Measuring M_{\rm BH} & L/L_{\rm Edd}

# Black Hole Growth to z=2 - I: Improved Virial Methods for Measuring MBH & L/LEdd

## Abstract

We analyze several large samples of Active Galactic Nuclei (AGN) in order to establish the best tools required to study the evolution of black hole mass () and normalized accretion rate (). The data include spectra from the SDSS, 2QZ and 2SLAQ public surveys at , and a compilation of smaller samples with . We critically evaluate the usage of the Mg ii and C iv lines, and adjacent continuum bands, as estimators of  and , by focusing on sources where one of these lines is observed together with H. We present a new, luminosity-dependent bolometric correction for the monochromatic luminosity at 3000Å, , which is lower by a factor of than those used in previous studies. We also re-calibrate the use of  as an indicator for the size of the broad emission line region () and find that , in agreement with previous results. We find that for all sources with . Beyond this FWHM, the Mg ii line width seems to saturate. The spectral region of the Mg ii line can thus be used to reproduce H-based estimates of  and , with negligible systematic differences and a scatter of 0.3 dex. The width of the C iv line, on the other hand, shows no correlation with either that of the H or the Mg ii lines and we could not identify the reason for this discrepancy. The scatter of (C iv), relative to (H) is of almost 0.5 dex. Moreover, 46% of the sources have , in contrast with the basic premise of the virial method, which predicts , based on reverberation mapping experiments. This fundamental discrepancy cannot be corrected based on the continuum slope or any C iv-related observable. Thus, the C iv line cannot be used to obtain precise estimates of . We conclude by presenting the observed evolution of  and  with cosmic epoch. The steep rise of  with redshift up to flattens towards the expected maximal value of , with lower- sources showing higher values of  at all redshifts. These trends will be further analyzed in a forthcoming paper.

###### keywords:
galaxies: active – galaxies: nuclei – quasars: general
12

## 1 Introduction

The growth of Super-Massive Black Holes (SMBHs), which reside in the centers of most large galaxies, progresses through episodes of radiatively-efficient accretion, during which such systems appear as Active Galactic Nuclei (AGN). Many details of this growth process are still unknown. In the local Universe most AGN are powered by lower-mass BHs, with , growing at very slow rates (e.g., Marconi et al., 2004; Hasinger et al., 2005; Netzer & Trakhtenbrot, 2007, hereafter NT07). It is thus clear that the more massive BHs accreted at higher rates in the past. Indeed, several studies suggest that the typical normalized accretion rates () increase with redshift (e.g., Fine et al., 2006, NT07). In contrast, the few luminous QSOs for which  was measured show extremely high masses, of about , and high accretion rates, near their Eddington limit. These systems could have grown to be the most massive BHs known () as early as (Willott et al., 2010; Kurk et al., 2007; De Rosa et al., 2011; Trakhtenbrot et al., 2011, hereafter T11).

For un-obscured, type-I AGN,  can be estimated by so-called “single-epoch” or “virial” estimators. These methods rely on the results of reverberation-mapping (RM) experiments, which provide empirical relations between the emissivity-weighted radius of the Broad Line Region (BLR) and the source luminosity . These relations are parametrized as . Assuming the motion of the BLR gas is virialized, the single-epoch  estimators take the general form

 MBH=fG−1(λLλ)αV2BLR, (1)

where is a probe of the BLR velocity field and is a factor that depends on the geometrical distribution of the BLR gas. A common estimator of this type is the “H method” (hereafter []). Here  is estimated from  at 5100Å (hereafter ), FWHM(H) is the velocity proxy, and Kaspi et al. (2000, 2005); Bentz et al. (2009). Another method is based on the Mg ii line and the adjacent continuum luminosity (). Although the few Mg ii-dedicated reverberation campaigns have not yet revealed a robust relation (e.g., Clavel et al., 1991; Metzroth et al., 2006), several studies calibrated the Mg ii relation by using UV spectra of sources that have H reverberation data. One particular example is the relation presented in McLure & Jarvis (2002), and later refined by McLure & Dunlop (2004, hereafter MD04). The “H” and “Mg ii” estimators were used in numerous papers to derive  for many thousands of sources. This means focusing on either or AGN (e.g., Croom et al., 2004; Fine et al., 2006; Shen et al., 2008; Rafiee & Hall, 2011). Other studies used these estimators for small samples of sources at higher redshifts, where the lines are observed in one of the NIR bands Shemmer et al. (2004); Netzer et al. (2007); Kurk et al. (2007); Marziani et al. (2009); Dietrich et al. (2009); Willott et al. (2010).

In principal,  can also be estimated from the broad C iv line3, since (C iv) is known from RM experiments (e.g., Kaspi et al., 2007). A specific calibration of this type is given in Vestergaard & Peterson (2006) and several other papers. Such methods would potentially enable the study of large samples of AGN at for which C iv is observed in large optical surveys (e.g., Vestergaard et al., 2008; Vestergaard & Osmer, 2009). However, there is evidence that such C iv-based estimates are highly uncertain. In particular, Baskin & Laor (2005, hereafter BL05) found that the C iv line is often blue-shifted with respect to the AGN rest-frame, and that FWHM(C iv) is often smaller than FWHM(H), both indicating that the dynamics of the C iv-emitting gas may be dominated by non-virial motion A later study by Vestergaard & Peterson (2006) claimed that at least some of these findings may be due to the inclusion of narrow-line objects and low-quality spectra in the BL05 sample. Despite this reservation, several subsequent studies of large, flux-limited samples clearly demonstrated that the relation between the widths of the C iv and Mg ii lines shows considerable scatter and, as a result, the C iv-based estimates of  can differ from those deduced from Mg ii by up to an order of magnitude Shen et al. (2008); Fine et al. (2010). Moreover, studies of small samples of AGN by, e.g., (Netzer et al., 2007, N07 hereafter), (Shemmer et al., 2004, S04 hereafter) and (Dietrich et al., 2009, D09 hereafter), show that the large discrepancies between C iv and H mass estimates persist even in high-quality spectra of broad-line AGN (i.e. where ). These issues were investigated in a recent study by Assef et al. (2011), which suggested an empirical correction for the C iv-based estimators that is based on the shape of the observed UV-optical spectral energy distribution (SED). This correction, however, may turn impractical for large optical surveys of high redshift AGN, where only the rest-frame UV regime is observed.

In this paper, we re-examine the methods used to derive  and  of high-redshift type-I AGN. We discuss both large and small samples, including some that were never presented in this context. The various samples are described in §2 and the measurement procedures in §3. We discuss the monochromatic luminosities and bolometric corrections in §4. We briefly discuss the premise of the virial assumption in for estimating  in §5. In §6 we examine how the Mg ii emission line complex can be used to measure , and in §7 we provide new evidence regarding the fundamental limitations of the C iv method. Finally, in §8 we briefly describe the observed evolution of  and , as measured with the tools developed in this paper, and summarize our conclusions. A more detailed analysis of the evolutionary trends is deferred to a forthcoming paper. Throughout this work we assume a cosmological model with , , and .

## 2 Samples Selection

The main goal of the present work is to test how the Mg ii and C iv emission line complexes can be used to estimate ,  and , and to apply these methods to probe the evolution of  to . While Mg ii can be compared to H or C iv within the same spectroscopic window (at or , respectively), the comparison between C iv and H has to be based on a combination of separate observations. This dictates two distinct types of samples: (1) large samples drawn from the Sloan Digital Sky Survey (SDSS; York et al., 2000), 2dF QSO Redshift survey (2QZ; Croom et al., 2004) and the 2dF SDSS LRG And QSO survey (2SLAQ; Richards et al., 2005; Croom et al., 2008) catalogs; and (2) several smaller AGN samples, with publicly available data. The following details all the samples used in the paper, which are summarized in Table 1. We note that the 2QZ and 2SLAQ surveys provided relatively few high-quality spectra, which are usable for the comparisons between the different lines (see below). All the spectra analyzed in the present work were corrected for galactic extinction using the maps of Schlegel et al. (1998) and the model of Cardelli et al. (1989).

### 2.1 Sdss Dr7

We queried the public catalogs of the seventh data release of the SDSS (DR7; Abazajian et al., 2009) for all sources that are classified as “QSO” and have a high confidence redshift determination (i.e. zconf) in the range of . This resulted in 74517 sources. First, we omitted from this sample 7845 objects for which the line fitting procedures resulted in poor-quality fits, similarly to our criteria in NT07 (see more details in §3). Next, we omitted from our analysis all radio-loud sources. The radio loudness of each source, (following Kellermann et al., 1989), was derived from the FIRST data Becker et al. (1995); White et al. (1997) incorporated in the SDSS public archive. For sources at , we estimated using the continuum flux density near 2200Å and by assuming (Vanden Berk et al., 2001, VdB01 hereafter). We removed all sources with from our sample.

Finally, we omitted 1764 sources that are classified as broad absorption line QSOs (BALQSOs), based on their Mg ii and/or C iv spectral regions. For this, we used the relevant flags in the SDSS/DR7 catalog of Shen et al. (2011, see below) that, in turn, is largely based on the SDSS/DR5 catalog of BALQSOs of Gibson et al. (2009), with further (manual and hence partial) classification of the post-DR5 spectra. Although the Shen et al. (2011) BALQSO classification is probably incomplete, its combination with our fit quality criteria provide a sample which is almost completely BALQSO-free.

The final SDSS sample comprises 20894 for which H is observed () and 43995 sources for which only the Mg ii line is observed (i.e. ). There are 6731 sources for which both the H and Mg ii lines are observed (; the SDSS “HMg ii” sub-sample hereafter) and 10910 sources where both the Mg ii and C iv lines are observed (; the SDSS “Mg iiC iv” sub-sample hereafter).

The general SDSS sample significantly overlaps with the sample studied by Shen et al. (2011). Virtually all our sources are found within the Shen et al. (2011) catalog, in particular about 98% of our SDSS “HMg ii” and “Mg iiC iv” sub-samples, which have  derived from more than one emission line in that catalog. At lower redshifts the fractions are lower, and only of our H-only SDSS sources can be found within the Shen et al. (2011) catalog. This is due to our choice to include relatively faint sources that were excluded from the Shen et al. (2011) catalog, namely sources with (see also Schneider et al., 2010). We verified that these faint sources have high-quality H fits, according to our criteria (see §3). In addition, we note that our SDSS “HMg ii” sub-sample includes 354 of the 495 sources () analyzed in the study by Wang et al. (2009). The discrepancy is due to the fact that Wang et al. (2009) chose to include in their sample sources with slightly lower redshifts (). Our general (H) SDSS sample includes 473 of their sources ().

### 2.2 2qz

We used the public 2QZ catalog of Croom et al. (2004), initially choosing all sources which are classified as “QSO” and have high confidence redshift determination (i.e. ZQUAL flag better than 22). Our query naturally omits BALQSOs, which are classified as such within the 2QZ catalog (based on visual inspection of the spectra, which is incomplete; see Croom et al., 2004). This query provided 20315 sources. After applying the fit quality criteria to this large Mg ii 2QZ sample, it comprises 8873 sources. Generally, spectra obtained as part of the 2QZ and 2SLAQ AAO-based surveys are not flux calibrated. Previous studies used single-band magnitudes, in either the or the bands, to derive monochromatic luminosities, assuming all spectra follow a uniform UV-to-optical SED of (e.g., Croom et al., 2004; Richards et al., 2005; Fine et al., 2008). Here we employed a more robust flux calibration scheme, which relies on all the photometric data available in the 2QZ and 2SLAQ catalogs. This procedure is discussed in Appendix A.

In order to expand the luminosity range of the “HMg ii” sample, we preformed a separate query of all 2QZ sources, ignoring their class and ZQUAL. For the selection of “Mg iiC iv” sources we focused on sources at (within the larger Mg ii sample) These choices were made in order to include sources which might be flagged as problematic due to the proximity of the H line to the telluric features near 7000Å or to the limit of the observed spectral range, and to avoid these problems from affecting the C iv line. The “HMg ii” sources which passed the fit quality criteria were manually inspected, to omit any non-QSO objects or otherwise unreliable spectra. These procedures resulted in merely 238 sources with reliable H fits and 489 sources with reliable C iv fits.

To summarize, we have 9111 2QZ sources in our final sample, 238 of which have both H and Mg ii, and 489 sources with both Mg ii and C iv. These sources span the magnitude range , which corresponds to the luminosity range at (see §4.2). The 2QZ sample thus reaches about a factor deeper than the SDSS sample.

### 2.3 2slaq

Sources from the 2SLAQ survey were selected by using the public 2SLAQ QSO catalog, compiled and presented in Croom et al. (2008), and further analyzed in Fine et al. (2010). Our query focused on sources which are classified as “QSO”, and resulted in 5873 sources. Here too, BALQSOs are largely omitted, due to their separate classification. Low-redshift 2SLAQ sources suffer from the same limitations affecting the 2QZ low-redshift sources. We preformed a separate search to locate “HMg ii” candidates at . Our careful manual inspection resulted in only 39 sources with reasonable fits to both the H and Mg ii lines. We chose not to include any of these sources in the present analysis, due to their poor S/N and small number. Sources with in the large Mg ii sample have both Mg ii and C iv. We have 217 such sources, out of a total of 2287. These sources span the magnitude range , which corresponds to at . This is almost an order of magnitude deeper than SDSS, and about a factor of 2.5 deeper than 2QZ.

### 2.4 Additional Samples

There are many smaller samples in the literature with measurements of more than one of the three emission lines discussed here. We choose to focus on samples for which the relevant line measurements are either preformed by procedures very similar to ours and publicly tabulated, or on samples for which such measurements could be preformed on publicly available spectra. In particular, we used the samples of BL05 (59 sources), N07 (and S04; 44 sources), Shang et al. (2007, hereafter Sh07 hereafter; 22 sources), Sulentic et al. (2007, hereafter Sul07; 90 sources), McGill et al. (2008, hereafter Mc08; 19 sources), Dietrich et al. (2009, hereafter D09; 10 sources), Marziani et al. (2009, hereafter M09; 30 sources), and T11 (40 sources). Basic information on these samples is given in Table 1, while Appendix B provides more details regarding the acquisition and contents of the samples. In particular, Appendix B details which line measurements we used cases where sources appeared in more than one sample.

These small samples probe a wide range of redshift and luminosity and most of them have higher S/N than those typically obtained within the large SDSS, 2QZ and 2SLAQ surveys. In many of these samples the different lines were not observed simultaneously. Thus, it is possible that line and/or continuum variability contributes to the scatter in the relationships discussed in this work.

## 3 Line and Continuum Fitting

We have developed a set of line fitting procedures to derive the physical parameters related to the H, Mg ii and C iv lines. These procedures are similar in many ways to those described in previous publications (e.g., S04, N07, NT07, S08, F08). In particular, the H and Mg ii fitting procedures follow those used in NT07 and T11, respectively. In short, a linear continuum model is fitted to the flux densities in two 20Å-wide “bands” on both side of the relevant emission line. The bands are centered around 4435 & 5110Å for H, 2655 & 3020Å for Mg ii and 1450 & 1690Å for C iv. For the H and Mg ii lines, we subtract a template of Fe ii and Fe iii emission lines. This is done by choosing the best-fit broadened and shifted template based on data published by Boroson & Green (1992) and Vestergaard (2004). The grid spans a broad range both in the width and (velocity) shift of the iron features. In the case of Mg ii, we add flux in the wavelength range under the emission line itself (see Appendix C.2) The main emission lines are modeled by a combination of two broad and one narrow Gaussians. In the cases of Mg ii and C iv this model is duplicated for each of the doublet components. In Appendix C we discuss several minor but important technical issues concerning the fitting of the lines, including: (1) the ranges of all relevant parameters; (2) the narrow () components of all lines; (3) the improved iron template for the Mg ii spectral complex and its scaling; (4) the difference between our fitted  and the “real” power-law continuum, and (5) the emission features surrounding the C iv line.

All fitting procedures were tested on several hundred spectra, selected from all samples used here. For these, we individually inspected the fitted spectra. In particular, we verified that the simplifications involved in the C iv fitting procedure (Appendix C.3) are justified. After fitting the SDSS, 2QZ and 2SLAQ samples, we filtered out low-quality fits by imposing the criteria and .4 Both these quantities were calculated over a range of centered on the peaks of the relevant emission lines, and adjusted for degrees of freedom. This choice is similar to the one we made in our previous analysis of large samples (NT07). The smaller samples were treated manually, securing high-quality fitting results for all sources where adequate spectral coverage of the various lines was available. We stress that even when two of the three main lines are observed in the same spectrum (i.e. for the “HMg ii” and “Mg iiC iv” sub-samples), the fitting procedures are ran separately for each emission line complex. Thus, the parameters of the different models are fully independent, even when it is possible to assume otherwise.

The line fitting procedures provide monochromatic luminosities (,  and/or ) and emission lines widths. For the latter, we calculated the FWHM, the line dispersion (following Peterson et al., 2004) and the inter-percentile velocity (IPV; following Croom et al., 2004). We estimated  for all the sources for which the “H method” is applicable, by:

 RBLR(Hβ)=538.2(L51001046ergs−1)0.65lt−days. (2)

This is an updated version of the correlation found by Kaspi et al. (2005), taking into account the improved measurements presented in Bentz et al. (2009), but fitting the RM measurements of only the sources with , typical of the luminosities of high-redshift sources. (H) is derived by assuming (Netzer & Marziani, 2010, and references therein) and is given by:

 MBH(Hβ)=1.05×108(L51001046ergs−1)0.65[FWHM(Hβ)103kms−1]2M⊙. (3)

We have also experimented with flatter - relations, of , as suggested by some studies (e.g. Bentz et al., 2009). We found, as in Kaspi et al. (2005), that these have little effect on our main results, provided that the constant in Eq. 2 is adjusted accordingly. In particular, for the median luminosity of the SDSS “HMg ii” sub-sample () the - relation of Bentz et al. (2009) would have resulted in  estimates that are larger by 0.03 dex than the ones obtained by Eq. 2. For the “HC iv” and “Mg iiC iv” sub-samples we assumed the reverberation-based relation of Kaspi et al. (2007):

 RBLR(C\textsciv)=107.2(L14501046ergs−1)0.55lt−days. (4)

We also experimented with the relation of Vestergaard & Peterson (2006), , and verified that none of the results that follow depend on this choice.

## 4 Luminosities and Bolometric Corrections

### 4.1 Monochromatic Luminosities and SED Shapes

The UV-optical continuum emission of type-I AGN, between 1300 and 5000Å is well described by (e.g., VdB01). The typical luminosity scaling values in our SDSS sample are and , with standard deviations of 0.12 and 0.14 dex, respectively. Assuming the average conversion of  to the “real” continuum (see Appendix C.2), these ratios become and , respectively. For comparison, the ratios implied by an power law are 1.30 and 1.44, respectively. Figure 1 shows that the smaller “HC iv” sub-samples generally follow the scalings of . However, the ratio of provides a somewhat better scaling for these sources, for which both  and  are directly observed. This ratio is consistent with the VdB01 result, which is based on a composite of sources for which either  or  were observed. In particular, all the PG quasars that are part of the reverberation-mapped sample of Kaspi et al. (2000) follow these luminosity scalings (these are part of the BL05, Sh07 and Sul07 samples). Thus, we do not expect that SED differences would play a major role in scaling any relation that is consistent with the RM experiments. This is a crucial point in single-epoch  determinations and is further discussed in §7.

We note that 87% of the sources in the SDSS “Mg iiC iv” sub-sample have , and 93% of the sources in the smaller “Mg iiC iv” sub-samples (Sh07, BL05, N07+S04, M09 & D09) have (i.e., ). In contrast, half of the sources (6/12) presented by Assef et al. (2011) have . Such extreme UV-optical SEDs may represent AGN with intrinsically different properties or, perhaps, are affected by a higher-than-typical reddening. These issues are further discussed in §7.

### 4.2 Bolometric Corrections

To determine the bolometric luminosity () one has to assign a bolometric correction factor, . Here we focus on . Several earlier studies assumed a constant , e.g., Elvis et al. (1994), or Richards et al. (2006a). There are two problems with this approach: (1) The constant  was based on the total, observed X-ray to mid-IR (MIR) SED of AGN. As such, it includes double-counting of part of the AGN radiation (the MIR flux originates from re-processing of the UV-optical radiation). This results in overestimation of  and thus  (e.g., Marconi et al., 2004). (2) The shape of the SED is known to be luminosity-dependent. This dependence is most significant at the X-ray regime, and perhaps also at UV wavelengths (e.g., Vignali et al., 2003). The general trend is of a decreasing with increasing . Thus,  should decrease with increasing UV luminosity. Marconi et al. (2004) provides a luminosity-dependent prescription for estimating (4400Å) that addresses these two issues. The prescription can be modified to other wavelengths, by assuming a certain UV-optical SED.

We derived a new prescription for  by using the SDSS “HMg ii” sub-sample. First, we converted the measured  of each source to  using the prescription of Marconi et al. (2004) and assuming . This provides, for each of the 6731 sources,  and /, which are presented in Figure 2. The derived bolometric corrections are systematically lower than the aforementioned fixed values. For example, the typical correction is just 3.4 for sources with , which is the median luminosity of the SDSS sources at . This is a factor of 1.5 lower than the value of 5.15 used in several other studies of  at (e.g., Fine et al., 2008). Fig. 2 further reveals that, despite the large scatter at low , there is a clear trend of decreasing  with increasing , as expected. To quantify this trend, we grouped the data in bins of 0.2 dex in  and assumed that the error on  is , for the i-th bin in . An ordinary least squares (OLS) fit to the binned data points gives:

 (5) +1.76L23000,45−0.54L33000,45,

where . As Fig. 2 clearly shows, this relation predicts relatively large bolometric corrections for low-luminosity sources, to the extreme of for . Similarly high values were reported in the past, for a small minority of sources (e.g., Richards et al., 2006a). We suspect that the high values predicted by Eq. 5 are the result of unrealistic extrapolation of the Marconi et al. (2004) relation towards low luminosities, where the number of measured points is small. Moreover, we note that low-luminosity sources may suffer from more significant host-galaxy contamination, which results in a systematic over-estimation of  and . We thus caution that the bolometric corrections we provide here should not be used for sources with , where the emission from the host galaxy is comparable to that of the AGN and in particular in cases where the spectroscopic aperture affects the determination of the host emission (see, e.g., the analysis in Stern & Laor, 2012).

Figure 3 compares the bolometric luminosities derived from  by using Eq. 5 to those calculated from . The two methods provide consistent estimates of  with a scatter of less than 0.09 dex (standard deviation of residuals). The small scatter is probably dominated by the range of UV-to-optical slopes of individual sources.

We finally note that the real uncertainties on such estimates of  are actually governed by the range of global SED variations between sources, as well as the assumed physical (or empirical) model for the UV SED. For example, the assumed exponent of the X-ray model and the relation may amount up to 0.2 dex in , and thus in the calculated  (see, e.g., Vignali et al., 2003; Bianchi et al., 2009). Two very recent studies further demonstrated these complications. The study by Runnoe et al. (2012) showed that even the uniform Elvis et al. (1994) and Richards et al. (2006a) SEDs may provide  as low as , given that the integration is limited to (thus neglecting re-processed emission. However, the best-fit trends of Runnoe et al. (2012) predict for sources with , consistent with the commonly used value of 5.15. Jin et al. (2012) used an accretion disk fitting method that results in a much higher (unobserved) far-UV luminosity for a given optical and/or near-UV luminosity. Naturally, this model produced very high bolometric corrections, with (and as high as 20-30) for several sources with , compared to the Marconi et al. (2004) prediction of . This completely new approach to the estimate of  in AGN will not be further discussed in the present paper. Instead, we advice the usage of the corrections given by Eq. 5, which supplement the optical corrections of Marconi et al. (2004).

## 5 Virial MBH Estimates: Basic Considerations

A main goal of this paper is to present a critical evaluation of the various ways to measure  by using the RM-based “virial” method. It is therefore important to review the basic premise of the method and the justifications for its use.

Four critical points should be considered:

1. The emissivity weighted radius of the BLR, , is known from direct RM-measurements almost exclusively for only the H and C iv emission lines, and to a much lesser extent for Mg ii (see §1). The expressions chosen for the present work are given in Eqs. 2 and 4. They depend on the measured  and , and involve the assumption that the derived luminosities require no reddening-related, or other, corrections. The slope of these correlations () is empirically determined to be in the range 0.5–0.7, by a simple regression analysis of the observational results. It is not known whether the fundamental dependence is on , , the ionizing luminosity or perhaps , although there are theoretical justifications for all these cases. It is also not clear whether the slope itself depends on luminosity (i.e. whether the same relation holds for all luminosities, see, e.g., Bentz et al., 2009; Netzer & Marziani, 2010). Thus, the approach adopted here is to use all quantities as measured.

2. The mean ratio of the RM-based H and C iv radii, assuming the relations, is about 3.7. The number depends on the measured lags and the mean luminosities of the objects in the RM samples used to derive the relations, and may significantly vary for individual sources (see, e.g., the range of ratios in Peterson et al. 2004). The mean ratio of the two involved luminosities in the two RM samples is . This is similar to the mean ratio in several much larger samples where the two wavelength regions are observed (e.g., S04, BL05, N07; see §4.1) and is also similar to the SDSS-based composite spectrum of VdB01. The working assumption, therefore, is that this ratio represents the population of un-reddened type-I AGN well. Given this, the virial method cannot be applied directly to sources where deviates significantly from this typical value, since in such sources  may not scale with the luminosity in the same way as in the two RM samples used to derive the equations. For example, may indicate significant continuum reddening, which will result in a systematic underestimation of , if one uses Eq. 4 or similar relations. Correcting for such reddening must be performed prior to the estimation of  or .

3. The virial motion of the BLR, combined with the adoption of line FWHMs as the gas velocity indicator, and the assumption that the global geometry of the H and C iv parts of the BLR are similar (e.g. two spheres of different radii) lead to the prediction that . This is confirmed in a small number of intermediate luminosity objects showing the expected (e.g., Peterson et al., 2004). Given this, the simplest way to proceed to measure  in large samples, which lack any additional information regarding the or ratios, is to assume the same is true for all sources.

4. The best value of the geometrical factor in the mass equation (Eq. 1) is 1.0. This is an average value obtained from the comparison of RM-based “virial products” and the relation, for about 30 low redshift type-I AGN (e.g., Onken et al., 2004; Woo et al., 2010; Graham et al., 2011). None of these sources show a large deviation from this value. These calibrations rely predominantly on H, while C iv-related observables contribute to the estimation of virial products in only 5 sources. Unfortunately, there are no direct estimates of that are based solely on C iv.

Given points (ii) and (iii) above, the value of used in  estimates must be the same for the two emission lines. This is not meant to imply that certain sources cannot have two different virialized regions, for H and C iv, with different geometries and values of . It only means that the method is based on certain samples with certain properties and hence should not be applied to objects with different properties. In objects where is not directly measured and the line width ratio deviate significantly from the above, e.g. objects with , at least one of the lines should not be used as a  indicator within the framework of the virial method. Since estimates of based on FWHM(H) are known to be correct in most of the H-RM sample, we prefer to adopt in such cases the assumption that FWHM(H) provides the more reliable mass indicator.

The above considerations suggest that a single epoch mass determination based on H and  is a reliable mass estimate provided there is no significant continuum reddening. In case the amount of reddening is known, it should be taken into account prior to the application of the method. In the absence of direct mass calibration based on lines other than H, there are only two alternatives: use theoretical conjectures, or a-priori knowledge about the line width. The first can be used for the Mg ii line that is thought to originate from the same part of the BLR as H. If , then the line can be used for estimating . The second can be used for the C iv line since is known. As explained, the requirement is . If this ratio is indeed found in the majority of objects with measured FWHM(H) and FWHM(C iv), then a single epoch mass estimate based on C iv can be safely used. In the following sections we use such considerations to assess the validity of the use of the Mg ii and C iv lines as mass indicators in type-I AGN.

## 6 Estimating MBH with Mg iiλ2798

As discussed in §5, the fact that there is no RM-based determination of (Mg ii) means that the only way to obtain a Mg ii-based estimator for  is to calibrate it against (H), based on the the assumption that (verified by photoionization calculations). We therefore have to show that  can be used to estimate (H), and that . In what follows, we discuss these relations separately, and evaluate the ability of the (combined) virial product to reproduce (H).

### 6.1 Using L3000 to Estimate RBLR

In Figure 4 we present the relation between the calculated values of (H) and , for all the sources in the different “HMg ii” sub-samples. There is a clear and highly significant correlation between these two quantities. Since  is calculated directly from , this relation reflects the narrow distribution of UV-optical continuum slopes. To quantify the relation, we bin the SDSS and 2QZ “HMg ii” sub-samples (separately) in bins of 0.2 dex in . The uncertainties on each binned data-point are assumed to be the standard deviations of the values included in the respective bin. The typical uncertainty is of 0.13 dex . This is a conservative choice, which attempts to account for the entire scatter in the data.5 All other “HMg ii” sub-samples remain un-binned. For these, we assume uncertainties of 0.1 dex on  and 0.05 dex on . These choices reflect the absolute flux calibration uncertainties and the uncertainties related to the continuum and line fitting processes. Since the uncertainties in both axes are comparable, we use the BCES Akritas & Bershady (1996) and FITEXY Press (2002) fitting methods, both designed to also take into account the scatter in the data. All the BCES correlations are tested by a bootstrapping procedure with 1000 realizations of the data under study. We used the more sophisticated version of the FITEXY method, as presented by Tremaine et al. (2002), where the error uncertainties on the data are scaled in order to account for the scatter. Thus, all our FITEXY correlations resulted in . The best-fit linear relations (also shown in Fig. 4), parametrized as

 logRBLR=αlog(L30001044ergs−1)+β, (6)

resulted in and for both the BCES and FITEXY methods, The exact values and associated uncertainties are given in Table 2. The standard deviation of the residuals is about 0.1 dex. We note that some of the few low luminosity sources in Fig. 4 ( sources from the Sh07 & Mc08 samples) appear to lie above our best-fit relation. This might be due to the contamination of their optical spectra by host light, which would cause their  (and hence ) to be slightly overestimated, although effect should be very small ( dex; see, e.g., Sh07 and Bentz et al. (2009)). In addition, the few extremely high luminosity sources ( sources from the D09 & M09 samples) also lie slightly above the best-fit relation of the combined dataset. Thus, the slope of the relation may be somewhat shallower or steeper, in the low- or high-luminosity regimes, respectively. We tested these scenarios by re-fitting the data after omitting data points either above or below , that is keeping most of the (binned) SDSS data but omitting the extreme sources from the “small” samples, on either side of the luminosity range. The results of this analysis are also presented in Table 2. In particular, we find that for sources with the best-fit slope is , while for sources with it is . Table 2 also lists best-fit parameters for other choices of sub-samples. In most of these cases, the derived slopes for the relations are between those reported by McLure & Jarvis (2002) () and by MD04 (). The intercepts are also very similar to those of McLure & Jarvis (2002), and the intercept derived from the entire dataset (1.33) differs from the one derived in MD04 by only 0.06 dex, in the sense that our best-fit relation predicts higher  values for a given value of . The MD04 study relation was derived using only sources with , as is the case for most of the sources used here. Therefore, the differences between the MD04 relation and our results are not due to different luminosity regimes.

As Table 2 and Fig. 4 demonstrate, the range of slopes and intercepts that describe the relation is relatively broad, and probably somewhat luminosity-dependent. This situation is present in virtually all previous attempts to calibrate relations (see discussion in, e.g. Kaspi et al., 2000, 2005; Vestergaard & Peterson, 2006). Notwithstanding this issue, in what follows we chose to estimate  by using the relation:

 RBLR(L3000)=21.38(L30001044ergs−1)0.62lt−days. (7)

We caution that this relation may not be suitable for low-luminosity sources (), where a shallower slope should be used.

### 6.2 Using L(Mg\textscii) to Estimate RBLR

The line luminosity  can also be used to estimate . Such an approach may be used to overcome the difficulties in determining  in NIR spectra of high-redshift sources.6 The relation between , as determined from , and , is presented in Figure 5. The scatter in this relation is larger than that of the - relation. In particular, the 2QZ “HMg ii” sub-sample shows considerable scatter and almost no correlation between  and , probably due to the low-quality of the data at this extreme redshift range (see §2.2). Fitting the data with a linear relation of the form

 logRBLR=αlog(L(Mg\textscii)1042ergs−1)+β (8)

gives the values of and presented in Table 2. We get for the SDSS and 2QZ “HMg ii” sub-samples, but a considerably steeper slope (or a higher intercept) in the case we include the high-luminosity sources from the “small” samples. The scatter between the resulting (best-fit) estimates of  and those based on  (Eq. 2), for the SDSS and 2QZ sub-samples, is about 0.13 dex, only slightly higher than that of the - relations. Despite the advantages of using this - relation, we draw attention to the significant differences in the best-fit parameters that were derived from the different sub-samples, as well as by the two fitting methods, and caution that this relation is not as robust as the - one. In addition, an accurate determination of  requires a reasonable determination of the Fe ii and Fe iii features adjacent to the Mg ii line, and thus still depends on the determination of the continuum.

### 6.3 The Width of the Mg iiλ2798 Line

Our measure of FWHM(Mg ii) is different from the ones used in several earlier studies that measured the width of the total (doublet) profile. At large widths (), the two widths are basically identical, since the width of the line is much larger than the separation between the two components. For relatively narrow lines () the two measures differ significantly, and . For example, the typical profiles with (total) in our SDSS sample correspond to a single-component width of only . This implies that  can be systematically overestimated by a factor of for such narrow-line objects. This issue is crucial for studies of sources with high accretion rates. To correct earlier results that used the entire profile, we suggest the following simple relation, which is based on a fit to the SDSS data:

 FWHM(Mg\textscii,single)=1.01×FWHM(Mg\textscii,total) (9) −304kms−1.

We used this prescription to correct the relevant tabulated values of FWHM(Mg ii) published in earlier papers (see Table 1).

Several studies showed that FWHM(Mg ii) is very similar to FWHM(H(e.g. Shen et al., 2008, and references therein). This justifies the use of FWHM(Mg ii) as a tracer of virial BLR cloud motion. Indeed, the distribution of for our SDSS “HMg ii” sub-sample peaks at -0.05 dex and the standard deviation is 0.15 dex. A similar comparison of the IPV line widths results in an almost identical distribution. In order to further test this issue, we present in Figure 6 a direct comparison between FWHM(Mg ii) and FWHM(H), for all the “HMg ii” sub-samples. As expected, there is a strong correlation between the widths of the two lines. However, in most of the cases with , the Mg ii line is narrower. This trend is also reflected in the binned data shown in Fig. 6. For example, for sources with , the typical value for the Mg ii line is merely . We find that the best-fit relation between these line widths is

 logFWHM(Mg\textscii)=(0.803±0.007)logFWHM(Hβ) (10) +(0.628±0.027),

based on the BCES bisector. This relation is also shown in Fig. 6. Our result is in excellent agreement, both in terms of slope and intercept, with that of Wang et al. (2009), which is based on a subset of our SDSS “HMg ii” sub-sample (about 10% of the sources; see §2.1). The general trend we find between FWHM(Mg ii) and FWHM(H) is captured, in essence, by the statistical correction factors suggested by Onken & Kollmeier (2008), which are useful for large samples. However, it is not at all clear whether the fit reflects a real, global trend. The alternative is that up to , beyond which there are some differences in the mean location of the strongest line emitting gas.

### 6.4 Determination of MBH(Mg\textscii)

For each source in the “HMg ii” sub-samples we calculated an “empirical scaling factor”:

 μ(Mg\textscii)≡1GRBLR(L3000)FWHM(Mg\textscii)2, (11)

where is calculated through Eq. 7. In Figure 7 we show the distribution of the relevant normalization factor, defined as , for the SDSS “HMg ii” sub-sample. The distribution has a clear peak at a median (mean) value is (1.42), and a standard deviation of 0.32 dex. This is in agreement with the expected value of . For comparison, the McLure & Dunlop (2004)  estimator was derived assuming .7 We further investigated whether  depends on other observables, such as source luminosity, SED shape or EW(Mg ii). However, we find no significant correlations of this type. The only exception, a marginal anti-correlation with  (see §4.2) is most probably driven by the strong dependence of both  and  on FWHM(H). In addition, the small scatter in the relation, in comparison with that of the relation, suggests that the range of derived  values is driven solely by the scatter in . This is supported by the fact that we find no correlation between  and while the significant correlation with (not shown here) tightly follows a power-law with the expected slope of about 2.

Next, we combine our best-fit relation (Eq. 7) with FWHM(Mg ii) and the median value of  to obtain the final form of the Mg ii-based  estimator,

 MBH=5.6×106[L30001044ergs−1]0.62[FWHM(Mg\textscii)103kms−1]2M⊙ . (12)

This estimator differs from the one given in MD04 in its overall normalization, which is higher by a factor of about 1.75 than the MD04 one. Thus, we find that the MD04  estimator causes an underestimation of  by dex. This result is consistent with the findings of Shen et al. (2011), where the slope of the relation was forced to be 0.62. Naturally, the choice of a different relation, according to the parameters listed in Table 2, would result in subtle luminosity-dependent differences between our  estimates and those of MD04. We repeated the above steps for several other choices of parameters ( and in Eq. 6), and list in Table 2 the resulting scaling of the final  estimator (i.e. the equivalents of 5.6 in Eq. 12 above).8 Thus, one can use a different -based virial  estimator, fully described by and in Table 2, according to the required luminosity range (see discussion in §6.1).

To evaluate the improvement we obtained in estimating (Mg ii), we compare in Figure 8 the masses obtained with the H (following Eq. 3) and the MD04 methods, for all the “HMg ii” sub-samples. Clearly, there is a systematic offset between the two estimators, in the sense that (Mg ii,MD04) is typically lower than (H). The median offset within the SDSS “HMg ii” sub-sample is, indeed, dex. In Figure 9 we preform a similar comparison, but this time using the new Mg ii-based  estimates (Eq. 12). The systematic shift seen in Fig. 8, particularly at the high mass end, has completely disappeared. The median difference between the two estimates is negligible, although the scatter (standard deviation of residuals) remains about 0.32 dex.

Several studies suggested to use  estimators where the exponent of the velocity term differs from 2 (e.g., Greene & Ho, 2005; Wang et al., 2009). Like any additional degree of freedom, it is expected that this approach may reduce the scatter between H- and Mg ii-based estimates of . The usage of this empirical approach abandons the fundamental assumption of virialized BLR dynamics, which is the basis for all mass determinations considered here. For the sake of completeness, we derived a relation by combining Eqs 7 and 10 and minimizing the systematic offset with respect to the the H-based  estimators. This process resulted in the relation . The scatter between this estimator and the one based on H, for the SDSS “HMg ii” sub-sample, is of 0.34 dex, almost identical to the one obtained following Eq. 12 above. Since this relation departs from the simple virial assumption, we chose not to use in what follows.

can also be obtained by using the - relation presented in §6.2. We have repeated the steps described above, assuming (see Table 2), and obtained

 MBH=6.79×106[L(Mg\textscii)1042ergs−1]0.5[FWHM(Mg\textscii)103kms−1]2M⊙ . (13)

estimates based on this relation are also consistent with those based on H, and the scatter for the SDSS “HMg ii” sub-sample is of 0.33 dex, indistinguishable from the one achieved by using . We note that the Mg ii line presents a clear “Baldwin effect”, i.e. an anti-correlation between EW(Mg ii) and  (e.g. Baldwin, 1977, BL05). Therefore, the usage of  probably incorporates some other, yet-unknown properties of the BLR.

We tested the consistency of our improved -based determinations of  with those based on  (and H). For this, we calculated  for the SDSS “HMg ii” sub-sample based on the two available approaches: either using the bolometric corrections given by Eq. 5 and  estimates given by Eq. 12, or using the bolometric corrections of Marconi et al. (2004) and  estimates given by Eq. 3. In all cases we assume , appropriate for solar metallicity gas. The comparison of the two estimates of  clearly shows that the two agree well, with a scatter of about 0.31 dex, and negligible systematic difference. On the other hand, using the bolometric correction of Richards et al. (2006a, 5.15) and the MD04 estimates of  results in a systematic overestimation of  by a factor of about 2.2.

## 7 C ivλ1549-based estimates of MBH

As discussed in §5, under the virial assumption, and the known ratio of FWHM(C iv)/FWHM(H) in a small number of sources with RM-based  measurements, it is possible to use a reliable estimator for the size of the C iv-emitting region (Eq. 4) to calibrate a C iv-based estimator for , by combining (), FWHM(C iv) and a known -factor ( in our case). Since for the samples where (C iv) is directly measured it is smaller than (H), by a factor of 3.7, we expect . In what follows, we address the different ingredients of a virial C iv-based estimator, and show that for a large number of the type-I AGN studied here the C iv measurements are not consistent with the virial assumption.

Figures 10 and 11 compare FWHM(C iv) with FWHM(Mg ii) and FWHM(H), respectively. The scatter in both figures is much larger than the scatter in the Mg ii-H comparison diagram (Fig. 6), and the line widths do not follow each other. In practical terms, for any observed (single) value of FWHM(C iv), the corresponding values of FWHM(H) covers almost the entire range of , in contrast with the value expected from the assumption of virialized motion and an identical . We verified that alternative measures of the line width, such as the IPV, do not reduce the scatter or present any more significant relations between the different lines. Most importantly, a significant fraction of the sources under study () exhibit (Fig. 11), and 26% of the sources in the SDSS, 2QZ & 2SLAQ “Mg iiC iv” sub-samples have (Fig. 10), in contrast to the expectations of the virial method. The large scatter, and the lack of any correlation between FWHM(C iv) and either FWHM(H) or FWHM(Mg ii), were identified in several earlier studies of local (e.g. Corbin & Boroson, 1996, BL05), intermediate- (e.g., S08, F08) and high-redshift (S04, N07, T11) samples. We note that the high fraction of sources where FWHM(C iv) seems to defy the expectations is not due to a specific population of narrow-line sources (i.e., NLSy1s), and/or low-quality UV spectra, as was suggested by Vestergaard & Peterson (2006). In particular, we find that 55% of the “HC iv” sources with