THE LICK AGN MONITORING PROJECT: THE M_BH–σ_* RELATION FOR REVERBERATION-MAPPED ACTIVE GALAXIES

In the past decade, supermassive black holes have been found to be virtually ubiquitous at the centers of local quiescent galaxies (for reviews, see Kormendy 2004; Ferrarese & Ford 2005). Remarkably, the mass of the central black hole (M_BH) correlates with the global properties of their host spheroids, primarily the stellar velocity dispersion (σ_*; Ferrarese & Merritt 2000; Gebhardt et al. 2000a). This M_BH–σ_*  relation is believed to be of fundamental importance, providing one of the strongest empirical links between galaxy formation/evolution and nuclear activity. Measuring the slope and scatter of the relation, and any possible dependence on additional parameters, it is essential to make progress in our understanding of the co-evolution of galaxies and black holes.

One key open question is whether active galactic nuclei (AGNs) share the same M_BH–σ_* relation as quiescent galaxies, as would be expected if the relation were universal and nuclear activity were just a random transient phase. Initial studies using several reverberation-mapped Seyfert 1 galaxies show that the M_BH–σ_* relation of active galaxies is consistent with that of quiescent galaxies (Gebhardt et al. 2000b; Ferrarese et al. 2001). Other studies using larger samples with single-epoch black hole masses suggest that the slope of the relation may be different than that of quiescent galaxies, albeit at low significance (e.g., Greene & Ho 2006; Shen et al. 2008). However, these later studies rely on an indirect estimator of black hole mass, based on the kinematics and size of the broad-line region (BLR) as inferred from single-epoch spectra. The method works as follows: the broad-line profile gives a characteristic velocity scale ΔV (measured from either line dispersion, σ_line, or the full width at half-maximum intensity, V_FWHM); the average size of the BLR (R_BLR) is determined from the empirical correlation with optical luminosity (Wandel et al. 1999; Kaspi et al. 2005; Bentz et al. 2009a); and the black hole mass is obtained as

$$\begin{equation} M_{\rm BH} = f \frac{(\Delta V)^2 R_{\rm{BLR}}}{G}, \end{equation} \tag{ 1 }$$

where G is the gravitational constant and f is a virial coefficient that depends on the kinematics and the geometry of the BLR. Although single-epoch mass estimates are believed to be accurate to ∼0.5 dex (e.g., Vestergaard & Peterson 2006), it is hard to quantify precisely their uncertainty and any possible biases (Marconi et al. 2008). Thus, the slope and intrinsic scatter of the M_BH–σ_*  relation of active galaxies remain highly uncertain.

Multi-epoch data provide direct measurements of the BLR size via reverberation-mapping time lags (R_BLR = cτ, where c is the speed of light and τ is the measured reverberation time scale), and more secure measurements of the broad-line kinematics as determined from the variable component of the broad lines (Peterson et al. 2004). Onken et al. (2004) combined reverberation black hole masses with measurements of stellar velocity dispersion for 14 objects to measure the slope of the M_BH–σ_* relation, showing that AGNs and quiescent galaxies lie on the same M_BH–σ_*  relation for an appropriate choice of the virial coefficient (f = 5.5 ± 1.8).¹⁵ Remarkably, the scatter in M_BH relative to the M_BH–σ_* relation is found to be less than a factor of 3. Unfortunately, their sample is very small (14 objects) and covers a limited dynamic range ($6.2 < {\rm {log}}\, M_\mathrm{BH}\ < 8.4$), making it difficult to simultaneously measure the intrinsic scatter and the slope and investigate any trends with black hole mass, or with other properties of the nucleus or the host galaxy (cf. Collin et al. 2006). Under the assumption that the relations should be the same for active and quiescent galaxies, the Onken et al. (2004) study provides some information on the geometry of the BLR, an absolute normalization of reverberation black hole mass, and an upper limit to the intrinsic scatter in the virial coefficient. Since the reverberation-mapped AGN sample is the "gold standard" used to calibrate all single-epoch mass estimates (e.g., Woo & Urry 2002; Vestergaard et al. 2002; McLure & Jarvis 2002; Vestergaard & Peterson 2006; McGill et al. 2008; Shen et al. 2008), this comparison is effectively a crucial link in establishing black hole masses for all AGNs across the universe and for all evolutionary studies (e.g., McLure & Dunlop 2004; Woo et al. 2006, 2008; Peng et al. 2006a, 2006b; Netzer et al. 2007; Bennert et al. 2010; Jahnke et al. 2009; Merloni et al. 2010). In this paper, we present new measurements of host-galaxy velocity dispersion obtained from deep, high-resolution spectroscopy at the Keck, Palomar, and Lick Observatories. The new measurements are combined with existing data (e.g., Onken et al. 2004; Nelson et al. 2004; Watson et al. 2008) and with recently determined black hole masses from reverberation mapping (Bentz et al. 2009b) to construct the M_BH–σ_* relation of broad-lined AGNs for an enlarged sample of 24 objects. For the first time, we are able to determine simultaneously the intrinsic scatter and the slope of the M_BH–σ_* relation of active galaxies. Also, by forcing the slope of the M_BH–σ_* relation to match that of quiescent galaxies, we determine the average virial coefficient and its non-zero intrinsic scatter. Finally, we study the residuals from the determined M_BH–σ_* relation to investigate possible trends in virial coefficient with properties of the active nucleus.

The paper is organized as follows. In Section 2, we describe the observations and data reduction of the optical and the near-infrared (near-IR) data. New velocity dispersion measurements are presented in Section 3, along with a summary of previous measurements in the literature. In Section 4, we investigate the M_BH–σ_* relation of the present-day AGNs, and determine the virial coefficient in measuring M_BH. Discussion and summary follow in Sections 5 and 6, respectively.

The Lick AGN Monitoring Project (LAMP) was designed to determine the reverberation time scales of a sample of 13 local Seyfert 1 galaxies, particularly with low black hole masses (<10⁷ M_☉). NGC 5548, the best-studied reverberation target, was included in our Lick monitoring campaign to test the consistency of our results with previous measurements. The 64-night spectroscopic monitoring campaign, along with nightly photometric monitoring, was carried out using the 3 m Shane reflector at Lick Observatory and other, smaller telescopes. The detailed photometric and spectroscopic results are described by Walsh et al. (2009), Bentz et al. (2009b), and Bentz et al. (2010), respectively. In summary, nine objects including NGC 5548 showed enough variability in the optical continuum and the Hβ line to obtain the reverberation time scales. This significantly increases the size of the reverberation sample, particularly for this low-mass range.

For the LAMP sample of 13 objects, we carried out spectroscopy using various telescopes, to measure stellar velocity dispersions. Here, we describe each observation and the data-reduction procedures. The date, total exposure time, and other parameters for each observation are listed in Table 1.

2.1. Optical Data

2.2. Near-infrared Data

To measure stellar velocity dispersions, we used the Ca ii triplet region (rest frame ∼8500 Å) in the optical spectra, and Mg i and the CO line region (rest frame ∼1.5–1.6 μm) in the near-IR spectra. Velocity dispersions were determined by comparing in pixel space the observed galaxy spectra with stellar templates convolved with a kernel representing Gaussian broadening. Velocity template stars of various spectral types were observed with the same instrumental setup at each telescope, minimizing any systematic errors due to the different spectral resolution of various observing modes. The minimum χ² fit was performed using the Gauss–Hermite Pixel Fitting software (van der Marel 1994). The merit of fitting in pixel space as opposed to Fourier space is that typical AGN narrow emission lines and any residuals from night-sky subtraction can be easily masked out (e.g., Treu et al. 2004; Woo et al. 2005, 2006), and the quality of the fit can be directly evaluated. We used low-order polynomials (order 2–4) to model the overall shape of the continuum. Extensive and careful comparison was performed using continua with various polynomials, and the measured velocity dispersions based on each continuum fit with various order polynomials are consistent within the errors (see, e.g., Woo et al. 2004, 2005, 2006). Thus, continuum fitting with low-order polynomials does not significantly affect the velocity dispersion measurements.

3.1. Optical Data

For galaxies with optical spectra, we mainly used the strong Ca ii triplet lines (8498, 8542, 8662 Å) to measure the stellar velocity dispersions. We used all velocity template stars with various spectral types (G8–K5 giants) observed with the same instrumental setups, in order to account for possible template mismatches. Fits were performed with each template and the mean of these measurements was taken as the final velocity dispersion measurement. The rms scatter around the mean of individual measurements was added in quadrature to the mean of the individual measurement errors from the χ² fit. Given the high S/N of the observed spectra, the uncertainties are typically ∼5%.

We were able to measure the stellar velocity dispersion of eight objects: SBS 1116+538A, Arp 151, Mrk 1310, Mrk 202, IC 4218, NGC 5548, NGC 6814, and MCG −6-30-15. Among these, IC 4218 and MCG −6-30-15 were excluded in the M_BH–σ_* relation analysis since we were not able to measure the reverberation time scales (Bentz et al. 2009b). Figure 1 shows examples of the best-fit template along with the observed galaxy spectra. Our measurements based on high-quality data are generally consistent within the errors with previous measurements from the literature for these galaxies, although many of the previously existing measurements had very large uncertainties (see Section 3.3 below).

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In the case of the other five objects (Mrk 871, NGC 4748, Mrk 766, Mrk 290, and Mrk 142), the Ca ii triplet lines were contaminated by AGN emission lines as shown in Figure 2; hence, we did not obtain reliable stellar velocity dispersion measurements. Three of these objects that have black hole mass measurements from reverberation mapping were observed in the near-IR, and we determined stellar velocity dispersions for two of them as described in the next section.

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3.2. Infrared Data

Because of the difficulty of measuring the stellar velocity dispersion in the optical, we obtained near-IR spectra for three galaxies with reverberation black hole masses (NGC 4748, Mrk 766, and Mrk 142) using OSIRIS. We also observed several velocity template stars with K and M spectral types. As noted by other studies (e.g., Watson et al. 2008), M giant stars give better fits compared to K giants since M giants are the representative stellar population in the near-IR bands. After experimenting with various spectral type velocity templates, we decided to use M1III and M2III stars for velocity dispersion measurements and took the mean as the final measurement with an uncertainty given by the quadrature sum of the fitting error and the rms scatter of results obtained from different templates. Figure 3 shows the observed galaxy spectra compared with the best-fit template. NGC 4748 shows a relatively good fit, while Mrk 766 presents stronger residuals due to template mismatch. All major H-band stellar lines, such as Mg i 1.488 μm, Mg i 1.503 μm, CO(3–0) 1.558 μm, CO(4–1) 1.578 μm, Si i 1.589 μm, and CO(5–2) 1.598 μm, are present in both objects. In the case of Mrk 142, which is fainter and at higher redshift than the other two objects, stellar absorption lines cannot be clearly identified and we could not obtain a reliable template fit or velocity dispersion measurement.

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3.3. Previous Velocity Dispersion Measurements

The M_BH–σ_* relation is expressed as

$$\begin{equation} \log (M_{\rm BH} / M_{\odot }) = \alpha + \beta \log (\sigma _{\ast } / {200\ \mathrm{km\,s^{-1}}}). \end{equation} \tag{ 2 }$$

For a sample of quiescent galaxies for which the sphere of influence of the black hole is spatially resolved in stellar or gas kinematics studies, Ferrarese & Ford (2005) measured α = 8.22 ± 0.06 and β = 4.86 ± 0.43 with no evidence for intrinsic scatter. Using a larger sample, including late-type galaxies, Gültekin et al. (2009) recently reported α = 8.12 ± 0.08 and β = 4.24 ± 0.41 with an intrinsic scatter σ_int = 0.44 ± 0.06.

In this section, we investigate the M_BH–σ_* relation of active galaxies, by combining our eight new velocity measurements of the LAMP sample with 16 existing measurements of reverberation black hole mass and velocity dispersion taken from the literature. The relevant properties of the resulting sample of 24 AGNs are listed in Table 2, along with the original references for the measurements. Note that the selection bias investigated by Lauer et al. (2007) is not relevant to these reverberation objects since the bias mainly affects black holes in high-mass galaxies where the host-galaxy luminosity function is steeply falling.

As a first step, we determine the slope of the M_BH–σ_* relation (Section 4.1) assuming that the virial coefficient f is unknown but does not depend on black hole mass. We also determine the intrinsic scatter in the relation. Then in Section 4.2, we determine the mean value and intrinsic scatter of the virial coefficient f by assuming that the slope and zero point of the M_BH–σ_* relation are the same for active and quiescent galaxies. Note that in the following analysis the virial coefficient f and the zero point α are degenerate; therefore, additional information is needed to determine what fraction of the intrinsic scatter is due to f and how much is due to α, as we discuss below.

4.1. The Slope

Active galaxies appear to obey an M_BH–σ_* relation with a slope consistent with that of quiescent galaxies (e.g., Gebhardt et al. 2000b; Ferrarese et al. 2001). However, the slope has not been well determined so far, because of limitations in sample size and in the dynamic range of M_BH for active galaxies with reverberation and stellar velocity dispersion measurements. The local AGN samples with single-epoch black hole masses selected from the SDSS are much larger than the reverberation sample (e.g., Greene & Ho 2006; Woo et al. 2008), but these mass estimates have larger uncertainties (∼0.5 dex) and may suffer from unknown systematic errors (Collin et al. 2006).

By adding our LAMP galaxies to the previous reverberation sample, we significantly increase the sample size, particularly at low-mass scales (M_BH <10⁷ M_☉), and extend the dynamic range to almost three decades in M_BH. This enables substantial progress in determining the slope of the M_BH–σ_* relation, independently for active galaxies. One important caveat is that the virial coefficient f may vary across the sample. Unfortunately, with existing data we cannot constrain the virial coefficient of each object. Therefore, in the analysis presented here we will assume that f is independent of black hole mass, although we allow for intrinsic scatter at fixed black hole mass. In the next section, we will take the opposite viewpoint and assume that the slope is known from quiescent samples when we investigate the intrinsic scatter in f and its dependence on M_BH and AGN properties.

In practice, our observable is the so-called virial product VP ≡ σ²_lineR_BLR/G, which is assumed to be related to the black hole mass by a constant virial coefficient M_BH=f×VP. Substituting into Equation (2), one obtains

$$\begin{equation} \log {\rm VP} = \alpha + \beta \ \log \left(\frac{\sigma _{\ast }}{200\ \mathrm{km\,s^{-1}}} \right) - \log f. \end{equation} \tag{ 3 }$$

In order to determine the slope β of the relation, we compute the posterior distribution function P. Assuming uniform priors on the parameters α' = α − log f, β, and intrinsic scatter σ_int, the posterior is equal to the likelihood given by

$$\begin{equation} -2\ln P \equiv \sum _{i=1}^{N} \frac{\left[\log \mathrm{VP}_i - \alpha ^{\prime } - \beta \log \left(\frac{\sigma _{\ast }}{200}\right)_i \right]^2}{\epsilon _{{\rm tot},i}^2} + 2 \ln \sqrt{2\pi \epsilon _{{\rm tot},i}^2}, \end{equation} \tag{ 4 }$$

where

$$\begin{equation} \epsilon _{{\rm tot},i}^2 \equiv \epsilon _{\log \mathrm{VP}_i}^2 + \beta ^{\,2} \epsilon _{\log \sigma _*i}^2 + \sigma _{\rm int}^2, \end{equation} \tag{ 5 }$$

and _log VP is the error on the logarithm of the virial product ($= \rm \epsilon _{\rm VP} / (\rm VP\, ln\, 10))$ using the average of the positive and negative errors on the virial product as $\rm \epsilon _{\rm VP}$, $\rm \epsilon _{\rm \log \sigma _*}$ is the error on the logarithm of the stellar velocity dispersion ($= \rm \epsilon _{\sigma _*} / (\rm \sigma _* ln\, 10$)), and $\rm \sigma _{\rm int}$ is the intrinsic scatter, which is treated as a free parameter. Note that our adopted likelihood is equivalent to that adopted as default by Gültekin et al. (2009) for their maximum likelihood analysis (see their Appendix A.2), with the inclusion of an additional Gaussian term to describe the uncertainties in velocity dispersion. After marginalizing over the other variables, the one-dimensional posterior distribution functions yield the following estimates of the free parameters: α' = 7.28 ± 0.14, β = 3.55 ± 0.60, and σ_int = 0.43 ± 0.08.

To facilitate comparison with earlier studies (e.g., Tremaine et al. 2002), we also determine the slope with a "normalized" χ² estimator defined as

$$\begin{equation} \chi ^{2} \equiv \sum _{i=1}^{N} \frac{\left[\log \mathrm{VP}_i - \alpha ^{\prime } - \beta \log \left(\frac{\sigma _{\ast }}{200}\right)_i \right]^2}{\epsilon _{{\rm tot},i}^2}, \end{equation} \tag{ 6 }$$

where the intrinsic scatter σ_int = 0.43 is set to yield χ² per degree of freedom equal to unity. The best estimate of the slope based on this normalized χ² estimator is β = 3.72^+0.40_−0.39. If we assume no intrinsic scatter, then the best-fit slope becomes slightly steeper as β = 3.97^+0.13_−0.13. However, confirming our previous result based on a full posterior analysis, the intrinsic scatter cannot be negligible since the χ² per degree of freedom would be much larger than unity (∼14) if the intrinsic scatter is assumed to be zero.

These results indicate that the slopes of the M_BH–σ_* relation for active and quiescent galaxies are consistent within the uncertainties. The total intrinsic scatter is 0.43 ± 0.08 dex, which is a combination of the scatter of the M_BH–σ_* relation and that of the virial coefficient. Without additional information, we cannot separate the contribution of the intrinsic scatter on the virial coefficient (i.e., the diversity of BLR kinematic structure among AGN) from the intrinsic scatter in black hole mass at fixed velocity dispersion (i.e., the diversity of black hole masses among galaxies). We will address this issue in Section 4.2.

4.2. The Virial Coefficient

In order to transform the virial product into an actual black hole mass, we need to know the value of the virial coefficient f. Our current understanding of the geometry and kinematics of the BLR is insufficient to compute f from first principles. This creates a systematic uncertainty in the determination of black hole mass from reverberation mapping. As an example, if a spherical isotropic velocity distribution is assumed, then the velocity dispersion of the gas is a factor of $\sqrt{3}$ larger than the measured line-of-sight velocity dispersion (σ_line), hence a value of f = 3 is often assumed for a spherically symmetric BLR (e.g., Wandel et al. 1999; Kaspi et al. 2000). If, instead, the BLR is described by a circular rotating disk, then the value of f can be a few times larger than the isotropic case, depending on the inclination angle and the scale height of the disk (see Collin et al. 2006 for more details).

As mentioned above, determining the virial coefficient for individual AGNs is not feasible due to the limited spatial resolution of the current and foreseeable future instruments since the angular size of the BLR is microarcseconds. Therefore, in practice, an average virial coefficient is usually determined by assuming that local Seyfert galaxies and quasar host galaxies follow the same M_BH–σ_* relation (Onken et al. 2004) and deriving the appropriate value for f. Using this method, Onken et al. (2004) measured 〈f〉 = 5.5 ± 1.8 based on a sample of 14 reverberation-mapped Seyfert galaxies with measured stellar velocity dispersions.

In this section, we apply the same methodology to our enlarged sample to determine the average value of f as well as the intrinsic scatter, assuming the intercept (α) and slope (β) of the M_BH–σ_* relation for quiescent galaxies. In practice, we modify the likelihood given in Equation (4) by fixing the slope and intercept to the values obtained by two independent groups, Ferrarese & Ford (2005) and Gültekin et al. (2009). Figures 4 and 5 show the best-fit M_BH–σ_* relations for the enlarged reverberation sample, respectively, assuming the quiescent M_BH–σ_* relation from Ferrarese & Ford (2005) and that from Gültekin et al. (2009). Adopting the M_BH–σ_* relation of Ferrarese & Ford (2005; α = 8.22 and β = 4.86), we determine log f = 0.71 ± 0.10 and σ_int = 0.46^+0.07_−0.09 dex. Adopting the M_BH–σ_* relation from Gültekin et al (2009; α = 8.12 and β = 4.24), we obtain log f = 0.72^+0.09_−0.10 and σ_int = 0.44 ± 0.07. The average values are in good agreement with f = 5.5 ± 1.8 (i.e., log f = 0.74^+0.10_−0.13) as found by Onken et al. (2004). The average value is inconsistent with that expected for a spherical BLR (f = 3), and closer to that expected for a disk-like geometry, although there may be a diversity of geometries and large-scale kinematics (e.g., Bentz et al. 2009b; Denney et al. 2009a).

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As discussed at the end of Section 4.1, the intrinsic scatter determined in this section is a combination of intrinsic scatter in f and in zero point of the M_BH–σ_* relation (which are degenerate in Equation (4)). We need additional information to break this degeneracy. If the intrinsic scatter of the active galaxy M_BH–σ_* relation were close to 0.31 dex (as determined for elliptical galaxies by Gültekin et al. 2009), then the residual intrinsic scatter of the virial coefficient would be 0.31 dex. In contrast, if the scatter of the M_BH–σ_* relation of active galaxies were closer to 0.44 dex (as found for all galaxies by Gültekin et al. 2009), then the intrinsic scatter in f would be close to zero. Although we cannot break this degeneracy with current data, the bottom line is the same as far as determining black hole masses from reverberation-mapping data is concerned (and from single-epoch data, as a consequence). Since for the purpose of the calibration of f, the two sources of scatter are indistinguishable, the uncertainty in our calibration of f (whether it is due to diversity in BLR physics or diversity in the galaxy–AGN connection) is the largest remaining uncertainty in black hole mass determinations that rely on this technique.

To investigate any dependence of the virial coefficient on AGN properties, we collected from the literature the Hβ line widths (V_FWHM and σ_line) and the optical AGN luminosities for reverberation-mapped AGNs, corrected for the host-galaxy contribution (Peterson et al. 2004; Bentz et al. 2009a). When multiple luminosity measurements are available for given objects, we used the weighted mean of the luminosity (Table 9 of Bentz et al. 2009a). For the seven LAMP objects with M_BH and velocity dispersion measurements, Hubble Space Telescope imaging is not yet available for measuring the starlight correction to the spectroscopic luminosity. Thus, we inferred the AGN luminosity from the measured time lag using the size–luminosity relation (Bentz et al. 2009a). The bolometric luminosity was calculated by multiplying the optical AGN luminosity by a factor of 10 (Woo & Urry 2002), and we assumed 0.3 dex error in the Eddington ratio.

In Figure 6, we plot the residuals from the M_BH–σ_* relation with α = 8.12, β = 4.24 taken from Gültekin et al. (2009), and log f = 0.72 as determined in Section 6.2. The residuals were determined as $\Delta \log M_\mathrm{BH}= \log \rm{VP} + \log f - (\alpha + \beta \log \sigma _*)$. To test for potential correlations between residuals and line properties or Eddington ratios, we fit each data set with a straight line by minimizing the normalized χ² estimator. We find no dependence of the residuals on the line width (either V_FWHM or σ_line). In the case of the Eddington ratio, we find a weak dependence with a non-zero slope (−2.3^+0.6_−1.1), significant at the 95% level; however, this weak trend is mainly due to one object with the lowest Eddington ratio in the sample. When we remove that object, the best-fit slope becomes consistent with no correlation. Further investigation using a larger sample, evenly distributed over the range of the Eddington ratio, is required to better constrain the residual dependence on the Eddington ratio. In the case of the line width, it is not straightforward to test the dependence since the line width of most of our objects is relatively small, σ_line < 2000 km s⁻¹. By dividing our sample at σ_line = 1500 km s⁻¹ into two groups of similar sample size, we separately measured the virial coefficient for narrower-line and broader-line AGNs. The difference in the virial coefficient is Δlog f = 0.1–0.2, which is not significant given the uncertainty of 0.15 dex on the virial coefficient.

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We present the M_BH–σ_* relation of the reverberation sample in Figure 7, using the slope, β = 3.55 ± 0.60, determined in Section 4.1 and the average virial coefficient, 〈log f〉 = 0.72 ± 0.10, determined in Section 4.2. Compared to the local quiescent galaxies, active galaxies follow a consistent M_BH–σ_* relation with a similar slope and scatter. Note that the mean black hole mass of the reverberation sample (〈log M_BH/M_☉〉 = 7.3 ± 0.74) is an order of magnitude smaller than that of quiescent galaxies (〈log M_BH/M_☉〉 = 8.2 ± 0.79). The slightly shallower slope of the reverberation sample is consistent with the trend in the quiescent galaxies that the slope is shallower for galaxies with lower velocity dispersion (σ_* < 200 km s⁻¹); see Gültekin et al. (2009). In contrast, the slope of late-type quiescent galaxies (4.58 ± 1.58) seems higher than the slope of our active galaxies, which are mainly late-type galaxies. However, given the uncertainty in the slope (β = 3.55 ± 0.60) of the active galaxy M_BH–σ_* relation, the difference between quiescent and active galaxies is only marginal. We did not attempt to divide our sample into various morphology groups or a few mass bins to test the dependence of the slope, since the sample size is still small and biased toward lower mass objects.

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The smaller mean M_BH of the reverberation sample is due to the relative difficulty of measuring the stellar velocity dispersions for the more luminous and higher redshift quasars that have the largest black hole masses. Although there are 17 quasars with measured reverberation black hole masses, the stellar velocity dispersions have not been measured for most of those high-luminosity objects (see Watson et al. 2008; Dasyra et al. 2007). In our current sample, there are only three objects above 3 × 10⁸ M_☉. In order to better constrain the M_BH–σ_* relation of active galaxies, it will be necessary to pursue further measurements of velocity dispersions for reverberation-mapped AGNs at high masses. Additional stellar velocity dispersion measurements will be available based on Keck OSIRIS spectra in the near future (J.-H. Woo & M. A. Malkan 2010, in preparation).

The key assumption in determining the slope of the M_BH–σ_* relation is that the virial coefficient does not systematically vary as a function of black hole mass. Currently, we do not have sufficient information to test whether this assumption is valid. In an empirical study of the systematic effects of emission-line profiles, Collin et al. (2006) showed that it was necessary to use a smaller value of the virial coefficient for AGNs with larger V_FWHM. In contrast, the change of the virial coefficient was not required if the line dispersion (σ_line) was used instead. These results reflect the fact that there is a large range of V_FWHM/σ_line ratio, and that systematic uncertainties are higher if V_FWHM is used.

With the enlarged sample of 24 AGNs, we investigated the dependence of the virial coefficient on the line properties and the Eddington ratio. However, we did not clearly detect any increasing trend of the intrinsic scatter with the line profile, line width, or Eddington ratio, implying that the M_BH based on the mean virial coefficient is not systematically over- or underestimated as a function of line width or the Eddington ratio. We note that since our sample is still small and biased to narrower-line and lower Eddington ratio objects, further investigation with a larger sample, particularly at higher mass scales, is required to better constrain the systematic errors.

For single-epoch black hole mass determination, we suggest using log f = 0.72 ± 0.10 when the gas velocity is determined from the line dispersion (σ_line). If the FWHM of an emission line (V_FWHM) is measured instead of the line dispersion, then additional information on the V_FWHM/σ_line ratio will be needed in the black hole mass calculation. The value of V_FWHM/σ_line is typically assumed to be 2 (Netzer 1990), although there is a range of values. For example, the mean of V_FWHM/σ_line of our reverberation sample is 2.09 with a standard deviation of 0.45 (also see Peterson et al. 2004 and McGill et al. 2008). We did not attempt to determine an average virial coefficient using the virial product based on V_FWHM. A correction factor depending on V_FWHM/σ_line should be applied in addition to the virial coefficient, log f = 0.72, when gas velocity is determined from V_FWHM. Our empirical measurement of the mean virial coefficient is not very different from that of Onken et al. (2004), although the sample size and the dynamical range of the reverberation sample is significantly improved.

We note that our results on the mean virial coefficient are based on broad-line dispersions measured from the rms spectra, which are not generally available for single-epoch black hole mass determinations. Detailed comparison of broad-line profiles and widths between rms and single-epoch spectra is necessary to understand the additional uncertainty of single-epoch black hole mass estimates (see Denney et al. 2009a), and we will pursue further investigations of this issue using LAMP data in the future.

We summarize the main results as follows.

• 1.  
  Using the high S/N optical and near-IR spectra obtained at the Keck, Palomar, and Lick Observatories, we measured the stellar velocity dispersion of 10 local Seyfert galaxies, including 7 objects (SBS 1116+ 583A, Arp 151, Mrk 1310, Mrk 202, NGC 4253, NGC 4748, and NGC 6814) with newly determined reverberation black hole masses from the LAMP project (and NGC 5548, IC 4218, and MCG −6-30-15).
• 2.  
  For a total sample of 24 local AGNs, combining our new stellar velocity dispersion measurements of the LAMP sample and the previous reverberation sample with the stellar velocity dispersion from the literature, we determined the slope and the intrinsic scatter of the M_BH–σ_* relation in the range of black hole mass 10⁶ < M_BH/M_☉ < 10⁹. The best-fit slope is β = 3.55 ± 0.60, consistent within the uncertainty with the slope of quiescent galaxies. The intrinsic scatter, 0.43 ± 0.08 dex, of active galaxies is also similar to that of quiescent galaxies. Thus, we find no evidence for dependence of the present-day M_BH–σ_* relation slope on the level of activity of the central black hole.
• 3.  
  We determined the virial coefficient using the slope and the intercept of the M_BH–σ_* relation of quiescent galaxies, taken from two groups (Ferrarese & Ford 2005; Gültekin et al. 2009). The best-fit virial coefficient (log f) is 0.71 ± 0.10 (0.72^+0.09_−0.10) with an intrinsic scatter, 0.46^+0.07_−0.09 (0.44 ± 0.07) with the slope, 4.86 (4.24) taken from Ferrarese & Ford (Gültekin et al. 2009). We take f = 5.2 (i.e., log f =  0.72) as the best value of the mean virial coefficient, which is a factor of ∼1.7 larger than the standard factor obtained for isotropic spatial and velocity distribution. The substantial intrinsic scatter indicates that the virial coefficient is the main source of uncertainties in determining black hole masses, either using reverberation mapping or single-epoch spectra.

J.H.W. acknowledges support from Seoul National University by the Research Settlement Fund for new faculty, and support from NASA through Hubble Fellowship grant HF-51249 awarded by the Space Telescope Science Institute. T.T. acknowledges support from the NSF through CAREER award NSF-0642621, by the Sloan Foundation through a Sloan Research Fellowship, and by the Packard Foundation through a Packard Fellowship. Research by A.J.B. and A.V.F. is supported by NSF grants AST-0548198 and AST-0908886, respectively. The work of D.S. was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. Research by G.C. is supported by NSF grant AST-0507450. We thank the referee for useful suggestions.

Data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among Caltech, the University of California, and NASA; it was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community; we are most fortunate to have the opportunity to conduct observations from this mountain. We are grateful for the assistance of the staffs at the Keck, Palomar, and Lick Observatories. We thank Kartik Sheth for assisting with some of the Palomar observations. We acknowledge the use of the HyperLeda database (http://leda.univ-lyon1.fr). This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.