redshift RM AGNs. For a high redshift sample of 30 SDSS RM AGNs with measured stellar velocity dispersions, we find that the SMBH mass scatter is larger than that for our sample of 34 low redshift RM AGNs. It implies the possibility of evolution of the $M_{\rm BH}-\sigma_*$ relation from high-redshift to low-redshift AGNs.

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

3 major / 2 minor

Summary. The paper compiles 34 low-redshift RM AGNs with Hβ lags and σ_* measurements, assumes these objects obey the same M_BH-σ_* relation as quiescent galaxies to solve for the virial factor f using four velocity tracers (Hβ FWHM and line dispersion from mean and rms spectra), reports correlations of f with line width that disappear when using line dispersion, models BLRs as a thick disk to argue that FWHM depends on inclination while line dispersion does not, and finds 0.39 dex scatter in SMBH masses for the low-z sample using the calibrated mean-spectrum FWHM-based f; the scatter is larger for a high-z sample of 30 SDSS RM AGNs, suggesting possible evolution of the M_BH-σ_* relation.

Significance. If the assumption holds, the work supplies an empirical calibration of f for single-epoch mass estimates and a quantitative comparison of velocity tracers that bears on BLR geometry; the thick-disk modeling and the reported 0.39 dex scatter would then be useful benchmarks for the field.

major comments (3)
  1. [Abstract] Abstract, first sentence, and the procedure described in §3: f is obtained by setting the reverberation mass M_RM = f (R ΔV²/G) equal to the mass predicted from each object's measured σ_* via the quiescent-galaxy M_BH-σ_* relation; this makes every reported correlation between f and FWHM (or other parameters) a direct consequence of the assumed relation rather than an independent empirical result.
  2. [Abstract] Abstract: the quoted scatter of 0.39 dex for the 34 low-z objects is the residual dispersion around the M_BH-σ_* relation that was enforced by construction when f was defined; the larger scatter found for the high-z sample is likewise measured relative to the same enforced zero-point, so the difference cannot be interpreted as evidence for evolution without an independent test of the assumption.
  3. [Abstract] Abstract and the high-z comparison paragraph: the claim that larger scatter at high redshift implies evolution of the M_BH-σ_* relation rests on both samples obeying the identical low-z quiescent relation used to define f; if the intrinsic normalization or slope differs for RM AGNs, the reported scatters and the evolution inference become adjustments required to restore the assumption rather than measurements of evolution.
minor comments (2)
  1. The four definitions of f (mean/rms spectrum, FWHM/σ_line) should be written explicitly with equations in the methods section so that the reader can reproduce the exact numerical values.
  2. The high-z SDSS sample selection, lag measurements, and σ_* determinations are not described in sufficient detail to allow the reader to assess whether the larger scatter arises from measurement differences or from the assumption itself.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and substantive comments on our manuscript. We address each major comment below, acknowledging where the points are valid and proposing revisions to improve clarity regarding our assumptions and interpretations.

read point-by-point responses

  1. Referee: [Abstract] Abstract, first sentence, and the procedure described in §3: f is obtained by setting the reverberation mass M_RM = f (R ΔV²/G) equal to the mass predicted from each object's measured σ_* via the quiescent-galaxy M_BH-σ_* relation; this makes every reported correlation between f and FWHM (or other parameters) a direct consequence of the assumed relation rather than an independent empirical result.

    Authors: The referee is correct that the individual f values are computed directly from the assumed M_BH-σ_* relation, so any correlation between f and a velocity tracer (such as FWHM) follows mathematically from f ∝ 1/(ΔV)². The manuscript's key comparative result—that these correlations are strong for FWHM-based f but weak or absent for line-dispersion-based f—remains informative under the assumption, as it is consistent with inclination effects in a thick-disk BLR. We will revise the abstract and the description in §3 to explicitly state that the correlations are induced by the assumption and to frame the analysis as a consistency test between velocity tracers rather than an independent empirical discovery. revision: yes

  2. Referee: [Abstract] Abstract: the quoted scatter of 0.39 dex for the 34 low-z objects is the residual dispersion around the M_BH-σ_* relation that was enforced by construction when f was defined; the larger scatter found for the high-z sample is likewise measured relative to the same enforced zero-point, so the difference cannot be interpreted as evidence for evolution without an independent test of the assumption.

    Authors: We partially agree. The 0.39 dex scatter is obtained after adopting a single calibrated f (the sample mean of the individually computed f values) and applying it uniformly, which is the relevant procedure for single-epoch mass estimates. This scatter therefore reflects the dispersion in the individual f factors rather than being identically zero. For the high-z sample the same fixed f is used, so the larger scatter indicates greater deviation from the low-z M_BH-σ_* relation under the calibration assumption. We will revise the abstract to clarify this distinction and to present the high-z comparison as a test under the stated assumption rather than direct evidence of evolution. revision: partial

  3. Referee: [Abstract] Abstract and the high-z comparison paragraph: the claim that larger scatter at high redshift implies evolution of the M_BH-σ_* relation rests on both samples obeying the identical low-z quiescent relation used to define f; if the intrinsic normalization or slope differs for RM AGNs, the reported scatters and the evolution inference become adjustments required to restore the assumption rather than measurements of evolution.

    Authors: We agree that any suggestion of evolution is conditional on the low-z quiescent M_BH-σ_* relation applying equally to the high-z RM AGNs. The larger scatter could reflect either increased intrinsic scatter or a shift in normalization/slope. We will revise the abstract and the high-z paragraph to state this caveat explicitly and to describe the result as indicating the possibility of evolution or differences in the relation, rather than as a direct measurement of evolution. revision: yes

Circularity Check

2 steps flagged

Calibration of f assumes RM AGNs obey the identical M_BH-σ_* relation as quiescent galaxies, directly solving for f and rendering reported scatters dependent on that assumption.

specific steps
  1. fitted input called prediction [Abstract]
    "we calculate the virial factor f by assuming that the RM AGNs intrinsically obey the same M_BH-σ_* relation as quiescent galaxies, where M_BH is the mass of the supermassive black hole (SMBH)."

    The reverberation mass is M_RM = f (R ΔV²/G). Setting M_RM equal to the mass from the external M_BH-σ_* relation directly solves for f = M_σ / (R ΔV²/G). All subsequent f values, f vs. FWHM correlations, and mass scatters are therefore computed relative to a zero-point and normalization that were imposed by the assumption.

  2. fitted input called prediction [Abstract]
    "Considering the calibrated FWHM-based factor f from the mean spectrum, the scatter of the SMBH mass is 0.39 dex for our sample of 34 low redshift RM AGNs. For a high redshift sample of 30 SDSS RM AGNs with measured stellar velocity dispersions, we find that the SMBH mass scatter is larger than that for our sample of 34 low redshift RM AGNs."

    The quoted scatters are the residuals of M_RM around the M_BH-σ_* relation after f has been chosen to force M_RM = M_σ_* for each object. The larger high-z scatter is likewise measured relative to the same enforced relation, so both numbers are consequences of the initial assumption rather than independent empirical results.

full rationale

The paper's core step solves for the virial factor f by setting the reverberation-mapping mass equal to the mass predicted from each AGN's measured σ_* via the quiescent-galaxy M_BH-σ_* relation. This is stated in the abstract and used to generate all f values, the reported correlations of f with FWHM/line dispersion, the 0.39 dex scatter for the low-z sample, and the larger scatter for the high-z sample. Because f is chosen to enforce equality with the assumed relation, the scatters and correlations are residuals around a relation imposed by construction rather than independent measurements. The thick-disk model is introduced after the correlations and does not alter the definitional step.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on one domain assumption (identical M_BH-σ_* relation) and one ad-hoc modeling choice (thick-disk geometry). No free parameters are fitted beyond the definition of f itself; no new particles or forces are introduced.

axioms (2)
  • domain assumption The 34 RM AGNs obey exactly the same M_BH-σ_* relation as quiescent galaxies.
    Invoked in the first sentence to solve for f; if false, all derived f values are forced rather than measured.
  • ad hoc to paper The broad-line region can be approximated as a thick disk whose projected velocities produce the observed line profiles.
    Introduced after the correlations are reported to explain why FWHM but not σ_line depends on inclination.

pith-pipeline@v0.9.0 · 5944 in / 1848 out tokens · 25882 ms · 2026-05-25T13:16:04.416578+00:00 · methodology

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Reference graph

Works this paper leans on

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