Virial Estimates Research Articles

AGNfitter-rx : Modeling the radio-to-X-ray spectral energy distributions of AGNs

We present new advancements in the modeling of the spectral energy distributions (SEDs) of active galaxies by introducing the radio-to-X-ray fitting capabilities of the publicly available Bayesian code AGNfitter . The new code release, called AGNfitter-rx models the broad-band photometry covering the radio, infrared (IR), optical, ultraviolet (UV), and X-ray bands consistently using a combination of theoretical and semi-empirical models of the active galactic nucleus (AGN) and host-galaxy emission. This framework enables the detailed characterization of four physical components of the active nuclei, namely the accretion disk, the hot dusty torus, the relativistic jets and core radio emission, and the hot corona, and can be used to model three components within the host galaxy: stellar populations, cold dust, and the radio emission from the star-forming regions. Applying AGNfitter-rx to a diverse sample of 36 AGN SEDs at $z 0.7$ from the AGN SED ATLAS, we investigated and compared the performance of state-of-the-art torus and accretion disk emission models in terms of fit quality and inferred physical parameters. We find that clumpy torus models that include polar winds and semi-empirical accretion disk templates including emission-line features significantly increase the fit quality in 67<!PCT!> of the sources by reducing by $2 fit residuals in the $1.5-5 1mm m$ and $0.7 1mm m$ regimes. We demonstrate that, by applying AGNfitter-rx to photometric data, we are able to estimate the inclination and opening angles of the torus, consistent with spectroscopic classifications within the AGN unified model, as well as black hole masses congruent with virial estimates based on Halpha . We investigate wavelength-dependent AGN fractions across the spectrum for Type 1 and Type 2 AGNs, finding dominant AGN fractions in radio, mid-infrared, and X-ray bands, which are in agreement with the findings from empirical methods for AGN selection. The wavelength coverage and the flexibility for the inclusion of state-of-the-art theoretical models make AGNfitter-rx a unique tool for the further development of SED modeling for AGNs in present and future radio-to-X-ray galaxy surveys.

Effects of multiplicative noise on the fractional Hartree equation

This paper is dedicated to radial solutions to the Cauchy problem for the fractional Hartree equation with multiplicative noise. First, we establish a stochastic Strichartz estimate related to the fractional Schrödinger propagator. Local well-posedness for the Cauchy problem is proved by using stochastic and radial deterministic Strichartz estimates. Then, based on Itô’s formula and stopping time arguments, the existence of a global solution is studied. Finally, we investigate the blow-up phenomenon and give a criterion via localized virial estimates.

Virial black hole mass estimates of quasars in the XQ-100 legacy survey

ABSTRACT The black hole (BH) mass and luminosity are key factors in determining how a quasar interacts with its environment. In this study, we utilize data from the European Southern Observatory Large Programme XQ-100, a high-quality sample of 100 X-shooter spectra of the most luminous quasars in the redshift range 3.5 < z < 4.5, and measure the properties of three prominent optical and ultraviolet broad emission lines present in the wide wavelength coverage of X-shooter: C iv, Mg ii, and H β. The line properties of all three broad lines are used for virial estimates of the BH mass and their resulting mass estimates for this sample are tightly correlated. The BH mass range is $\log {(\rm {\mathit{ M}_{BH}}/\rm {M_\odot })} = 8.6{\!-\!}10.3$ with bolometric luminosities estimated from the 3000 Å continuum in the range $\log {(\rm {\mathit{ L}_{bol}}/\rm {erg\, s^{-1}})} = 46.7{\!-\!}48.0$. Robustly determined properties of these quasars enable a variety of follow-up research in quasar astrophysics, from chemical abundance and evolution in the broad-line region to radiatively driven quasar outflows.

MODERN ESTIMATES OF THE MASS OF THE MILKY WAY

An overview of various methods used to estimate the mass of the Galaxy is given. The results obtained on the basis of the analysis of the galactic rotation curve, on the kinematics of dwarf satellite galaxies of the Milky Way, on globular clusters, on the streams of dwarf galaxies, on distant halo stars, on escape velocities, on the dynamics of the Local group of galaxies, as well as results of N-body modeling are included. Estimates of the mass of the Galaxy \(M\) of the form (i) \(M( r)\), i.e. the mass enclosed inside a sphere with a radius of \(r\), (ii) \({{M}_{{200}}}\), where r 0 kpc, and (iii) virial estimates, \({{M}_{{{\text{vir}}}}}\). Such estimates were selected by us based on literary data,a significant part of which was obtained in the Gaia era. According to 20 individual estimates, the average value was found \({{\bar {M}}_{{200}}} = 0.88 \times {{10}^{{12}}} {{M}_{ \odot }}\) with a variance of \(0.24 \times {{10}^{{12}}} {{M}_{ \odot }}\) and a weighted average error of \(0.06 \times {{10}^{{12}}} {{M}_{ \odot }}\). According to 28 individual estimates we obtained \({{\bar {M}}_{{{\text{vir}}}}} = 1.05 \times {{10}^{{12}}} {{M}_{ \odot }}\) with variance \(0.44 \times {{10}^{{12}}} {{M}_{ \odot }}\) and a weighted average error of \(0.09 \times {{10}^{{12}}} {{M}_{ \odot }}\).

Long time dynamics and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with spatially growing nonlinearity

We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation i∂tu + Δu = −|x|b|u|p−1u in the radial Sobolev space Hr1(RN), where b > 0 and p > 1. We show the global existence and energy scattering in the intercritical regime, i.e., p>N+4+2bN and p<N+2+2bN−2 if N ≥ 3. We also obtain blowing-up solutions for the mass-critical and mass-supercritical nonlinearities. The main difficulty, coming from the spatial growing nonlinearity, is overcome by refined Gagliardo–Nirenberg-type inequalities. Our proofs are based on improved Gagliardo–Nirenberg inequalities, the Morawetz–Sobolev approach of Dodson and Murphy [Proc. Am. Math. Soc. 145(11), 4859–4867 (2017)], radial Sobolev embeddings, and localized virial estimates.

The virial theorem for retarded gravity

Galaxy Clusters are huge physical systems having a generic size of tens of millions of light years. Thus any modification at the center of the cluster will affect the outskirts only tens of millions of years afterwards. Those retardation considerations seem to be neglected in present day analysis used to estimate the total mass of the galaxy cluster, including those estimates which are based on the virial theorem. The significant differences between the velocities predicted by Newtonian action at a distance theory and observations are usually dealt with by either assuming an unobservable type of matter or by modifying gravity. Here we demonstrate that considering general relativistic effects one can explain the apparent excess matter appearing in galaxy clusters virial estimates.

Some results for a system of NLS arising in optical materials with χ3 nonlinear response

In this paper, we investigate the nonlinear Schödinger equations with cubic interactions, arising in nonlinear optics. To begin, we prove the existence results for normalized ground state solutions in the subcritical case and ‐supercritical case, respectively. Our proofs relies on the Concentration‐compactness principle, Pohozaev manifold, and rearrangement technique. Then, we establish the nonexistence of normalized ground state solutions in the ‐critical case by finding that there exists a threshold. In addition, based on the existence of the normalized solutions, we also establish the blow‐up results by using localized virial estimates as well as a new blow‐up criterion which is related to normalized solutions.

On Asymptotic Stability of the Sine-Gordon Kink in the Energy Space

We consider the sine-Gordon (SG) equation in 1+1 dimensions. The kink is a static, non symmetric exact solution to SG, stable in the energy space H^1times L^2. It is well-known that the linearized operator around the kink has a simple kernel and no internal modes. However, it possesses an odd resonance at the bottom of the continuum spectrum, deeply related to the existence of the (in)famous wobbling kink, an explicit periodic-in-time solution of SG around the kink that contradicts the asymptotic stability of the kink in the energy space. In this paper we further investigate the influence of resonances in the asymptotic stability question. We also discuss the relationship between breathers, wobbling kinks and resonances in the SG setting. By gathering Bäcklund transformations (BT) as in Hoffman and Wayne (Differ Int Equ 26(3–4):303–320, 2013), Muñoz and Palacios (Ann. IHP C Analyse Nonlinéaire 36(4):977–1034, 2019) and Virial estimates around odd perturbations of the vacuum solution, in the spirit of Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), we first identify the manifold of initial data around zero under which BTs are related to the wobbling kink solution. It turns out that (even) small breathers are deeply related to odd perturbations around the kink, including the wobbling kink itself. As a consequence of this result and Kowalczyk et al. (Lett Math Phys 107(5):921–931, 2017), using BTs we can construct a smooth manifold of initial data close to the kink, for which there is asymptotic stability in the energy space. The initial data has spatial symmetry of the form (kink + odd, even), non resonant in principle, and not preserved by the flow. This asymptotic stability property holds despite the existence of wobbling kinks in SG. We also show that wobbling kinks are orbitally stable under odd data, and clarify some interesting connections between SG and phi ^4 at the level of linear Bäcklund transformations.

Blow-up criteria and instability of standing waves for the fractional Schrodinger Poisson equation

In this article, we consider blow-up criteria and instability of standing waves for the fractional Schrodinger-Poisson equation. By using the localized virial estimates, we establish the blow-up criteria for non-radial solutions in both mass-critical and mass-supercritical cases. Based on these blow-up criteria and three variational characterizations of the ground state, we prove that the standing waves are strongly unstable. These obtained results extend the corresponding ones presented in the literature.

Statistical analysis of Al III and C III] emission lines as virial black hole mass estimators in quasars

Aims. We test the usefulness of the intermediate ionisation lines Al IIIλ1860 and C III]λ1909 as reliable virial mass estimators for quasars. Methods. We identified a sample of 309 quasars from the Sloan Digital Sky Survey Data Release 16 (SDSS DR16) in the redshift range 1.2 ≤ z ≤ 1.4 that have [O II]λ3728 recorded on the same spectrum as Al IIIλ1860, Si III]λ1892, and C III]λ1909. We set the systemic quasar redshift using careful measurements of [O II]λ3728. We then classified the sources as Population A (Pop. A), extreme Population A (Pop. xA), and Population B (Pop. B), and analysed the 1900 Å blend using multi-component models to look for systematic line shifts of Al IIIλ1860 and C III]λ1909 along the quasar main sequence. Results. We do not find significant shifts of the Al IIIλ1860 line peak in Pop. B or in the vast majority of Pop. A. For Pop. xA, a small median blueshift of −250 km s−1 was observed, motivating a decomposition of the Al III line profile into a virialised component centred at rest frame and a blueshifted component for an outflow emission. For Pop. B objects, we proved the empirical necessity to fit a redshifted very broad component, clearly seen in C III], and analysed the physical implications on a Pop. B composite spectrum using CLOUDY simulations. We find consistent black hole mass estimations using Al III and C III] as virial estimators for the bulk of Pop. A. Conclusions. Al III (and even C III]) is a reliable virial black hole mass estimator for Pop. A and Pop. B objects. The Pop. xA sources deserve special attention due to the significant blueshifted excess observed in the line profile of Al III, although it is not as large as the excess observed in C IVλ1549.

Decay for Skyrme wave maps

We consider the decay problem for global solutions of the Skyrme and Adkins–Nappi equations. We prove that the energy associated to any bounded energy solution of the Skyrme (or Adkins–Nappi) equation decays to zero outside the light cone (in the radial coordinate). Furthermore, we prove that suitable polynomial weighted energies of any small solution decays to zero when these energies are bounded. The proof consists of finding three new virial type estimates, one for the exterior of the light cone, based on the energy of the solution, and a more subtle virial identity for the weighted energies, based on a modification of momentum-type quantities.

Discovery of the most distant luminous quasar to date: J0313-1806

<p indent="0mm">Distant quasars are unique media for studying topics such as the formation mechanisms of first-generation seed black holes, the interaction of active galactic nuclei and their host galaxies, and the history of cosmic reionization. So far, there have been found only six quasars with <italic>z</italic> &gt; 7, and two of them (ULAS J1342 + 0928, <italic>z</italic> = 7.54; J100758.264+211529.207, <italic>z </italic>= 7.52) have a redshift larger than 7.5. On January 20, 2021, in <italic>the Astrophysical Journal Letters</italic>, an international collaborative team of astronomers led by the University of Arizona reported a bright quasar, named J0313-1806, with a bolometric luminosity of 1.4 × <sc>10⁴⁷ erg s^–1.</sc> The process of discovering the quasar, the technique, and the scientific significance of this discovery are explained in this paper. Their work is based on a data set obtained mainly from the Pan-STARRS1, the DESI Legacy Imaging Surveys, the UKIRT Hemisphere Survey, the VISTA Hemisphere Survey, and the Wide-field Infrared Survey Explorer Survey. To measure the redshift of this quasar more accurately, the dust continuum and the CII emission from the quasar’s host galaxy were observed using the Atacama Large Millimeter/Submillimeter Array in Chile. Via spectral fitting, they calculated the quasar’s redshift of 7.642, that is more than 13 billion light-years from Earth and dating back to only 670 million years after the Big Bang, which makes the quasar the most distant and earliest one. Based on the spectral fitting and the MgII virial estimator, the mass of the supermassive black hole was calculated to be 1.6×10⁹<italic>M</italic>_⊙. The formation of such a massive supermassive black hole in a short period time imposes strict constraints on the mass and the formation model of the seed black hole. Suppose the black hole accreted at the Eddington rate, if it started its growth at redshift <italic>z</italic> = 15 – 30 (i.e., 400–570 Ma growth time), to what we observe today, it requires a 10⁴–10⁵<italic>M</italic>_⊙ seed black hole, that might be more supportive of direct-collapse black holes forming in pre-galactic dark matter halos. The quasar’s spectra also exhibit evident broad absorption line features in CIV and SiIV, together with a strongly blueshifted CIV emission line, indicating the relativistic outflows with the maximum speed of up to 20% of the speed of light. In addition, the ALMA observations imply a strong star formation of 200 <italic>M</italic>_⊙ a⁻¹ taking place in the host galaxy of J0313-1806. The continuum observations indicate that substantial dust (7 × 10⁷<italic>M</italic>_⊙) was already built up in the quasar host galaxy. Those makes J0313-1806 an ideal target for probing the effects of active galactic nuclei on their host galaxies. The number of high-redshift quasars is still small, making it difficult for constraining the formation model of the seed black holes, understanding the relation of the supermassive black hole and its host galaxy and so on. We look forward to those ongoing and upcoming optical/infrared sky surveys, such as the James Webb Space Telescope (JWST) and the Large Synoptic Survey Telescope (LSST), which will bring breakthroughs in astronomy.

The Intermediate-ionization Lines as Virial Broadening Estimators for Population A Quasars* *Based in part on observations made with ESO Telescopes at the Paranal Observatory under programs 082.B-0572(A) and 083.B-0273(A).

The identification of a virial broadening estimator in the quasar UV rest frame suitable for black hole mass computation at high redshift has become an important issue. We compare the H i Balmer H β line width to the ones of two intermediate-ionization lines: the Al iii λ1860 doublet and the C iii] λ1909 line, over a wide interval of redshift and luminosity (0 ≲ z ≲ 3.5; [erg s−1]), for 48 sources belonging to the quasar population characterized by intermediate to high values of the Eddington ratio (Population A). The present analysis indicates that the line widths of Al iii λ1860 and H β are highly correlated and can be considered equivalent for most Population A quasars over five orders of magnitude in luminosity; for C iii] λ1909, multiplication by a constant correction factor ξ ≈ 1.25 is sufficient to bring the FWHM of C iii] in agreement with that of H β. The statistical concordance between low-ionization and intermediate-ionization lines suggests that they predominantly arise from the same virialized part of the broad-line region. However, blueshifts of modest amplitude (few hundred kilometers per second) with respect to the quasar rest frame and an excess (≲1.1) Al iii broadening with respect to H β are found in a fraction of our sample. Scaling laws to estimate M BH of high-redshift quasars using the Al iii and C iii] line widths have rms scatter ≈0.3 dex. The Al iii scaling law takes the form [M ⊙].

BASS. XXV. DR2 Broad-line-based Black Hole Mass Estimates and Biases from Obscuration

We present measurements of broad emission lines and virial estimates of supermassive black hole masses (M BH) for a large sample of ultrahard X-ray-selected active galactic nuclei (AGNs) as part of the second data release of the BAT AGN Spectroscopic Survey (BASS/DR2). Our catalog includes M BH estimates for a total of 689 AGNs, determined from the Hα, Hβ, Mg ii λ2798, and/or C iv λ1549 broad emission lines. The core sample includes a total of 512 AGNs drawn from the 70 month Swift/BAT all-sky catalog. We also provide measurements for 177 additional AGNs that are drawn from deeper Swift/BAT survey data. We study the links between M BH estimates and line-of-sight obscuration measured from X-ray spectral analysis. We find that broad Hα emission lines in obscured AGNs ( log(NH/cm−2)>22.0 ) are on average a factor of 8.0−2.4+4.1 weaker relative to ultrahard X-ray emission and about 35−12+7 % narrower than those in unobscured sources (i.e., log(NH/cm−2)<21.5 ). This indicates that the innermost part of the broad-line region is preferentially absorbed. Consequently, current single-epoch M BH prescriptions result in severely underestimated (>1 dex) masses for Type 1.9 sources (AGNs with broad Hα but no broad Hβ) and/or sources with log(NH/cm−2)≳22.0 . We provide simple multiplicative corrections for the observed luminosity and width of the broad Hα component (L[bHα] and FWHM[bHα]) in such sources to account for this effect and to (partially) remedy M BH estimates for Type 1.9 objects. As a key ingredient of BASS/DR2, our work provides the community with the data needed to further study powerful AGNs in the low-redshift universe.

The initial conditions for young massive cluster formation in the Galactic Centre: convergence of large-scale gas flows

ABSTRACT Young massive clusters (YMCs) are compact (≲1 pc), high-mass (&amp;gt;104 M⊙) stellar systems of significant scientific interest. Due to their rarity and rapid formation, we have very few examples of YMC progenitor gas clouds before star formation has begun. As a result, the initial conditions required for YMC formation are uncertain. We present high resolution (0.13 arcsec, ∼1000 au) ALMA observations and Mopra single-dish data, showing that Galactic Centre dust ridge ‘Cloud d’ (G0.412 + 0.052, mass = 7.6 × 104 M⊙, radius = 3.2 pc) has the potential to become an Arches-like YMC (104 M⊙, r ∼ 1 pc), but is not yet forming stars. This would mean it is the youngest known pre-star-forming massive cluster and therefore could be an ideal laboratory for studying the initial conditions of YMC formation. We find 96 sources in the dust continuum, with masses ≲3 M⊙ and radii of ∼103 au. The source masses and separations are more consistent with thermal rather than turbulent fragmentation. It is not possible to unambiguously determine the dynamical state of most of the sources, as the uncertainty on virial parameter estimates is large. We find evidence for large-scale (∼1 pc) converging gas flows, which could cause the cloud to grow rapidly, gaining 104 M⊙ within 105 yr. The highest density gas is found at the convergent point of the large-scale flows. We expect this cloud to form many high-mass stars, but find no high-mass starless cores. If the sources represent the initial conditions for star formation, the resulting initial mass function will be bottom heavy.

Benjamin--Ono Soliton Dynamics in a Slowly Varying Potential Revisited

The Benjamin Ono equation with a slowly varying potential is $$ \text{(pBO)} \qquad u_t + (Hu_x-Vu + \tfrac12 u^2)_x=0 $$ with $V(x)=W(hx)$, $0 0$ are parameters. For initial condition $u_0(x)$ to (pBO) close to $Q_{0,1}(x)$, it was shown in a previous work by Z. Zhang that the solution $u(x,t)$ to (pBO) remains close to $Q_{a(t),c(t)}(x)$ and approximate parameter dynamics for $(a,c)$ were provided, on a dynamically relevant time scale. In this paper, we prove exact $(a,c)$ parameter dynamics. This is achieved using the basic framework of the previous work by Z. Zhang but adding a local virial estimate for the linearization of (pBO) around the soliton. This is a local-in-space estimate averaged in time, often called a local smoothing estimate, showing that effectively the remainder function in the perturbation analysis is smaller near the soliton than globally in space. A weaker version of this estimate is proved in a paper by Kenig & Martel as part of a ``linear Liouville'' result, and we have adapted and extended their proof for our application.

Non-radial finite time blow-up for the fourth-order nonlinear Schrödinger equations

We revisit the finite time blow-up for the fourth-order Schrödinger equation with focusing inhomogeneous nonlinearity −|x|−2|u|4nu. By exploiting localized virial estimates and spatial decay of the nonlinearity, we prove the finite time blow-up of non-radial solutions with negative energy. Our result is the first one dealing with the existence of non-radial blow-up solutions to the fourth-order Schrödinger equations.

Magnetic Imprints of Eruptive and Noneruptive Solar Flares as Observed by Solar Dynamics Observatory

Abstract The abrupt and permanent changes of the photospheric magnetic field in the localized regions of active regions during solar flares, called magnetic imprints (MIs), have been observed for nearly the past three decades. The well-known coronal implosion model is assumed to explain such flare-associated changes but the complete physical understanding is still missing and debatable. In this study, we made a systematic analysis of flare-related changes of the photospheric magnetic field during 21 flares (14 eruptive and seven noneruptive) using the 135 s cadence vector magnetogram data obtained from the Helioseismic and Magnetic Imager. The MI regions for eruptive flares are found to be strongly localized, whereas the majority of noneruptive events (&gt;70%) have scattered imprint regions. To quantify the strength of the MIs, we derived the integrated change of horizontal field and the total change of Lorentz force over an area. These quantities correlate well with the flare strength, irrespective of whether flares are eruptive or not, or have a short or long duration. Further, the free energy (FE), determined from virial theorem estimates, exhibits a statistically significant downward trend that starts around the flare time and is observed in the majority of flares. The change of FE during flares does not depend on eruptivity but has a strong positive correlation (≈0.8) with the Lorentz force change, indicating that part of the FE released would penetrate the photosphere. While these results strongly favor the idea of significant feedback from the corona on the photospheric magnetic field, the characteristics of MIs are quite indistinguishable from flares being eruptive or not.

Quadratic Klein-Gordon equations with a potential in one dimension

Abstract This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely only on Strichartz or virial estimates and is therefore able to treat low-power nonlinearities (hence also nonlocalised solitons) and capture the global (in space and time) behaviour of solutions. More specifically, we consider quadratic nonlinear Klein-Gordon equations with a regular and decaying potential in one space dimension. Additional assumptions are made so that the distorted Fourier transform of the solution vanishes at zero frequency. Assuming also that the associated Schrödinger operator has no negative eigenvalues, we obtain global-in-time bounds, including sharp pointwise decay and modified asymptotics, for small solutions. These results have some direct applications to the asymptotic stability of (topological) solitons, as well as several other potential applications to a variety of related problems. For instance, we obtain full asymptotic stability of kinks with respect to odd perturbations for the double sine-Gordon problem (in an appropriate range of the deformation parameter). For the $\phi ^4$ problem, we obtain asymptotic stability for small odd solutions, provided the nonlinearity is projected on the continuous spectrum. Our results also go beyond these examples since our framework allows for the presence of a fully coherent phenomenon (a space-time resonance) at the level of quadratic interactions, which creates a degeneracy in distorted Fourier space. We devise a suitable framework that incorporates this and use multilinear harmonic analysis in the distorted setting to control all nonlinear interactions.

A Local Baseline of the Black Hole Mass Scaling Relations for Active Galaxies. IV. Correlations Between M BH and Host Galaxy σ, Stellar Mass, and Luminosity

Abstract The tight correlations between the mass of supermassive black holes (M BH) and their host-galaxy properties have been of great interest to the astrophysical community, but a clear understanding of their origin and fundamental drivers still eludes us. The local relations for active galaxies are interesting in their own right and form the foundation for any evolutionary study over cosmic time. We present Hubble Space Telescope optical imaging of a sample of 66 local active galactic nuclei (AGNs); for 14 objects, we also obtained Gemini near-infrared images. We use state-of-the-art methods to perform surface photometry of the AGN host galaxies, decomposing them into spheroid, disk, and bar (when present), and inferring the luminosity and stellar mass of the components. We combine this information with spatially resolved kinematics obtained at the Keck Telescopes to study the correlations between M BH (determined from single-epoch virial estimators) and host galaxy properties. The correlations are uniformly tight for our AGN sample, with intrinsic scatter 0.2–0.4 dex, smaller than or equal to that of quiescent galaxies. We find no difference between pseudo and classical bulges or barred and nonbarred galaxies. We show that all the tight correlations can be simultaneously satisfied by AGN hosts in the 107–109 M ⊙ regime, with data of sufficient quality. The M BH–σ relation is also in agreement with that of AGN with M BH obtained from reverberation mapping, providing an indirect validation of single-epoch virial estimators of M BH.