Strong long-term variability in active galactic nuclei affects virial black hole mass measurements

Domain: neuroscience
Published: 2026-02-05
Editorial outcome: major_revision

Abstract
Active galactic nuclei (AGN) with broad emission line regions (BLRs) contain rapidly growing supermassive black holes. Their masses are typically derived from BLR gas motions inferred from spectra and BLR sizes estimated from the AGN luminosity. The breathing BLR model predicts that the spectral line width should change as luminosity varies, keeping mass estimates constant. Here, we present observations of an AGN sample at redshift < 0.1 with two epochs separated by 20 years. On average, we find that luminosity and mass estimates vary by a factor of 2, showing a long-term mean-reversion pattern, while the line widths remain constant. A constant line width suggests that full BLRs do not change size during continuum or line flux variability, naturally explaining why scaling relations depend on instantaneously measured Eddington ratios. As a result, we find that single-epoch masses with the best repeatability after 20 years are obtained when sizes are estimated from the narrow [OIII] λ5007 Å emission line luminosity.

Introduction
The origin of supermassive black holes (SMBHs) and the history of their growth in the Universe is one of the most highly debated questions in cosmology1,2. Robust measurements of black hole masses are crucial to this topic and almost always rely on diagnosing the motion of material in orbit around the SMBH. In practice, nearly all known SMBHs are so distant that mass measurements rely on estimates of the orbital sizes of the broad emission line region (BLR) located light-days from the SMBH, and on the velocities of the gas emitting in the BLR, while assuming approximately Keplerian motion3–5. The gas motions are probed through the projected line-of-sight velocity that is reflected in the width of the broad emission lines in active galactic nuclei (AGN) spectra and likely depends on viewing angle6.

The traditional method to estimate the size of the BLR, R_BLR, is reverberation mapping (RM), which measures a time delay between a variable continuum band (used as a proxy for the ionising continuum) and its response in the flux of a spectral emission line emitted from the BLR7,8. The RM method relies on long-term monitoring, which is resource-intensive or even impractical in the large BLRs of rapidly growing SMBHs observed at high redshift. Hence, a much simpler method, known as the virial single-epoch (SE) method9–15, has become the standard and has been applied to several hundred thousand objects accounting for over 99% of all measured SMBHs16–18. While observationally efficient, requiring only a single spectrum, it is less precise, since R_BLR is inferred from AGN luminosity such as the continuum at 5100Å, L_5100, or broad Hβ λ4861Å emission, L_Hβ, using empirical relations calibrated against low-redshift RM measurements. In either method, the SMBH mass is

M_BH = f \frac{v^2 R_BLR}{G}, (1)

where v is the projected line-of-sight velocity and R_BLR is the radius of the BLR, G is the gravitational constant, and f is a dimensionless virial factor that accounts for the uncertain geometry and orientation of the BLR, as well as the relation between the measured RM lags and line widths19. Many studies have sought to refine size-luminosity (R_BLR − L) relations and improve mass estimates by using the luminosity of broad emission lines or the accretion disc continuum as proxies for the ionising radiation20–24. Generally, a large scatter is observed in empirical R_BLR − L relations25, prompting searches for secondary parameters that drive the offsets and methods to calibrate their influence on mass estimates26,27. Here, we present evidence for an alternative approach to SE mass estimation, motivated by global population observations, that does not require invoking a second parameter.

In addition to the global R_BLR − L relation defined by RM samples, individual AGN often show larger lags during epochs of higher continuum luminosity28,29. This behaviour, called virial breathing, arises when increased luminosity shifts the responsivity-weighted BLR radius outward, producing longer lags but with a slope shallower than that of the ensemble R_BLR − L relation25,30. If the BLR response centroid does not vary linearly with the ionising luminosity, deviations from the R_BLR − L relation and corresponding variations in mass estimates will occur. The best-studied case, NGC 5548, has been monitored with nearly 1600 spectra over 43 years31. Evidence for breathing has been reported4,32, but this AGN has also exhibited BLR holidays, during which emission lines and continuum flux variations were decorrelated33,34. One study claims a time delay of 3.5 years in the response of R_BLR and the FWHM of the broad emission lines to changes in luminosity35. Since luminosity varies stochastically on much shorter timescales, the AGN in NGC 5548 is rarely in equilibrium, instead fluctuating around its long-term mean R_BLR − L relation. Observations by the SDSS RM project30 over much shorter periods than for NGC 5548 have produced a broad range of behaviour, but the median width-luminosity slope is shallower than the slope predicted by virial breathing.

It is important to recognise that RM and SE methods probed different parts of the BLR. A single spectrum captures all photons from the BLR, reflecting the integrated velocity distribution, whereas RM isolates only the variable component of an emission line, via an RMS profile from multiple epochs8. This variable component is the one observed to breathe across campaigns. RMS line widths are typically broader than the mean FWHM, though strongly correlated, consistent with the total line emission being more Lorentzian in shape36; this implies that the inner BLR responds more strongly than the outer regions. Therefore, the RM-based lag traces a response-weighted R_BLR, distinct from the emission-weighted R_BLR inferred from SE spectra, and breathing behaviour of the latter has not been conclusively established. Moreover, RM-based R_BLR values depend on the characteristic variability timescale of the continuum, introducing additional scatter into the R_BLR − L relations37,38. Differences in these observational techniques are often cited as the reason for the scatter in SE mass estimates.

In this study, we investigate the repeatability of SE mass measurements in a volume-complete AGN sample. From archival spectra of over 125,000 nearby galaxies in the hemispheric 6-degree Field Galaxy Survey (6dFGS39), a set of 574 broad line AGN complete at K < 12.65 mag was identified40. Of these, 323 AGN at z < 0.1 were observed after a mean time separation of 18.5 years (17.5 years rest frame) with the ANU 2.3 m telescope at Siding Spring Observatory. See Methods for our sample selection criteria, primarily involving the detection of broad Hβ emission in both epochs of observations. This sample has a median bolometric AGN luminosity log(L_bol = erg s^-1) of 44.2, a median SMBH mass log(M_BH = M_sun) of 7.5, and a median Eddington ratio of about 0.05 (Fig. 1; Table 1). The 20-year time separation allows us to see AGN in the least correlated state currently possible, while keeping their true SMBH masses effectively constant. This time separation is similar (for f = 1) or longer than the orbital period (for f > 1) for the median BLR radius and orbital velocity of the 6dFGS AGN sample (see Table 1). In addition, we analyse high-cadence spectroscopic data of NGC 5548, generating 2389 pairs of spectra separated by 17–20 years to mimic the two-epoch design. While these pairs capture the behaviour of a single AGN in detail, the 6dFGS sample represents the broader AGN population.

Results
A 20-year comparison of black hole mass diagnostics
Comparing the pairs of spectra of AGN in the 6dFGS sample separated by about 20 years, we find strong changes in the strength of the continuum and broad emission lines (see example spectra in Fig. 1). In Fig. 2, we show the change in broad Hβ flux for the whole sample, ordered by the relative strength of broad to narrow lines in the 6dFGS epoch as expressed in the ratio Hβ/[OIII]. The average AGN is seen to change by nearly a factor of 2 (Δlog L_Hβ = 0.26 dex) over the 20 years, even after excluding more than 60 changing-look AGN in which the broad Hβ line is absent in one of the epochs41. While it is not unexpected for a single AGN to show variability of this amplitude, it is perhaps surprising that the global AGN population shows on average this level of variability in just 20 years. We note the pattern of mean reversion, whereby previously weak AGN have preferentially strengthened and vice versa, which is expected in a bounded distribution. Otherwise, if all members of a population were evolving along a symmetric random walk, the population would rapidly disperse.

In addition to the Hβ/[OIII] ratio changes, we compare changes in the FWHM of the broad Hβ line, FWHM(Hβ), as a function of changes in the broad-line luminosity, L_Hβ, and the continuum luminosity, L_5100.

We find that the majority of AGN undergo strong luminosity variations without the associated variation in FWHM(Hβ) that would be the signature of virial breathing (see Fig. 3). As the luminosity variability fails to drive the expected change in R_BLR inferred from an instantaneous spectrum, a behaviour we call size inertia of the BLR, AGN in general appear to deviate from the mean R_BLR − L relation continuously and stochastically. This expected change in R_BLR assumes that the BLR gas emitting Hβ is virialised at all radii, that variations in the ionising flux merely shift the effective emitting radius, and that the single-epoch FWHM reliably traces R_BLR.

The change in FWHM(Hβ) has a standard deviation of 0.12 dex in the 6dFGS sample, with a reduced χ2 of 6.6 for a model of ΔFWHM(Hβ) = 0, indicating real variation in FWHM(Hβ). Changes in luminosity and line widths that do not correlate in the direction expected in the breathing model due to the size inertia of the BLR lead to changes in the mass estimate. The 20-year repeatability of the Hβ-derived SE mass has a 68% confidence interval of 0.47 dex width and a 95% confidence interval of 1.29 dex, suggesting a 1σ error of 0.33 dex for any individual snapshot measurement (0.47 = \sqrt{2} * 0.33 dex). This uncertainty reflects only the statistical error from long-term repeatability and does not account for calibration uncertainties of the R_BLR − L relation itself. In NGC 5548 (Fig. 3b, d), we see the same behaviour in the dominant core of the distribution, although an outer contour of rarer behaviour hints at episodes of virial breathing. Figure 4 shows the time evolution of spectral measurements and mass estimates: from 1986 to 2010, the virial SE black hole mass estimate appears to rise by over 0.5 dex.

In Fig. 5, we compare the single-epoch (SE) mass estimates in the 6dFGS sample, separated by two decades, using different proxies for R_BLR. The intrinsic scatter in the differences is 0.47 dex for L_Hβ-based and 0.45 dex for L_5100-based SE masses (16th-84th percentile). Beyond the scatter in L_Hβ and L_5100-derived SE mass estimates, the difference across the two epochs correlate with the ratio between broad and narrow line strength (Fig. 5b); an often used ratio to denote Seyfert subtypes42 (from 1.0 to 1.8) is Hβ/[OIII].

If the R_BLR − L relation does exist across a sample, what is the best proxy for BLR size that pairs with the observed linewidth to yield the most accurate black hole mass? Both the line luminosity, L_Hβ, and the host galaxy-subtracted AGN continuum luminosity, L_5100, vary rapidly. In contrast, the narrow-line luminosity L_[OIII] is a diagnostic of AGN power where short-term variability is suppressed by blending signals with a broad range of time delays, given that the [OIII] emission originates in regions spanning hundreds of parsecs43,44. Note that, although the size of the most distant detectable [OIII] emission increases with AGN power, the light-weighted mean emission distance from the nucleus and the effective width of the time filtering remain mostly unknown. Moreover, [OIII] is emitted more isotropically45 than the likely highly viewing-angle dependent L_Hβ and L_5100, emitted from a potentially disk-like BLR and a geometrically thin accretion disc, respectively. Estimating SE masses from narrow emissions lines has been proposed before, and R_BLR − L relations from L_[OIII] do hold on a global scale46. The larger intrinsic scatter in the L_[OIII] from the RM-derived R_BLR − L relation (0.25 dex) compared to L_Hβ (0.17 dex) and L_5100 (0.18 dex)25 has discouraged use of L_[OIII] for SE mass estimation, but the unique dataset analysed in this work demonstrate the value of the longer time filtering provided by the narrow line region.

Our prescription for a modified virial SE mass estimator thus combines the FWHM of the broad Hβ emission line with a longer-term average of the AGN power, represented by the narrow-line luminosity L_[OIII]. Hence,

log M_BH = 2 log_10 (FWHM(Hβ) / 1000 km s^-1) + 0.49 log_10 (L_[OIII] / 10^41.69 erg s^-1) + C (2)

where β = 0.49 ± 0.04 was obtained from a RM based R_BLR − L calibration for L_[OIII]25. We calibrate the zero-point, C = 6.85 ± 0.17, from RM-based estimates of R_BLR and BH masses for 13 AGN listed in Table 2, hereafter called the RM sample. This calibration is limited by how much the RM sample represents the broader AGN population and could improve with a larger sample, but does not affect our result on the repeatability of SE masses.

Although estimating black hole masses from L_[OIII] may seem physically indirect, our results justify this approach: SE masses based on L_[OIII] show the lowest intrinsic scatter among all estimators, with a 16th-84th percentile range of 0.40 dex across two-decade separations (Fig. 5). Under the assumption that L_[OIII] remains constant across these epochs, the scatter arises solely from variations in FWHM(Hβ), with measurement errors accounting for roughly one-third of the observed spread. By contrast, SE masses based on L_Hβ or L_5100 show residuals proportional to the Hβ/[OIII] ratio, with slopes of 0.6 and 0.3 respectively. The L_[OIII]-based approach effectively removes this dependence by subtracting a unity-slope linear trend from the L_Hβ-based residuals, thereby flattening the relation and reducing uncertainties.

Methods
Sample and observations
The parent sample in our study is the 6-degree Field Galaxy Survey (6dFGS39,63), which includes spectra of 125,071 galaxies at a median redshift of 0.053. These galaxies were selected from the Two Micron All Sky Survey (2MASS) Extended Source Catalogue64, complete to K < 12.65 mag, and includes miscellaneous auxiliary targets. The survey covered the Southern sky at Galactic latitude |b| ≥ 10°, with observations dating between 2001 and 2009. From the spectra of this flux-limited galaxy sample, a sample of broad emission line (BEL) AGN was identified by spectral fitting and visual inspection (6dFGS AGN40). Here, we focus on 574 objects at z < 0.1, which appear to be an near-complete sample of AGN in large galaxies at K < 12.65 mag in the Southern Sky, and use that as our historic reference sample. The objects not in our sample include AGN in dwarf galaxies below the K-band flux limit, weak AGN with BELs undetectable with 6dFGS sensitivity, or high-luminosity AGN that appear as point sources in 2MASS.

We re-observed 560 of the AGN from the 6dFGS sample with the Australian National University (ANU) 2.3 m Telescope at Siding Spring Observatory (SSO) near Coonabarabran in Australia, mostly between 2022 and 2024. These 560 objects include 137 type 1.9 Seyferts, which feature a broad Hα line without a broad Hβ line. Observations were conducted with the Wide Field Spectrograph (WiFeS65,66), an integral field spectrograph with a 38 × 25 arcsec field of view and 1 arcsec^2 spaxels. The B3000 and R3000 diffraction gratings provided coverage over 3200−9800 Å. Data reduction was performed using PyWiFeS67, a Python-based pipeline designed for WiFeS. The WiFeS spectra of the 6dFGS AGN will be separately published as a legacy resource, once observations of missing objects are complete. In addition to the 6dFGS AGN, we also observed a sample of 13 AGN with WiFeS, which have been previously reverberation-mapped to get a black hole mass estimate; references for mass estimates from the reverberation mapping studies are listed in Table 2.

The 6dFGS spectra were obtained through fibres with a 6.7 arcsec aperture, corresponding to 12.4 kpc at z = 0.1, and include narrow line emission from star formation in the host galaxy, although the [OIII] emission is usually dominated by ionisation from the AGN (> 90%)68. These spectra were observed at a spectral resolution of R~1000. We match the WiFeS observations to 6dFGS and avoid instrumental biases in the two-epoch comparison by integrating light within a 6.7 arcsec diameter region in the field of view of the WiFeS integral field unit and smoothing the spectra to the 6dFGS wavelength resolution.

Spectral decomposition
We applied the Bayesian AGN Decomposition Analysis for SDSS Spectra (BADASS369) code package to fit multiple Gaussian components to emission lines and extract broad Hβ λ4861Å and [OIII] λ5007Å luminosities from spectra, which are de-reddened for Milky Way extinction. For consistency, we fit both 6dFGS and WiFeS spectra within the same rest-frame wavelength range, 4120−5500 Å, masking 4270−4400 Å wavelength range to avoid Hγ λ4340Å emission. The spectral fitting included free parameters for an AGN power law continuum, single-stellar population host models70, and a broad and narrow optical Fe II continuum71. Broad and narrow emission lines were modelled using two Gaussian components each. The dispersion of each component was restricted to < 500 km s^-1 for the narrow Hβ and [OIII] lines, and > 500 km s^-1 for the broad Hβ component. Particularly, the second Gaussian component for the narrow emission lines aims to capture blue-shifted outflows often found in AGN69,72. The sum of the broad Hβ components was used to calculate the Hβ/[OIII] line ratio.

BADASS3 fits all spectral components simultaneously, estimating parameters and uncertainties using a Markov chain Monte Carlo (MCMC) approach. In this work, uncertainties are derived from the posterior distributions of the MCMC chains, with the 16–50th and 50–84th percentile ranges reported as the lower and upper errors, respectively. These uncertainties are propagated when computing derived quantities such as the Hβ/[OIII] ratio and SE mass estimates.

AGN were classified as broad-lined or narrow-lined based on whether the fitting algorithm identified a broad Gaussian component for Hβ with an amplitude above the mean noise in the region. To ensure a consistent BEL comparison, we included only AGN with broad Hβ emission detectable above the noise level in both epochs. This criterion excludes type 1.9 Seyferts and the turn-off changing-look AGN identified in previous work41, which show broad Hβ emission in the 6dFGS epoch but fall below the detection threshold in the later WiFeS spectra. This final sample consists of 323 AGN. On average, the peak of the broad line is greater than the mean per-pixel noise in the Hβ fitting region by a factor of 5.6 for the WiFeS spectra, and 5.1 for the 6dFGS spectra.

Correcting for instrumental differences
Since 6dFGS was primarily a redshift survey, its spectra were not flux-calibrated, leading to spurious spectral features including those associated with the detector sensitivity, fibre interference patterns or poor sky subtraction40,63. In 28% (90/324) of the 6dFGS spectra in our sample, these features were fit with an unrealistically large broad Hβ component, mostly hitting the maximum FWHM limit of 25,000 km s^-1 allowed in the fitting algorithm, along with strong broad optical Fe II continuum, which summed up to a constant flux across the wavelength range. A similar effect, albeit much less pronounced, was observed in 7% (22/324) of the WiFeS spectra, primarily due to degeneracies in disentangling the host galaxy, optical Fe II, and AGN power-law continua. To mitigate these overestimated broad Hβ components in both datasets, we excluded such components from the combined luminosity by filtering out individual Gaussian components with FWHM > 12,000 km s^-1, a threshold well separated from the core of the FWHM distribution, which is centred at about 4500 km s^-1 and has a standard deviation of about 2000 km s^-1. The 12,000 km s^-1 limit is consistent with the FWHM(Hβ) distribution observed in a much larger AGN sample16. Particularly for the 6dFGS spectra, this broad component captures spectral artefacts, and we allow fits to reach such high line widths to effectively subtract these features.

We flux-calibrate the 6dFGS spectra such that the narrow [OIII] λ5007Å emission line flux reproduces the value from the flux-calibrated WiFeS spectra, assuming that the narrow [OIII] emission has not changed in intrinsic strength; this is a standard approach for AGN, though we acknowledge the possibility of L_[OIII] variability in a compact narrow-line region57. Any wavelength dependent offset in the continuum due to detector sensitivity is negligible due to the proximity of Hβ λ4861Å and continuum at λ5100Å to [OIII] λ5007Å.

The instruments used for the 6dFGS and WiFeS epoch observations of the AGN in our sample have never operated simultaneously, preventing a direct comparison of the spectra for identical objects. The intrinsic variability of AGN further complicates comparisons between non-simultaneous epochs. Among the spectral features, only the [OIII] λ5007Å emission line is sufficiently prominent and isolated to test for systematic differences. However, this line is also used to calibrate the 6dFGS spectra and therefore cannot serve as an independent comparison. The narrow Hβ emission, which is not expected to vary as it primarily traces star formation, shows a mean inter-epoch difference of -0.02 dex, indicating a lack of systematic offsets between the two instruments. The scatter in this difference is not directly interpretable, as the narrow line may be degenerate with the broad Hβ component.

To test whether the flat ΔFWHM(Hβ)-ΔL relation observed in our AGN sample (Fig. 3) is genuine or driven by correlated uncertainties, we simulate a control AGN sample with similar properties. We begin with 323 AGN drawn from a normal luminosity distribution with a scatter of 0.5 dex. Radii are then assigned using an R_BLR − L relation with slope 0.5 and intrinsic scatter of 0.25 dex25. For each object, we simulate a random luminosity change, ΔL, sampled from a normal distribution with scatter 0.26 dex, consistent with the observed change in the Hβ/[OIII] ratio, and compute a second-epoch R_BLR using the R_BLR − L relation. Two epochs of FWHM values are then estimated under the assumption FWHM \propto R_BLR^-0.5, i.e. a breathing relation. Measurement noise is added using the median observed scatter, rounded up to 0.05 dex for both FWHM and L.

We perform 10,000 random realisations of this distribution and compute the median slope in each case. The median slope is shown in Fig. 8 as a function of the correlation between FWHM and L measurement errors. While correlated uncertainties do increase the observed slope, even at maximum correlation the simulated slopes do not approach the observed flat distribution in Fig. 3.

NGC 5548 comparison
The legacy NGC 5548 dataset31 includes a compilation of about 1600 spectra observed from 1972 to 2015, making it the best studied Seyfert galaxy. This dataset includes fits to the Hβ, [OIII] emission lines and L_5100 continuum, which we directly use in this study. The large number of measurements with varying cadence introduces scatter in the light curves. To reduce noise, we computed the median in 30-day bins and estimated the uncertainties using the scatter per bin. The 30-day binning period is justified, as we focus solely on variability over decadal timescales, where short-term fluctuations within a month can be neglected. To compare the high-cadence dataset with our two-epoch sample, we grouped the binned NGC 5548 measurements into 2389 pairs separated by 18.5 ± 0.9 years, mirroring the separation distribution of the 6dFGS-WiFeS observations. For consistency, we adopted a constant log L_[OIII] of 41.57 (erg s^-1), derived from the flux used to normalise the spectra for the entire dataset31. The dataset has a median and 16th-84th percentile range Hβ/[OIII] ratio of 1.2 ± 0.5 over the full monitoring period. The SE mass distributions in Figure 4 uses only unbinned data between MJD 47500 and 52000, as the sparse sampling beyond this range hinders reliable spread comparisons, although the trend does not change when using the full monitoring period, with 16-84th percentile scatter of 0.245, 0.221 and 0.192 dex for L_Hβ-, L_5100- and L_[OIII]-derived masses, respectively.

Single-epoch mass estimates
We compare the reverberation-mapped black hole masses of the 13 AGN in the RM sample (Table 2) with SE mass estimates derived with L_Hβ and L_5100 as proxies for the BLR radius. The respective R_BLR − L slopes25 used for these proxies are 0.45 and 0.43, with mass estimates computed with the following equations:

log M_BH = 2 log_10 (FWHM(Hβ) / 1000 km s^-1) + 0.45 log_10 (L_Hβ / 10^42.33 erg s^-1) + 6.98 (3)

log M_BH = 2 log_10 (FWHM(Hβ) / 1000 km s^-1) + 0.43 log_10 (L_5100 / 10^44.056 erg s^-1) + 6.93 (4)

We also include a corrected relation for L_5100 using the ratio of optical Fe II and broad Hβ luminosity27:

log M_BH = 2 log_10 (FWHM(Hβ) / 1000 km s^-1) + 0.45 log_10 (L_5100 / 10^44.056 erg s^-1) + 0.35 log_10 (L_FeII / L_Hβ) + 7.05 (5)

where L_FeII is the luminosity of Fe II continuum from 4340 Å to 4680 Å as obtained from the decomposed spectral fits. The intercepts for the single-epoch mass estimators are calibrated using the RM masses of AGN in the RM sample (see Fig. 7).

SE mass estimates for the RM sample are derived from the WiFeS spectra, which were flux-calibrated using standard stars and with the full spectral resolution of R ~ 3000. However, we incorporate an additional 10% uncertainty in the luminosities to account for uncorrected variation in weather conditions. For the Hβ/[OIII] ratios shown in Fig. 7, we do not include this additional error, as the ratio remains unaffected by flux calibration uncertainties. We estimated the uncertainty of the RMS around the 1:1 line using a bootstrap approach that incorporates asymmetric uncertainties.

For SE mass estimates, the FWHM is the standard measure for the width of the full line16,46,73, whereas RM masses typically use the line dispersion, σ, of the RMS spectrum8, as it more directly traces the response-weighted BLR emission. Comparisons of the two metrics suggest that σ yields more accurate SE mass estimates, although FWHM remains broadly reliable74. We test this in our sample by examining changes in σ and in line shape, defined as S = FWHM/σ, relative to luminosity. The Hβ line dispersion is measured within a window of ± 2.5 × the median absolute deviation around the half-flux wavelength75. We find consistent trends for both FWHM and σ with respect to L_5100, but in the case of L_Hβ, the dispersion increases slightly with luminosity, corresponding to a decrease in line shape, S. This behaviour is expected, as broad emission lines tend to be better described by Lorentzian rather than Gaussian profiles at higher luminosities76. We adopt FWHM as our preferred metric, both because it remains the widely accepted measure for SE mass estimates and because our results are unchanged when using σ.

In Fig. 7, we compare the mass estimates from L_[OIII] with other estimates in the RM sample. We find that L_[OIII]-derived masses exhibit the lowest scatter among all SE estimators with 0.26 dex RMS compared to 0.30 dex for the L_Hβ-derived masses. These RMS values are calculated from the deviation from the one-to-one line after fitting for the zero-point in SE mass estimates using the RM masses. The RMS deviation is in part inflated due to uncertainties in the RM masses itself. Subtracting the measurement errors from both RM and SE masses, the intrinsic RMS deviation for the masses around the one-to-one line are 0.28, 0.29, and 0.25 dex for L_Hβ-, L_5100-, and L_[OIII]-derived masses, respectively.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "I have carefully read the manuscript submitted by Amrutha et al.\nAssuming a virialized distribution of line-emitting \"clouds\" within the\nbroad emission line regions of AGN, this authors investigate the\nrepeatability of single-epoch (SE) supermassive black hole (SMBH) mass\nestimates in a new and volume-complete AGN sample (6-degree Field Galaxy\nSurvey, 6dFGS). 324 AGN with z < 0.1 were observed at least twice with\na mean time separation of ~20 years to measure changes in the broad\nHbeta and 5100A continuum luminosities as well as that in the broad\nHbeta full-width-half-maximum (FWHM). The broad Hbeta to the narrow\nemission line luminosity ratio (assuming a constant value in the latter)\nwas also measured.\nTwo additional interesting features brought into this paper's analysis\nare that (1) optical spectral measurements of the type 1.5 AGN NGC 5548\nspanning nearly 40 years were similarly analyzed, and (2) SE SMBH mass\nestimates were determined from optical spectra (acquired on the same\nspectrograph) of 13 AGN that have appeared in the reverberation-mapping\nliterature with mass estimates.\nThe overall goal is to determine which measurable AGN luminosity\nquantity is the most stable / robust estimator of the characteristic\nsize R(BLR) (as associated with the Hbeta broad emission line), which\nthen appears within the virial SMBH mass estimation equation. Such\nwould hopefully minimize the systematic scatter in our SMBH mass\nestimates. The authors' analyses find that L([OIII}) is a more\nstable/robust estimator of the characteristic size of R(BLR), and\nsuggest a few reasons as to why, relative to L(5100) and L(Hbeta).\nWith a unique approach, this is an important contribution to the\nAGN field pertaining to estimating the masses of SMBH, particularly\nas associated with single-epoch estimates (in contrast to those\narising from reverberation-mapping observing campaigns). I find\nthe general methodology and analyses to be sufficiently explained\nand sound.\nHowever, before I can recommend this work for publication, is one\nissue of significance that I would like to see addressed (part A)\nThere are also a few general predictions of photoionization\nmodels of BLRs that I hope the authors' would find helpful in their\ninterpretations of their findings (part B). I also have some comments,\nsuggestions and requests related to improvements in clarity in\ncommunication (part C).\nI would be happy to review a revised version of this paper.\nA) An issue of significance\nWhile this paper references the work of Wang & Woo (2024) and\nGreene et al. (2010), both of which explore the primary topic\nof investigation in the present paper, I find that it does so\nin a cursory manner at best.\nGreene et al. explores comparing contrasting R(BLR) indicators\nthrough the implementation of several AGN luminosity indicators\n(L5100) continuum, L(Hbeta), L([OIII] 5007, L[OIV] 26 micron,\nand L(2-10keV), within their Lick AGN Monitoring Project (LAMP)\nsample.\nWang and Woo do the same (excluding the [OIV] IR forbidden line)\nusing archival reverberation mapping data of 212 AGNs with light\ncurves. They perform a uniform spectral measurement and light\ncurve analysis across these AGN and also discuss potential reasons\nfor the differences among the luminosity proxies for R(BLR).\nIt seems to me that the methods, analyses, and findings of these\ntwo papers deserve head-to-head comparison to those of the present\nwork. The present work (as noted near Eq. 1) even adopts the\nR(BLR)-L([OIII]) relation derived by Greene et al., and Wu & Wang\nalso discuss \"several possibilities for the insignificant deviation\nbetween sub- and super-Eddington AGNs in the R(BLR)-L[OIII] relation.\n---> Please clearly acknowledge the approaches and key findings of\nthese previous published work, and how this present work and its\nfindings compare to and fit in/confirm and/or expand our\nunderstanding.\nThese comparisons could be conducted in separate sections,\nwith the methods and analysis approaches discussed along side those\nof the present paper in an appropriately related section, and a\ncomparison of findings and interpretations thereof in a discussion\nof findings type section. However, I leave this to the authors as\nto how best accomplish this task of comparing this work to that\nof these two highly-related published works.\nB) Issues pertaining to photoionization models and reverberation\nwithin the BLR\n1) A measured broad emission line (Hbeta) lag (c*tau) relative to\nthe 5100A continuum flux variations is *not* a simple measure (or\neven a constant scale factor) of some physical, characteristic size\nof the line-emitting region -- such as one that might be plugged\ninto: f*v^{2}*R = G*M_{SMBH}.\nThis topic arises within the text on pages 1, 2 of the manuscript.\nIn addition to the fact that surfaces of constant time delay cut\nthrough different geometries differently (for a fixed line of sight):\nThe cross-correlation function between emission line and continuum\nlight curves has long been known to be equivalent to the convolution\nof the transfer function with the continuum (driver) auto-correlation\nfunction. Thus the broad emission line's measured amplitude of response\nand lag depend on the *temporal / amplitude characteristics of the\ndriving continuum light curve* within the observing window of an\nobservational campaign -- in addition to the radially-distributed\nresponsivity and responsivity-weighted radius of that particular\nemission line within a given BLR geometry.\n(The above is analogous to driving a mechanical system consisting\nof 1 or more natural vibrational frequencies.)\nA continuum driver whose flux variations are temporally short\nrelative to an emission line's responsivity-weighted radius (and\nassociated light travel time given a particular geometry) will\nproduce emission line variations of lower amplitude and shorter\ndelays (relative to that expected from the transfer function\ncentroid or approximately the responsivity-weighted radius,\nfor example). See, for example, the discussion surrounding\nFigure 9 in Goad & Korista (2014)\nhttps://ui.adsabs.harvard.edu/abs/2014MNRAS.444...43G/abstract\nand the noted references related to this topic therein.\nSummarizing....\nA particular broad emission line emanating from a BLR with fixed\nboundaries and geometry and \"average\" responsivity will reverberate\nwith differing amplitudes and lags depending on the temporal/amplitude\nnature of the driving continuum (of the same average luminosity)\nspanned within the observational campaign's duration. In general\na measured lag tau does not map simply to some physical size R/c.\n---> This certainly must contribute to scatter in the Hbeta lag-luminosity\nrelationship, as well as to the \"size inertia\" discussed in this paper\nand is also related to Figure 6.\n2) The \"breathing\" of a broad emission line's characteristic size R\nwith variations of an AGN's continuum luminosity, R ~ L^{beta}, is\nrelated to the emission line's radial responsivity distribution in the\nBLR: eta(r) = dlog(F_line)(r) / dlog(Phi_H),\nwhere F_line is the local radial (r) emission line flux and Phi_H is\nthe flux of ionizing photons. First, as noted in O'Brien et al. (1995)\nand Korista & Goad (2004) (see references, below), \"breathing\" occurs\nif the emission line's responsivity varies radially within the BLR\n(a constant responsivity in space and wrt source luminosity level would\nresult in no breathing). Second, and more importantly, the expectation\nthat a physical radius (e.g., the responsivity-weighted radius) associated\nwith the size of the region emitting a particular emission line scales as\nthe square-root of the ionizing source luminosity (aka \"breathing\") is\nexpected ONLY for emission lines with responsivities near unity (1)\n(linearly responding) over all but the inner-most regions of the BLR.\nThis would include the HeII recombination lines, as well as the UV\nintercombination lines, such as C III] 1909.\nBut emission lines with less than linear responsivities, such as the\nHydrogen Balmer emission lines will \"breath\" more shallowly, in that\nthe responsivity-weighted radius will grow more slowly than the\nsquare-root of the magnitude of the source's ionizing photon luminosity\nQ(H).\n(This dependence of breathing amplitude on the emission line's\nresponsivity can be understood with the following example --\nan emission line with a very small responsivity, e.g., MgII 2800, will\nnot only vary with a small amplitude in flux, but is also not expected\nto undergo significant breathing, either. In an unrealistic limit of\nzero emission line responsivity, nothing about the emission line\nchanges due to changes in the source ionizing photon luminosity.)\nTwo implications to the present paper:\n(i) Wang et al. (2020) and the present work assume erroneously that\n\"standard\" breathing necessarily means R ~ sqrt(L) within an individual\nAGN, and thus the resulting \"standard\" virial line width W should scale\nas: W ~ L^{-0.25}.\n{By the way, as originally defined in the literature (see references below),\nbreathing describes the behavior of a characteristic size of a line-emitting\nregion as the source ionizing luminosity changes, R ~ L^{beta}. What the\nwidth of a broad emission line does depends at minimum on the particular\nvelocity field of the line-emitting gas, which may or may not be\nvirial-dominated. I do not understand why Wang et al. (2020) and possibly\nothers have associated breathing with W and L.}\n(ii) Additionally, the UV continuum, energetically closer to the ionizing\nLyman continuum responsible for most of Q(H), is known to vary with a\ngreater amplitude than the optical 5100A continuum.\nIn the well-studied NGC 5548:\nL(1350) ~ L(5100)^{0.75},\ne.g., see Goad & Korista (2015)\nhttps://ui.adsabs.harvard.edu/abs/2015MNRAS.453.3662G/abstract\n(Section 2.2, footnote 4) and the noted references by Peterson et al.\nand Bentz et al. within.\nIn any case it is not surprising that broad Hbeta shows a significantly\nshallower breathing amplitude in log W vs. log L (Figure 3) than\n\"expected\" (log slopes of -0.25) for both NGC 5548 and the 6dFGS\nsample measurements.\nIn my investigation of the literature, the concept of broad emission\nline \"breathing\" was introduced within an observational context\n(lag tau scaling with source luminosity) in a study involving the 1988-89\nIUE/optical campaign of NGC 5548) by Netzer & Maoz (1990):\nhttps://ui.adsabs.harvard.edu/abs/1990ApJ...365L...5N/abstract .\nBut, again, we should keep in mind that c*tau does not map simply to a\nphysical R, nor even to a constant (in time) scale factor of R. This fact\nwill introduce noise to any breathing effects appearing in the data.\nO'Brien et al. (1995)\nhttps://ui.adsabs.harvard.edu/abs/1995MNRAS.275.1125O/abstract\nintroduced the concept within the context of photoionization models\nof the BLR (Responsivity-weighted radius shifting with source luminosity).\nNeither of the above, however, referred to the process as \"breathing\".\nThe term \"breathing\" in this context may have been introduced to\nthe literature in Korista & Goad (2004)\nhttps://ui.adsabs.harvard.edu/abs/2004ApJ...606..749K/abstract\nwhich explores the responsivities of the broad H, He optical emission\nlines, and the relationship of an emission line's responsivity to\nbreathing and an emission line's intrinsic Baldwin Effect.\nThe intrinsic Baldwin Effect in an emission line merely requires\nthat an emission line's responsivity is non-linear with respect to\nthe driving continuum variations.\n\"Breathing\" requires that the responsivity varies radially within the\nBLR and thus in time as the source of ionizing photons changes luminosity\n*significantly over time scales substantially longer than the light-crossing\ntime of the BLR*. Such a delay associated with the manifestation of\nbreathing in the broad emission line spectra of NGC 5548 has been possibly\nidentified in Lu et al., as noted in this paper (Reference #33).\nFinally, I would like to emphasize that normal \"breathing\" of the BLR does\nnot involve a physical change in size of the BLR, but rather for a given\nemission line a change in its responsivity-weighted radius or radial\ncentroid of response (or similar). I ask that the authors keep this in mind\nin their wording associated with \"breathing\" BLRs.\n---> Upshot\nThe discussion of Figure 3 and discussions concerning \"lack of virial\nbreathing\" in this paper would benefit with the inclusion of the above\ncontext. Please also mention the \"expected\" *value*, log slope -0.25 in\nlog W vs. Log Lcont, which assumes an emission line whose responsivity is\nnearly linear over most of the BLR. The broad Hbeta emission line is not\nexpected to respond linearly (responsivity ~ 1) to variations in the\nionizing continuum Q(H), but with substantially lower response amplitudes.\nThe findings in Figure 3 bear this out.\n3) It has been long recognized, and Wang & Woo (2024) most recently confirm,\nthat [OIII]5007 (as well as other narrow emission lines) exhibits a substantial\nglobal Baldwin Effect (Baldwin 1977) -- its EW scales downward as a function of\nAGN luminosity. At the same time the broad Balmer lines show only a weak\n(at best) Baldwin Effect. It then seems a bit odd that L[OIII] should be a more\nrobust/reliable/stable indicator of the characteristic size of the BLR (as\nassociated with Hbeta emission). Perhaps the authors might want to mention and\naddress this, perhaps within their \"Caveats\" section.\nC) Items of a more minor nature: suggestions, comments, requests for\nimprovements in clarity. I will list these in order of their appearance in the\ntext.\n1) Within the abstract, there is a sentence that begins:\n\"Variability without breathing...\"\n---> \"Continuum -- emission line flux variability without breathing...\"\n2) The last sentence within the abstract, \"Finally, more precise...\"\nIs precise the appropriate word? \"more accurate\"?\nCould the authors be a bit more specific -- in what way?\n3) First paragraph of page 1:\n\"...the broad emission line region (BLR) in the outer accretion discs of\nSMBHs...\"\nWhile we do not know well the geometry/structure of the BLR or its\nrelationship to the central accretion disks surrounding the SMBHs of AGN,\nit is unlikely to simply be a continuation of these accretion discs.\n4) First sentence of paragraph 2 on page 1:\n\"...which measures a time delay between the variable light curve of the\nionising accretion disc continuum...\"\nNot all of the ionizing continuum arises within the accretion disk and we\nrarely observe those wavelengths responsible for most of it. So I would\nsuggest something like:\n--->\n\"...which measures a time delay between a measurable continuum band\nacting as a proxy of the variable ionising continuum...\"\n5) Last sentence of paragraph 2 on page 1:\n\"...using an empirical relation calibrated with measured sizes.\"\n--->\n\"...using an empirical relation calibrated with measured sizes estimated\nfrom their emission line lags in low-z AGN.\"\n6) First sentence of first paragraph on page 2:\nf tries to account for a lot of ignorance on our part. Besides the unknown\ngeometry of the BLR (and our LOS orientation to it), there are also the\nunknowns concerning the virial relation between the R we estimate from\nc*tau and some measure of \"v\" -- either the FWHM or the profile dispersion\nsigma.\n7) Second sentence of second paragraph on page 2:\n\"In this model, an increase in the luminosity of an individual\nAGN drives the BLR to larger distances, along the sample R−L relation.\"\nThe luminosity drives the emission line's response centroid to larger\ndistances and thus larger delays. And as mentioned above, unless the emission\nline's responsivity is near 1 (linear response to incident continuum flux\nvariations), this relation between \"R\" and source ionizing luminosity L will\nbe substantially shallower than R ~ L^{0.5}, and thus NOT \"along the sample\nR-L relation\".\n--->\n\"In this model, an increase in the luminosity of an individual\nAGN drives the BLR response centroid to larger distances and lags, but with a\nshallower relation than the sample R−L relation.\"\n8) Third paragraph:\nKeep in mind that SE mass estimates are still utilizing an R-L relationship\nthat is calibrated from reverberation mapping observations. Also relevant\nhere is that the mean broad emission line profile and its RMS profile are\nrarely similar, with the latter often broader than the former.\n9) First paragraph within section \"Is L[OIII] a better proxy for BLR size?\":\nIn the second sentence, it's probably worth noting that L5100 refers to the\n*central source* L5100 -- i.e., that corrected for light of the host galaxy.\n10) I don't have a specific suggestion to address this, but please consider\nthat \"R\" is often associated with L[OIII] in the text -- an emission line\nthat arises in a much larger region (NLR) than the BLR. The reader may lose\ntrack as to what \"R\" is being discussed, particularly since the luminosity\nof the forbidden [OIII] line is physically disconnected from the physics of\nthe BLR. You may wish to consider using R_{BLR} everywhere, or some other\nremedy.\n11) Within the Discussion, as pertains to Figure 6: please elaborate within\nthe text what is being communicated within Figure 6. This reader may be dense\nand need some assistance, but I suspect others may need help as well. The\nimpact of Figure 6 would greatly benefit with a more explicitly description of\nwhat is being communicated. The intended information may be altered somewhat\nby the context I've provided in Section B, above .\n12) In the 3rd paragraph of the Discussion, I found this sentence to be opaque,\nboth internally and as it relates to information presented in the previous\nsentence(s):\n\"In that case, however, the correction is derived from the continuum luminosity\nitself and largely suppresses the contribution of luminosity emitted by regions\nclose to the central engine, such as LHβ and L5100.\"\nPlease help me and other readers out, here.\n13) In the last sentence of the 4th paragraph within the Discussion:\n\"...and the lack of virial breathing necessitates the adoption of an R−L\nrelation that uses an L that varies only slowly.\"\nIs this sentence referring to the \"Size Inertia\" referred to in the title of\nthis paper?\nI would like to point out that this phrasing of \"size inertia\" does not\nappear anywhere within the text of this paper, and it would be helpful to\nthe reader if the connection can be made explicitly.\nReminder: normal \"breathing\" of the BLR does not involve a physical change in\nsize (or structural distribution) of the BLR, but rather for a given emission\nline a change in its responsivity-weighted radius or radial centroid of\nresponse (or similar).\n14) Second to last sentence of the Discussion,\n\"...compared to the fast varying disc luminosity.\"\nAs mentioned earlier, the accretion disk itself isn't the only structure\ninvolved in the ionizing radiation (there is also some kind of hot \"corona\").\n--->\n\"...compared to the fast varying central source of ionising radiation.\"\n(or something like that)\n15) Second paragraph within the Caveats Section.\na) The end of sentence 1 seems to be discussing a broad emission line region\nthat may or may not be changing size. This is misleading in the context of\n\"breathing\" that is discussed in this paper. And why would there be\n\"accompanying changes in the orbital velocity of the BLR\"? Again, in the\ncontext of \"breathing\" discussed in this paper, we're just moving the\nemissivity- and responsivity-weighted radii inward and outward as the average\nL_ion changes substantially over time periods substantially longer than the\nlight-crossing time of the BLR. Please reword.\nb) The sentence: \"...i.e., fluctuations in the Eddington ratio.\"\nThis is incorrect.\nThe observed short time scale continuum variations ARE NOT due to changes\nin the mass accretion rate or associated Eddington ratio (or at least not in\nany standard paradigm of AGN accretion). These variations are much too short\nin time scale for that. The mass accretion rate (and so Eddington ratio) can\nchange, but on much longer time scales, and we may see evidence of such in\nextreme high and low continuum states exhibited in well-observed AGN with\nlong time-lines (e.g., NGC 4151, NGC 5548) or perhaps in some of the\n\"changing look\" AGN.\n16) Third paragraph within the Caveats Section:\n\"That said, it is not clear that the mean emission-weighted radius of a BLR\nis the same as its mean variability-weighted radius; these two radii might\nwell be different, if the BLR does not reverberate with equal fractional\namplitude at every radius.\"\nAll PI models of broad emission line gas show that for many BELs (the Balmer\nlines among them), the emissivity-weighted and responsivity-weighted radii\ntypically differ significantly, and that the radial responsivity is not a\nsimple power law in radius: typically larger at larger radii, smaller at\nsmaller radii.\nAgain, the observation within RM mapping spectra that the RMS flux BEL\nprofile differs significantly in shape (often broader) than the mean flux\nBEL profile is relevant, here. So is the observation of \"breathing\".\n17) Within the NGC 5548 Section\nIt would be helpful, I think, to include a mention of the\n<L(Hbeta)> / L[OIII] ratio in NGC 5548 (I think it is about 1.5).\nRelatedly, in Figure 2, it would be helpful to show the full (or some kind\nof) range in this luminosity ratio for NGC 5548 to see where it lies and\nspans on the diagram, relative to the 6dF AGN sample variability.\nAlso pertaining to Figure 2, perhaps in its caption and/or with a bit more\nelaboration within the text near the bottom of page 2, it would be helpful\nto describe carefully what the quantity that is plotted along its horizontal\naxis: \"6dF BELAGN Index\". The name certainly is not transparent in its meaning.\nAnd are the objects ordered by their mean luminosity ratio?", "reviewer_2": "Referee Report\nTitle: Correcting Black Hole Masses Biased by Size Inertia in Broad Line Regions\nRecommendation: Reject\nSummary\nThis manuscript presents a study based on a two-epoch spectroscopic analysis of a sample of low-redshift AGNs, with the\ngoal of improving single-epoch black hole mass estimates. The approach centers on using [O III] luminosity as a proxy for\nthe ionizing luminosity, under the assumption that it provides a more stable indicator of the BLR radius over time.\nThe topic is timely and relevant, as improving the reliability of mass estimators is crucial for understanding AGN\ndemographics and evolution. However, there are serious concerns about both the methodology and the physical\ninterpretation adopted by the authors. The manuscript does not, in its current form, provide sufficient justification for its core\nassumptions and does not meet the standard for publication in Nature Communications.\nMajor Comments\nUse of FWHM and Line Width Assumptions\nThe most fundamental issue is the use of FWHM as a velocity width indicator in the virial mass formula, and the assumption\nthat black hole mass scales with the square of FWHM. This is particularly problematic in the context of a study on variability,\nwhere FWHM responds differently than the line dispersion (sigma_line) to continuum flux changes. Since line width enters\nt enters\nthe mass estimate as a power-law (typically squared), a poorly behaved velocity measure like FWHM introduces significant\nuncertainty. The authors do not explore how FWHM compares to sigma_line under varying luminosity, nor do they justify\ntheir choice. Simply assuming FWHM is an appropriate measure in this context is not acceptable.\nUse of [O III] as a Proxy for Ionizing Flux\nThe use of [O III] flux as a proxy for the ionizing flux also raises concerns. While the authors mention potential advantages,\nthey do not consider significant drawbacks. First, [O III] luminosity reflects the time-averaged state of the continuum over\nyears to decades, due to both light-travel and recombination timescales. In contrast, H-beta responds on reverberation\ntimescales, which are much shorter and more relevant for BLR dynamics. Second, [O III] is produced by much higher energy\nphotons than those driving Balmer line emission. Empirically, the study by Greene et al. (2010) shows that the BLR radius\nscales more tightly with H-beta luminosity than with [O III], and with less scatter. The authors have not provided any evidence\nthat [O III] is a better or even comparable proxy to H-beta.\nConceptual Clarifications\nIn the introduction, the manuscript refers to virial breathing as an “assumption,” when it is actually a predicted consequence\nof the standard reverberation mapping interpretation under virial dynamics.\nThe statement that the variable part of the line may have a different velocity distribution than the mean line is technically\ncorrect, but empirical studies show that line widths in mean and variable spectra are highly correlated.\n--------------\nPresentation of NGC 5548 and Figures\nThe analysis of NGC 5548 is reduced to a single data point in Figure 1, despite the long history of monitoring and variability.\nIt would be more informative to show a range or distribution.\nFigure 2 shows changes in H-beta to [O III] ratios over a 20-year interval. The figure is difficult to interpret quantitatively as\nthe x-axis is not clearly defined. Moreover, a change of 0.26 dex in flux over two decades is not surprising and likely\ncommon among AGNs.\nOn [O III] Variability Timescales\nThe claim that [O III] luminosity remains constant over human timescales is incorrect. Long-term monitoring has shown that\n[O III] can vary. The more accurate assumption in prior work is that [O III] remains approximately constant over BLR\nreverberation timescales, not over decades.\nTerminology and References\nTerminology should follow standard spectroscopic conventions. For example, the line should be referred to as [O III]\nlambda5007, not lambda 5007 [O III].\nTable 2 relies on outdated virial product estimates. More recent results are available in Dalla Bonta et al. (2020) for most\nobjects, as well as in Bentz et al. (2023, ApJ, 944, 29) for IC 4329A, Du et al. (2015, ApJ, 806, 22) for Mrk 1044, and Bentz et\nal. (2021, ApJ, 906, 50) for NGC 3783.\nConclusion\nThe manuscript does not provide a convincing justification for its methodological choices, particularly the use of FWHM and\n[O III] luminosity as primary inputs to the mass scaling relation. The results do not offer clear new insight into the physics of\nAGN mass estimation and do not meet the threshold of novelty or rigor expected for this journal.", "reviewer_3": "We have read “Correcting Black Hole Masses Biased by Size Inertia in Broad Line Regions” by Amrutha et al. This\nmanuscript reports analysis of spectral data of active galactic nuclei (AGN) that had previously been observed roughly two\ndecades earlier. The authors study the joint variability in the width of the H-beta line and the bolometric luminosity (as\nmeasured by 5100 Angstrom continuum measurement and others). Based on their analysis the authors find that the joint\nvariability is not well described by the prevailing model (the “breathing” broad line region [BLR] model). This, in particular,\nhas implications for mass estimations of the AGN black holes. The authors then advocate for use of the [O III] luminosity\ninstead of other luminosity indicators (5100 continuum, H-beta, or combinations with Fe line complex luminosities).\nWe feel that the potential result is interesting, but there are a number of concerns that prevent us from being able to\ndetermine whether the conclusions are sound. The authors will need to address our major concerns before we can\nrecommend publication.\n*** MAJOR CONCERNS ***\nA key concern is the experimental setup. The authors are comparing 6dFGS spectra taken on Schmidt with WiFeS spectra\ntaken on the ANU 2.3-m telescope. Small systematic differences in the different data sets (even though they were reduced\nand fitted with the same procedure) could lead to the appearance of a problem in the model when it is in the data. Is it\npossible to directly compare the instrumental setups for consistency? While the authors have the NGC 5548 data that should\nbe self-consistent, the trend within NGC 5548 is far less convincing.\nRelated to the above, measurement uncertainties are not addressed in the manuscript at all. It is essential to understand\nhow measurement uncertainties are incorporated in the analysis. This is especially important when some of the effects that\nare being reported are slight (e.g., the difference between 5100 and [O III] scatter in mass measurements is only 0.035 dex).\nThe correlations of uncertainties between continuum and line widths is a particularly problematic aspect that needs to be\naddressed. The fact that the Baldwin effect was not mentioned nor any attempt to mitigate it is particularly worrisome. It is not\npossible to publish this work without a thorough examination of how this effect plays a role in the analysis.\nA key test of whether what the authors see is real would be to simulate observations with realistic correlated uncertainties\nthat vary according to the R–L relation with an intrinsic scatter. How often does the simulated Delta FWHM vs Delta L\ncorrelation show the same trend (or more extreme) than the data do?\nMeasurement uncertainties also need to be taken into account when measuring 16–18 percentiles in Fig 4 and rms in Fig 5.\nThat is, which of the relations has the smallest intrinsic scatter may be different from which having the smallest total scatter.\nThe latter is better for estimating black hole masses more precisely; the former is better for understanding the physics.\n*** OTHER CONCERNS ***\nINTRODUCTION\nThird paragraph\n- “Drops an assumption” → What assumption?\n- “Current formulations” does this refer to the two aforementioned methods or does it include others?\nFourth paragraph\n- “The AGN is rarely in equilibrium” → Is this referring to NGC 5548 or AGN in general?\n- “Where a few objects follow the slope [...] while others do not” → quantify this more\nFinal paragraph\n- In what sense is the 6dFGS AGN sample “complete”? How were the 324 AGN in the ANU sub-sample selected? No\ncomparison of the ANU sub-sample to the parent is presented to demonstrate that selection effects are not behind the trends\nseen by the authors.\n- “This sample has a median bolometric luminosity [...]” → Is this referring to the sub sample? If so, specify. Here you state\nmedian ~ 44.2, later you use mean ~ 44, choose one\n- The last few sentences explaining the two-epoch simulated sample from NGC 5548 data is a little confusing and could\nbenefit from more details.\nA 20-YEAR COMPARISON OF BLACK-HOLE MASS DIAGNOSTICS\n- This is the first time Hbeta is mentioned, this should be introduced earlier.\n- “The 20-year repeatability of [...]” How do these numbers change/improve when using the proposed L_[OIII] scaling\nrelation?\n- The authors write “The change in FWHM_Hbeta has a standard deviation of ~0.12 dex … with a reduced chi^2 of 6.6\naround the median change.” What is the model for which a reduced chi^2 was calculated? How many degrees of freedom in\nthe model? From context clues I think it might be that the authors are suggesting a no-change model. But this was very\npoorly written and nothing is explained in how it was calculated. This must be addressed in order to assert that there is a\n“real variation in FWHM_Hbeta”. Perhaps the model is the dashed line in Figure 3? None of this is clear as written.\nIS L_[OIII] A BETTER PROXY FOR BLR SIZE?\n- Add citation for sentence starting “In contrast, the narrow-line luminosity [...]”\n- “These RMS values are in part due to [...]” Details of the calculation of the RMS are required.\n- “The absence of mean FWHM changes [...] will remain stable over time [...] will vary by [...]” Can this be calculated and\nquantified using the larger sample?\n- “The residuals appear proportional to [...]” Quantify the significance of this result.\n- “By design, the L_[OIII]-based masses subtract a linear function [...]” expand on this\nMETHODS\n- “Appear to be a near-complete” → By what metric are you determining completeness?\nFIGURES\n- Fig. 4: What does M_L_[OIII] vs time look like? How do the lower histograms change when using the full time span of the\ndata? Can an analog of the lower panels be made for the 324 AGN? What about the 13 RM AGN (e.g. by using Delta M_SE\ninstead of M_SE)?", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "We thank the reviewers for their detailed comments and suggestions, which we found very helpful. In\nresponse, we have made several adjustments to the wording, added new figures, and performed a test on\nwhether our results can be explained by the uncertainties in the figure. All changes are marked in bold\ntext. These changes do not affect the overall results and conclusions of the paper. Below, we address the\nreviewers’ comments in order.\nOne major change is that we have opted to use Wang & Woo (2024) R L relations for all (L ,\nBLR Hβ\n−\nL , and L ) SE mass estimates. This does not change the previous result, but makes the SE mass\n5100 [OIII]\nestimate comparison using R L measurements based on the most recent, comprehensive data set.\nReviewer 1 (Remarks to the Author):\nI have carefully read the manuscript submitted by Amrutha et al. Assuming a virialized distribution\nof line-emitting ”clouds” within the broad emission line regions of AGN, this authors investigate the\nrepeatabilityofsingle-epoch(SE)supermassiveblackhole(SMBH)massestimatesinanewandvolume-\ncomplete AGN sample (6-degree Field Galaxy Survey, 6dFGS). 324 AGN with z < 0.1 were observed\nat least twice with a mean time separation of 20 years to measure changes in the broad Hbeta and\n5100A continuum luminosities as well as that in the broad Hbeta full-width-half-maximum (FWHM).\nThe broad Hbeta to the narrow emission line luminosity ratio (assuming a constant value in the latter)\nwas also measured. Two additional interesting features brought into this paper’s analysis are that (1)\noptical spectral measurements of the type 1.5 AGN NGC 5548 spanning nearly 40 years were similarly\nanalyzed, and (2) SE SMBH mass estimates were determined from optical spectra (acquired on the\nsame spectrograph) of 13 AGN that have appeared in the reverberation-mapping\nliterature with mass estimates.\nTheoverallgoalistodeterminewhichmeasurableAGNluminosityquantityisthemoststable\n/ robust estimator of the characteristic size R(BLR) (as associated with the Hbeta broad emission line),\nwhich then appears within the virial SMBH mass estimation equation. Such would hopefully minimize\nthe systematic scatter in our SMBH mass estimates. The authors’ analyses find that L([OIII]) is a more\nstable/robustestimatorofthecharacteristicsizeofR(BLR),andsuggestafewreasonsastowhy,relative\nto L(5100) and L(Hbeta).\nWith a unique approach, this is an important contribution to the AGN field pertaining to estimating\nthe masses of SMBH, particularly as associated with single-epoch estimates (in contrast to those arising\nfrom reverberation-mapping observing campaigns). I find the general methodology and analyses to be\nsufficiently explained and sound.\nHowever, before I can recommend this work for publication, is one issue of significance that I would like\nto see addressed (part A) There are also a few general predictions of photoionization models of BLRs\nthat I hope the authors’ would find helpful in their interpretations of their findings (part B). I also have\nsome comments, suggestions and requests related to improvements in clarity in communication (part C).\nI would be happy to review a revised version of this paper.\nWhile this paper references the work of Wang & Woo (2024) and\nGreene et al. (2010), both of which\nexploretheprimarytopicofinvestigationinthepresentpaper, Ifindthatitdoessoinacursorymanner\nat best. Greene et al. explores comparing contrasting R(BLR) indicators through the implementation of\nseveral AGN luminosity indicators (L5100) continuum, L(Hbeta), L([OIII] 5007, L[OIV] 26 micron, and\nL(2-10keV), within their Lick AGN Monitoring Project (LAMP) sample. Wang and Woo do the same\n(excluding the [OIV] IR forbidden line) using archival reverberation mapping data of 212 AGNs with\nlight curves. They perform a uniform spectral measurement and light curve analysis across these AGN\nand also discuss potential reasons for the differences among the luminosity proxies for R(BLR).\nIt seems to me that the methods, analyses, and findings of these two papers deserve head-to-head com-\nparison to those of the present work. The present work (as noted near Eq. 1) even adopts the R(BLR)-\nL([OIII]) relation derived by Greene et al., and Wu & Wang also discuss ”several possibilities for the\ninsignificant deviation between sub- and super-Eddington AGNs in the R(BLR)-L[OIII] relation.\n---> Please clearly acknowledge the approaches and key findings of these previous published work, and\nhow this present work and its findings compare to and fit in/confirm and/or expand our understanding.\nThese comparisons could be conducted in separate sections,\nwith the methods and analysis approaches discussed along side those\nof the present paper in an appropriately related section, and a\ncomparison of findings and interpretations thereof in a discussion\nof findings type section. However, I leave this to the authors as\nto how best accomplish this task of comparing this work to that\nof these two highly-related published works.\nWe have made explicit references to these two published works in our results section in the paragraph\nbefore Equation (1) and added two paragraphs in the discussion addressing the larger scatter observed\nin R(BLR)-L([OIII]) relations including discussions from previous work regarding the Eddington ratio\ndependence.\nB) Issues pertaining to photoionization models and reverberation within the\nBLR\n1) A measured broad emission line (Hbeta) lag (c*tau) relative to the 5100A continuum flux variations\nis *not* a simple measure (or even a constant scale factor) of some physical, characteristic size of the\nline-emitting region – such as one that might be plugged into: f*v^{2}*R = G*M_{SMBH}. This topic\narises within the text on pages 1, 2 of the manuscript.\nIn addition to the fact that surfaces of constant time delay cut through different geometries differently\n(forafixedlineofsight): Thecross-correlationfunctionbetweenemissionlineandcontinuum\nlightcurves\nhas long been known to be equivalent to the convolution of the transfer function with the continuum\n(driver) auto-correlation function. Thus the broad emission line’s measured amplitude of response and\nlag depend on the *temporal / amplitude characteristics of the driving continuum light curve*\nwithin the observing window of an observational campaign – in addition to the radially-distributed\nresponsivity and responsivity-weighted radius of that particular emission line within a given BLR geometry.\n(The above is analogous to driving a mechanical system consisting\nof 1 or more natural vibrational frequencies.)\nAcontinuumdriverwhosefluxvariationsaretemporallyshortrelativetoanemissionline’sresponsivity-\nweighted radius (and associated light travel time given a particular geometry) will produce emission\nlinevariationsofloweramplitudeandshorterdelays(relativetothatexpectedfromthetransferfunction\ncentroidorapproximatelytheresponsivity-weightedradius,forexample). See,forexample,thediscussion\nsurrounding Figure 9 in Goad & Korista (2014)\nhttps://ui.adsabs.harvard.edu/abs/2014MNRAS.444...43G/abstract and the noted references related to\nthis topic therein.\nSummarizing....\n”average” responsivity will reverberate with differing amplitudes and lags depending on the tempo-\nral/amplitude nature of the driving continuum (of the same average luminosity) spanned within the\nobservational campaign’s duration. In general a measured lag tau does not map simply to some physical\nsize R/c.\n---> This certainly must contribute to scatter in the Hbeta lag-luminosity\nrelationship, as well as to the ”size inertia” discussed in this paper\nand is also related to Figure 6.\nA sentence incorporating this information citing Goad & Korista (2014) in the introduction, and a\nmodification to Figure 6 has been made. We have also added text explaining the scatter in RM-lag to\nR conversion in the discussion when explaining Figure 6.\nBLR\n2)The”breathing”ofabroademissionline’scharacteristicsizeRwithvariationsofanAGN’scontinuum\nluminosity, R ~ L^{beta}, is related to the emission line’s radial responsivity distribution in the BLR:\neta(r) = dlog(F_line)(r) / dlog(Phi_H), where F_line is the local radial (r) emission line flux and\nPhi_Histhefluxofionizingphotons. First, asnotedinO’Brienetal. (1995)andKorista&Goad(2004)\n(see references, below), ”breathing” occurs if the emission line’s responsivity varies radially within the\nBLR (a constant responsivity in space and wrt source luminosity level would result in no breathing).\nSecond, and more importantly, the expectation that a physical radius (e.g., the responsivity-weighted\nradius) associated with the size of the region emitting a particular emission line scales as the square-root\noftheionizingsourceluminosity(aka”breathing”)isexpectedONLYforemissionlineswithresponsivities\nnear unity (1) (linearly responding) over all but the inner-most regions of the BLR. This would include\ntheHeIIrecombinationlines, aswellastheUVintercombinationlines, suchasCIII]1909. Butemission\nlines with less than linear responsivities, such as the Hydrogen Balmer emission lines will ”breath” more\nshallowly, in that the responsivity-weighted radius will grow more slowly than the square-root of the\nmagnitude of the source’s ionizing photon luminosity Q(H). (This dependence of breathing amplitude on\nthe emission line’s responsivity can be understood with the following example – an emission line with a\nvery small responsivity, e.g., MgII 2800, will not only vary with a small amplitude in flux, but is also not\nexpectedtoundergosignificantbreathing,either. Inanunrealisticlimitofzeroemissionlineresponsivity,\nnothing about the emission line changes due to changes in the source ionizing photon luminosity.)\nTwo implications to the present paper:\n(i) Wang et al. (2020) and the present work assume erroneously that ”standard” breathing necessarily\nmeans R sqrt(L) within an individual AGN, and thus the resulting ”standard” virial line width W\nshouldscaleas: W ~ L^{-0.25}. Bytheway,asoriginallydefinedintheliterature(seereferencesbelow),\nbreathing describes the behavior of a characteristic size of a line-emitting region as the source ionizing\nluminosity changes, R ~ L^{beta}. What the width of a broad emission line does depends at minimum\non the particular velocity field of the line-emitting gas, which may or may not be virial-dominated. I do\nnot understand why Wang et al. (2020) and possibly others have associated breathing with W and L.\n(ii) Additionally, the UV continuum, energetically closer to the ionizing Lyman continuum responsible\nfor most of Q(H), is known to vary with a greater amplitude than the optical 5100A continuum. In the\nwell-studied NGC 5548:\nL(1350) ~ L(5100)^{0.75},\ne.g., see Goad & Korista (2015) https://ui.adsabs.harvard.edu/abs/2015MNRAS.453.3662G/abstract\n(Section 2.2, footnote 4) and the noted references by Peterson et al. and Bentz et al. within.\nIn any case it is not surprising that broad Hbeta shows a significantly shallower breathing amplitude in\nlog W vs. log L (Figure 3) than ”expected” (log slopes of -0.25) for both NGC 5548 and the 6dFGS\nsample measurements.\nIn my investigation of the literature, the concept of broad emission line ”breathing” was introduced\nwithin an observational context (lag tau scaling with source luminosity) in a study involving the 1988-89\nIUE/optical campaign of NGC 5548) by Netzer & Maoz (1990):\nhttps://ui.adsabs.harvard.edu/abs/1990ApJ...365L...5N/abstract . But, again, we should keep in mind\nthat c*tau does not map simply to a physical R, nor even to a constant (in time) scale factor of R. This\n3\nfact will introduce noise to any breathing effects appearing in the data.\nO’Brien et al. (1995) https://ui.adsabs.harvard.edu/abs/1995MNRAS.275.1125O/abstract introduced\nthe concept within the context of photoionization models of the BLR (Responsivity-weighted radius\nshifting with source luminosity). Neither of the above, however, referred to the process as ”breathing”.\nTheterm”breathing”inthiscontextmayhavebeenintroducedtotheliteratureinKorista&Goad(2004)\nhttps://ui.adsabs.harvard.edu/abs/2004ApJ...606..749K/abstractwhichexplorestheresponsivitiesofthe\nbroad H, He optical emission lines, and the relationship of an emission line’s responsivity to breathing\nand an emission line’s intrinsic Baldwin Effect. The intrinsic Baldwin Effect in an emission line merely\nrequiresthatanemissionline’sresponsivityisnon-linearwithrespecttothedrivingcontinuumvariations.\n”Breathing” requires that the responsivity varies radially within the BLR and thus in time as the source\nof ionizing photons changes luminosity *significantly over time scales substantially longer than the light-\ncrossing time of the BLR*. Such a delay associated with the manifestation of breathing in the broad\nemission line spectra of NGC 5548 has been possibly identified in Lu et al., as noted in this paper\n(Reference 33).\nFinally,Iwouldliketoemphasizethatnormal”breathing”oftheBLRdoesnotinvolveaphysicalchange\nin size of the BLR, but rather for a given emission line a change in its responsivity-weighted radius\nor radial centroid of response (or similar). I ask that the authors keep this in mind in their wording\nassociated with ”breathing” BLRs.\n--->UpshotThediscussionofFigure3anddiscussionsconcerning”lackofvirialbreathing”inthispaper\nwould benefit with the inclusion of the above context. Please also mention the ”expected” *value*, log\nslope -0.25 in log W vs. Log Lcont, which assumes an emission line whose responsivity is nearly linear\nover most of the BLR. The broad Hbeta emission line is not expected to respond linearly (responsivity\n1) to variations in the ionizing continuum Q(H), but with substantially lower response amplitudes. The\nfindings in Figure 3 bear this out.\nWe completely agree that the profound difference between these two sizes, the emission-weighted and\nthe responsivity-weighted R , is an important concept, which has kept our minds busy for long while\nBLR\nwe did not appreciate how little we had made this distinction clear in the original version of the text.\nHence, we now make this point much more prominent, particularly in the introduction and discussion.\nCorrespondingly, the non-linear responsivity Hβ and the assumptions regarding breathing have been\nadded to the discussion.\n3)Ithasbeenlongrecognized,andWang&Woo(2024)mostrecentlyconfirm,that[OIII]5007(aswellas\nother narrow emission lines) exhibits a substantial global Baldwin Effect (Baldwin 1977) – its EW scales\ndownward as a function of AGN luminosity. At the same time the broad Balmer lines show only a weak\n(at best) Baldwin Effect. It then seems a bit odd that L[OIII] should be a more robust/reliable/stable\nindicator of the characteristic size of the BLR (as associated with Hbeta emission). Perhaps the authors\nmight want to mention and address this, perhaps within their ”Caveats” section.\nWe do indeed see that both the global and intrinsic Baldwin effect is more pronounced in [OIII] than\nHβ, directly related to the lack of [OIII] emission variability in response to continuum variability. The\nR\nBLR\nL relations for [OIII] are usually steeper than those for Hβ or L\n5100\n, to counter the stronger\n−\nBaldwin effect.\nInadditiontotheglobaleffect,theintrinsiceffectarguesforourcasethatiftheemission weightedradius\ndoes not change in our observation timescale, then neither should our luminosity proxy for the radius in\norder to estimate the same black hole mass, which in this case is L .\n[OIII]\nA paragraph discussing the Baldwin effect and the above points has been added to the Discussion.\nC) Items of a more minor nature: suggestions, comments, requests for im-\nprovements in clarity. I will list these in order of their appearance in the\ntext.\n1) Within the abstract, there is a sentence that begins: ”Variability without breathing...”\n4\n---> ”Continuum – emission line flux variability without breathing...”\nWe have changed the sentence to ”Continuum and emission-line variability without breathing...”\n2)Thelastsentencewithintheabstract,”Finally,moreprecise...”Isprecisetheappropriateword? ”more\naccurate”? Could the authors be a bit more specific – in what way?\nThe last sentence has been replaced, to accommodate changes made in this revision.\n3) First paragraph of page 1: ”...the broad emission line region (BLR) in the outer accretion discs of\nSMBHs...”\nWhilewedonotknowwellthegeometry/structureoftheBLRoritsrelationshiptothecentralaccretion\ndisks surrounding the SMBHs of AGN, it is unlikely to simply be a continuation of these accretion discs.\nThe ”outer accretion discs” has been replaced with ”which resides at distances of light days away from\nthe SMBH” to remove any definitive reference to the structure of the BLR.\n4) First sentence of paragraph 2 on page 1: ”...which measures a time delay between the variable light\ncurve of the ionising accretion disc continuum...” Not all of the ionizing continuum arises within the\naccretion disk and we rarely observe those wavelengths responsible for most of it. So I would suggest\nsomething like: ---> ”...which measures a time delay between a measurable continuum band acting as a\nproxy of the variable ionising continuum...”\nFollowed the reviewer’s suggestion in replacing the text.\n5)Lastsentenceofparagraph2onpage1: ”...usinganempiricalrelationcalibratedwithmeasuredsizes.”\n--->”...usinganempiricalrelationcalibratedwithmeasuredsizesestimatedfromtheiremissionlinelags\nin low-z AGN.”\nReplaced with ”derived from low-redshift RM studies.”\n6) First sentence of first paragraph on page 2:\nf tries to account for a lot of ignorance on our part. Besides the unknown geometry of the BLR (and\nour LOS orientation to it), there are also the unknowns concerning the virial relation between the R we\nestimate from c*tau and some measure of ”v” – either the FWHM or the profile dispersion sigma.\nThe unknown relation between the lag and the line-width has been mentioned in the same sentence.\n7) Second sentence of second paragraph on page 2: ”In this model, an increase in the luminosity of an\nindividual AGN drives the BLR to larger distances, along the sample R-L relation.”\nTheluminositydrivestheemissionline’sresponsecentroidtolargerdistancesandthuslargerdelays. And\nasmentionedabove,unlesstheemissionline’sresponsivityisnear1(linearresponsetoincidentcontinuum\nfluxvariations),thisrelationbetween”R”andsourceionizingluminosityLwillbesubstantiallyshallower\nthan R ~ L^{0.5}, and thus NOT ”along the sample R-L relation”.\n--->”Inthismodel,anincreaseintheluminosityofanindividualAGNdrivestheBLRresponsecentroid\nto larger distances and lags, but with a shallower relation than the sample R-L relation.”\nThe reviewer’s suggestion on replacing the text has been implemented.\n8) Third paragraph: Keep in mind that SE mass estimates are still utilizing an R-L relationship that is\ncalibrated from reverberation mapping observations. Also relevant here is that the mean broad emission\nline profile and its RMS profile are rarely similar, with the latter often broader than the former.\nThe entire paragraph has been modified to include details suggested by the reviewer here, to make the\ndifference explicit.\n9) First paragraph within section ”Is L[OIII] a better proxy for BLR size?”: In the second sentence, it’s\nprobably worth noting that L5100 refers to the *central source* L5100 – i.e., that corrected for light of\n5\nthe host galaxy.\nThe definition of L has been modified to ”host galaxy-subtracted AGN continuum luminosity”\n5100\n10) I don’t have a specific suggestion to address this, but please consider that ”R” is often associated\nwith L[OIII] in the text – an emission line that arises in a much larger region (NLR) than the BLR.\nThe reader may lose track as to what ”R” is being discussed, particularly since the luminosity of the\nforbidden [OIII] line is physically disconnected from the physics of the BLR. You may wish to consider\nusing R_{BLR} everywhere, or some other remedy.\nWe now use R to refer to the radius of the BLR throughout the paper.\nBLR\n11) Within the Discussion, as pertains to Figure 6: please elaborate within the text what is being\ncommunicatedwithinFigure6. Thisreadermaybedenseandneedsomeassistance,butIsuspectothers\nmay need help as well. The impact of Figure 6 would greatly benefit with a more explicitly description\nof what is being communicated. The intended information may be altered somewhat\nby the context I’ve provided in Section B, above .\nWe are trying to convey the different contributors of intrinsic scatter in R L relations, with the\nBLR\n−\nfocus on luminosity variations, which account for roughly a quarter of the scatter in just 20 years. The\nparagraph referring to Figure 6 has been modified to hopefully communicate the message better.\n12) In the 3rd paragraph of the Discussion, I found this sentence to be opaque, both internally and as it\nrelates to information presented in the previous sentence(s):\n”In that case, however, the correction is derived from the continuum luminosity itself and largely sup-\npresses the contribution of luminosity emitted by regions close to the central engine, such as LHbeta and\nL5100.”\nPlease help me and other readers out, here.\nThe sentence was trying to convey that the Eddington ratio has been used to correct for luminosity\nvariations, but the luminosity itself is a measure of the instantaneous Eddington ratio, which is round-\nabout way of removing the contribution of L or L . The sentence has been removed and the final\nsentence of the paragraph has been modified to include this information.\n13) In the last sentence of the 4th paragraph within the Discussion: ”...and the lack of virial breathing\nnecessitates the adoption of an R-L relation that uses an L that varies only slowly.”\nIs this sentence referring to the ”Size Inertia” referred to in the title of this paper?\nI would like to point out that this phrasing of ”size inertia” does not appear anywhere within the text of\nthis paper, and it would be helpful to the reader if the connection can be made explicitly.\nReminder: normal ”breathing” of the BLR does not involve a physical change in size (or structural\ndistribution)oftheBLR,butratherforagivenemissionlineachangeinitsresponsivity-weightedradius\nor radial centroid of response (or similar).\nThis paragraph and the following paragraph have been modified/added to include a link to size inertia,\nand added clarity regarding breathing and different radii as per suggestions in Part B.\n14) Second to last sentence of the Discussion, ”...compared to the fast varying disc luminosity.”\nAs mentioned earlier, the accretion disk itself isn’t the only structure involved in the ionizing radiation\n(thereisalsosomekindofhot”corona”). --->”...comparedtothefastvaryingcentralsourceofionising\nradiation.” (or something like that)\nThe reviewer’s suggestion on replacing the text has been implemented.\n15) Second paragraph within the Caveats Section.\n6\na) The end of sentence 1 seems to be discussing a broad emission line region that may or may not be\nchanging size. This is misleading in the context of ”breathing” that is discussed in this paper. And why\nwould there be ”accompanying changes in the orbital velocity of the BLR”? Again, in the context of\n”breathing” discussed in this paper, we’re just moving the emissivity- and responsivity-weighted radii\ninward and outward as the average L_ion changes substantially over time periods substantially longer\nthan the light-crossing time of the BLR. Please reword.\nThe sentence has been modified with reference to ”inferred orbital velocity of the BLR, derived from the\nwidth of the observed total emission line” rather than ”BLR size.”\nb) The sentence: ”...i.e., fluctuations in the Eddington ratio.” This is incorrect. The observed short time\nscalecontinuumvariationsARENOTduetochangesinthemassaccretionrateorassociatedEddington\nratio (or at least not in any standard paradigm of AGN accretion). These variations are much too short\nin time scale for that. The mass accretion rate (and so Eddington ratio) can change, but on much longer\ntimescales,andwemayseeevidenceofsuchinextremehighandlowcontinuumstatesexhibitedinwell-\nobserved AGN with long time-lines (e.g., NGC 4151, NGC 5548) or perhaps in some of the ”changing\nlook” AGN.\nThe Eddington ratio we refer to (instantaneous measurement L/M which varies with L) might not be\nBH\nwhat the reviewer refers to (true mass accretion rate which should not vary on luminosity fluctuation\ntimescales). Taking into account the observed amplitude of Hβ/[OIII] ratio changes in Figure 2, we can\nexpect moderate fluctuations in the observed R over decadal timescales.\nEdd\nWehavechangedthetextinawaythatsuggeststheluminosity-derivedR isaninstantaneousmeasure-\nEdd\nmentthatvaries(noisily)withluminosity,ratherthanatruechangeinmassaccretion(fourthparagraph\nof the Discussion).\n16) Third paragraph within the Caveats Section: ”That said, it is not clear that the mean emission-\nweighted radius of a BLR is the same as its mean variability-weighted radius; these two radii might\nwell be different, if the BLR does not reverberate with equal fractional amplitude at every radius.”\nAll PI models of broad emission line gas show that for many BELs (the Balmer lines among them),\nthe emissivity-weighted and responsivity-weighted radii typically differ significantly, and that the radial\nresponsivity is not a simple power law in radius: typically larger at larger radii, smaller at smaller radii.\nAgain, the observation within RM mapping spectra that the RMS flux BEL profile differs significantly\nin shape (often broader) than the mean flux BEL profile is relevant, here. So is the observation of\n”breathing”.\nThedistinctionbetweentheresponsivity-weightedandemission-weightedradiiisnowstatedexplicitlyin\nthe text. In addition, we have added a new paragraph in the Discussion (paragraph 5) that incorporates\nresultsfromphotoionizationmodels(includingthereferencesnotedbythereviewerinpartB)toregarding\nthe radial and non-linear responsivity of Hβ.\n17) Within the NGC 5548 Section It would be helpful, I think, to include a mention of the <L(Hbeta)>\n/ L[OIII] ratio in NGC 5548 (I think it is about 1.5). Relatedly, in Figure 2, it would be helpful to show\nthe full (or some kind of) range in this luminosity ratio for NGC 5548 to see where it lies and spans on\nthe diagram, relative to the 6dF AGN sample variability.\nThemeanHβ/[OIII]ratiohasbeenaddedtothetext,andFigure2hasbeenmodifiedtoshowthemedian\nratio with one- and two-sigma percentile ranges.\nAlso pertaining to Figure 2, perhaps in its caption and/or with a bit more elaboration within the text\nnear the bottom of page 2, it would be helpful to describe carefully what the quantity that is plotted\nalong its horizontal axis: ”6dF BELAGN Index”. The name certainly is not transparent in its meaning.\nAnd are the objects ordered by their mean luminosity ratio?\nThe objects are ordered based on their initial Hβ/[OIII] ratio, i.e., their 6dFGS epoch. The axis label\nhas been changed to make this more clear.\nReferee Report\nTitle: Correcting Black Hole Masses Biased by Size Inertia in Broad Line Regions\nRecommendation: Reject\nSummary\nThismanuscriptpresentsastudybasedonatwo-epochspectroscopicanalysisofasampleoflow-redshift\nAGNs,withthegoalofimprovingsingle-epochblackholemassestimates. Theapproachcentersonusing\n[O III] luminosity as a proxy for the ionizing luminosity, under the assumption that it provides a more\nstable indicator of the BLR radius over time.\nThetopicistimelyandrelevant,asimprovingthereliabilityofmassestimatorsiscrucialforunderstanding\nAGNdemographicsandevolution. However, thereareseriousconcernsaboutboththemethodologyand\nthephysicalinterpretationadoptedbytheauthors. Themanuscriptdoesnot,initscurrentform,provide\nsufficient justification for its core assumptions and does not meet the standard for publication in Nature\nCommunications.\nMajor Comments\nUse of FWHM and Line Width Assumptions\nThemostfundamentalissueistheuseofFWHMasavelocitywidthindicatorinthevirialmassformula,\nandtheassumptionthatblackholemassscaleswiththesquareofFWHM.Thisisparticularlyproblematic\nin the context of a study on variability, where FWHM responds differently than the line dispersion\n(sigma_line) to continuum flux changes. Since line width enters the mass estimate as a power-law\n(typically squared), a poorly behaved velocity measure like FWHM introduces significant uncertainty.\nThe authors do not explore how FWHM compares to sigma_line under varying luminosity, nor do they\njustifytheirchoice. SimplyassumingFWHMisanappropriatemeasureinthiscontextisnotacceptable.\nInthesub-sectionofmethodspertainingtoSEmassestimates,wehaveaddedaparagraphjustifyingthe\nuse the FWHM as a line width in our SE estimates. The FWHM is the most commonly used velocity\nwidth indicator for SE mass estimates, although the σ is used for RM mass estimates. Work by\nline,rms\nDalla Bont`a et al. (2020) compares the two, and finds that although σ works better at estimating\nline\nmasses, using FWHM is not unreasonable. For comparison, we present a version of Figure 3 in the\nmanuscript below, but using σ instead, and our results remain largely the same. We note that the\nline\ngoal of this work is to examine how SE mass estimates depend on variability, and to present evidence for\nthe resulting mass differences in observations separated by 20 years. Since FWHM is the most widely\nadoptedline-widthmeasureinexistingSEmassprescriptions,weuseFWHMconsistentlyinourestimates\nto ensure direct comparability with standard practice.\nUse of [O III] as a Proxy for Ionizing Flux\nThe use of [O III] flux as a proxy for the ionizing flux also raises concerns. While the authors mention\npotential advantages, they do not consider significant drawbacks. First, [O III] luminosity reflects the\ntime-averaged state of the continuum over years to decades, due to both light-travel and recombination\ntimescales. In contrast, H-beta responds on reverberation timescales, which are much shorter and more\nrelevant for BLR dynamics. Second, [O III] is produced by much higher energy photons than those\ndriving Balmer line emission. Empirically, the study by Greene et al. (2010) shows that the BLR radius\nscalesmoretightlywithH-betaluminositythanwith[OIII],andwithlessscatter. Theauthorshavenot\nprovided any evidence that [O III] is a better or even comparable proxy to H-beta.\nThe points made by the reviewer regarding [OIII] are very important and directly relate to the main\nresults of the paper. It has become clear to us that we have not sufficiently well explained the difference\n8\n0.5\n0.0\n0.5\n−\nβHgol∆ σ\n(a)\nL\nHβ\nβH/\nβHgol∆\nσ\nMHWF\n(b)\nL\nHβ\n0.5\n0.0\n0.5\n−\n1.0 0.5 0.0 0.5 1.0\n− −\n∆logL\nβHgol∆ σ\n(c)\nL\n5100\n1.0 0.5 0.0 0.5 1.0\n− −\n∆logL\nβH/\nβHgol∆\nσ\nMHWF\n(d)\nL\nchange in Hβσ vs. change in LHβ and L5100 for the 6dFGS AGN sample. (b, d): Change in line shape, HβFWHM/Hβσ\nvs. changeinLHβ andL5100.\nbetween the RM-measured responsivity-weighted BLR radius and the overall mean radius of the BLR\nemitting the full emission line. We believe that we have made changes in the text to now explain this\npart and what role it plays better. In summary:\nIndeed, the larger scatter of the R L relation when using L , compared to L or L , arises\nBLR [OIII] Hβ 5100\n−\nbecauseR LrelationscalibratedfromRMcampaignsprobethevariable,responsivity-weightedBLR\nBLR\n−\nradius, rather than the total, emission-weighted radius measured from a single spectrum. Now, since the\nsingle-epoch virial mass estimates work from the full broad emission line, their emission-weighted BLR\nradii are not only different from the radii found in RM campaigns and thus in R L relations (which\nBLR\n−\ncould easily be calibrated out), but they will also vary in a different way (which cannot be calibrated\nunless we know exactly how).\nIn this work we find empirically that, when comparing SMBH mass estimates separated by 20 years,\nthe L -derived SE masses exhibit the smallest differences. This is evident in Figure 5 (repeat mea-\n[OIII]\nsurements of a large AGN sample) and Figure 7 (a smaller sample compared to RM-derived masses).\nEvidently, the problem is that the variability in L or L does NOT lead to a size change in the\nHβ 5100\noverall emission-weighted BLR such that the propagating impact on a virial single-epoch BH mass esti-\nmate would be compensated. We simply observe that the slowly varying L luminosity can provide\n[OIII]\na more stable basis for SE mass estimates (and thus presumably for overall BLR sizes as opposed to\nresponse-weightedsizes)overlongtimescales. Therewasnoreasonthishadtobetheoreticallyexpected,\nbut this is now the new observational evidence from our large and complete AGN sample.\nConceptual Clarifications\nIn the introduction, the manuscript refers to virial breathing as an “assumption,” when it is actually a\npredicted consequence of the standard reverberation mapping interpretation under virial dynamics.\nThedefinitionofvirialbreathinghasbeenmodified,andtextregardingtheoriginofbreathingbehaviour\nhas been added to the introduction.\nThe statement that the variable part of the line may have a different velocity distribution than the mean\n9\nline is technically correct, but empirical studies show that line widths in mean and variable spectra are\nhighly correlated.\nThis has been explicitly stated in the text in the introduction, with reference to Shen et al. (2024).\nPresentation of NGC 5548 and Figures\nThe analysis of NGC 5548 is reduced to a single data point in Figure 1, despite the long history of\nmonitoring and variability. It would be more informative to show a range or distribution.\nFigure 1 shows the luminosity range of NGC 5548 over 43 years as error bars. This was previously\nincorrectly labelled as ”uncertainty”, but has now been corrected.\nFigure 2 shows changes in H-beta to [O III] ratios over a 20-year interval. The figure is difficult to\ninterpret quantitatively as the x-axis is not clearly defined. Moreover, a change of 0.26 dex in flux over\ntwo decades is not surprising and likely\ncommon among AGNs.\nThex-axislabelhasbeenmodifiedtomakethefigureclear. Figure2aimstoconveythesamplebehaviour\nspectroscopically, along with global trends.\nOn [O III] Variability Timescales The claim that [O III] luminosity remains constant over human\ntimescales is incorrect. Long-term monitoring has shown that [O III] can vary. The more accurate as-\nsumptioninpriorworkisthat[OIII]remainsapproximatelyconstantoverBLRreverberationtimescales,\nnot over decades.\nCaveats regarding using L as a proxy for R had been discussed. It is unclear at this stage,\n[OIII] BLR\nwhether any slow variability in L has negatively contributed to the estimated mass residuals or\n[OIII]\nhappens to align constructively with the timescales of a slow BLR adjustment to longer-term variations\nin AGN luminosity.\nTerminology and References\nTerminology should follow standard spectroscopic conventions. For example, the line should be referred\nto as [O III] lambda5007, not lambda 5007 [O III].\nThis has been fixed and the lines are now referred to with the format suggested by the reviewer.\nTable 2 relies on outdated virial product estimates. More recent results are available in Dalla Bonta et\nal. (2020) for most objects, as well as in Bentz et al. (2023, ApJ, 944, 29) for IC 4329A, Du et al. (2015,\nApJ, 806, 22) for Mrk 1044, and Bentz et al. (2021, ApJ, 906, 50) for NGC 3783.\nThe reference for NGC 3783 has been updated. The remaining virial products already had estimates\nwith references to the reviewer’s list or with references within Dalla Bont`a et al. (2020).\nConclusion\nThe manuscript does not provide a convincing justification for its methodological choices, particularly\nthe use of FWHM and [O III] luminosity as primary inputs to the mass scaling relation. The results do\nnot offer clear new insight into the physics of AGN mass estimation and do not meet the threshold of\nnovelty or rigor expected for this journal.\nAs per the detailed reply above, FWHM is a widely adopted standard for virial single-epoch masses.\nThe use of OIII as a primary input into a relation is validated by our large sample showing superior\nrobustness of mass estimates; this is an empirical finding offered in this manuscript for the first time,\nwhich we believe to be a valuable stimulation for theoretical modelling experts to explain.\nWehaveread“CorrectingBlackHoleMassesBiasedbySizeInertiainBroadLineRegions”byAmrutha\netal. Thismanuscriptreportsanalysisofspectraldataofactivegalacticnuclei(AGN)thathadpreviously\nbeen observed roughly two decades earlier. The authors study the joint variability in the width of the\nH-beta line and the bolometric luminosity (as measured by 5100 Angstrom continuum measurement and\nothers). Based on their analysis the authors find that the joint variability is not well described by the\nprevailing model (the “breathing” broad line region [BLR] model). This, in particular, has implications\nformassestimationsoftheAGNblackholes. Theauthorsthenadvocateforuseofthe[OIII]luminosity\ninstead of other luminosity indicators (5100 continuum, H-beta, or combinations with Fe line complex\nluminosities).\nWe feel that the potential result is interesting, but there are a number of concerns that prevent us from\nbeing able to determine whether the conclusions are sound. The authors will need to address our major\nconcerns before we can recommend publication.\n*** MAJOR CONCERNS ***\nA key concern is the experimental setup. The authors are comparing 6dFGS spectra taken on Schmidt\nwithWiFeSspectratakenontheANU2.3-mtelescope. Smallsystematicdifferencesinthedifferentdata\nsets (even though they were reduced and fitted with the same procedure) could lead to the appearance\nof a problem in the model when it is in the data. Is it possible to directly compare the instrumental\nsetups for consistency? While the authors have the NGC 5548 data that should be self-consistent, the\ntrend within NGC 5548 is far less convincing.\nThe two instruments were not active during the same period, so unfortunately they could never observe\nthe same object at the same time to compare and calibrate any systematic differences. We have now\nadded to the text attempts to mitigate any differences in the data set, see Methods: Correcting for\ninstrumental differences. We agree that the effect in NGC5548, which itself was observed with several\ninstruments, is quantitatively less convincing, but appears qualitatively similar.\nRelated to the above, measurement uncertainties are not addressed in the manuscript at all. It is essen-\ntial to understand how measurement uncertainties are incorporated in the analysis. This is especially\nimportant when some of the effects that are being reported are slight (e.g., the difference between 5100\nand [O III] scatter in mass measurements is only 0.035 dex). The correlations of uncertainties between\ncontinuum and line widths is a particularly problematic aspect that needs to be addressed. The fact\nthat the Baldwin effect was not mentioned nor any attempt to mitigate it is particularly worrisome. It\nis not possible to publish this work without a thorough examination of how this effect plays a role in the\nanalysis.\nWethankthereviewerforthesuggestiontodiscusstheBaldwineffect,whichwehaveincludednowinthe\nDiscussion section. Also, we have added information on the measurement uncertainties in the Methods\nsection. Briefly here, uncertainties for the spectral decomposition were estimated using a Markov chain\nMonte Carlo (MCMC) approach and are derived from the posterior distributions of the MCMC chains,\nwith the 16–50th and 50–84th percentile ranges reported as the lower and upper errors, respectively.\nA key test of whether what the authors see is real would be to simulate observations with realistic\ncorrelateduncertaintiesthatvaryaccordingtotheR–Lrelationwithanintrinsicscatter. Howoftendoes\nthesimulatedDeltaFWHMvsDeltaLcorrelationshowthesametrend(ormoreextreme)thanthedata\ndo?\nThe test regarding the correlated uncertainties is a good suggestion by the reviewers that we have now\nimplemented and explained in ”Methods: Correcting for instrumental differences” along with Figure 8\nshowing results of our simulation.\nMeasurement uncertainties also need to be taken into account when measuring 16–18 percentiles in Fig\n4 and rms in Fig 5. That is, which of the relations has the smallest intrinsic scatter may be different\n11\nfrom which having the smallest total scatter. The latter is better for estimating black hole masses more\nprecisely; the former is better for understanding the physics.\nThe text and figures have been updated to differentiate between measured scatter and intrinsic scatter.\nThis is done for our AGN sample. For the NGC 5548 sample, the FWHM errors measurements are not\navailable, and hence only the total scatter is reported.\n*** OTHER CONCERNS ***\nINTRODUCTION\nThird paragraph\n- “Drops an assumption” → What assumption?\n- “Current formulations” does this refer to the two aforementioned methods or does it include others?\nThe sentence has been replaced with ”In this paper, we present evidence for an alternative approach to\nSE mass estimates, motivated by new observations, without modelling a second parameter.”\nFourth paragraph\n- “The AGN is rarely in equilibrium” → Is this referring to NGC 5548 or AGN in general?\nThis is in reference to NGC 5548, and the text has been updated to mention this.\n- “Where a few objects follow the slope [...] while others do not” → quantify this more\nThe sentence has been replaced to be more specific.\nFinal paragraph - In what sense is the 6dFGS AGN sample “complete”? How were the 324 AGN in\nthe ANU sub-sample selected? No comparison of the ANU sub-sample to the parent is presented to\ndemonstrate that selection effects are not behind the trends seen by the authors.\n- “This sample has a median bolometric luminosity [...]” → Is this referring to the sub sample? If so,\nspecify. Here you state median 44.2, later you use mean 44, choose one\n- The last few sentences explaining the two-epoch simulated sample from NGC 5548 data is a little\nconfusing and could benefit from more details.\nThe paragraph has been modified to provide more details regarding 6dFGS completeness (K < 12.65\nmag and z < 0.1) and reference to the Methods Section has been added to a more detailed explanation\nregarding sample selection. The text regarding NGC 5548 has also been edited to elaborate on the\ntwo-epoch pairing.\nA 20-YEAR COMPARISON OF BLACK-HOLE MASS DIAGNOSTICS\n- This is the first time Hbeta is mentioned, this should be introduced earlier.\nHβ is now introduced in the second paragraph of the introduction.\n- “The 20-year repeatability of [...]” How do these numbers change/improve when using the proposed\nL_[OIII] scaling relation?\nFigure 5 now shows the 20-year repeatability for our AGN sample. We have modified the text to make\nthis our key result instead of the comparison with RM masses, as it better shows the sample behaviour.\n-Theauthorswrite“ThechangeinFWHM_Hbetahasastandarddeviationof 0.12dex... withareduced\nchi^2 of 6.6 around the median change.” What is the model for which a reduced chi^2 was calculated?\nHow many degrees of freedom in the model? From context clues I think it might be that the authors\nare suggesting a no-change model. But this was very poorly written and nothing is explained in how it\nwas calculated. This must be addressed in order to assert that there is a “real variation in FWHM_Hbeta”.\nPerhaps the model is the dashed line in Figure 3? None of this is clear as written.\nIS L_[OIII] A BETTER PROXY FOR BLR SIZE?\n- Add citation for sentence starting “In contrast, the narrow-line luminosity [...]”\nWe have cited Capetti et al. (1996) and Dopita et al. (2015).\n- “These RMS values are in part due to [...]” Details of the calculation of the RMS are required.\nThe details are written in the text (RMS deviation from the one-to-one line for RM vs SE masses). This\nparagraph has been moved to the end of the Methods section.\n- “The absence of mean FWHM changes [...] will remain stable over time [...] will vary by [...]” Can this\nbe calculated and quantified using the larger sample?\n- “The residuals appear proportional to [...]” Quantify the significance of this result.\n- “By design, the L_[OIII]-based masses subtract a linear function [...]” expand on this\nResults from the larger sample has now been explicitly presented and discussed in the paper.\nMETHODS\n- “Appear to be a near-complete” → By what metric are you determining completeness?\nWe are using a K magnitude limit, from the 2MASS Extended Source Catalogue. We have made explicit\nreference to this in the text, adding where our sample is incomplete.\nFIGURES-Fig. 4: WhatdoesM_L_[OIII]vstimelooklike? Howdothelowerhistogramschangewhen\nusingthefulltimespanofthedata? Canananalogofthelowerpanelsbemadeforthe324AGN?What\nabout the 13 RM AGN (e.g. by using Delta M_SE instead of M_SE)?\nForaconstantL , theL -derivedmasswillfollowthesametrendastheFWHM,withaconstant\n[OIII] [OIII]\nscaling and offset in log space. We now have a second axis for the FWHM panel showing the [OIII] mass.\nUsing the full time span provides a similar trend, but the scatter is increased in all mass estimates. We\nhave mentioned this in the NGC 5548 comparison sub-section in Methods.\nThe current figure 5 shows the difference for our AGN sample, which we felt better presented the results\nwe are trying to convey.\nFor the RM sample, we do not have two epoch spectra for all of them, as they were separately observed\nto calibrate our SE masses.\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part\nof the Nature Communications initiative to facilitate training in peer review and to provide appropriate\nrecognition for Early Career Researchers who co-review manuscripts.\nReferences\nCapetti A., Axon D. J., Macchetto F., Sparks W. B., Boksenberg A., 1996, ApJ, 469, 554\nDalla Bont`a E., et al., 2020, ApJ, 903, 112\nDopita M. A., et al., 2015, ApJS, 217, 12\nGoad M. R., Korista K. T., 2014, MNRAS, 444, 43\nShen Y., et al., 2024, ApJS, 272, 26\nWang S., Woo J.-H., 2024, ApJS, 275, 13\nLine Regions\nResponse to reviewer comments (2nd revision)\nNovember 14, 2025\nWe thank the reviewers for their comments, in response to which we have revised the manuscript. All\nchanges are marked in bold text. These changes do not affect the overall results and conclusions of\nthe paper. We also change the title of paper to “Strong long-term variability in active galactic\nnuclei affects virial black hole mass measurements” as we feel it better represents the focus of the\nmanuscript in its current form. Below, we address the reviewers’ comments in order.\nReviewer 1 (Remarks to the Author):\nI am content with the authors’ substantial revision of this manuscript and happy to recommend publi-\ncation. I’ve a single suggestion: In Figure 8, please use a different color (e.g., green) or red or something\nother than two similar colors of blue to distinguish between ”Observed” and ”Expected”.\nFigure 8 now uses blue and red to mark ”Observed” and ”Expected”.\nReviewer 3 (Remarks to the Author):\nI have read the revised manuscript “Correcting Black Hole Masses Biased by Size Inertia in Broad\nLine Regions” by Amrutha et al. The manuscript is much improved in terms of presentation and analysis of\nthe work. I am convinced of the soundness of their results. With the clearer presentation of the results,\nhowever, it is more obvious to me that what the authors have found is a very small effect.\nAt the top of page 5, they note that the intrinsic scatter introduced by variability in L accounts for\nHβ\nroughly one quarter of the observed intrinsic scatter in the R -L relation. First, I do not see how\nBLR Hβ\nthey came to a value of one quarter. Assuming Gaussian scatter, variances add. Thus a total intrinsic\nscatter of 0.17 dex is composed of an L -variability component (0.08 dex) and the rest (sigma rest).\nHβ\nThen sigma rest**2 = (0.17)**2 - (0.08)**2. So, sigma rest = 0.15 dex.\nThe “roughly one quarter” estimate referred to the contribution in variance, not in standard deviation.\nAs an illustration, if the variance contributions from variability, orientation, and the remaining effects\nrelated to responsivity and characteristic timescales are each on the order of 0.1 dex, which is a realistic\nassumption, their combined effect would account for the full observed scatter:\n(cid:112)\n0.12+0.12+0.12 ≈0.173. (1)\nTo avoid confusion, we have revised the text to remove the “quarter” phrasing and now state that the\nscatter due to variability is comparable to, and therefore non-negligible relative to, other sources of\nscatter.\nSo if one could completely control for L variability, it would decrease the intrinsic scatter by 0.02 dex.\nHβ\nSimilarly, Figs 4b and 5a, show that this is a minor effect. This is at odds with the authors’ claim that\n1\n“the dominant scatter is due to luminosity variability” or that “viewing angles and dust extinction [are]\nsub-dominant”.\nWe thank the referee for picking up on these statements. We agree that controlling for L variability\nHβ\nwouldreducetheintrinsicscatterbyonly0.02dex, andwedidnotintendtoimplythatvariabilityisthe\nsingle most dominant contributor to the observed scatter. The earlier wording was a remnant from the\ninitial manuscript that used parameters from Kaspi et al. (2005), where adopting an R -L slope of\n√ BLR Hβ\n0.63 and an intrinsic scatter of 0.14 dex yields σ = 0.26/ 2×0.63 = 0.116 dex, corresponding to\nlogRHβ\n(0.116/0.14)2 ≈68% of the total variance. This fraction no longer holds when using the newer R −L\nBLR\nrelations from Wang & Woo (2024), and we missed removing these statements in our previous draft.\nWe have now revised all relevant statements, in the summary paragraph after the Discussion and in\nFigure 6, to state that the scatter in R −L relations due to luminosity variability is “comparable” to\nBLR\nother sources, rather than “dominant”.\nBecause the magnitude of the effect is so small, I do not find this effect to be important enough for this\njournal. I recommend against publication.\nWethanktherefereefortheirassessment. Wewouldliketohighlightthatthemainresultofthisworkis\nnot just about the small magnitude of the scatter reduction, but rather the identification of a systematic\nand population-level behaviour in AGN variability and its implications for black hole mass estimates.\nOur analysis, based on a flux-limited AGN sample with repeat spectroscopy over a 20-year baseline,\nrevealssignificantluminosityvariationsthatfollowaclearmean-reversionbehaviour: AGNtendtoreturn\ntoward an average activity level after stochastic excursions, and this reversion is already evident within\ntheobservedtimescale(Figure2). However,theAGNpopulationdoesnotexhibitcorrespondingchanges\nin line width in response to luminosity variations, as would be expected if the BLR strictly followed\nvirialmotion(Figure3). Thisindicatesthatthecommonlyassumedcancellationbetweenluminosityand\nvelocityvariationsdoesnotholduniversally,andthatsuchvariabilityintroducessystematicoffsetsinSE\nmass estimates rather than simply contributing additional scatter.\nFurthermore,wefindthatusingL -basedR estimatesyieldsa0.075dexreductionintheintrinsic\n[OIII] BLR\nscatter of repeated SE mass measurements compared to L , even without invoking a second-order\nHβ\ncorrection based on the Eddington ratio. This is particularly notable because the intrinsic scatter in the\nR −L relation for L (0.25 dex) is larger than that for L (0.17 dex; Wang & Woo 2024), yet\nBLR [OIII] Hβ\nthe repeatability of L -based SE masses is higher. Consistent with this interpretation, the observed\n[OIII]\nHβ/[OIII] line ratio appears to serve as a diagnostic of an AGN’s offset from the R\nBLR\n−L relation, or\nequivalently, its offset in SE mass estimate. This ratio therefore carries physical meaning regardless of\ncalibrationandisbestunderstoodastracingthecurrentaccretionstaterelativetothelong-termaverage,\na perspective that, to our knowledge, has not been proposed previously.\nWe believe these results are novel and significant for understanding long-term AGN variability and its\nimpact on SMBH mass estimates, revealing effects that have been implicitly assumed to average out in\nsingle-epoch methods.\nReviewer 4 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part\nof the Nature Communications initiative to facilitate training in peer review and to provide appropriate\nrecognition for Early Career Researchers who co-review manuscripts.\nReferences\nKaspi S., Maoz D., Netzer H., Peterson B. M., Vestergaard M., Jannuzi B. T., 2005, ApJ, 629, 61\nWang S., Woo J.-H., 2024, ApJS, 275, 13\nblack hole mass measurements\nResponse to reviewer comments (3rd revision)\nDecember 12, 2025\nWe thank the reviewers for their comments, in response to which we have revised the manuscript. All\nchanges to the text are marked in bold text. All figure captions are also edited to reflect changes based\non the style guide. These changes do not affect the overall results and conclusions of the paper. Below,\nwe address the reviewers’ comments in order.\nReviewer 3 (Remarks to the Author):\nThe authors have addressed all of my technical considerations. I noticed a couple of minor things that\nshould be fixed. Minor issues page 2, graph 2: authors write “NGC 5448”\nThis has been revised to NGC 5548.\nFig. 1 caption: authors write “L 5200 + log(9.26)” but must mean “log(L 5200) + log(9.26)”\nWe did indeed mean log L + log(9.26), and we have corrected this.\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part\nof the Nature Communications initiative to facilitate training in peer review and to provide appropriate\nrecognition for Early Career Researchers who co-review manuscripts.\n1"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I am content with the authors' substantial revision of this manuscript and happy to recommend publication.\nI've a single suggestion: In Figure 8, please use a different color (e.g., green) or red or something other than two similar colors of blue to distinguish between \"Observed\" and \"Expected\".", "reviewer_3": "I have read the revised manuscript “Correcting Black Hole Masses Biased by Size Inertia in Broad Line Regions” by Amrutha et al. The manuscript is much improved in terms of presentation and analysis of the work. I am convinced of the soundness of their results. With the clearer presentation of the results, however, it is more obvious to me that what the authors have found is a very small effect.\nAt the top of page 5, they note that the intrinsic scatter introduced by variability in L_Hbeta accounts for roughly one quarter of the observed intrinsic scatter in the R_BLR–L_Hbeta relation. First, I do not see how they came to a value of one quarter. Assuming Gaussian scatter, variances add. Thus a total intrinsic scatter of 0.17 dex is composed of an L_Hbeta-variability component (0.08 dex) and the rest (sigma_rest). Then sigma_rest**2 = (0.17)**2 - (0.08)**2. So, sigma_rest = 0.15 dex.\nSo if one could completely control for L_Hbeta variability, it would decrease the intrinsic scatter by 0.02 dex. Similarly, Figs 4b and 5a, show that this is a minor effect. This is at odds with the authors’ claim that “the dominant scatter is due to luminosity variability” or that “viewing angles and dust extinction [are] sub-dominant”.\nBecause the magnitude of the effect is so small, I do not find this effect to be important enough for this journal. I recommend against publication.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Correcting Black Hole Masses Biased by Size Inertia in Broad\nLine Regions\nResponse to reviewer comments (2nd revision)\nNovember 14, 2025\nWe thank the reviewers for their comments, in response to which we have revised the manuscript. All\nchanges are marked in bold text. These changes do not affect the overall results and conclusions of\nthe paper. We also change the title of paper to “Strong long-term variability in active galactic\nnuclei affects virial black hole mass measurements” as we feel it better represents the focus of the\nmanuscript in its current form. Below, we address the reviewers’ comments in order.\nReviewer 1 (Remarks to the Author):\nI am content with the authors’ substantial revision of this manuscript and happy to recommend publi-\ncation. I’ve a single suggestion: In Figure 8, please use a different color (e.g., green) or red or something\nother than two similar colors of blue to distinguish between ”Observed” and ”Expected”.\nFigure 8 now uses blue and red to mark ”Observed” and ”Expected”.\nReviewer 3 (Remarks to the Author):\nI have read the revised manuscript “Correcting Black Hole Masses Biased by Size Inertia in Broad\nLine Regions” by Amrutha et al. The manuscript is much improved in terms of presentation and analysis of\nthe work. I am convinced of the soundness of their results. With the clearer presentation of the results,\nhowever, it is more obvious to me that what the authors have found is a very small effect.\nAt the top of page 5, they note that the intrinsic scatter introduced by variability in L accounts for\nHβ\nroughly one quarter of the observed intrinsic scatter in the R -L relation. First, I do not see how\nBLR Hβ\nthey came to a value of one quarter. Assuming Gaussian scatter, variances add. Thus a total intrinsic\nscatter of 0.17 dex is composed of an L -variability component (0.08 dex) and the rest (sigma rest).\nHβ\nThen sigma rest**2 = (0.17)**2 - (0.08)**2. So, sigma rest = 0.15 dex.\nThe “roughly one quarter” estimate referred to the contribution in variance, not in standard deviation.\nAs an illustration, if the variance contributions from variability, orientation, and the remaining effects\nrelated to responsivity and characteristic timescales are each on the order of 0.1 dex, which is a realistic\nassumption, their combined effect would account for the full observed scatter:\n(cid:112)\n0.12+0.12+0.12 ≈0.173. (1)\nTo avoid confusion, we have revised the text to remove the “quarter” phrasing and now state that the\nscatter due to variability is comparable to, and therefore non-negligible relative to, other sources of\nscatter.\nSo if one could completely control for L variability, it would decrease the intrinsic scatter by 0.02 dex.\nHβ\nSimilarly, Figs 4b and 5a, show that this is a minor effect. This is at odds with the authors’ claim that\n1\n“the dominant scatter is due to luminosity variability” or that “viewing angles and dust extinction [are]\nsub-dominant”.\nWe thank the referee for picking up on these statements. We agree that controlling for L variability\nHβ\nwouldreducetheintrinsicscatterbyonly0.02dex, andwedidnotintendtoimplythatvariabilityisthe\nsingle most dominant contributor to the observed scatter. The earlier wording was a remnant from the\ninitial manuscript that used parameters from Kaspi et al. (2005), where adopting an R -L slope of\n√ BLR Hβ\n0.63 and an intrinsic scatter of 0.14 dex yields σ = 0.26/ 2×0.63 = 0.116 dex, corresponding to\nlogRHβ\n(0.116/0.14)2 ≈68% of the total variance. This fraction no longer holds when using the newer R −L\nBLR\nrelations from Wang & Woo (2024), and we missed removing these statements in our previous draft.\nWe have now revised all relevant statements, in the summary paragraph after the Discussion and in\nFigure 6, to state that the scatter in R −L relations due to luminosity variability is “comparable” to\nBLR\nother sources, rather than “dominant”.\nBecause the magnitude of the effect is so small, I do not find this effect to be important enough for this\njournal. I recommend against publication.\nWethanktherefereefortheirassessment. Wewouldliketohighlightthatthemainresultofthisworkis\nnot just about the small magnitude of the scatter reduction, but rather the identification of a systematic\nand population-level behaviour in AGN variability and its implications for black hole mass estimates.\nOur analysis, based on a flux-limited AGN sample with repeat spectroscopy over a 20-year baseline,\nrevealssignificantluminosityvariationsthatfollowaclearmean-reversionbehaviour: AGNtendtoreturn\ntoward an average activity level after stochastic excursions, and this reversion is already evident within\ntheobservedtimescale(Figure2). However,theAGNpopulationdoesnotexhibitcorrespondingchanges\nin line width in response to luminosity variations, as would be expected if the BLR strictly followed\nvirialmotion(Figure3). Thisindicatesthatthecommonlyassumedcancellationbetweenluminosityand\nvelocityvariationsdoesnotholduniversally,andthatsuchvariabilityintroducessystematicoffsetsinSE\nmass estimates rather than simply contributing additional scatter.\nFurthermore,wefindthatusingL -basedR estimatesyieldsa0.075dexreductionintheintrinsic\n[OIII] BLR\nscatter of repeated SE mass measurements compared to L , even without invoking a second-order\nHβ\ncorrection based on the Eddington ratio. This is particularly notable because the intrinsic scatter in the\nR −L relation for L (0.25 dex) is larger than that for L (0.17 dex; Wang & Woo 2024), yet\nBLR [OIII] Hβ\nthe repeatability of L -based SE masses is higher. Consistent with this interpretation, the observed\n[OIII]\nHβ/[OIII] line ratio appears to serve as a diagnostic of an AGN’s offset from the R\nBLR\n−L relation, or\nequivalently, its offset in SE mass estimate. This ratio therefore carries physical meaning regardless of\ncalibrationandisbestunderstoodastracingthecurrentaccretionstaterelativetothelong-termaverage,\na perspective that, to our knowledge, has not been proposed previously.\nWe believe these results are novel and significant for understanding long-term AGN variability and its\nimpact on SMBH mass estimates, revealing effects that have been implicitly assumed to average out in\nsingle-epoch methods.\nReviewer 4 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part\nof the Nature Communications initiative to facilitate training in peer review and to provide appropriate\nrecognition for Early Career Researchers who co-review manuscripts.\nReferences\nKaspi S., Maoz D., Netzer H., Peterson B. M., Vestergaard M., Jannuzi B. T., 2005, ApJ, 629, 61\nWang S., Woo J.-H., 2024, ApJS, 275, 13\n2"}, {"round": 3, "reviewer_comments": {"reviewer_3": "The authors have addressed all of my technical considerations. I noticed a couple of minor things that should be fixed.\nMinor issues\npage 2, graph 2: authors write “NGC 5448”\nFig. 1 caption: authors write “L_5200 + log(9.26)” but must mean “log(L_5200) + log(9.26)”", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Strong long-term variability in active galactic nuclei affects virial\nblack hole mass measurements\nResponse to reviewer comments (3rd revision)\nDecember 12, 2025\nWe thank the reviewers for their comments, in response to which we have revised the manuscript. All\nchanges to the text are marked in bold text. All figure captions are also edited to reflect changes based\non the style guide. These changes do not affect the overall results and conclusions of the paper. Below,\nwe address the reviewers’ comments in order.\nReviewer 3 (Remarks to the Author):\nThe authors have addressed all of my technical considerations. I noticed a couple of minor things that\nshould be fixed. Minor issues page 2, graph 2: authors write “NGC 5448”\nThis has been revised to NGC 5548.\nFig. 1 caption: authors write “L 5200 + log(9.26)” but must mean “log(L 5200) + log(9.26)”\nWe did indeed mean log L + log(9.26), and we have corrected this.\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part\nof the Nature Communications initiative to facilitate training in peer review and to provide appropriate\nrecognition for Early Career Researchers who co-review manuscripts.\n1"}]

Decision letter