EXPLORING THE LOW-MASS END OF THE M_BH–σ_* RELATION WITH ACTIVE GALAXIES

The stellar velocity dispersions were measured by a direct fitting method (Burbidge et al. 1961; Rix & White 1992), in which the spectra of velocity template stars are broadened and fitted to the galaxy spectra locally in a specific spectral region. A Gaussian profile was assumed as the line-of-sight velocity distribution. We follow Barth et al. (2002) and Greene & Ho (2006a) and express the fitted model spectrum, M(x), as

$$\begin{equation} M(x) = \lbrace [T(x) \otimes \ G(x)] + C(x)\rbrace P(x), \end{equation} \tag{ 1 }$$

where T(x) is the stellar template spectrum, G(x) is the Gaussian broadening function, C(x) represents a featureless continuum, and P(x) is a polynomial factor. Since our fits are performed across a small wavelength region, we adopt a quadratic polynomial for C(x), which could also account for some other additive components, such as the "pseudo-continuum" due to Fe ii emission originating from the BLR of the AGN. The low-order multiplicative polynomial P allows for the differences between the template and the galaxy, including continuum shape, reddening in the galaxy spectrum, and wavelength-dependent flux calibration errors. We use a quadratic polynomial for P, since higher-order polynomials will tend to fit the absorption features and adversely affect the dispersion measurements (Barth et al. 2002). Besides the velocity dispersion and six parameters for these two polynomials, the redshift of the galaxy is also a free parameter. We determined the best-fit parameters by minimizing χ², using the Levenberg–Marquardt least-squares fitting routine provided by the mpfit package in IDL (Markwardt 2009).

Our collection of stellar templates includes 19 G and K giant stars observed with ESI and 22 F, G, K, and M giant stars observed with MagE. For each galaxy, we inspect the fitting and discard the templates that fail to fit. Then we list χ² of the remaining templates in ascending order, and select the first two-thirds to be well-fit templates. The measured velocity dispersion derived with the best-fitting template (minimum χ²) is adopted to be the best estimate of σ_*. Generally, the best fits were obtained with K-giant templates. The uncertainty in measurement is calculated as the quadrature sum of the fitting uncertainty of the best-fit template and the standard deviation of the measurements of all the selected templates. A minimum of six templates are used in each calculation, except for several objects, marked by ":" behind the value of σ_* in Table 1, that were not well fit by six or more templates. In these cases, the results of all well-fit templates will be accounted for in estimating the uncertainty.

The spectral region around the Ca ii λλ8498, 8542, 8662 triplet (∼8470–8700 Å, hereafter the CaT region) is ideal for measuring velocity dispersions because the CaT lines are strong, not blended with other strong lines, and relatively insensitive to stellar population variations. They are also less strongly diluted by AGN continuum contamination than stellar features at blue wavelengths. However, for objects with redshift higher than 0.05, the CaT absorption features can be affected by both night sky emission residuals and telluric absorption bands, making it difficult to obtain useful measurements. Stellar features at bluer wavelengths are unaffected by telluric absorption, and we also carried out measurements in the Mg ib region (∼5050–5430 Å) and the "Fe region" (∼5250–5820 Å) redward of Mg ib for measurements, following Barth et al. (2002) and Greene & Ho (2006a). Greene & Ho (2006a) ran a series of simulations to evaluate the contamination of narrow emission lines, including [Fe vii] λ5158, [Fe vi] λ5176, and [N i] λλ5197, 5200, around the Mg ib features, as well as the pseudo-continuum of broad Fe ii extending in this region. They found that the Fe region containing strong Fe i absorption features resulted in better recovery of σ_*, since it includes less contamination by coronal emission lines.

We tested these two regions by fitting stellar templates to a set of simulated spectra, composed as a linear combination of a K2 star, an A0 star, and a featureless linear continuum. The combined spectra were then broadened by a Gaussian velocity-broadening kernel with width ranging from σ_* = 30 to 100 km s⁻¹ in increments of 10 km s⁻¹. Slight variations in the shape of the spectra, representing calibration errors and random errors, were imposed on the broadened model spectra, and the models were degraded to S/N = 10, 30, 50, and 100 per pixel. We also redshifted the model spectra by an arbitrary value comparable to the redshifts of our observed sample. Then the modeled spectra were fitted using the methods described above, and the measured σ_* was compared with the input value. We find, for δσ_* ≡ [σ(output) − σ(input)]/σ(input), that 〈δσ_*〉 = 0.02 ± 0.11 from the Mg ib region and 〈δσ_*〉 = 0.03 ± 0.02 from the Fe region for S/N = 30, the median S/N for our observed sample. Reassuringly, the results from these test measurements are consistent with input values. We find that the scatter in results from the Mg ib region is larger than that from Fe region. The scatter for both fitting regions decreases with increasing S/N, to 0.04 and 0.01 for S/N = 100, respectively. The larger scatter of the Mg ib region might be attributed to template mismatch. We checked the template spectra and found that for different types of stars the variations in the width of the Mg ib lines are slightly larger than those for the Fe i features. For a different type of star, for example, in the sequence of G8 → K3 → K5, the Mg ib line widths increase, while Fe absorption widths are consistent; this may explain at least in part why the dispersion measurements obtained from the Mg ib region have larger scatter.

In practice, when fitting to actual galaxy spectra, we excluded wavelength regions covering emission lines including [N i] λλ5197, 5200 and the high-ionization Fe lines. The majority of the objects have several high-ionization Fe lines, including [Fe vii] λ5158, [Fe vi] λ5176, [Fe vii] λ5278, [Fe xiv] λ5303 and [Fe vi] λ5335. We tested model fits that excluded and included the Mg ib absorption features themselves, and found that the best-fit template could generally fit the Mg ib features well for galaxies not dominated by AGN Fe emission. This is because most objects in this sample have relatively low σ_*, so the mismatch of the [Mg/Fe] abundance ratio that sometimes affects template fits to elliptical galaxies with high σ_* (Worthey et al. 1992) is apparently not a significant issue for this sample.

In order to avoid issues of mismatch in flux calibration or spectral resolution across echelle orders, we prefer to measure velocity dispersions from a single echelle order (rather than from multiple orders that have been "stitched" together), and the specific fitting region for each individual galaxy was adjusted to remain within one echelle order. For the MagE data, we first measured σ_* from both the Mg ib region and the Fe region individually. We found that if these two regions are in the same order, we obtained highly consistent results from both regions. This is partly due to the limited wavelength range of each order for the MagE data and the fact that there is substantial overlap between the Mg ib and Fe regions. Therefore, if these two are on the same order for a given galaxy, we carry out a single fit extending across both of these regions, and list the result as σ_*(Mg ib) in Table 1. When a separate value is listed in the σ_*(Fe) column of the table, this denotes that the Mg ib and Fe regions fell in adjacent echelle orders and were fitted separately. Figure 2 illustrates some examples of both ESI and MagE spectra in the Fe and CaT regions.

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We were able to obtain useful measurements of σ_* for 56 of the 76 newly observed galaxies. The two galaxies with both ESI and MagE observations have consistent velocity dispersions within the uncertainty measured by two instruments. The remaining 20 galaxies are either dominated by AGN emission or have S/N too low to permit a successful fit. We also re-measured σ_* for 15 of the 17 objects from BGH05 (excluding the two objects from that sample that were highly AGN-dominated). The results are consistent with the BGH05 measurements, within the uncertainties. If we define the deviation between our new measurements and BGH05 both from the Mg ib region, as Δσ_* = log σ_*(new) − log σ_*(BGH05), the mean value of Δσ_* is less than 0.001 dex and the rms difference is only 0.02 dex. All of the σ_* measurements are summarized in Table 1. For the galaxies with more than one measurement of σ_* from different fitting regions, the corresponding results are mostly consistent within the uncertainties. We take the average of the available measurements as the best estimate of the stellar velocity dispersion in each object, and list as σ_* in Table 1.

To decompose the Hα+[N ii] lines, we first subtracted the underlying continuum. The continuum model is essentially the same as in Equation (1) used to measure σ_*, but fitted over the spectral region surrounding Hα, covering the rest-frame region 6100–7100 Å. For this model spectrum (M(x) in Equation (1)), the velocity dispersion was constrained to lie within three times the 1σ uncertainties around the measured velocity dispersion as described in the preceding section. Emission lines were masked out from the calculation of χ² for the fits. The continuum fits were carried out on spectra in which the echelle orders were "stitched" together, since in some cases the Hα line falls near the end of an echelle order. The best-fit continuum model was then subtracted from the spectra to yield a pure emission-line spectrum.

We used multiple-Gaussian models to fit the Hα and [N ii] lines as well as [S ii] λλ6717, 6731. We followed Greene & Ho (2007a), using a multi-component Gaussian fit to the [S ii] doublet to model other blended lines. Up to four Gaussians were used to model the [S ii] doublet. The velocities and widths of the two [S ii] lines were constrained to be the same, while the intensity ratio was allowed to vary. For objects with very weak or absent [S ii] emission, we used the core of the [O iii] λ5007 line as the narrow-line model instead.

The [N ii] doublet lines were constrained to have their relative rest-frame wavelengths and intensity ratio fixed to their laboratory values of 35.42 Å and 2.96, respectively. We then fit as many Gaussian components to the broad component of Hα as needed to achieve an acceptable fit: starting with a single Gaussian model, new components were added one at a time if they resulted in a 20% decrease in χ². Generally, one or two Gaussians proved sufficient to fit the broad Hα emission adequately. Figure 3 shows some examples of fits to Hα. Only a small number of galaxies with asymmetric profiles or very broad wings required a third Gaussian component. In most cases we were able to achieve an acceptable fit to the Hα+[N ii] blend, but there are a few cases (∼6) for which no acceptable fit to the narrow Hα line could be obtained with the [S ii] model. For those cases, we relaxed the width constrains on the narrow lines. In each case, the width of the narrow Hα in the best-fit model remained quite close to the [S ii] line width but the Hα fit was significantly improved by allowing its width to be a free parameter.

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We also modeled the Hβ + [O iii] region in continuum-subtracted spectra, in the rest-frame range of 4800–5150 Å, following procedures similar to those described above for the Hα+[N ii] region. We used double-Gaussian models for the broad component of Hβ as well as for [O iii], for which the double Gaussians represent a wing and a core component. The corresponding components in the two [O iii] lines are constrained to have the same velocity width, and the wavelength separations and intensity ratios are fixed at their laboratory values. For those galaxies with significant broad Fe ii emission (mainly Fe ii λλ4924, 5018), we adopted an analytical model for the broad Lorentzian system "L1" in Véron-Cetty et al. (2004, see their Appendix A). Each Fe ii line was fitted with a Lorentzian profile, allowing the line center to vary within a small range around the expected value to account for shifting from the systemic redshift. The widths of both Fe ii lines were constrained to be the same, while their flux ratio was allowed to vary. In order to model the narrow Hβ, we follow Greene & Ho (2005b) in using the profile of [S ii]. The Hβ centroid was fixed to the relative wavelength of narrow Hα if necessary, and its flux was limited to be no larger than the value for Case B recombination (Hα = 3.1Hβ; Osterbrock & Ferland 1989). The procedure produced an acceptable fit to Hβ in most cases, except for some objects (∼13) in which the profile of [S ii] does not seem to be a good model for narrow Hβ, possibly resulting from small changes in spectral resolution across different echelle orders of the spectra. In these cases we used a single Gaussian to model narrow Hβ, and we relaxed the width and flux constraints. In Figure 3 we show some examples of best-fit models for the Hβ + [O iii] region. Individual components for Fe ii emission are also shown if they are present. It turns out that the detected Fe ii lines in this sample are relatively narrow, typically 700 km s⁻¹ FWHM, significantly narrower than the commonly used Fe ii template I ZW 1, whose broad line system "L1" has FWHM = 1100 km s⁻¹ (Véron-Cetty et al. 2004).

The FWHM of the overall profile of broad Hα is used for tracing the velocity dispersion of gas in the BLR, in the close environment of the black hole. We continue to use the FWHM(Hα) as a measurement of the line width for the whole sample in a consistent way, because the commonly used Hβ lines suffer from low S/N. Broad Hβ is often weak and sometimes not even detectable for objects in our sample.

To estimate errors on the line width measurements, we should include statistical uncertainties from profile fits caused by noise and other random errors from spectral extraction and calibration, and uncertainties in continuum subtraction and deblending of broad components from narrow emission lines. In general, the formal fitting uncertainties from the profile fits are quite small, typically only 11 km s⁻¹. Since the broad lines for this low-mass sample are narrow (mostly with FWHM < 2000 km s⁻¹), the measurements of widths are much less affected by continuum subtraction than broad-lined AGNs (see Section 2.5 of Dong et al. 2008). Due to the small wavelength range over which the continuum-subtraction fits were performed and the good quality of the fits, the continuum subtraction does not add substantially to the error budget for the line widths. To explore the uncertainty from the emission-line profile deblending in more detail, we create a set of artificial spectra from the best-fit model, following Greene & Ho (2005b). For each galaxy, we created a realization of the combined emission lines from the best-fit model parameters, and added Gaussian noise to match the S/N of the data. The artificial spectra created this way suffer from the same deblending difficulty as the original data. Then the artificial spectra are fitted with multiple-Gaussian models using the fitting procedure described above. The difference between the model and measured FWHM(Hα) is typically ∼5% (or 0.02 dex). With these artificial spectra, we also investigated the uncertainty associated with the choice of model for the narrow-line profile, which often dominates the uncertainty in the decomposition of narrow/broad lines. We substituted the narrow-line model with the [S ii] profile, the core of [O iii], and narrow Hβ, respectively, in the fitting procedure with the multiple-Gaussian model above. The typical standard deviation in the FWHM of broad Hα is ∼8% (or 0.03 dex). These two sources of error are combined to be typically ∼9% (or 0.04 dex), and taken as the estimated uncertainty of FWHM(Hα).

It is instructive to compare our broad Hα measurements with those obtained from SDSS spectra by GH07. The smaller spectroscopic apertures for the Keck and Magellan data result in a smaller degree of starlight dilution of the AGN features, and the higher spectral resolution of the new data should permit more accurate deblending of the emission lines. Thus, we expect that the Keck and Magellan data should generally yield more accurate measurements of the broad Hα widths, particularly for very weak Hα emission lines. The SDSS spectra, on the other hand, have a more reliable flux calibration. In a small fraction of the objects from the GH07 sample, the broad Hα emission in the SDSS spectra is so weak that it is uncertain whether a distinct broad component is genuinely present or not, and our new spectra are particularly useful for testing the reality of these features tentatively seen in the SDSS data.

We examined how our new measurements of the broad Hα FWHM compare with the results that GH07 obtained from fitting SDSS spectra, to test whether the objects with very weak broad Hα in GH07's low-mass AGN sample were genuine Seyfert 1 galaxies or not. We visually inspected the best-fit models for all the objects and divided them into two categories: objects with "definite" and "possible" broad Hα. In the following, we refer to the two sub-categories as the d and p subsamples, as indicated in Table 1. For the d sample, the wings of broad Hα generally extend beyond the [N ii] emission lines, and the broad Hα is significantly wider than the narrow emission lines such as [N ii]. We classified objects as belonging to the p subsample if the peak amplitude of broad Hα was less than twice the rms pixel-to-pixel deviation in the continuum-subtracted spectrum in the region surrounding Hα, or if the ratio of the flux of broad Hα to the rms deviation of the continuum-subtracted spectrum was below 200. Some examples of "possible" broad Hα are shown in Figure 4.

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Among the 93 galaxies in our sample, 14 (15%, 1 galaxy observed by both ESI and MagE) had ambiguous "possible" broad Hα emission. To investigate whether this was simply due to these 15 spectra having low S/N, we examined the average S/N in the Hα+ [N ii] region, and found that only three p objects had very low S/N of <11. Overall, the objects classified as being in the p category have a median S/N of 25, compared to a median value of 40 for our whole sample, so some of the p objects may simply be suffering from low S/N. However, some of the p objects have very high S/N spectra, so we cannot attribute all the cases of ambiguity in broad Hα to low S/N. There are six p objects which show only very weak or possible broad Hα in the SDSS spectra (these are denoted as the c subsample in GH07). The other 8 out of the 14 p galaxies are classified as broad-line AGNs by GH07. The difference between the SDSS results and our new fitting results could possibly be due to intrinsic AGN variability or variable AGN obscuration. Among the eight p galaxies, SDSS J093147.25+063503.2 is confirmed to have outflow components in the narrow lines, which might have mimicked part of the broad component in SDSS spectra. There are also five galaxies in the c ("candidate") sample of GH07 which we now classify as having definite broad Hα based on our new high-resolution spectra. A likely explanation is that the better quality of our new spectra makes the decompositions more accurate and better reveals the intrinsic properties of the broad components. Therefore, 8 (9%) out of 82 galaxies previously classified as broad-line AGNs by GH07 are found to have only possible broad Hα from the high-resolution spectra. This indicates the likely rate of false positive detections of broad Hα in the SDSS sample for objects near the threshold of detectability for broad Hα. In our sample, there is one object, SDSS J145045.54−014752.8, showing obvious double-peaked features in narrow lines, and also a definite broad component in Hα (see the last object of the right panel in Figure 3).

We compare our new measurements of broad Hα FHWM to those of GH07 by defining ΔFWHM ≡ log FWHM(Hα)_new − log FWHM(Hα)_GH07. Figure 5 displays ΔFWHM versus FWHM(Hα)_new. These two measurements are in reasonable agreement up to FWHM ∼ 1000 km s⁻¹, but the difference increases as FWHM increases beyond 1000 km s⁻¹. This is probably because the higher resolution of the new data enables us to fit the emission lines more accurately with more complex multi-Gaussian models, although some of the difference may be due to intrinsic source variability as well.

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The black hole mass is estimated via the virial relationship M_BH = fR_BLRΔV²/G, where R_BLR is the radius of the BLR, and the orbital velocity at that radius is estimated by the velocity width of the broad emission line, ΔV. Assuming that the gas in the BLR is virialized, the gas velocity traces the central mass in the AGN, which is dominated by the black hole within the BLR radius.

The broad-line width can be measured with the FWHM or σ_line, which is the second moment of the profile, or the line dispersion. Both have merits and difficulties (Peterson et al. 2004). The line dispersion σ_line has been suggested to be a more robust and precise estimator of viral velocity when measured from the rms spectra of reverberation mapping data sets (Peterson et al. 2004; Collin et al. 2006). However, it is highly sensitive to the contribution from the extended line wings, and therefore less robust in single-epoch spectra when there is blending of other emission lines on the line wings (Denney et al. 2009). For measurements from single-epoch data, the FWHM is more commonly adopted (see a review in McGill et al. 2008); it is sensitive to the line core and to the decomposition of the broad and narrow components, but it is relatively insensitive to the accuracy of measurement of faint extended wings on the broad-line profile. Denney et al. (2009) suggest that when the S/N is lower than 10–20, both line width measurements would become unreliable, and line-profile fits would introduce systematic errors to single-epoch masses (e.g., ∼0.17 dex offset in M_BH estimated with FWHM; their Table 5). In practice, we generally fit the broad Hα with two Gaussians. This is a purely empirical procedure and no specific physical meaning is assigned to the two components separately. Thus, we measure FWHM(Hα) from the overall profile of the Gaussian components used to fit the broad component of Hα.

We follow GH07's approach (see Greene & Ho 2007b; see their Appendix) in calculating M_BH, which is estimated by using the line width of Hα and the BLR radius inferred from the broad Hα luminosity (Greene & Ho 2005b). In this work, the width of the broad Hα emission line is measured by decomposing the Hα+[N ii] lines in the ESI and MagE data. The method relies on the broad-line region radius–luminosity (R_BLR–L) relation derived from reverberation mapping (Kaspi et al. 2005; Bentz et al. 2006, 2009) to determine the BLR radius from the estimated continuum luminosity, which in turn is estimated from the broad Hα luminosity following Greene & Ho (2007b), since this is more accurately determined than the nonstellar continuum from our spectroscopic data. We update GH07's "recipe" for determining BH masses with the revised R_BLR–L relation presented by Bentz et al. (2009), who analyzed high-resolution HST images of 34 reverberation-mapped (RM) AGNs to obtain more accurate measurements of AGN continuum luminosity. The revised R_BLR–L relation from Bentz et al. (2009) is

$$\begin{equation} \log \left(\frac{R_{\mathrm{BLR}}}{\hbox{lt-days}} \right) = 1.50^{+0.05}_{-0.02} + 0.519^{+0.063}_{-0.066} \log \left(\frac{L_{5100}}{10^{44} \,\hbox{erg s$^{-1}$}} \right). \end{equation} \tag{ 2 }$$

We follow GH07 in assuming f = 0.75 (Netzer 1990) although we note that this f value is not derived for any specific physical model of the BLR. The choice of this particular f factor aids in the comparison of our results with prior work, as described below. From Greene & Ho (2005b), the empirical relation between the line widths of broad Hα and Hβ is

$$\begin{eqnarray} \mathrm{FWHM}(\hbox{H$\beta $}) &=& (1.07 \pm 0.07)\nonumber\\ &&\times 10^3 \left(\frac{\mathrm{FWHM}(\hbox{H$\alpha $})}{10^3 \hbox{km s$^{-1}$}} \right)^{1.03 \pm 0.03} \,\hbox{km s$^{-1}$}. \end{eqnarray} \tag{ 3 }$$

Combining these results with the virial relationship gives the BH mass as

$$\begin{eqnarray} \log \left(\frac{M_\mathrm{BH}}{M_{\odot }} \right) &=& 6.72^{+0.08}_{-0.06} + 0.519^{+0.063}_{-0.066} \log \left(\frac{L_{5100}}{10^{44} \,\hbox{erg s$^{-1}$}} \right)\nonumber\\ &&+ (2.06 \pm 0.06) \log \left(\frac{\mathrm{FWHM}(\hbox{H$\alpha $})}{10^3 \,\hbox{km s$^{-1}$}} \right). \end{eqnarray} \tag{ 4 }$$

If we substitute the continuum luminosity with luminosity of Hα using the following empirical relation from Greene & Ho (2005b),

$$\begin{equation} L_{\mathrm{H{\alpha }}}= (5.25 \pm 0.02) \times 10^{42} \left(\frac{L_{5100}}{10^{44} \,\hbox{erg s$^{-1}$}} \right) ^ {1.157 \pm 0.005} \,\hbox{erg s$^{-1}$}, \end{equation} \tag{ 5 }$$

then we obtain the BH mass as

$$\begin{eqnarray} \log \left(\frac{M_\mathrm{BH}}{M_{\odot }} \right) &=& 6.40^{+0.09}_{-0.07} + (0.45 \pm 0.05) \log \left(\frac{L_{\mathrm{H{\alpha }}}}{10^{42} \,\hbox{erg s$^{-1}$}} \right)\nonumber\\ &&+ (2.06 \pm 0.06) \log \left(\frac{\mathrm{FWHM}(\mathrm{H}\alpha)}{10^3 \,\hbox{km s$^{-1}$}} \right). \end{eqnarray} \tag{ 6 }$$

Our Keck and Magellan spectra were not all taken under photometric conditions, and since the observations were obtained through narrow spectroscopic apertures, slit losses can be significant. The SDSS spectra have a more consistent flux calibration, so it is preferable to use L(Hα) measured from the SDSS data, even if this does introduce some additional uncertainty due to the fact that the Hα line widths and luminosities are measured from non-simultaneous observations. Most objects in our sample are in the GH07 sample of active galaxies containing low-mass BHs, so we can obtain L_Hα for most of our sample from GH07. For the five objects in the BGH05 sample that were not included in GH07, we use the Hα luminosity L_Hα from Greene & Ho (2004). One object in the BGH05 sample was not part of either the GH07 or Greene & Ho (2004) catalogs, and for this object we estimated L_Hα from L₅₁₀₀ measured by Barth et al. (2005) using the L_Hα–L₅₁₀₀ relation (Equation (5)) of Greene & Ho (2005b).

The BH masses estimated by Equation (6) are systematically lower by 0.08 dex than those obtained using the method described in the Appendix of Greene & Ho (2007b), since they used the earlier version of the R_BLR–L relationship from Bentz et al. (2006). We still refer to our mass estimator as M_GH07, since it follows their basic method with only the R_BLR–L relationship updated. The rms scatter in the L_Hα–L₅₁₀₀ relation is ∼0.2 dex (Greene & Ho 2005b). If we take the uncertainties on the Hα luminosity as discussed in Greene & Ho (2005b), which are typically 0.13 dex, the typical formal uncertainties on the BH masses are 0.14 dex based on the propagation of the measurement errors on the Hα luminosity and width. This random error does not include the important systematic uncertainty in the normalization factor f. This is comparable to the observable error for BH masses in Seyfert galaxies (0.12–0.16 dex) estimated by Denney et al. (2009) based on single-epoch data, considering the effects of AGN variability and random measurement errors.

In the discussion above, we followed GH07 and updated their recipe to calculate M_BH. The method is based on the Hα emission lines only (not requiring a continuum measurement) and it allows us to compare our results in a consistent way with previous work on SDSS AGNs presented by GH07. One particular point to consider is that our mass estimates assume a normalization factor f = 0.75 for the virial masses, for consistency with GH07 and other previous work, but this is not a unique choice for f. The virial factor has been the subject of much discussion in the literature (e.g., Onken et al. 2004; Collin et al. 2006), and different f values have been used in various recipes to obtain M_BH estimates. McGill et al. (2008) compared M_BH estimators based on different emission-line and continuum measurements and showed that systematic errors could be as large as 0.38 ± 0.05 dex.

We would like to examine how different recipes change the M_BH values of our sample and the M_BH–σ_* relation that will be discussed in Section 4.2. Most of the recipes make use of continuum luminosity, but we lack direct measurements of the optical AGN continuum luminosity for our sample. An alternative way is to substitute L₅₁₀₀ with the broad Hα luminosity using the empirical relationship given by Greene & Ho (2005b). Some recipes rely on measurements of FWHM(Hβ), and to test those relationships we substitute FWHM(Hβ) with FWHM(Hα) using the empirical relationship between Hα and Hβ widths from Greene & Ho (2005b). If we use the recipe of Vestergaard & Peterson (2006), the masses would be higher by about 0.25 dex. The formalism presented by Wang et al. (2009) gives a relatively larger BH mass than our estimator in the low-mass end, typically about 0.6 dex higher at the BH mass of 〈log M_GH07/M_☉〉 = 6. They calibrated their BH mass estimator by fitting the reverberation-based masses for 35 AGNs using the continuum luminosity and Hβ FWHM, as log (M_BH/M_☉) = α + γlog (L₅₁₀₀/10⁴² erg s⁻¹) + βlog (FWHM(Hβ)/10³ km s⁻¹) and fixed γ to be 0.5. Their fits yield values of α = 1.39 and β = 1.09. This value of β is less than β = 2, which is commonly adopted for mass estimators. The 35 RM AGNs used in their fits mostly have BH masses in the range from 10⁷ M_☉ to 10⁹ M_☉, and α tends to compensate for the mass difference in the fitting. This results in larger BH masses in the low-mass range, i.e., for objects below about 10⁷ M_☉.

There is some lower limit to the BH masses that we are able to detect in SDSS data, due to a combination of factors. A possibly low BH occupation fraction in very low-mass galaxies would result in a low detection rate. The low luminosity of AGNs with small black holes makes the AGNs hard to detect. The host galaxy has to be bright enough to be spectroscopically targeted by SDSS, and the S/N of SDSS spectra is limited. Moreover, the large aperture of SDSS spectra can mix AGN emission with H ii regions and dilute the AGN signal. Since we have identified the "definite" broad-lined AGNs that have genuine broad Hα in our sample, we obtain a cleaner sample that illustrates how low we can really go in selecting objects with low M_BH using SDSS. Our definite broad-lined sample with successful measurements of σ_* has a median BH mass 〈M_BH〉 = 9.5 × 10⁵ M_☉, and a minimum of 2 × 10⁵ M_☉.

Figure 6 shows the M_BH–σ_* relation for our sample of AGNs with low BH masses, and for active galaxies with higher black hole masses as well as nearby galaxies with direct dynamical measurements. The comparison sample of 56 active galaxies with higher BH masses based on single-epoch spectroscopy was selected from SDSS DR3 with z ⩽ 0.05 by Greene & Ho (2006b, hereafter GH06 sample), but the M_BH values have been re-calculated with our updated mass recipe (Equation (6)). Literature data on 24 RM AGNs with stellar velocity dispersion measurements presented by Woo et al. (2010 and references therein) were included. The M_BH for these RM AGNs were calculated from the virial products (VPs) listed in Table 2 of Woo et al. (2010). Since σ_line was used in VP, an isotropic velocity distribution gives f_σ = 3, assuming σ_line = FWHM/2 (Onken et al. 2004). The ratio between FWHM and σ_line is different for different line profiles, i.e., FWHM/σ_line = 2.35 for a Gaussian profile, and it varies around an average of two (Collin et al. 2006). Instead of the virial factor f obtained by Woo et al., we assume f_σ = 3 for consistency with the GH07 sample. This would decrease the masses by 0.24 dex compared to those listed in their work. We also included the well-known intermediate-mass BHs in NGC 4395 with reverberation mass M_BH = (3.6 ± 1.1) × 10⁵ M_☉ (Peterson et al. 2005) and the velocity dispersion with an upper limit σ_* ⩽ 30 km s⁻¹ from Filippenko & Ho (2003), and POX 52 with virial mass M_BH ≈ (3.1–4.2) × 10⁵ M_☉ based on broad Hβ line width and L₅₁₀₀ (Barth et al. 2004; Thornton et al. 2008) and velocity dispersion σ_* = 36 ± 5 km s⁻¹ from Barth et al. (2004). Masses for both objects were calculated utilizing the virial coefficient 〈f_σ〉 = 5.5 (Onken et al. 2004; Peterson et al. 2004); we also scaled down both masses by 0.26 dex for consistency with other objects shown in the plot. Note that the virial normalization factor we assumed is arbitrary, and the derived M_BH was decreased by about 0.26 dex compared to masses derived using the f factor from Onken et al. (2004). But the slope of M_BH–σ_* relation is independent of f. Our low-mass sample appears to smoothly follow the extension of the M_BH–σ_* relation for inactive galaxies, and there seems to be no strong change in slope across the whole mass range including active and inactive galaxies. We will quantify the slope for all the active galaxies described above.

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Assuming a log–linear form log M_BH = α + βlog (σ_*/200 km s⁻¹), we fit the slope and zero point of the M_BH–σ_* relation for all the active galaxies with two regression methods: the symmetric least-squares fitting method, fitexy (Press et al. 1992) modified following Tremaine et al. (2002), and the maximum-likelihood estimate (MLE) method linmix_err (Kelly 2007), both implemented in IDL. The former method accounts for uncertainties in both coordinates as well as the intrinsic scatter by adding a constant to the error in the dependent variable, and solves for the best linear fit by minimizing the reduced χ² to unity. Linmix_err uses a Bayesian method to account for measurement errors and intrinsic scatter, and computes a posterior probability distribution function of parameters. Since there is no significant difference between the results from the two regression methods, we will only quote results from the MLE method linmix_err. We obtain α = 7.68 ± 0.08 and β = 3.32 ± 0.22, with ₀ = 0.46 ± 0.03 dex. The slope is a bit flatter than that for nearby galaxies, which have β = 4.24 ± 0.41 (Gültekin et al. 2009), but consistent with Greene & Ho (2006b) and Woo et al. (2010), who both showed evidence of a shallower slope for active galaxies than the inactive M_BH–σ_* relation. If we fix the slope β to the best-fit value of 4.24 for nearby galaxies, we obtain, for the full sample of active galaxies, a zero point of α = 7.99 ± 0.04, which is −0.13 ± 0.09 offset from the value of α = 8.12 ± 0.08 from Gültekin et al. (2009).

We now consider residuals in the M_BH–σ_* relation, defined as ΔM_BH ≡ log (M_BH/M_☉) − log (M_BH/M_☉)_fit, where log (M_BH/M_☉)_fit is calculated from σ_*, as a function of the bolometric luminosity and Eddington ratio (Figure 7). We follow GH07's method for bolometric correction to estimate the bolometric luminosity from L_Hα, according to L_bol = 2.34 × 10⁴⁴(L_Hα/10⁴²)^0.86 erg s⁻¹. Among the 93 objects in our sample, the median value of the Eddington ratio is 〈L_bol/L_Edd〉 = 0.3, where L_Edd ≡ 1.26 × 10³⁸(M_BH/M_☉). The sample is dominated by objects radiating at substantial fractions of their Eddington limits, since the SDSS selection favors identification of the most luminous AGNs in any mass range. We find that the residual ΔM_BH is significantly correlated with L_bol (Figure 7(a)). The Spearman rank correlation coefficient is r_s = 0.33, with a probability P < 10⁻³ that no correlation is present. Moreover, ΔM_BH shows a strong anti-correlation with L_bol/L_Edd (Figure 7(b), r_s = −0.53, P < 10⁻⁹). We are wary of overinterpreting these correlations, since L_bol/L_Edd is formally anti-correlated with M_BH  and M_BH is correlated with L_bol because both of them are deduced based on L_Hα.

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Note that for our M_BH estimates we are using propagated measurement uncertainties based on the errors in Hα width and luminosity, while the true uncertainties in M_BH are probably dominated by the uncertainty in the BLR geometry and the chosen value of f. If we adopt 3σ uncertainties in M_BH as measurement errors and do the regression again, there is little change in either the slope or zero point of the derived M_BH–σ_* relation, but the intrinsic scatters decrease by about 35% to ₀ = 0.28 ± 0.05 dex. The regression parameters for different samples and different uncertainties considered are listed in Table 2. Woo et al. (2010) recently reported the M_BH–σ_* relation for 24 RM active galaxies with BH mass 10⁶ < M_BH/M_☉ < 10⁹. They obtained a slope β = 3.55 ± 0.60 and intrinsic scatter ₀ = 0.43 ± 0.08, which are also consistent with our result. In the following, we will examine whether the intrinsic scatter varies as a function of mass.

Table 2. Regression Parameters

Sample	N	Uncertainty	α	β	0
(1)	(2)	(3)	(4)	(5)	(6)
Full	155	1σ	7.68 ± 0.08	3.32 ± 0.22	0.46 ± 0.03
Full	155	3σ	7.69 ± 0.08	3.28 ± 0.22	0.28 ± 0.05
Full w/o "p"	142	1σ	7.75 ± 0.08	3.48 ± 0.21	0.41 ± 0.03

Notes. Column 1: data set considered; the "Full" sample comprises objects with both M_BH and σ_* available, including the SDSS sample and 24 RM AGNs with stellar velocity dispersion measurements presented by Woo et al. (2010), NGC 4395, and POX 52. The "Full w/o p" subsample denotes the full sample minus the objects classified as having possible broad Hα. Column 2: number of objects in each sample. Column 3: adopted uncertainties for M_BH. Column 4: zero point assuming log M_BH = α + β log (σ_*/200 km s⁻¹), for f = 0.75. Column 5: M_BH–σ_* slope. Column 6: intrinsic scatter.

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We divide the active galaxies mentioned above into two data sets, with lower and higher M_BH, and analyze the intrinsic scatter ₀ for the two data sets. The lower-M_BH data set includes our sample and the Lick AGN Monitoring Project (LAMP) sample presented by Woo et al. (2010). The higher-M_BH data set includes the GH06 SDSS sample and 17 previous RM active galaxies (collected also in Woo et al. 2010). Assuming that the M_BH recipe we used is equally valid at all masses (which is not necessarily the case), we fit the two data sets and quantify the intrinsic scatter to be ₀ = 0.38 ± 0.04 dex for the low-M_BH sample presented here and ₀ = 0.40 ± 0.04 dex for the higher-M_BH objects. The scatter for low-mass data set slightly changes when including NGC 4395 and POX 52, or excluding the LAMP sample. The scatter decreases a little to ₀ = 0.34 ± 0.04 dex if we exclude the "possible" broad Hα objects. The derived scatters for the two data sets are consistent, indicating that the intrinsic scatter in virial BH masses is not strongly mass-dependent. We also examine the standard deviation of residuals from the best-fit M_BH–σ_* relation for the two data sets, and get 0.50 dex and 0.44 dex for the low and high mass range, respectively.

The intrinsic scatter in M_BH–σ_* across the entire mass range is about 0.46 dex, corresponding to a factor of ∼3. That is close to the intrinsic scatter for inactive galaxies, ₀ = 0.44 ± 0.06 (Gültekin et al. 2009). We will show in Section 4.5 that the intrinsic scatter persists or even increases if we scale the active galaxies to follow the M_BH–σ_* relation of nearby galaxies with direct measurements of BH masses and then fit the relation for the combined sample of the active galaxies and nearby galaxies. Some of the dispersion in the M_BH–σ_* relation may result from different galaxy morphological types following different M_BH–σ_* loci, as recent evidence has suggested (Hu 2008; Greene et al. 2008, 2010; Graham & Li 2009; Gültekin et al. 2009; Gadotti & Kauffmann 2009). For example, the intrinsic scatter in the M_BH–σ_* relation of early-type galaxies (referring to elliptical galaxies and S0 galaxies) is smaller than that for late-type galaxies (Graham 2008; Gültekin et al. 2009). There is tentative evidence for a similar trend in the Greene & Ho (2004) sample, whose host galaxy morphologies have been investigated by Greene et al. (2008) using HST images. The difference could either be due to an unaccounted systematic error in the M_BH measurements for spirals, or because the scatter in M_BH for spirals is actually larger, as might happen if the behavior of BHs hosted by pseudobulges (see Kormendy & Kennicutt 2004 for a review) and classical bulges is different. This has been suggested by Hu (2008), who found that pseudobulges tend to host lower M_BH than classical bulges at a given σ_*. Similarly, a recent study by Greene et al. (2010) presents stellar velocity dispersions for a sample of nine megamaser disk galaxies with accurately measured BH masses in the range of ∼10⁷M_☉. They find that the maser galaxies fall below the M_BH–σ_* relation of elliptical galaxies defined by Gültekin et al. (2009). Based on morphology and stellar population properties, they speculate that most of the nine megamaser galaxies are likely to contain pseudobulges, providing further support for the idea that the pseudobulge M_BH–σ_* relation is offset below the relation for classical bulges (Hu 2008; Gadotti & Kauffmann 2009). In Section 4.3, we investigate the possible dependence of the M_BH–σ_* relation fit on the host galaxy morphological type for our sample.

Simulations of massive BH growth indicate that the scatter in the M_BH–σ_* relation should increase toward the low-mass end. In models with high-mass seeds, galaxies evolve from well above the present-day M_BH–σ_* relation toward the relation, while galaxies starting with low-mass seeds first lie far below the relation and their BHs grow to move toward the relation (Volonteri 2010 and references therein). Furthermore, Volonteri (2010) suggested that for heavy seeds of ∼10⁵ M_☉, the low-mass end of the M_BH–σ_* relation will flatten to an asymptotic value. In our sample, we do not find clear evidence for increasing scatter in the M_BH–σ_* relation at low masses, and we cannot discriminate among possible seed models based on our sample. Increasing scatter at the low-mass end should nevertheless be a generic result regardless of the seed masses or the details of the BH growth mechanism. As discussed by Peng (2007), major mergers decrease the scatter of the M_BH–M_gal relation because of a central-limit tendency of the BH-to-galaxy mass relation to eventually converge to some mean value.

Given the previous expectation that low-mass galaxies should show a larger scatter in the M_BH–σ_* relation than high-mass galaxies, as well as the recent observational suggestions of an offset M_BH–σ_* relation for pseudobulges relative to classical bulges, it remains somewhat puzzling that the Type 1 AGN samples do not show an obvious trend of either increasing scatter or changes in slope toward lower masses. This appears to be the case both for the small sample of AGNs with reverberation mapping (Woo et al. 2010), and for the larger sample of objects with single-epoch masses presented here. It could be that the random errors on individual AGN virial BH mass estimates are so large that the scatter in the mass estimates (resulting from variations in BLR kinematics, inclination, or other properties) simply dominates over the intrinsic scatter in the M_BH–σ_* relation for these objects, preventing us from detecting mass-dependent variations in the intrinsic scatter. Direct stellar-dynamical measurements of BH masses in nearby RM AGNs can be an important cross-check on the derived masses. However, while observations so far do not reveal evidence for any dramatic offsets between stellar-dynamical and reverberation masses, only a few objects are presently amenable to both types of measurements (Davies et al. 2006; Onken et al. 2007; Hicks & Malkan 2008). For the vast majority of reverberation-mapped AGNs, the angular size of the BH's gravitational sphere of influence is too small to be resolved by either HST or ground-based telescopes employing adaptive optics, and only a small number of broad-lined AGNs are currently suitable targets for stellar-dynamical measurements of M_BH.

Recent work by Graham et al. (2011) suggests that the M_BH–σ_* relation is different for AGNs with different host galaxy morphological types (i.e., barred disks or unbarred disks). For our sample, we can carry out a preliminary check for any offsets in the M_BH–σ_* plane as a function of host morphology. The SDSS images are not sufficient to carry out a full census of host galaxy types for this sample, but there is morphological information for a subset of our sample from an HST Wide Field Planetary Camera 2 (WFPC2) imaging snapshot survey with the F814W filter (Jiang et al. 2011). Among the galaxies in our sample having HST images, there are 38 objects which are morphologically classified as disk galaxies and which also have M_BH and velocity dispersion measurements, including 12 barred and 26 unbarred galaxies. Only one galaxy from the GH06 higher-mass SDSS sample is included in the HST imaging survey; it is classified as an unbarred disk galaxy. Including the reverberation-mapped AGN sample compiled by Woo et al. (2010), as well as NGC 4395 and LAMP objects, we have morphological information for a total of 63 disk galaxy AGN hosts (25 barred and 38 unbarred). The morphological classifications for the reverberation-mapped objects are from the NASA/IPAC Extragalactic Database where available, and otherwise adopted from classification by Bentz et al. (2009), while the classification for LAMP objects are provided by M. C. Bentz et al. (2011, in preparation). If we remove the galaxies with a "possible" broad Hα component, there are 56 disk galaxies with morphological information (22 barred and 34 unbarred). One caveat in our division is that it is based on optical images, while it has been suggested that some galaxies appear to be unbarred in optical bands, but are very clearly barred when viewed in the infrared (e.g., Block & Wainscoat 1991; Mulchaey & Regan 1997). Despite this caveat, we are limited by the data presently available and we carry out a preliminary examination based on the optical host-galaxy morphologies.

We re-do the regression fits for the M_BH–σ_* relation as in Section 4.2 for different subsamples, and the results are listed in Table 3. The best fit of the M_BH–σ_* relation for the full subsample of disk galaxies (α = 7.50 ± 0.13, β = 3.04 ± 0.30) is consistent with that for the entire sample of active galaxies described previously (α = 7.68 ± 0.08, β = 3.32 ± 0.22). Within the uncertainties, the slope is consistent with the slope of 4.05 ± 0.83 for 24 nonellipticals reported by Gültekin et al. (2009). Fixing the slope to 4.05 yields an intercept of α = 7.89 ± 0.06, which also agrees well with α = 8.01 ± 0.16 for nonellipticals (Gültekin et al. 2009). The fit for barred disks only gives α = 7.81 ± 0.27 and β = 4.13 ± 0.72, similar to the M_BH–σ_* relation (α = 7.80 ± 0.10, β = 4.34 ± 0.56) for barred inactive galaxies presented by Graham et al. (2011).

Table 3. Regression Parameters for Barred/Unbarred Disk Galaxies

		Free Fit				Fit with Fixed Slope
Sample	N	α	β	0		α	β	0
(1)	(2)	(3)	(4)	(5)		(6)	(7)	(8)
Disk	63	7.50 ± 0.13	3.04 ± 0.30	0.43 ± 0.05		7.96 ± 0.07	4.24	0.50
Barred	25	7.81 ± 0.27	4.13 ± 0.72	0.46 ± 0.09		7.86 ± 0.09	4.24	0.40
Unbarred	38	7.44 ± 0.15	2.79 ± 0.34	0.42 ± 0.06		8.04 ± 0.09	4.24	0.54
Disk w/o "p"	56	7.56 ± 0.13	3.19 ± 0.31	0.42 ± 0.05		7.95 ± 0.07	4.24	0.46
Barred w/o "p"	22	7.80 ± 0.29	4.03 ± 0.80	0.46 ± 0.09		7.87 ± 0.09	4.24	0.40
Unbarred w/o "p"	34	7.52 ± 0.15	3.01 ± 0.33	0.40 ± 0.06		8.01 ± 0.09	4.24	0.48

Notes. Column 1: data set considered; the "Disk" ("Barred"/"Unbarred") sample comprises the "Full" sample with morphological type available and classified as (barred/unbarred) disk galaxies. The "Disk w/o p" subsample denotes the "Disk" sample minus the objects classified as having "possible" broad Hα. Column 2: number of objects in each subsample. Column 3: zero point assuming log M_BH = α + β log (σ_*/200 km s⁻¹), for f = 0.75. Column 4: the M_BH–σ_* slope. Column 5: intrinsic scatter. Column 6: zero point of the M_BH–σ_* relation with the slope fixed to that of inactive galaxies listed in Column 7, and the upper limits on the intrinsic scatter listed in Column 8.

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Figure 8 shows the morphological type for the disk galaxies on the M_BH–σ_* plot, and also the best fits of the M_BH–σ_* relations for barred and unbarred disks (see Table 3). We consider our most reliable results to come from the subset of our sample that excludes the p objects with uncertain detections of broad Hα. For this subsample, we find α = 7.80 ± 0.29 for barred disks versus 7.52 ± 0.15 for unbarred disks, and the slope β = 4.03 ± 0.80 for barred disks versus 3.01 ± 0.33 for the unbarred subset. Thus, the barred disks have a marginally larger zero point and steeper slope than unbarred disks, but the difference is not significant given the substantial uncertainties on the fits. We also fit the M_BH–σ_* relation with the slope fixed to that of inactive galaxies, 4.24 (Gültekin et al. 2009), as an additional test for any offset. With fixed slope, the distinction between the barred and unbarred subsamples almost vanishes (see Table 3). Alternatively, if we were to adopt the high slope value proposed by Graham et al. (2011) (β = 5.13), then the zero points would be all increased by about 0.35.

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We have measured σ_* for our sample from the ESI and MagE spectra obtained with slit widths of 075 and 1'', respectively. At the median redshift (〈z〉 = 0.08, or luminosity distance D_L = 300 Mpc) for our sample, 1'' corresponds to a scale of 1.5 kpc. If the bulge is not the dominant component of the host galaxy over that scale, then the slit spectra will be contaminated by the light from the disk, and the measured σ_* will include contributions from the rotational velocity of stars orbiting on the disk. Specifically, σ_* will be increased artificially in an edge-on disk, and might be slightly decreased in a face-on disk. In order to examine the effect of disk inclination on our sample, we use the axis ratio b/a to indicate inclination, where b/a close to 1 implies a face-on (low-inclination) system and b/a close to 0 denotes an edge-on (high-inclination) system. The disk axis ratio is obtained from fits to HST images. For the subset of our sample having HST imaging, the axis ratio measurements are based on the analysis of the WFPC2 snapshot imaging survey data (Jiang et al. 2011). For reverberation-mapped AGNs, axis ratios measured from HST imaging data are taken from Bentz et al. (2009, their Table 4) and from M. C. Bentz et al. (2011, in preparation). We divide the sample into three axis-ratio bins: b/a > 0.88, 0.72 < b/a < 0.88, and b/a < 0.72.

Figure 9 shows the disk galaxies compiled in Section 4.3 in the M_BH–σ_* plot, labeled by axis ratio. The low-inclination objects (large blue triangles) lie apparently to the left of the medium- (large orange squares) and high-inclination (large red circles) objects. We define ΔM_BH as the M_BH deviation from the best-fit M_BH–σ_* relation reported by Gültekin et al. (2009). The distributions of ΔM_BH for the three bins are shown in Figure 10. A Kolmogorov–Smirnov (K-S) test on the distributions of ΔM_BH for the low- and high-inclination subsamples yields a probability of 0.011, indicating that the distribution of the offset ΔM_BH is likely to be significantly different for these two subsamples. Two extra PG objects (PG 1229+204 and PG2130+099) with M_BH and σ_* measurements are also shown in Figure 9. Their axis ratios of disk components are 0.62 and 0.55, respectively, and they are classified as high-inclination objects. Including these two objects in the K-S test above does not alter the result. We also carry out fits to the M_BH–σ_* relation for each subsample, and list the best-fit parameters in Table 4. With the slope fixed to 4.24, the low-inclination subsample shows offsets from the other two subsamples by more than 0.4 dex. This gives additional evidence that the low-inclination and high-inclination systems are significantly offset in the M_BH–σ_* plane, such that the more highly inclined host galaxies tend to have higher σ_* at a given value of M_BH. The offset is consistent with the expectation that disk rotation contributes more significantly to the measured values of σ_* in the more highly inclined systems. This may be one source of error contributing to the scatter in the relation, and it may also introduce a bias to the zero point of the relation. Although it would be preferable to plot an M_BH–σ_* relation that did not include this possible bias, there is no simple way to correct for the rotational contribution to the velocity dispersions from our data.

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Table 4. Regression Parameters for Different Disk Inclination

Sample	Axis Ratio		Free Fit				Fit with Fixed Slope
		N	α	β	0		α	β	0
(1)	(2)	(3)	(4)	(5)	(6)		(7)	(8)	(9)
High	b/a < 0.72	20	7.42 ± 0.25	3.24 ± 0.70	0.50 ± 0.10		7.74 ± 0.11	4.24	0.48
Medium	0.72 < b/a < 0.88	20	7.70 ± 0.20	3.86 ± 0.49	0.38 ± 0.10		7.85 ± 0.09	4.24	0.36
Low	b/a > 0.88	15	7.59 ± 0.32	2.76 ± 0.67	0.44 ± 0.12		8.25 ± 0.13	4.24	0.48

Notes. Column 1: data set considered; the "High" ("Medium"/"Low") sample comprises the "Disk" sample with the axis ratio of the disk component available, divided into three bins of the inclination angle. Column 2: axis ratio range of each subsample. Column 3: number of objects in each subsample. Column 4: zero point assuming log M_BH = α + β log (σ_*/200 km s⁻¹), for f = 0.75. Column 5: the M_BH–σ_* slope. Column 6: intrinsic scatter. Column 7: zero point of the M_BH–σ_* relation with the slope fixed to that of inactive galaxies listed in Column 8, and the upper limits on the intrinsic scatter listed in Column 9.

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In the discussions above, we have adopted an arbitrary virial factor f = 0.75 for consistency with most previous work on the SDSS-selected GH07 sample. Alternatively, we can also fit for the value of f that brings our sample into the best agreement with the local M_BH–σ_* relation, following the same method used by Onken et al. (2004) and Woo et al. (2010) for reverberation-mapped AGNs. The values reported by Gültekin et al. (2009) are α = 8.12 ± 0.08 and β = 4.24 ± 0.41. Fixing the slope to 4.24 for our full sample, we obtain α = 7.99 ± 0.04 for f = 0.75 as presented in Section 4.2. Therefore, the mean virial factor determined from this fit would be 〈f〉 = 1.01 ± 0.21. Alternatively, fitting our sample to the M_BH–σ_* relation from Graham et al. (2011) would give a lower value of 〈f〉 = 0.52 ± 0.08. If we take the subsample of the 15 low-inclination galaxies above in Section 4.4 and scale to the M_BH–σ_* relation from Gültekin et al. (2009), we get 〈f〉 = 0.56 ± 0.17. These results are reasonably compatible with previous calibrations of 〈f〉 for reverberation-mapped AGNs, taking into account the fact that our work uses the FWHM as the measure of line width while the reverberation-based mass scale is based on the second moment of the Hβ line (e.g., Onken et al. 2004; Woo et al. 2010). Fundamentally, the calibration of 〈f〉 via the M_BH–σ_* relation is still limited by small-number statistics, and improving the determination of the virial normalization factor will require increasing the number of AGNs having high-quality reverberation mapping data as well as the number of spiral galaxies having direct measurements of M_BH from spatially resolved dynamics, and equally important, obtaining accurate σ_* measurements for them.

For completeness of our analysis, we fit the M_BH–σ_* relation for the combined sample of active and nearby galaxies. First, we use the virial factor derived above to recompute the BH masses for the active galaxies. Then we re-do the regression fit for the combined sample. Using the virial factor 〈f〉 = 1.01 derived above, which is based on normalizing the AGNs to the M_BH–σ_* relation of Gültekin et al. (2009), we combine the AGNs with the 49 nearby galaxies with direct dynamical measurements of BH masses presented by the same authors. Then the regression for the combined sample yields α = 8.02 ± 0.05 and β = 3.81 ± 0.16 with an intrinsic scatter of ₀ = 0.46 ± 0.03. If the alternative 〈f〉 = 0.52 is used, then the fit yields a zero point of 7.89 ± 0.06 and a slope of 4.14 ± 0.16 with an intrinsic scatter of 0.49 ± 0.03.⁷ The intrinsic scatter for the combined sample is the same as or even larger than that for active galaxies only. We note that the slope and the intrinsic scatter depend on how we select the virial factor, because the active galaxies are scaled to the local M_BH–σ_* relation first and then included for the fit.

It is well known that the velocity dispersion of the ionized gas in the narrow-line region (NLR), most often measured from the [O iii] λ5007 line, correlates with the stellar velocity dispersion of the host galaxy bulge, suggesting that gravity dominates the global kinematics of the NLR for active galaxies (e.g., Whittle 1985; Nelson & Whittle 1996; Boroson 2003; Greene & Ho 2005a; Bian et al. 2006; Komossa & Xu 2007; Barth et al. 2008). The high-resolution ESI and MagE data provide an opportunity to examine this relationship in galaxies of small σ_*. In addition to the [O iii] line, we also consider lines from lower ionization species such as [N ii] λ6583 and the [S ii] λλ6716, 6731 doublet, and check their relationship to σ_* for our sample.

For the [O iii] λλ4959, 5007 lines, each was fitted by a double-Gaussian model, representing a core and often (but not always) a blueshifted wing component. Generally, the core component has a higher peak flux than the wing component. In some cases, the two-component model sometimes overfits noise fluctuations near the line peak, resulting in individual Gaussian components with very narrow widths. We set an empirical criterion for the separation of the two components (>80 km s⁻¹) to define the blue- and redshifts. The blue- or redshifted wing component was removed from the FWHM_core measurements if the separation was >80 km s⁻¹. For the other cases, where the centroids of the two components were separated by less than 80 km s⁻¹, we measured the FWHM from the [O iii] profile and took the resulting value to be FWHM_core. Detailed studies of some galaxies where the NLR is spatially resolved reveal the possibility that the entire [O iii] line may be outflowing (Fischer et al. 2011 and references therein). We use the relative velocity of the narrow component of Hβ to help determine the core component, since the Balmer recombination lines are less affected by the outflow when present.

Figure 11 illustrates the relationships between FWHM([O iii])/2.35 and σ_* for the core and wing components of the [O iii] line as well as for the entire [O iii] profile (Figure 11(c)). The FWHM of the [O iii] core correlates with σ_*, with a Spearman rank correlation coefficient r_s = 0.61, and a probability P_null less than 10⁻⁴ for the null hypothesis of no correlation (Figure 11(a)). The relationship between the FWHM of the entire [O iii] profile and σ_* has a larger scatter, consistent with previous results (e.g., Nelson & Whittle 1996; Greene & Ho 2005a; Barth et al. 2008). This illustrates that if we use the FWHM of [O iii] as a proxy for σ_* in active galaxies, the high-velocity wing component should be removed from the profile. The velocity shifts between the centroids of the wing to core components, defined as ΔVel, are mainly negative (Figure 12, left panel), representing blueshifts of the wing components, while the centroid velocity of the core components are generally close to the systemic velocity measured from the stellar absorption lines. If the blueshifted wing component of [O iii] is driven by winds or outflows (Whittle 1985; Komossa et al. 2008), then the strength or velocity offset of the wing component could correlate with some measure of AGN activity, such as the Eddington luminosity ratio, L_bol/L_Edd. Thus, we investigate how the equivalent width (EW) ratio of the [O iii] wing to core components correlates with Eddington ratio, but we find no correlation between them. We find that ΔVel correlates with L_bol/L_Edd only weakly (r_s = −0.24, Figure 12, middle panel). We also find that ΔVel correlates well with the FWHM of the wing component (r_s = 0.56, P_null < 0.001) and less strongly with the FWHM of the core component (r_s = 0.41, P_null = 0.03). This confirms that the previously established correlation between [O iii] blueshift and line width continues to hold in this low-mass regime.

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We also examine the correlation between gas and stellar velocity dispersions for the [S ii] doublet and the [N ii] λ6583 emission line, which has been decomposed from broad Hα. Since we use the [S ii] profile as the model for the narrow lines for most cases (discussed in Section 3.2), we can use either the [S ii] or [N ii] line. The [N ii] line is adopted here because there are a few cases wherein [S ii] is too weak to measure its width reliably. The FWHM is measured from the overall, combined profile of Gaussians fitted to each narrow emission line. As shown in Figure 11, the FWHM of [N ii] correlates well with σ_*, with a Spearman rank correlation coefficient r_s = 0.72 and P_null < 10⁻⁴. We follow Ho (2009) to define the gaseous velocity dispersion σ_g ≡ FWHM/2.35 and the residual of σ_g − σ_* to be Δσ ≡ log σ_g − log σ_*. While Ho (2009) finds a correlation between Δσ and Eddington ratio for nearby active galaxies, we do not see such a correlation in our sample (Figure 12, right panel). Our sample mostly lies below the relation found by Ho (2009), but is generally consistent with Greene & Ho (2005a). The objects scatter around Δσ = 0 (σ_g = σ_*), regardless of increasing Eddington ratio.

We have measured emission lines of Hβ, [O iii] λ5007, Hα, [N ii] λ6583, and [S ii] λλ6716, 6731, which enable us to plot our sample on line-ratio diagnostic diagrams (e.g., BPT (Baldwin–Phillips–Terlevich) diagrams; Baldwin et al. 1981; Veilleux & Osterbrock 1987), as shown in Figure 13, and to compare our measurements with the SDSS measurements obtained with larger spectroscopic apertures (fiber diameter of 3''). The flux of [O i] λ6300 was also measured from the starlight-subtracted spectra by fitting with a single Gaussian. From the new data (with aperture sizes of 075 × 10 and 10 × (15–3'') for the ESI and MagE data, respectively), we see that a few objects move out of the H ii region of the BPT diagrams and into the Seyfert region above the maximum starburst line (Kewley et al. 2006). With the small-aperture data, most objects lie in the Seyfert region of the diagram, with some objects extending to relatively low values of [N ii]/Hα which are indicative of metallicity lower than typical for classical Seyferts (Groves et al. 2006).

We investigate whether the line ratios measured from the small-aperture spectra systematically change relative to measurements based on the SDSS spectra. The distributions of line ratios are shown on the top and right panels in Figure 13. These distributions for the low-ionization ratios [N ii]/Hα, [S ii]/Hα, and [O i]/Hα only change slightly as a function of aperture size, with systematic offsets less than 0.08 dex between the median values for the small-aperture and SDSS measurements. This could be the effect of various parameters, due to the complex NLR conditions, including metallicity, ionization parameter, electron density, and dust reddening. But the [O iii]/Hβ ratio measured from the small-aperture data is systematically 0.3 dex higher than the values measured from the SDSS fiber aperture. The simplest interpretation is that in the larger SDSS aperture the emission lines included a larger contribution of emission from H ii regions, which would have lower [O iii]/Hβ than the AGNs. This would also imply that the host galaxies are actively star-forming, and we may be witnessing growth of the host galaxy bulge or pseudobulge coevally with the BH growth. Integral-field observations at high angular resolution would be particularly useful for examining the distribution and luminosity of H ii regions in the host galaxies.

These line-ratio diagrams are generally used to discriminate between star-forming galaxies and AGNs (e.g., Ho et al. 1997; Kauffmann et al. 2003; Hao et al. 2005; Barth et al. 2008). However, as noted by GH07, the broad-line AGNs do not lie exclusively above the maximum starburst line of Kewley et al. (2006) on the BPT diagrams, and some objects even fall below the empirical line of Kauffmann et al. (2003) in the region of pure star-forming galaxies. That is, judging by the narrow-line ratios alone, these objects would not be classified as AGNs. For these objects, the detection of the broad Hα emission is the primary clue that an AGN is present, and if the broad emission lines were obscured by a dusty parsec-scale torus or by dust lanes in the host galaxy, then these objects would not have been identified as AGNs at all.

We explore whether the objects which fall below the maximum starburst line are unusual in terms of any of the measured parameters including σ_*, σ_gas (indicated as FWHM([N ii])/2.35), the flux ratio between [O iii] wing and core, the Fe ii strength (relative to [O iii]), FWHM(Hα), or the flux ratio between the narrow and broad components of Hα (F_n/b). We find that this sub-population of objects does not exhibit any unusual properties in terms of these parameters, except in F_n/b, which shows a higher relative strength of the narrow components in this sub-population. Further investigation shows that the median redshift of the sub-population (z ∼ 0.1) is larger than the whole sample (z ∼ 0.08). Thus, the higher F_n/b ratio could be explained by the dilution of extended star formation contained in the aperture. It is unfortunately difficult to derive any detailed information on the stellar populations of the hosts from the spectra because of the strong AGN contribution to the spectra. We examined the ESI and MagE spectra, finding that higher-order Balmer absorption of significant strength is visually apparent in 18 objects (about 20% of the sample), of which about 10 objects are in the sub-population below the maximum starburst line. The absorption features imply contributions from A stars. Thus, compared with the sample as a whole, a larger fraction of objects in the H ii sub-population appear to have significant contributions from intermediate-age stellar populations.