A&A 456, 929-939 (2006)
DOI: 10.1051/0004-6361:20054153

VLT/ISAAC spectra of the H$\beta$ region in intermediate-redshift quasars

II. Black hole mass and Eddington ratio^,

J. W. Sulentic¹ - P. Repetto^2,3 - G. M. Stirpe⁴ - P. Marziani² - D. Dultzin-Hacyan³ - M. Calvani²

1 - Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA
2 - Osservatorio Astronomico di Padova, INAF, Vicolo dell' Osservatorio 5, 35122 Padova, Italy
3 - Instituto de Astronomía, UNAM, Aptdo. Postal 70-264, México, D. F. 04510, México
4 - Osservatorio Astronomico di Bologna, INAF, via Ranzani 1, 40127 Bologna, Italy

Received 5 September 2005 / Accepted 16 June 2006

Abstract
We derive black hole masses for a sample of about 300 AGNs in the redshift range 0 < z < 2.5. We use the same virial velocity measure (FWHM H $\beta _{\rm BC}$) for all sources which represents a significant improvement over previous studies. We review methods and caveats for determining AGN black hole masses via the virial assumption for motions in the gas producing low ionization broad emission lines. We derive a corrected FWHM(H $\beta _{\rm BC}$) measure for the broad component of H$\beta$ that better estimates the virialized line emitting component by comparing our FWHM measures with a sample of reverberated sources with H$\beta$ radial velocity dispersion measures. We also consider the FWHM of the Fe II$\lambda$4570 blend as a potential alternative velocity estimator. We find a range of black hole mass between $\log$  ${M}{_{\rm BH}}$ $\sim$ 6.0-10.0, where ${M}{_{\rm BH}}$ is in solar masses. Estimates using corrected FWHM(H$\beta$), as well as FWHM(Fe II) measures, reduce the number of sources with $\log$ ${M}{_{\rm BH}}$ > 9.5 and suggest that extremely large ${M}{_{\rm BH}}$ values ($\log$ ${M}{_{\rm BH}}$ $\ga$ 10) may not be realistic. Derived $L/L{_{\rm Edd}}$ values show no evidence for a significant population of super-Eddington radiators especially after correction is made for sources with extreme orientation to our line of sight. Sources with FWHM(H $\beta _{\rm BC}$) $\la$ 4000 km s^-1 show systematically higher $L/L{_{\rm Edd}}$ and lower ${M}{_{\rm BH}}$ values than broader lined AGNs (including almost all radio-loud sources).

Key words: galaxies: quasars: emission lines - galaxies: quasars: general - line: profiles - black hole physics

   
1 Introduction

Gravitational accretion onto supermassive black holes is generally accepted as the ultimate energy source of Active Galactic Nuclei (AGNs). The last decade has seen a major effort to derive reasonable estimates of black hole masses ( ${M}{_{\rm BH}}$) by assuming virialized motions in the broad line emitting gas:

$$M_{\rm BH} = \frac{v^2 r_{\rm BLR}}{G}$$ (1)

where v is the velocity dispersion of the emitting gas at distance  $r_{\rm BLR}$. The velocity dispersion can be written as $v = f {\it FWHM}$ where FWHM is the full width half maximum measured for a suitable emission line. The factor f depends on the geometry and details of the kinematics (Onken et al. 2004; Krolik 2001; McLure & Dunlop 2001). The usefulness of the virial assumption for ${M}{_{\rm BH}}$ determination is best exploited using reverberation-mapping studies, especially for the Balmer lines (Koratkar & Gaskell 1991; Peterson et al. 2004; Kaspi et al. 2000; Wandel et al. 1999; Peterson & Wandel 1999). Kaspi et al. (2005,2000) derive a relation between Broad Line Region (BLR) distance $r_{\rm BLR}$ and continuum luminosity

$$r_{\rm BLR} \propto (\lambda L_\lambda )^{\alpha}$$ (2)

where the exponent $\alpha$ is constrained between 0.5 and 1.0, and is most likely $\approx$0.6-0.7 for broad H$\beta$ (H $\beta _{\rm BC}$) and optical continuum luminosity.

There are caveats associated with this method. The virial assumption is not likely to be generally valid for the emission line gas in AGNs. It has been known for several decades that different emission lines in a source can show different width and profile shape (e.g., Sulentic 1989; de Robertis 1985). The virial assumption implies that the velocity dispersion will steadily decrease with distance from the central black hole ${\propto} r^{-\frac{1}{2}}$. The time lag between continuum fluctuations and corresponding emission line responses will therefore anti-correlate with line width. This trend has been confirmed in a few objects (Peterson & Wandel 2000).

Observations however show that profile width and shape depend on the ionization potential. The strongest high-ionization lines (HILs; e.g. C IV$\lambda$1549) often display blueward asymmetric profiles, or even centroid blueshifts (up to several 1000 km s^-1) with respect to the best estimates of the rest frame of the source (e.g., Marziani et al. 1996; Gaskell 1982; Bachev et al. 2004; Richards et al. 2002). Blueshifts like those observed for C IV$\lambda$1549 are indicative of obscuration and radial motions which invalidate the virial assumption.

Low ionization lines (LILs) like H$\beta$ are the best candidates for emission arising from a virialized medium. All or much of the Balmer (and Fe II) line emission is thought to arise from an accretion disk or a flattened cloud distribution near the disk (e.g., Marziani et al. 1996; Collin-Souffrin et al. 1988). A major caveat, especially for the Balmer lines, involves the possibility that: (1) there are two or more emission components in a line, and (2) only one of them may arise in a region where the virial assumption holds. The Balmer lines do not usually show very large line shifts (i.e., shift $\leq$ FWHM) although profiles can be very asymmetric. Both red and blue asymmetries are observed for H $\beta _{\rm BC}$ which is the most studied line because it is relatively unblended and is observable with optical spectrometers up to $z \approx 1$ (Osterbrock & Shuder 1982; Sulentic 1989; Sulentic et al. 1990; Corbin & Francis 1994; Corbin 1991; Stirpe 1990).

The most ambiguous sources from the point of view of ${M}{_{\rm BH}}$ determination show FWHM(H $\beta _{\rm BC}$) $\ga$ 4000 km s^-1 (Population B, following Sulentic et al. 2000b) and redshifted profiles and/or red asymmetries. Not all parts of the H$\beta$ profile respond to continuum changes in the same way implying that some of the line emitting gas may be optically thin or, less likely, is not exposed to the variable H I ionizing radiation (we will return to this issue in Sect. 5.2). The broad line profile in Pop. B sources may be due to two distinct emitting regions: (1) an optically thick classical BLR and (2) a broader and redshifted very broad component that may be optically thin or marginally optically thick to the Lyman continuum, originating in a distinct Very Broad Line Region (VBLR) (Marziani & Sulentic 1993; Shields et al. 1995; Sulentic et al. 2000c). The redshift of the VBLR component raises doubts that it arises from virialized gas. A strong BLR response to continuum fluctuations, coupled with a weak or absent response of the VBLR component, can lead to an overestimate of FWHM for the virialized BLR component resulting in an overestimate of ${M}{_{\rm BH}}$ (Vestergaard 2002; Kaspi et al. 2000; Wandel et al. 1999).

Sources with FWHM(H $\beta _{\rm BC}$) $\la$ 4000 km s^-1 (Population A, Sulentic et al. 2000b) should provide more reliable ${M}{_{\rm BH}}$ estimates. The H $\beta _{\rm BC}$ profile is usually well fit with a symmetric function (Véron-Cetty et al. 2001; Sulentic et al. 2002) and the BLR emission is thought to arise from a Keplerian disk. The most unreliable Pop. A sources, in a disk emission scenario, should be those observed near face-on where the rotational (i.e. virial) contribution to FWHM(H $\beta _{\rm BC}$) is minimal. At least some of the face-on sources may be identified as the so-called "blue outliers'' which show a weak and significantly blueshifted [OIII] $\lambda\lambda$4959,5007 lines (Boroson 2005; Aoki et al. 2005; Marziani et al. 2003b; Zamanov et al. 2002). The inferences in this, and the preceding paragraph have emerged from the Eigenvector 1 (E1) scenario that we have pursued for the past 5+ years (Sulentic et al. 2000a,b, following Boroson & Green 1992). The results reported in this paper appear to confirm them.

There are additional caveats connected with using FWHM(H $\beta _{\rm BC}$) for ${M}{_{\rm BH}}$ derivations. Line asymmetries can affect ${M}{_{\rm BH}}$ estimates for both Pop. A and B sources. The $v_{\rm r}$ displacement of the line centroid at fractional intensities (at half maximum, $c(\frac{1}{2}$)) can give useful information about the uncertainty of FWHM(H $\beta _{\rm BC}$) measurements. The average displacement value is a few hundred km s^-1 with median $c(\frac{1}{2}$) $\approx 400$ km s^-1 (Marziani et al. 2003a). This implies that deviations from a symmetric profile can affect ${M}{_{\rm BH}}$ estimates by $\approx$40%. Uncertainties are also introduced by: (1) contamination from overlapping/nearby lines such as Fe II, He II$\lambda$4686, and [OIII] $\lambda\lambda$4959,5007 (Osterbrock & Shuder 1982; Joly 1988; Jackson et al. 1991; de Robertis 1985), (2) FWHM measures based on single-epoch observations, (3) low S/N spectra and (4) spectra without H$\beta$ narrow component (H $\beta_{\rm NC}$) subtraction. The $r_{\rm BLR}$ - $L_\lambda$ relation is also not free from uncertainties. Reverberation mapping-based ${M}{_{\rm BH}}$ determinations are certainly affected by the non-negligible radial extent of the optically thick BLR. The derived $r_{\rm BLR}$ is not a very well defined quantity and $\alpha$ is also somewhat uncertain because of the intrinsic scatter in the correlation. Finally, reverberation data does not exist for high luminosity/redshift quasars requiring an extrapolation of the $r_{\rm BLR}$ - $L_\lambda$ relation in order to estimate ${M}{_{\rm BH}}$ for these sources.

Uncertainties for ${M}{_{\rm BH}}$ derivations using single profile observation of H $\beta _{\rm BC}$ are estimated to be a factor of 2-3 (at a 1$\sigma$ confidence level), but may be as low as 30% if the velocity dispersion of the variable part of H $\beta _{\rm BC}$ profile is employed as a virial estimator and systematic effects are taken into account (Peterson et al. 2004). ${M}{_{\rm BH}}$ estimates based on the virial relation retain a statistical validity considering that AGNs span a 5 dex range in ${M}{_{\rm BH}}$ (10⁵$M_\odot$ $\la$ ${M}{_{\rm BH}}$  $\la 10^{10}$ $M_\odot$) and that the relation has now been applied to large samples of objects ($\sim$10³; McLure & Dunlop 2004).

With these considerations in mind, and supported by previous results, we use the virial relation to compute ${M}{_{\rm BH}}$ for $\approx$280 AGNs in our E1 sample of low redshift ( $0 \la z \la 1$) sources (Marziani et al. 2003b) supplemented with 25 intermediate redshift/high luminosity ( $1 \la z \la 2.5$) quasars. We derive ${M}{_{\rm BH}}$ and $L/L{_{\rm Edd}}$ values in several ways. New VLT-ISAAC data are presented for 9 sources (Sect. 2) which supplement data already published for sixteen quasars (Sulentic et al. 2004). Line measures are presented in Sect. 4. We use H $\beta _{\rm BC}$ and Fe II$\lambda$4570 line widths in a consistent way over the redshift range $0 \la z \la 2.5$ (a range of 10⁵ in luminosity). We compute black hole mass ${M}{_{\rm BH}}$, Eddington ratio $L/L{_{\rm Edd}}$ (Sect. 5) and we discuss how mass determinations might be improved (Sect. 5.2) so that the evolution of ${M}{_{\rm BH}}$ and $L/L{_{\rm Edd}}$ with redshift can be considered (Sects. 5.3 and 6).

   
2 Observations and data reduction

New intermediate redshift data were obtained between 10/2003-03/2004 in service mode with the infrared spectrometer ISAAC mounted on VLT1 (ANTU) at the European Southern Observatory. Each spectrum corresponds to a wavelength range (IR windows sZ, J, sH) that covers the region of redshifted H $\beta _{\rm BC}$ and Fe II$\lambda$4570 or Fe II$\lambda$5130 at least in part. Reduction of quasar spectra and standard stars followed exactly the same procedures described in Sulentic et al. (2004). Wavelength calibration yielded rms residuals of 0.4, 0.6 and 0.9 Å in the sZ, J and sH windows, respectively. Absolute flux scales of the spectra will be inaccurate because atmospheric seeing almost always exceeded the slit width ($\approx$0 $.\!\!^{\prime\prime}$6) resulting in significant light loss.

Table 1 summarizes the new observations and the basic format is given below the table. All sources come from the Hamburg-ESO (HE) quasar survey, which is a flux limited (with limiting $m_{\rm B } \approx 17.5$), color-selected survey (Wisotzki et al. 2000 ). Column 2 of Table 1 lists the blue apparent magnitudes from the HE survey papers (Wisotzki et al. 2000 ; Reimers et al. 1996) while Col. 3 lists the source redshift z computed as described in Sect. 3. The brightest sources of the HE at intermediate redshift were preferentially selected. Column 4 indicates whether [O III]$\lambda$5007 was used to compute z as indicative of the source rest frame. The absolute magnitude $M_{\rm B}$ reported in Col. 5 was computed by assuming H₀ = 70 km s^-1 Mpc^-1, and relative energy density $\Omega_{\rm M} = 0.3$ and $\Omega_{\Lambda} = 0.7$. The k correction was computed for a spectral index a = 0.6 ( $S \propto \nu^{-a}$). Column 6 gives the ratio of log specific fluxes at 6 cm and 4400 Å ( $\log R_ {\rm k}$). In most cases only NRAO VLA Sky Survey (NVSS) upper limits are available. Columns 7-12 give details of the observations explained in the footnotes. The continuum S/N estimate given in Col. 12 was measured using a small region of the spectrum that was as flat and free of line emission as possible.

   
3 Data analysis

3.1 Redshift determination and rest frame corrections

Small offsets are present in the wavelength calibration, because the arc lamp frames were obtained in daytime, and therefore usually after grism movement. A correction for these shifts was obtained by measuring the centroids of 2-3 OH sky lines against the arc calibration and calculating the average difference, which reached at most 6.5 Å or 2.5 pixels in either direction. Rest frame determination for the 9 new sources was usually estimated from the H$\beta$ peak redshift (assumed rest frame $\lambda = 4861.33$ Å). [O III]$\lambda$5007 (assumed rest frame $\lambda$ = 5006.85 Å) yielded a consistent measurement in only two sources (where the results were averaged. Two sources show no clear detection of [OIII] $\lambda\lambda$4959,5007 while the remaining five show a significant disagreement between [O III]$\lambda$5007 and H $\beta_{\rm NC}$. In these sources (we call the extreme examples of them "blue outliers'') it is not advisable to use [O III]$\lambda$5007 for redshift determination (Boroson 2005; Aoki et al. 2005; Marziani et al. 2003b; Zamanov et al. 2002). The adopted estimate was used to deredshift the spectra while the dopcor IRAF task applied a (1+z)³ correction to convert observed specific fluxes into rest frame values. Figure 1 shows the deredshifted spectra.

3.2 Continuum and Fe II subtraction

Our spectral analysis made use of standard IRAF tasks with the first step involving continuum modelling and subtraction. Using Chebyshev polynomials of 3${\rm rd}$ or 2${\rm nd}$ order, a reasonably smooth continuum subtraction was obtained for all sources. To estimate errors in the continuum assessment introduced by noise, we also defined a minimum and a maximum continuum. Continuum fluxes were chosen at about -3$\sigma$ (minimum) and +3$\sigma$ (maximum), where $\sigma$ is the noise standard deviation from the most likely continuum choice. Errors in continuum placement defined by difference between the extreme continua and the most probable one were then propagated according to standard error theory. The results of this procedure are consistent with continuum fits employing very simple models (Shang et al. 2005; Malkan & Sargent 1982). We assumed that the continuum underlying the H$\beta$ spectral regions is due to two components: either a blackbody of temperature 25 000 K or a power law of slope b = 0.7 ( $f_\nu \propto \nu^{-b}$; assumed to be valid only locally around H$\beta$). In 5 sources the sole black body component produces a good fit; in 2 the blackbody component is dominant, and only in the remaining 2 cases the power-law alone can provide a good fit. This method has some limitations due to the small spectral bandwidth covered by our spectra, to the relative strength of Fe II and to internal reddening effects. Since we did not attempt to change the blackbody temperature nor the slope of the power-law, we adopt the empirical continuum which is visually more accurate.

The emission blends of Fe II were subtracted using the template method based upon the spectrum of I Zw 1 (Marziani et al. 2003a; Boroson & Green 1992). The strongest Fe II blends fall in the wavelenghth ranges 4450-4600 Å (blue blend: Fe II$\lambda$4570) and 5200-5600 Å (red blend: Fe II$\lambda$5130). The method includes the following steps: (1) the template intensity is scaled to roughly match the observed spectrum, (2) a Gaussian broadening factor is estimated from FWHM H $\beta _{\rm BC}$  (3) a careful estimate of minimum and maximum plausible broadening factor is made to set a reliable ${\pm} 3\sigma$ uncertainty and (4) the template intensity is adjusted as necessary after broadening. The resultant template was subtracted from the continuum-subtracted spectra. The blue side of the spectrum including Fe II$\lambda$4570 is missing, or not fully covered, in several sources (HE 0946-0500, HE 1003+0149, HE 1017-0009, HE 1249-0648 and HE 1258-0823). In these cases the best template fit was achieved for the red blend and a fixed ratio between the red and blue blends was assumed to reproduce Fe II$\lambda$4570. Figure 2 shows the estimated Fe II emission (green lines). Fe II was subtracted before continuum fitting in sources with strong Fe II emission.

The [OIII] $\lambda\lambda$4959,5007 lines were measured after Fe II subtraction and taking into account the following considerations: (1) the flux ratio between [O III]$\lambda$5007 and [O III]$\lambda$5007$\lambda$4959 should be $\approx$3, (2) both lines should show identical profiles and (3) any H $\beta _{\rm BC}$ emission underlying [OIII] $\lambda\lambda$4959,5007 is expected to have a smooth shape. Sources HE 1249-0648 and HE 1258-0823 (see Fig. 2) show a small bump at $\lambda \approx 5016$ Å which is either an Fe II subtraction residual or emission from He I$\lambda$5016 (weak redshifted [O III]$\lambda$5007 would be almost unprecedented). After subtraction of H $\beta_{\rm NC}$ (following Marziani et al. 2003a), the H $\beta _{\rm BC}$ profile was fit with a high order spline function (IRAF task sfit). This procedure does not yield a model fit but only an empirical fit that smooths the noise and reproduces the main features and inflections in H $\beta _{\rm BC}$.

Table 2: Measurements of fluxes, equivalent widths and FWHM of strongest lines.

   
4 Immediate results

4.1 Line measurements and uncertainties

Line fluxes and equivalent widths

Table 2 gives line measurements for the new VLT spectra with the basic format given in the footnote. Columns 3 and 4 give equivalent width (EW) measures for H $\beta _{\rm BC}$ and Fe II$\lambda$4570 respectively. We evaluated uncertainties associated with the continuum level (derived from the minimum and maximum reasonable continuum estimates) and line flux errors estimated from the ${\it S/N}$. These estimates were combined quadratically to obtain uncertainties for EW measures. A similar procedure was applied to obtain uncertainty estimates for H $\beta_{\rm NC}$ and [OIII] $\lambda\lambda$4959,5007. The EW uncertainty for H $\beta_{\rm NC}$ was derived from the estimated maximum and minimum possible H $\beta_{\rm NC}$ component in the H$\beta$ line. The relative error of the H $\beta_{\rm NC}$ flux can be large (see Table 3) and in some cases an H $\beta_{\rm NC}$ component may not be present.

Table 3: Measurements of fluxes and equivalent widths of narrow lines.

Fe  II $_{\rm opt}$ width

Fe II emission is heavily blended so that widths of individual lines must be obtained from the best broadening parameter that was used for the template fit. This requires that we assume a constant width for all Fe II lines which so far appears to be reasonable. FWHM(Fe II$\lambda$4570) values derived from the template broadening factor are reported in Col. 5 of Table 2. Uncertainty estimates for FWHM(Fe II$\lambda$4570) were obtained by increasing/decreasing the broadening factor until we could detect significant changes in the best fit. Simulated data reveal that it is possible to estimate the Fe II$\lambda$4570 width up to FWHM(Fe II$\lambda$4570) $\approx$ 6000 km s^-1. Due to the very large uncertainty of Fe II$\lambda$4570 width determination for individual sources, ${M}{_{\rm BH}}$ estimates based on FWHM(Fe II$\lambda$4570) measures are used mainly for confirmatory purposes of statistical trends detected with H$\beta$.

Table 4: H $\beta _{\rm BC}$ line profile measurements.

Table 5: H $\beta _{\rm BC}$ line centroids at different fractional heights.

H $\beta _{\rm BC}$ line profiles

Measurements of FWHM(H$\beta$) together with other important line parameters like asymmetry index, kurtosis and line centroid at various fractional intensities were derived using a FORTRAN program developed for that purpose. These parameters are the same as defined in several previous papers (Marziani et al. 1996; Sulentic et al. 2004; Marziani et al. 2003a) and are given in Tables 4 and 5. Each line measure is followed in the next column by its appropriate uncertainty. Columns 2 and 4 of Table 4 give the Full Width at Zero Intensity (FWZI) and FWHM. Column 6 gives the asymmetry index (AI) as defined in Sulentic et al. (2004). Column 8 lists kurtosis values. Table 5 lists measures of the H $\beta _{\rm BC}$ centroid at various fractional intensities (in km s^-1). All uncertainties represent the $2 \sigma$ confidence level.

The dichotomy in H $\beta _{\rm BC}$ profile shape (and many other properties) between Population A and B (Sulentic et al. 2002) is seen in the new source measures and in the rest of our higher redshift sample (Sulentic et al. 2004). Redward asymmetries (A.I. $\ga$ 0.2) are most often found in Pop. B sources. They also show H $\beta _{\rm BC}$ profiles that are best fit with Gaussian functions, and some profile appear composite. A few sources appear to deviate from the trend found in previous work that FWHM of H $\beta _{\rm BC}$ and the Fe II lines are very similar (Marziani et al. 2003c,a). HE 1249-0648 and HE 1258-0823 show FWHM(H $\beta _{\rm BC}$) $\gg$ FWHM(Fe II$\lambda$4570). While FWHM(Fe II$\lambda$4570) is subject to large uncertainty, the difference is confirmed by careful reinspection of these spectra. This condition is seen in only 2/215 sources in the Marziani et al. (2003a) sample. Both (IRAS 07598+6508 and Mkn 235) are BAL QSOs which are also FIR ultra-luminous (Sulentic et al. 2006). In addition to a C IV$\lambda$1549 BAL with high terminal velocity, these objects have a strong, blueshifted C IV$\lambda$1549 emission line component. Blueshifted Balmer emission is probably associated with the high-ionization gas emitting C IV$\lambda$1549. The absence of any detectable [OIII] $\lambda\lambda$4959,5007 emission, along with a possible He I feature at $\lambda$5016 Å support the possibility that HE 1249-0648 and HE 1258-0823 could be BAL QSOs similar to IRAS 07598+6508 and Mkn 235. Further support comes from an inspection of the HE survey spectra of both objects which indeed show blueshifted broad absorption in the Mg II$\lambda$2800 line.

   
5 Black hole mass and Eddington ratio

5.1 Basic equations

One can write the velocity dispersion v in Eq. (1) as $v \approx \sqrt{3}/2$ FWHM(H $\beta _{\rm BC}$) in the case of randomly oriented velocities projected along the line-of-sight. The expression for black hole mass is then:

$$M_{\rm BH} \approx \frac{3}{4} \frac{r_{\rm BLR} {\it FWHM}{(\rm H\beta_{\rm BC})}^2}{G}\cdot$$ (3)

In the absence of reverberation data one must rely on the correlation between reverberation radius and source luminosity $r_{\rm BLR} \propto \left(L_{5100}\right)^{\alpha}$, where L₅₁₀₀ is the specific luminosity at $\lambda \approx 5100$ Å (Kaspi et al. 2005,2000). This formula relates BLR distance from the central continuum source and specific luminosity (erg s^-1 Å^-1) near 5100 Å. Following Kaspi et al. (2005) one can write:

$$r_{\rm BLR} \approx 0.697 \times 10^{17} \cdot \left[\frac{ ... ...\AA\right)}{10^{44} ~\rm erg ~ s^{-1}}\right]^{0.67} {\rm cm}.$$ (4)

We can make two estimates of ${M}{_{\rm BH}}$ using these relationships assuming two different values for $\lambda L_\lambda$. The first method derives $M_{\rm B}$ from tabulated values of V or B apparent magnitudes:

$$\lambda L_{\lambda}\left(5100~{\rm\AA}\right) \approx 3.14 \times 10^{35 - 0.4 (M_B)} ~{\rm erg~s^{-1}}.$$ (5)

A second possibility involves estimating the specific luminosity near 5100 Å directly from flux measures using the adopted continuum fits for our spectra. More explicitly $\lambda L_\lambda$ is computed as follows:

$$\lambda L_{\lambda}\left(5100 ~{\rm\AA}\right) = 4 \pi d_{\rm C}^2 \lambda f_\lambda (5100~ \rm\AA)~ \rm erg~s^{-1},$$ (6)

where $d_{\rm C}$ is the transverse comoving distance and $f_\lambda$ is the specific flux in the rest frame at 5100 Å (after correction for Galactic extinction $A_{\rm B}$ at the observed wavelength). The second choice has the advantage that the continuum $f_\lambda$ and FWHM(H $\beta _{\rm BC}$) are measured from the same spectrum. Expected light losses are estimated to be $\approx$35% of the quasar flux with average Paranal seeing (0 $.\!\!^{\prime\prime}$6). We apply this average correction to the observed flux before computing ${M}{_{\rm BH}}$.

Substituting the expression for $r_{\rm BLR}$ in the mass formula one obtains the following relation:

$$M_{\rm BH} \approx 5.48 \times 10^6 \left[\frac{\lambda L_{\... ...c{{\it FWHM}\rm (H\beta_{BC})}{1000 ~\rm km ~ s^{-1}}\right)^2$$ (7)

where ${M}{_{\rm BH}}$ is in solar mass units ($M_\odot$).

	Figure 3: Distribution of ${M}{_{\rm BH}}$ estimates as a function of z using uncorrected FWHM(H $\beta _{\rm BC}$) values. Intermediate z sources observed with ISAAC are at $z \protect\ga 0.8$. The solid curve estimates the minimum detectable ${M}{_{\rm BH}}$ for our sample - see text.
Open with DEXTER

Figure 3 shows the distribution of ${M}{_{\rm BH}}$ as a function of z. ${M}{_{\rm BH}}$ estimates are based on the FWHM(H $\beta _{\rm BC}$) measures reported in Table 4. All sources with $z \ga 0.8$ are from the VLT sample. We do not show ${M}{_{\rm BH}}$ values derived from $M_{\rm B}$ as in Marziani et al. (2003b) for clarity, since they basically confirm the same trends obtained from the specific fluxes.

Figure 4: FWHM measured from the variable part of the H $\beta _{\rm BC}$ profile for low-z sources with reverberation data (Peterson et al. 2004) versus FWHM(H $\beta _{\rm BC}$) (km s-1) measures from Marziani et al. (2003c). Solid straight lines show best fits for FWHM(H $\beta _{\rm BC}$) $\le$ 4000 km s-1 (Pop. A) and for sources with broader lines (Pop. B). The solid curve shows a weighted least-square fit of all data with a second order polynomial. The diagonal dot-dashed line indicates the location of equal ordinate and abscissa values.

Open with DEXTER

   
5.2 Improved ${M}{_{\rm BH}}$ estimators?

As briefly summarized in Sect. 1, there is some evidence suggesting that the virial assumption is reasonable for LILs in a significant fraction of quasars (Population A; about 50-60% of low z quasars). Our VLT spectra confirm that any dependence of FWHM(H $\beta _{\rm BC}$) on source luminosity is weak (Sulentic et al. 2004). Tables 4 and 5 suggest that low redshift trends for asymmetries and line shifts are preserved in the intermediate z sample. It is therefore not certain that FWHM(H $\beta _{\rm BC}$) is a valid estimator of the virial velocity for all sources even after proper H $\beta_{\rm NC}$, [OIII] $\lambda\lambda$4959,5007, Fe II subtraction.

A more physical approach to ${M}{_{\rm BH}}$ estimation uses the FWHM of the variable part of the H $\beta _{\rm BC}$ profile (Peterson et al. 2004). Figure 4 plots FWHM $_{{\rm rms}}$ values from Peterson et al. (2004) versus our FWHM(H $\beta _{\rm BC}$) measures (Marziani et al. 2003a) for all sources in common. Sources with FWHM $\la$ 4000 km s^-1 show a correlation while the situation is less clear for sources with ${\it FWHM}\ga 4000$ km s^-1. We recall that this value indicates the nominal population A-B boundary (Sulentic et al. 2000b) that emerged in our E1 studies. A least-square best fit analysis yields a "corrected'' FWHM(H $\beta _{\rm BC}$) estimate: FWHM $^{{\rm corr}}$(H $\beta _{\rm BC}$)  $\approx -710 (\pm 800) + 1.13 (\pm 0.28) ~$ FWHM(H $\beta _{\rm BC}$) for ${\it FWHM} \la 4000$ km s^-1 and FWHM $^{{\rm corr}}$(H $\beta _{\rm BC}$)  $\approx 650(\pm 1000) + 0.79(\pm 0.14) ~$FWHM(H $\beta _{\rm BC}$) for ${\it FWHM}\ga 4000$ km s^-1. The slope depends somewhat on the fitting method (a robust fit yields 0.94 and 0.72) but is always less steep for ${\it FWHM}\ga 4000$ km s^-1. An immediate implication is that optically thick BLR gas responding to continuum changes shows a velocity dispersion correlated with - but slightly lower than that of the integrated profile at all FWHM.

A break in the linear fit at 4000 km s^-1 is consistent with several previous findings: (a) mean and possibly systematic H $\beta _{\rm BC}$ profile differences between Pop. A and B sources (Sulentic et al. 2002); (b) the lack of strong profile changes in Pop. A sources (Giveon et al. 1999); (c) profile asymmetries frequently observed in Pop. B sources. They might represent a distinct redshifted emission component (Sulentic et al. 2002) which may arise in less optically thick gas than the rest of the H $\beta _{\rm BC}$ profile (Sulentic et al. 2000c). If the redshifted component is emitted in an innermost VBLR, the FWHM of the whole H $\beta _{\rm BC}$ profile is obviously increased over the value due to the line component that is actually responding, and that is most likely located farther away from the central continuum source. Even if the virial assumption holds for the VBLR (but the frequent asymmetries warn us that this might not be the case), the use of the FWHM from the whole H $\beta _{\rm BC}$ profile and of $r_{\rm BLR}$ from the reverberating part, yields an ${M}{_{\rm BH}}$ overestimate. This interpretation of the correlations in Fig. 4 is advanced with caution because of the small sample size and especially because of the poor statistics for sources with FWHM $\ga$ 4000 km s^-1. It is also possible to produce a meaningful fit with a second-order polynomial. We apply a tentative correction to the FWHM measures FWHM $^{{\rm corr}}$(H $\beta _{\rm BC}$) and therefore to resultant ${M}{_{\rm BH}}$ estimates using the second-order fit shown in Fig. 4 which approximates very well the linear trends.

If an optically thin/nonvirialized component is present in the H $\beta _{\rm BC}$ profile of many sources then we might use as a virial estimator the FWHM of a line, or lines, arising in BLR gas but not likely to be present in the VBLR region. FWHM(Fe II$\lambda$4570) is an obvious alternative because there is no evidence for a VBLR emission component in the broad Fe II blends. It is not strictly correct to use it because the $r_{\rm BLR}$ - $L_\lambda$ relation was deduced for H$\beta$. The most serious difficulty lies in obtaining a reliable FWHM(Fe II$\lambda$4570) estimate from the heavily blended Fe II emission. Considering the EW and FWHM limits for detection of Fe  II $_{\rm opt}$ emission (Marziani et al. 2003a) we conclude that a reasonable FWHM measurement is possible for $\approx$120 sources in our low z spectral atlas. Measurement uncertainties for FWHM(Fe II) will be larger than for FWHM(H $\beta _{\rm BC}$) and are estimated to lie between 20-50%. The best fit of FWHM(Fe II$\lambda$4570) vs. FWHM(H $\beta _{\rm BC}$) (km s^-1) is consistent with FWHM(Fe II$\lambda$4570) $\approx$ FWHM(H $\beta _{\rm BC}$) for FWHM(H $\beta _{\rm BC}$) $\la$ 4000 km s^-1, while it is FWHM(Fe II$\lambda$4570) $\approx$ 0.67 $\cdot$ FWHM(H $\beta _{\rm BC}$) + 820 km s^-1 if FWHM(H $\beta _{\rm BC}$) $\ga$4000 km s^-1.

The relationship between FWHM(H $\beta _{\rm BC}$) and FWHM (Fe II) confirms that individual Fe II lines show approximately the same width as H $\beta _{\rm BC}$ and as the rms H $\beta _{\rm BC}$ component implying a common kinematic environment if FWHM(H $\beta _{\rm BC}$) $\la$ 4000 km s^-1. If FWHM(H $\beta _{\rm BC}$) $\ga$4000 km s^-1, FWHM(Fe II) follows a trend closer to that of the rms H $\beta _{\rm BC}$ component. Therefore if the rms H $\beta _{\rm BC}$ component arises from gas in virialized motion, the same can be reasonably assumed for the whole Fe II$\lambda$4570 emission.

	Figure 5: ${M}{_{\rm BH}}$ computed from corrected FWHM $^{{\rm corr}}$(H $\beta _{\rm BC}$) as a function of z (filled circles). ${M}{_{\rm BH}}$ estimates computed from FWHM(Fe II$\lambda$4570) measures are shown as open symbols. Symbol types and solid curve are as described for Fig. 3.
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5.3 ${M}{_{\rm BH}}$ and $L/L{_{\rm Edd}}$ dependence on redshift

5.3.1 ${M}{_{\rm BH}}$

Figure 3 shows the distribution with z of ${M}{_{\rm BH}}$ estimates derived using uncorrected FWHM(H $\beta _{\rm BC}$) measures. This can be compared with Fig. 5 where we show corresponding distributions of ${M}{_{\rm BH}}$ derived using corrected FWHM $^{{\rm corr}}$(H $\beta _{\rm BC}$) measures and using FWHM(Fe II$\lambda$4570). FWHM $^{{\rm corr}}$(H $\beta _{\rm BC}$) values were derived from the second-order relation in Fig. 4.

Figure 6: ${M}{_{\rm BH}}$ computations using same samples as Fig. 3, where RQ and RL sources are indicated by filled and open symbols respectively ( left panel). Triangles indicate sources for which radio data are insufficient to compute a meaningful $R{\rm _K}$. The right panel distinguishes ${M}{_{\rm BH}}$ for Pop. A (filled symbols) and Pop. B (open symbols). The solid curve is as described for Fig. 3.

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The distribution of data points in the $\log$ ${M}{_{\rm BH}}$ vs. z plane reflects selection effects intrinsic to any flux limited sample. If we consider AGNs that are radiating at a given $L/L{_{\rm Edd}}$ we easily obtain ${M}{_{\rm BH}}$ for a given apparent magnitude:

  $$\begin{eqnarray*}{M_{{{\rm BH}},9\odot}} = & & {3.85 \times 10^6 h^{-2}} {\left(... ...frac{z}{1.266}} \right) \right]^2 \cdot (1 + z)^{(1 - {\rm a})}. \end{eqnarray*}$$

Appendix B details how Eq. (8) was derived. If we adopt a limiting magnitude $m_{\rm B } \approx 17.5$ (appropriate for the Hamburg-ESO survey) and consider a maximum Eddington ratio $L/L{_{\rm Edd}}$ $\approx$ 1 we obtain the minimum ${M}{_{\rm BH}}$ detectable as a function of z. Limiting ${M}{_{\rm BH}}$ curves are shown in Fig. 3, Fig. 5, and Fig. 6.

All sources show ${M}{_{\rm BH}}$ $\ga 10^6$ $M_\odot$. In the range $1 \la z \la 2.5$ we begin to find a significant black hole population in the range $\sim$10^9.5 $M_\odot$ $\la$ ${M}{_{\rm BH}}$  $\la 10^{10}$ $M_\odot$. If we consider ${M}{_{\rm BH}}$ values derived from Fe II$\lambda$4570 as well as from FWHM $^{{\rm corr}}$(H $\beta _{\rm BC}$) (Fig. 5, open and filled symbols respectively) we get a somewhat different picture. Almost all sources lie below ${\approx} 5 \times 10^{9}$ $M_\odot$ (all sources with $z \la 1$); only three sources whose ${M}{_{\rm BH}}$ has been computed with from FWHM $^{{\rm corr}}$(H $\beta _{\rm BC}$) lie significantly above this limit. Our sample of intermediate z quasars with compatible quality data is still small so caution is needed. It should be pointed out that the use of H $\beta _{\rm BC}$ $^{{\rm corr}}$ is self-consistent, since no H$\beta$ surrogate line was used.

The upper panel of Fig. 6 identifies sources on the basis of radio-loudness where open and solid symbols denote radio-loud (RL; see definition in Sulentic et al. 2003) and radio-quiet (RQ) sources respectively. At low z RQ sources are distributed across the full range of ${M}{_{\rm BH}}$ up to ${M}{_{\rm BH}}$ $\ga 10^9$$M_\odot$, although RL sources tend to have larger ${M}{_{\rm BH}}$ (Metcalf & Magliocchetti 2006; Marziani et al. 2003b, and references therein). The same consideration apply to Pop. A and Pop. B sources (lower panel of Fig. 6), and this is not surprising since most Pop. A sources are RQ. At intermediate z, selection effects limit the detectable black holes to ${M}{_{\rm BH}}$ $\ga 10^9$ $M_\odot$, so that only the high end of the quasar ${M}{_{\rm BH}}$ distribution can be traced. It is too early to decide whether systematic differences in the ${M}{_{\rm BH}}$ distribution of RQ and RL sources may still exist at intermediate z, at least from the present data.

5.3.2 $L/L{_{\rm Edd}}$

Figure 7 shows the distribution of $L/L{_{\rm Edd}}$ estimates as a function of z. A preliminary analysis in terms of $L/L{_{\rm Edd}}$, and including discussion of bolometric luminosity estimation, was presented in Marziani et al. (2003b).

The existence of apparently super-Eddington radiators in the low-z part of Fig. 7 involves the most extreme NLSy1 sources. One caveat about interpreting any of them as super-Eddington involves the significant uncertainties associated with these estimates. If we assume a typical uncertainty of $\pm$50% for the bolometric correction (neglecting beaming or lensing) and $\pm$10% for the virial estimator (basically the uncertainty in the FWHM measurement) we find a $\Delta \log ~$ $L/L{_{\rm Edd}}$ $\approx$ 0.13 (neglecting the scatter in Eq. (2)). A serious source of uncertainty ignored until now involves source orientation with respect to the line-of-sight. The virial velocity dispersion that H $\beta _{\rm BC}$ is assumed to measure is now widely assumed to involve Keplerian rotation in an accretion disk or in a flattened distribution near the disk. If our candidate super-Eddington sources are mostly viewed with (face-on disk and/or pole-on jet) then ${M}{_{\rm BH}}$ can be significantly underestimated. An orientation correction applied to extreme NLSy1 sources (we call them Pop. A "blue outliers'') interpreted as face-on, will move them below $L/L{_{\rm Edd}}$ $\approx$ 1 (Marziani et al. 2003b, see arrows in their Fig. 12). At high z we so far find no strong evidence for any super-Eddington sources. The thick solid line in Fig. 7 shows the expected $L/L{_{\rm Edd}}$ detection limit deduced from Eq. (8) for a flux-limited survey ( $m_{\rm B } \approx 17.5$) and a quasar with ${M}{_{\rm BH}}$  $\approx 4 \times 10^9$ $M_\odot$. All quasars in our sample fall above the minimum detectable $L/L{_{\rm Edd}}$, suggesting that masses larger than $\approx 4 \times 10^9$ $M_\odot$ are not strictly necessary from our data (see Sect. 6.1 for further discussion).

A physical basis for the dichotomy between Pop. A and Pop. B (Marziani et al. 2001; Sulentic et al. 2000b; Marziani et al. 2003b) is supported by this analysis in the sense that Fig. 4 shows evidence for a change at about FWHM(H $\beta _{\rm BC}$) $\approx$ 4000 km s^-1. Figure 7 identifies Pop. A and B sources as filled and open symbols respectively. Here we apply the luminosity-dependent definition of the Pop. A-B boundary as defined in Fig. 6 of Sulentic et al. (2004). It shows that the Eddington ratio of Pop. A sources is systematically larger than that of Pop. B, and that the apparent boundary between the two populations may increase with redshift. If we focus on the intermediate z quasars then Pop. A sources show an average $L/L{_{\rm Edd}}$ $\approx$ 0.78 compared to 0.27 for 10 Pop. B sources. Even these small samples of Pop. A and B sources show significantly different $L/L{_{\rm Edd}}$ distributions according to a Kolmogorov-Smirnov (K-S) test. RL AGNs are systematically low $L/L{_{\rm Edd}}$ radiators since they are almost entirely Pop. B sources.

It is interesting to note that with: (1) a limiting magnitude $m_{\rm B } \approx 16.5$, (2) $L/L{_{\rm Edd}}$ $\la$ 1 and (3) ${M}{_{\rm BH}}$  $\la 4 \times 10^9$ $M_\odot$ we should detect fewer sources beyond $z \approx 2$ (no source below the dot-dashed curve of Fig. 7). If our assumptions about the quasar bolometric correction are valid up to that redshift, selection effects may influence the relative frequency of Pop. A and B sources (the two Populations have different $L/L{_{\rm Edd}}$ distributions) rather than the intrinsic properties of LILs. In this case selection effects on $L/L{_{\rm Edd}}$ should strongly influence the so-called "Baldwin effect'' involving C IV$\lambda$1549 and other HILs (Bachev et al. 2004, and references therein) because C IV$\lambda$1549 is more prominent in Pop. B, and Pop. B sources are more easily lost at high z.

Figure 7: Distribution of $L/L{_{\rm Edd}}$ with z following Fig. 3 with $\lambda L_\lambda$ values derived from the 5100 Å rest-frame flux and with $\alpha = 0.67$. E1 Populations A and B are indicated by filled and open symbols respectively. The solid line traces the minimum $L/L{_{\rm Edd}}$ for a fixed mass value of $4 \times 10^{9}$ $M_\odot$ to observe a quasar above the limiting magnitude of the HE quasar survey $m_{\rm B } \approx 17.5$. The dot-dashed lines indicate the same limit but assuming limiting magnitude $m_{\rm B } \approx 16.5$.

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6 Discussion

The present paper provides ${M}{_{\rm BH}}$ estimates that have three advantages: (1) a consistent data analysis procedure is employed over the entire redshift range $0.0 \la z \la 2.5$ by using the same ${M}{_{\rm BH}}$ tracer, H $\beta _{\rm BC}$; (2) ${\it S/N}$ and resolution of the spectroscopic data sample are high enough to permit a careful study of the H $\beta _{\rm BC}$ profile, and (3) the data quality allows reasonable estimates of FWHM Fe II in many of the sources.

   
6.1 What are the largest black hole masses?

Netzer (2003) discussed several problems with ${M}{_{\rm BH}}$  $\ga 10^{10}$ $M_\odot$. If the black hole mass vs. bulge mass ( ${M}{_{\rm bulge}}$) relation (Ferrarese & Merritt 2000) is valid at high z then ${M}{_{\rm BH}}$  $\sim 10^{10}$ $M_\odot$ would imply stellar velocity dispersion $\sigma_\star \approx 700$ km s^-1 (following Gebhardt et al. 2000) and resultant bulge masses ${M}{_{\rm bulge}}$ $\ga$ 10¹³ $M_\odot$ which are not observed at low-z (Netzer 2003; Wang 2003; McLure & Jarvis 2004). Recent results for the fundamental plane of elliptical galaxies, and the most massive spheroids at $z \la 0.3$ from Sloan Digital Sky Survey measures, confirm that $\sigma_\star \la 500$ km s^-1 (SDSS, Bernardi et al. 2005,2003, all galaxies with $\sigma \ga 500$ km s^-1 are likely due to chance superposition).

There are several proposed interpretations of this problem: (1) the ${M}{_{\rm BH}}$ - ${M}{_{\rm bulge}}$ relationship may not strictly hold for all hosts, (2) the virial assumption is not applicable, (3) results are plagued by such large uncertainties - including the one of the luminosity index $\alpha$ - that very large mass estimates are not real (Vestergaard 2002), and (4) some systematic effects may not have been considered.

6.1.1 The ${M}{_{\rm BH}}$ - ${M}{_{\rm bulge}}$ relationship

Nuclei with ${M}{_{\rm BH}}$ $\ga$ $5 \times 10^9$ $M_\odot$ are not observed in galaxies of the local Universe if a direct black hole mass determination is possible from circum-nuclear kinematics (Marconi & Hunt 2003). They are expected to be rare and difficult to find, considering also that they should be in a dormant or nearly dormant stage at the present epoch. Integrating the quasar luminosity function at $z \approx 1.5$ (Boyle et al. 2000), we find a comoving density of quasars above the HE limiting magnitude (which corresponds to $M_{\rm B}$  $\approx -27.1$ for a k-correction $a = 0.6) \sim 3 \times 10^{-8}$ Mpc^-3. This indicates that the present-day density of the most massive black holes that were once luminous quasars should be very low, ${\sim} 2\times 10^{-9}$ Mpc^-3. Even so, they should be much more frequent than the very massive spheroids that would host them if the ${M}{_{\rm BH}}$ - ${M}{_{\rm bulge}}$ relationship is valid. To estimate the density of spheroids with $\sigma_\star \ga 500$ km s^-1, one must extrapolate the $\sigma$ distribution function provided by Sheth et al. (2003) to unobserved domains in $\sigma$. Integrating the Sheth et al. (2003) function for $\sigma_\star \ga 500$km s^-1, one obtains that the comoving density of all local spheroids is three orders of magnitude lower ( ${\sim} 6 \times 10^{-13}$ Mpc^-3) than that of the most massive black holes. A possible implication is that the ${M}{_{\rm BH}}$ - ${M}{_{\rm bulge}}$ relation is not linear or of universal validity, i.e. some galaxies host larger black hole masses than expected. This conclusion should remain valid also at intermediate redshifts. A constant ${M}{_{\rm BH}}$/ ${M}{_{\rm bulge}}$ ratio cannot hold forever if bulge mass grows by secular evolutionary processes. Evolution goes in the sense that the spheroids may increase their masses at a later cosmic age ($z \la2$; Peng et al. 2006; Treu et al. 2004) significantly more than their central black hole: the most massive black hole should be already "brightly shining'' at $z \approx 2$ (McLure et al. 2006, and references therein). A nonlinear ${M}{_{\rm BH}}$ - ${M}{_{\rm bulge}}$ relation has been proposed (Laor 2001) for local hosts, with ${M}{_{\rm BH}}$ accounting for only 0.05% of the bulge mass in low-luminosity galaxies and 0.5% in giant ellipticals. In this case, ${M}{_{\rm BH}}$  $\sim 10^{10}$ $M_\odot$ would imply ${M}{_{\rm bulge}}$  $\sim 10^{12}$ $M_\odot$, which is at the upper end of the spheroid masses measured locally (Marconi & Hunt 2003).

The problem can be made to disappear if ${M}{_{\rm BH}}$  $\la 3 \times 10^9$ $M_\odot$. Bernardi et al. (2005) find a maximum $\sigma_\star \approx 500$ km s^-1. These local galaxies (with expected ${M}{_{\rm BH}}$ $\approx$ $3 \times 10^9$ $M_\odot$) may have been the hosts of the luminous quasars at $1 \la z \la 2$ since the number density of spheroids with $\sigma_\star \ga 350$ km s^-1 is ${\sim} 4 \times 10^{-7}$ Mpc^-3, still somewhat larger than the present-day density of supermassive black holes that were radiating at $M_{\rm B}$ $\la -25$ ( ${\sim} 10^{-7}$ Mpc^-3).

The curves in Fig. 7, where ${M}{_{\rm BH}}$  $\sim 4 \times 10^9$$M_\odot$ has been assumed, suggest that there is no need for ${M}{_{\rm BH}}$  $\ga 10^{10}$ $M_\odot$ at $z \la 2.5$. Similarly the brightest HE sources in the redshift range $1 \la z \la 2$ ( $m_{\rm B} \approx 15.0$ at $z \approx 1.7$) are consistent with $L/L{_{\rm Edd}}$ $\approx$ 1 if ${M}{_{\rm BH}}$  $\sim 4 \times 10^9$ $M_\odot$. In other words, if ${M}{_{\rm BH}}$  $\sim 10^{10}$ $M_\odot$, then a source at $L/L{_{\rm Edd}}$ $\approx$ 1 would be brighter than the brightest quasars observed in the range $1 \la z \la 2$ (if H₀ is close to the value assumed in this paper).

6.1.2 Could a non-virial component yield a huge ${M}{_{\rm BH}}$?

One might envision a non-virial component that increases with source luminosity and systematically broadens the H $\beta _{\rm BC}$ profile. In our so-called Pop. A sources, the high ionization wind that can dominate C IV$\lambda$1549 emission might reasonably be expected to produce Balmer line emission as well. Such an additional (blueshifted) component on the H $\beta _{\rm BC}$ profile would increase FWHM measures (Sulentic et al. 2006, HE 1249-0648 and HE 1258-0823 may show this effect). One intriguing possibility is that high-luminosity sources are extreme Pop. A sources which is supported by models invoking a radiation pressure driven wind. Such an effect would make the correction ($\sim$500 km s^-1) deduced for low-z Pop. A sources inadequate. At the other extreme, strong redward asymmetries observed in Pop. B sources indicate that the integrated profile may be affected by gravitational redshift and non-virial motions (Marziani et al. 2003b). A quantitative analysis of these suggestions needs careful analysis beyond the scope of this paper, but in both cases any correction would lower ${M}{_{\rm BH}}$.

6.1.3 Statistical errors

It is interesting to consider in more detail the sources that show ${M}{_{\rm BH}}$ $\sim$ 10¹⁰ $M_\odot$ in our sample. HE 0248-3628 with the largest estimated ${M}{_{\rm BH}}$ is also one of the highest $L/L{_{\rm Edd}}$ radiators with $L/L{_{\rm Edd}}$ $\sim$ 1. It shows an anomalously large (by a factor $\sim$10) flux with respect to our other VLT sources which makes it an extremely luminous quasar ($M_{\rm B}$ $\la -30$; see Sulentic et al. 2004). It falls near the Pop. A-B boundary in Eigenvector 1 space and is a borderline radio-loud source by our definition (Sulentic et al. 2003). It disappears as an outlier if the ${M}{_{\rm BH}}$ estimated from the tabulated $M_{\rm B}$ is adopted. HE 2355-4621 is a second source with ${M}{_{\rm BH}}$  $\sim 10^{10}$ $M_\odot$. It behaves like a normal radio-quiet Pop. B source. The H $\beta _{\rm BC}$ profile shows a prominent redward asymmetry, which is strongly affecting the width at half maximum. It is also interesting to consider HE 1104-1805, which is the most luminous source in our new sample with ($M_{\rm B}$ $\approx$ -29.5). It behaves like an ordinary Pop. A source and shows ${M}{_{\rm BH}}$ $\approx$ $5 \times 10^9$ $M_\odot$ which may be overestimated if the continuum is lens brightened (Appendix A).

The high-mass wing of the ${M}{_{\rm BH}}$ distribution in the redshift range $1 \la z \la 2$ is consistent with the wing of a Gaussian peaked at $\approx$ $3 \times 10^9$ $M_\odot$ and dispersion $\Delta \log$ ${M}{_{\rm BH}}$ $\approx$ 0.3 if ${M}{_{\rm BH}}$ is computed from FWHM $^{{\rm corr}}$(H $\beta _{\rm BC}$) ( $4 \times 10^{9}$ $M_\odot$ if no correction to H $\beta _{\rm BC}$ is applied). The Gaussian dispersion is consistent with the estimated errors of individual measurements. K-S tests do not favor significantly different peak masses or a much different dispersion. This suggests that sources with ${M}{_{\rm BH}}$  $\ga 3 \times 10^9$ $M_\odot$ of our data might be mostly due to random errors associated to the uncertainty in individual ${M}{_{\rm BH}}$ measurements.

6.1.4 Systematic effects

A first systematic effect considers the uncertainty in the index $\alpha$. Our ${M}{_{\rm BH}}$ estimates show the onset of ${M}{_{\rm BH}}$  $\ga 3 \times 10^9$ $M_\odot$ at $z \sim0.8$; at $z \ga 0.8$ we observe some of the most luminous HE quasars. The high luminosity range of the Kaspi et al. (2005) relationship remains poorly sampled: there are just 2-3 sources in the range $45 \la \log \lambda L_\lambda \la 46$ (to which most of our intermediate z sources belong), creating a sample bias and making the correlation analysis intrinsically unstable. In addition, a recent reanalysis of the $r_{\rm BLR}$ - $\lambda L_\lambda$ correlation suggests a value of $\alpha$ as low as 0.5 (Vestergaard & Peterson 2006). If $\alpha$ is overestimated by $\Delta\alpha \approx 0.17$ over a luminosity range of $\sim$10, ${M}{_{\rm BH}}$ may be overestimated by $\Delta \log$ ${M}{_{\rm BH}}$ $\approx$ 0.2 in our intermediate z sources.

Pop. A sources show good evidence that LILs are emitted in a strongly flattened system, probably an accretion disk or gas co-planar with the disk. C IV$\lambda$1549 in these sources seems to be dominated by a wind component (Bachev et al. 2004). If these considerations apply also to Pop. B sources (but it is by no means clear, given the large FWHM(H $\beta _{\rm BC}$) of sources believed to be observed pole-on) and if the maximum angle between disk axis and line-of-sight is ${\approx} 45^\circ$, a correction could imply a factor $\la$2 systematic increase in  ${M}{_{\rm BH}}$. However, these consideration of orientation effects may not even reopen the problem of very large ${M}{_{\rm BH}}$ values: if our suggestion of a maximum ${M}{_{\rm BH}}$ $\approx$ $3 \times 10^9$ $M_\odot$ is appropriate, taking into account the systematic orientation effects would yield a maximum ${M}{_{\rm BH}}$ $\approx$ $6 \times 10^9$ $M_\odot$ which is still plausible. The result that $L/L{_{\rm Edd}}$ $\la$ 1 at $z \ga 0.8$ would be reinforced by systematic orientation effects.

Summing up, our data suggest that very large masses $\sim 10^{10}$ $M_\odot$ may be not be real, and may be predominantly due to statistical errors and emission line profile broadening that is in part non-virial. The data presented in this paper are consistent with ${M}{_{\rm BH}}$ not exceeding $3 \times 10^9$ $M_\odot$ for our sources if the correction to FWHM(H $\beta _{\rm BC}$) described earlier is applied.

6.2 The best ${M}{_{\rm BH}}$ estimators

LILs like Mg II$\lambda$2800 and Fe II may yield more reliable results than H $\beta _{\rm BC}$. H I Balmer line emission can be substantial from gas in a variety of physical conditions. In Pop. B sources, a very broad component may increase the FWHM of the integrated H $\beta _{\rm BC}$ profile (also mimicked by low S/N data; Shemmer et al. 2004; McIntosh et al. 1999). This very broad component may be optically thin to the ionizing continuum, and therefore non-responsive to continuum changes. The H $\beta _{\rm BC}$ profile of Pop. A sources may be affected by a high-ionization component mentioned earlier. Fe II is thought to be emitted in a region very optically thick to Lyman continuum which is probably photoionized (Vestergaard & Peterson 2005; Wang et al. 2005), as the part of H $\beta _{\rm BC}$ responding to continuum changes should be. It is not surprising that the reverberating part of H $\beta _{\rm BC}$ and Fe II provides width estimates which are consistent, since they are expected to measure the width of a similar sub-region within the BLR. Similar considerations apply to Mg II$\lambda$2800 (Wills et al. 1985) since Mg II$\lambda$2800 should be mainly emitted in the same zone as Fe II. McLure & Dunlop (2004) present virial ${M}{_{\rm BH}}$ estimates for $\approx$13 000 quasars in the redshift interval $0.1 \la z \la 2.1$ based on spectra from the SDSS first data release. The mean ${M}{_{\rm BH}}$ increases with increasing redshift basically as shown in Fig. 3. The mass values found by them are also consistent with a limiting ${M}{_{\rm BH}}$ around $3 \times 10^9$ $M_\odot$, with large scatter. They use FWHM(H $\beta _{\rm BC}$) or FWHM(Mg II$\lambda$2800) and find a consistency between the most massive at $z \approx 2$ and those at $z \approx 0$. Given measurement difficulties, and doubts about the virial assumption for most other lines, corrected measures for H $\beta _{\rm BC}$, Fe II, and Mg II$\lambda$2800 may offer the best hope for reliable ${M}{_{\rm BH}}$ and $L/L{_{\rm Edd}}$ estimates out to $z \approx$ 2.5.

7 Conclusions

Nine intermediate z VLT/ISAAC spectra with high resolution and S/N supplement an earlier sample of 17 sources. Emission line measurements on H $\beta _{\rm BC}$ and Fe II presented in this paper strengthen the conclusion of Sulentic et al. (2004) that luminosity effects are weak or absent in the low-ionization lines of AGNs. Results on the H $\beta _{\rm BC}$ profile are consistent with the population A-B hypothesis and Eigenvector 1 parameter space concept developed for low z AGNs. We computed virial masses and Eddington ratios for the 25 intermediate-z objects plus about 280 lower z sources, using the same emission line over the entire redshift range for the first time.

We also have how the distributions of ${M}{_{\rm BH}}$ and $L/L{_{\rm Edd}}$ vs. z are shaped by selection effects intrinsic to any flux-limited survey at the low ${M}{_{\rm BH}}$ end. At the high ${M}{_{\rm BH}}$ end, masses exceeding a few 10⁹ $M_\odot$ may be rare if corrections for non-virial broadening and statistical errors are taken into account. This suggestion is based on just 25 objects distributed over the entire redshift range $0.9 \la z \la 2.5$. Confirmation from a larger sample of intermediate-z observations is needed.

Acknowledgements

We thank a referee for many useful suggestions and the tenacity to ensure that they were taken into account. We also thank Lutz Wisotzki for providing us with the HE optical spectra. D.D.-H., and J.S. acknowledge financial support from grant IN100703 from PAPIIT, DGAPA, UNAM.

   
Appendix A: Notes on individual sources

A.1 HE 0512-3329

HE 0512-3329 was discovered as a probable gravitationally lensed quasar with the Space Telescope Imaging Spectrograph (STIS). It is a doubly imaged QSO with a source redshift of z = 1.58 and an image separation of 0 $.\!\!^{\prime\prime}$644. The flux ratio $\frac{A}{B}$ of the lensed images shows a strong dependence on wavelength. In the R and I bands, A is brighter than B by about 0.45 mag while the two are almost equal in the band B. For smaller wavelengths, especially close to the limit near 2000 Å, B becomes much brighter than A by about 1.3 mag. A natural explanation for this effect is differential reddening caused by different extinction effects in the two lines of sight (Gregg et al. 2000). Microlensing by stars and other compact object in the lensing galaxy also plays a role in this source (Wucknitz et al. 2003). Unfortunately, the small separation does not allow to us to distinguish component A and B on the ISAAC spectrum. The HE 0512-3329 acquisition image is compatible with an unresolved source. Considering the flux ratios in the R and I band, it is reasonable to assume that our spectrum is dominated by the A component.

A.2 HE 1104-1805

HE 1104-1805 is a double-image lensed quasar discovered by Wisotzki et al. (1993). The image separation is $\Delta \theta = 3\hbox{$.\!\!^{\prime\prime}$ }19$, the source redshift is $z_{\rm s} = 2.319$, and the lens redshift is $z_{\rm l} = 0.729$. Wisotzki et al. (1995 ) reported that the continuum flux in both images is highly variable but that the line fluxes do not change, as expected if microlensing is operating. On the acquisition image of HE 1104 there is a second source at 3.5'' but it is completely off-slit.

   
Appendix B: Minimum ${M}{_{\rm BH}}$ and $L/L{_{\rm Edd}}$ as a function of z

The absolute B magnitude $M_{\rm B}$ can be related to the specific luminosity at 5100 Å assuming an average spectral shape between 4400 (effective wavelength of B band) and 5100 Å. If the spectral shape is described by a power-law ( $f_\nu \propto \nu^{\rm -b}$) with b = 0.3 (Marziani et al. 2003b):

$$\log~[\lambda L_\lambda (\lambda = 5100~ {\rm\AA})] = -0.4 M_{\rm B} + 35.497.$$ (B.1)

If we further assume that $L = 10\lambda \cdot L_\lambda$, with $\lambda=5100$ Å i.e., a bolometric correction factor 10, we can write:

$$\log~[\lambda L_\lambda (\lambda = 5100~ {\rm\AA})] = \frac{L... ...cdot L_{\rm Edd}} \cdot 1.3 \times 10^{47} M_{{\rm BH},9\odot}$$ (B.2)

where the black hole mass ${M}{_{\rm BH}}$ has been written in units of 10⁹ $M_\odot$. The absolute magnitude $M_{\rm B}$ is:

$$M_{\rm B} = m_{\rm B} + 5 - 5 \log d_{\rm L} - 2.5 (a - 1) \log~(1 + z)$$ (B.3)

where the luminosity distance $d_{\rm L}$ is in parsec, and the last term is the K(z) correction.

Assuming $\Omega_{\rm M} \neq\ 0, \Omega_{\Lambda} \neq\ 0$, and $\Omega_{\rm k} = 0$, we have for the transverse comoving distance:

$$d_{\rm C} = \frac{c}{H_0} \int_0^z \frac{{\rm d}z'}{\sqrt{\Omega_{\rm M} (1+z)^3 + \Omega_\Lambda}}\cdot$$ (B.4)

This integral can be approximated with residuals less than 3% at all z (maximum error for $z \rightarrow 0$).

$$d_{\rm C} \approx \frac{c}{H_0} \left[ (1.500 ) \left( 1 - {\... ...0.996 \left( 1 - {\rm e}^{-\frac{z}{1.266 }} \right) \right].$$ (B.5)

The luminosity distance is:

$$d_{\rm L} = d_{\rm C} (1 + z)$$ (B.6)

and we can use the new relationship to derive a working relationship for ${M}{_{\rm BH}}$:

  $$\begin{eqnarray*}{M_{{{\rm BH}},9\odot}} &= & {3.85 \times 10^6 h^{-2}} {\left( ... ...-\frac{z}{1.266}} \right) \right]^2 (1 + z)^{(1 - {\rm a})}, \\ \end{eqnarray*}$$

where h = H₀/75. Curves shown in Figs. 3, 5-7 assume a = 0.6. Differences for $0.3 \la a \la 0.6$ are minor.

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Online Material

	Figure 1: Calibrated VLT-ISAAC spectra for 9 new intermediate-redshift quasars. Abscissæ are rest-frame wavelength in Å, ordinates the rest-frame specific flux in units of 10-15 erg s-1 cm-1 Å-1.
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Figure 2: Spectral Atlas for the new intermediate-redshift quasars. Left-hand panels show the continuum-subtracted H$\beta$ spectral region. Axes are the same as Fig. 1. The best-fit Fe II emission model is traced as a thin (green) line. Right-hand panels show the enlarged H$\beta$ profiles after continuum and Fe II subtraction. The (blue and red) thick line shows a spline fit of the short and long wavelength sides of the H $\beta _{\rm BC}$ profile, respectively.

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Table 1: Basic properties of sources and Log of observations.