Recalibration of the M_BH–σ_⋆ Relation for AGN

RM allows accurate determination of the VP, given by $\mathrm{VP}=\ {V}^{2}{R}_{\mathrm{BLR}}/G$, where G is the gravitational constant, V is measured from the width of a broad emission line, and R_BLR is the size of the BLR. VPs are drawn from the AGN BH Mass Database⁵ (Bentz & Katz 2015) and from Bentz et al. (2016), and are listed in column (7) of Table 1. In all cases, VP is determined from the ${\rm{H}}\beta$ line. Individual references are available from the database.

Table 1.  AGN Sample

Object	re	References	σ⋆	${\sigma }_{{\star }_{\mathrm{int}}}$	Std Deviation	References	VP	Morphological Type
	('')		(km s−1)	(km s−1)	(km s−1)		(107 M⊙)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
Mrk 79	2.0	1	120 ± 9	125 ± 15	21	6	${0.951}_{-0.256}^{+0.267}$	barred late
NGC 3227	2.7	5	114 ± 3	136 ± 6	13	6	${0.139}_{-0.032}^{+0.029}$	barred late
NGC 3516	2.1	5	139 ± 4	143 ± 4	12	6	${0.577}_{-0.076}^{+0.051}$	barred late
NGC 4051	1.0	5	74 ± 2	69 ± 4	4	6	${0.031}_{-0.009}^{+0.010}$	barred late
NGC 4151	2.1	5	105 ± 5	110 ± 8	15	6	${0.923}_{-0.115}^{+0.163}$	barred late
NGC 4253	1.4	2	84 ± 4	85 ± 9	9	6	${0.032}_{-0.028}^{+0.028}$	barred late
NGC 4593	11.5	5	113 ± 3	144 ± 5	14	6	${0.177}_{-0.038}^{+0.038}$	barred late
NGC 5273	6.8	3	62 ± 3	69 ± 5	9	8	${0.103}_{-0.076}^{+0.057}$	early
Mrk 279	1.6	1	153 ± 7	156 ± 17	26	6	${0.657}_{-0.177}^{+0.177}$	early
PG 1411+442	3.1	1	⋯	208 ± 30	⋯	7	${6.263}_{-3.376}^{+3.344}$	early
NGC 5548	11.2	5	131 ± 3	162 ± 12	34	6	${1.212}_{-0.050}^{+0.052}$	late
PG 1617+175	1.7	1	⋯	201 ± 37	⋯	7	${9.620}_{-4.790}^{+4.272}$	early
NGC 6814	1.7	2	71 ± 3	69 ± 3	5	6	${0.336}_{-0.064}^{+0.063}$	barred late
Mrk 509	2.8	1	⋯	183 ± 12	⋯	7	${2.529}_{-0.204}^{+0.223}$	early
PG 2130+099	0.32	1	⋯	165 ± 19	⋯	7	${0.630}_{-0.086}^{+0.086}$	late
MCG-06-30-15	1.01	4	95 ± 5	91 ± 5	22	8	${0.037}_{-0.009}^{+0.010}$	early

Notes. Column 1: galaxy name. Column 2: r_e. Column 3: reference for r_e. Column 4: σ_⋆ within r_e with associated 1σ uncertainty. Column 5: σ_⋆ measured from a single spectrum integrated within a circular aperture of radius r_e, with associated 1σ uncertainty. Column 6: standard deviation for the set of σ_⋆ values averaged to determine the value in column 4. No value is included for galaxies where an integrated spectrum was used. Column 7: reference for σ_⋆. Column 8: VP. Column 9: morphological type, based on surface brightness decompositions of Bentz et al. (2009, 2013) and M. C. Bentz et al. (2017, in preparation).

References. (1) Bentz et al. (2009), (2) Bentz et al. (2013), (3) Bentz et al. (2014), (4) Bentz et al. (2016), (5) M. C. Bentz et al. (2017, in preparation), (6) Batiste et al. (2017), (7) Grier et al. (2013), (8) this work.

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Spatially resolved stellar kinematics are available for NGC 5273, from the ATLAS^3D survey of early-type galaxies in the northern hemisphere⁶ (Cappellari et al. 2011, 2013), and for MGC-06-30-15, from the work of Raimundo et al. (2013). Following the method of B17, σ_⋆ is determined for these galaxies by taking an error-weighted average of the values for each spaxel within a circular aperture defined by the effective radius, r_e.

Accurate measurements of r_e are available from the works of Bentz et al. (2014) for NGC 5273 and Bentz et al. (2016) for MGC-06-30-15. They are determined from detailed surface brightness decompositions of Hubble Space Telescope (HST) images, using GALFIT (Peng et al. 2002, 2010). The AGN is isolated from the galaxy surface brightness features, and substructure such as bars and disks are accounted for.

For MCG-06-30-15, we follow Raimundo et al. (2013) and exclude the central 01 from our calculation of σ_⋆, as the noise associated with the AGN continuum precludes secure measurement of the stellar kinematics. Furthermore, the kinematic map in Figure 1 shows that a small region within r_e (101) was cut off (y ≤ −06, the bottom of the map), due to an illumination artifact in the SINFONI data (see Raimundo et al. 2013 for details). We assume that the kinematics within r_e are well represented by the region that remains and determine σ_⋆ without applying any correction.

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Two estimates of uncertainty are provided for each σ_⋆ determination. The statistical uncertainty, based on the measurement error, is shown in column (4) of Table 1 along with σ_⋆. Column (6) shows the standard deviation among the set of σ_⋆ values that have been averaged to determine the overall σ_⋆. The standard deviation provides a measure of the spatial variation in the kinematics within r_e and may be more physically meaningful as an estimate of uncertainty.

Finally, σ_⋆ determinations from IFS are available for four high-luminosity quasar hosts from Grier et al. (2013). This study employs a different definition of σ_⋆, which includes a contribution from the rotational velocity. Rather than averaging the kinematics within a chosen aperture, the spectra within that aperture are instead co-added, and σ_⋆ is measured from the resulting rotationally broadened spectrum. While this method differs from that of B17, it has been favored in some recent studies, including those using IFS (e.g., Gültekin et al. 2009; Cappellari et al. 2013; van den Bosch 2016; KH13). For elliptical or near face-on disk galaxies the difference between the methods should be minimal, since rotation along the line of sight will not dominate the stellar kinematics (W13). Based on the GALFIT decompositions of Bentz et al. (2009), two of these quasar hosts are elliptical (i.e., fitted with only a bulge component) and two are low-inclination disk galaxies. Consequently, we do not expect any bias to arise from including these measurements.

We can directly test this expectation because our sample contains galaxies qualitatively similar to the quasar hosts, including the low-inclination disk galaxies NGC 5273 and NGC 6814. We measure σ_⋆ for the whole sample via the method of Grier et al. (2013) (column (5) of Table 1). The greatest variation between σ_⋆ estimates from the two methods occurs for inclined spiral galaxies with significant substructure (e.g., NGC 4593) or evidence of ongoing or recent interactions (e.g., NGC 3227, MCG-06-30-15, and NGC 5548; M. C. Bentz et al. 2017, in preparation). As expected, σ_⋆ varies minimally between the two methods for NGC 5273 and NGC 6814, suggesting that including the quasars is unlikely to introduce significant bias.

Measurements of σ_⋆ for the full sample are given in Table 1.

Using σ_⋆ from Cappellari et al. (2013), we find a best-fitting relation for the quiescent sample of

$$\begin{eqnarray}\mathrm{log}\ \left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{M}_{\odot }}\right) & = & (8.55\pm 0.09)\\ & & +\ (5.32\pm 0.63)\ \mathrm{log}\left(\displaystyle \frac{{\sigma }_{\star }}{200\ \mathrm{km}\,{{\rm{s}}}^{-1}}\right).\end{eqnarray} \tag{ 2 }$$

This agrees remarkably well with the parameterization of W13, who find a slope of 5.31 ± 0.33, and is consistent with that of Grier et al. (2013; 5.04 ± 0.19) and that of Savorgnan & Graham (2015; 6.34 ± 0.8).

We find a slightly shallower slope when we use our own determinations of σ_⋆, more consistent with that of KH13 (4.38 ± 0.29):

$$\begin{eqnarray}\mathrm{log}\,\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{M}_{\odot }}\right) & = & (8.66\pm 0.09)\\ & & +(4.76\pm 0.60)\,\mathrm{log}\left(\displaystyle \frac{{\sigma }_{\star }}{200\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right).\end{eqnarray} \tag{ 3 }$$

While the slopes are consistent with the literature, the intercepts are higher. Comparing Equation (2) with the parameterization by W13 (who find α = 8.37 ± 0.05) is particularly instructive, since the slopes are almost identical. Differences between the intercepts arise from systematic differences between the sample of σ_⋆ measurements. On average, the σ_⋆ measurements from Atlas^3D data are lower than the literature values, causing the relation to be shifted left, thus increasing the intercept. While lower σ_⋆ values are expected from IFS (see, e.g., KH13; B17), these measurements may be biased low if r_e are overestimated (Section 3). Thus, the intercepts that we measure are likely too high, while the intercepts quoted in the literature are probably too low.

To determine the best-fitting relation for AGN, VP is used in place of M_BH:

$$\begin{eqnarray}&&\mathrm{log}\,\left(\displaystyle \frac{\mathrm{VP}}{{M}_{\odot }}\right)={\alpha }_{\mathrm{AGN}}\ +\ \beta \ \mathrm{log}\left(\displaystyle \frac{{\sigma }_{\star }}{200\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right),\end{eqnarray} \tag{ 4 }$$

where, since ${M}_{\mathrm{BH}}=f\ \mathrm{VP}$, α_AGN includes $\mathrm{log}\ f$.

The relation is parameterized with both error estimates (statistical uncertainty is used when a standard deviation is not available), and the best-fit parameters are shown in Table 3. In general, parameterizations that include the standard deviation (column (6) of Table 1) as the error in σ_⋆ are steeper; however, they are all consistent with each other. We adopt as our best fit that which uses the statistical measurement error:

$$\begin{eqnarray}\mathrm{log}\,\left(\displaystyle \frac{\mathrm{VP}}{{M}_{\odot }}\right) & = & (7.53\pm 0.26)\\ & & +(3.90\pm 0.93)\mathrm{log}\left(\displaystyle \frac{{\sigma }_{\star }}{200\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right).\end{eqnarray} \tag{ 5 }$$

This agrees with the parameterization for AGN found by W13 (3.46 ± 0.61), as well as for two of their quiescent galaxy subsamples: late-type galaxies (4.23 ± 1.26) and galaxies with pseudo-bulges (3.28 ± 1.11). It is also consistent with the quiescent galaxy parameterizations found in Section 4.1. The scatter in the relation is found to be 0.30 ± 0.15 dex, which is similarly consistent with previous studies.

Table 3.  Fits to the ${M}_{\mathrm{BH}}\mbox{--}{\sigma }_{\star }$ Relation

Sample/Fit	σ⋆	σ⋆ Error	α	β	f
(1)	(2)	(3)	(4)	(5)	(6)	(7)
Quiescent	integrated	measurement error	8.55 ± 0.09	5.32 ± 0.63	⋯	0.16 ± 0.06
	spatially resolved	standard deviation	8.66 ± 0.09	4.76 ± 0.60	⋯	0.11 ± 0.05
Active	spatially resolved	measurement error	7.53 ± 0.26	3.90 ± 0.93	⋯	0.30 ± 0.15
	spatially resolved	standard deviation	7.55 ± 0.26	4.00 ± 0.94	⋯	0.27 ± 0.16
	integrated	measurement error	7.38 ± 0.25	3.53 ± 0.93	⋯	0.34 ± 0.18
Equation (2)	spatially resolved	measurement error	7.87 ± 0.15	5.32	4.82 ± 1.67	0.33 ± 0.17
Equation (2)	spatially resolved	standard deviation	7.86 ± 0.15	5.32	4.94 ± 1.75	0.27 ± 0.16
Equation (2)	integrated	measurement error	7.77 ± 0.17	5.32	6.05 ± 2.45	0.43 ± 0.21
Equation (3)	spatially resolved	measurement error	7.73 ± 0.14	4.76	8.49 ± 2.77	0.29 ± 0.16
Equation (3)	spatially resolved	standard deviation	7.72 ± 0.14	4.76	8.67 ± 2.89	0.24 ± 0.14
Equation (3)	integrated	measurement error	7.64 ± 0.16	4.76	10.37 ± 3.86	0.37 ± 0.18
W13	spatially resolved	measurement error	7.86 ± 0.15	5.31	3.23 ± 1.14	0.33 ± 0.16
W13	spatially resolved	standard deviation	7.86 ± 0.15	5.31	3.27 ± 1.17	0.27 ± 0.16
W13	integrated	measurement error	7.76 ± 0.17	5.31	4.03 ± 1.6	0.42 ± 0.21

Notes. Column 1: the first five rows give the sample being fitted, either quiescent or active, the rest show the parameterization being used. Column 2: method by which σ_⋆ was determined, either from an average of spatially resolved spectra, or from a single integrated spectrum. Column 3: uncertainty in σ_⋆ used in the fit. Column 4: the intercepts. Column 5: the slopes. Column 6: calculated f value. Column 7: scatter in the relation.

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Table 3 also shows the best-fit parameters when the alternative definition of σ_⋆ is used. This gives a slightly shallower slope; however, the fits are consistent between the two measures of σ_⋆.

As with previous studies, our best fits to the AGN M_BH–σ_⋆ relation are shallower than those for the quiescent sample. However, this does not necessarily indicate a fundamental difference between the relations for active and quiescent galaxies. M_BH is determined via different methods for the two samples, so different selection criteria apply. For quiescent galaxies it is necessary to resolve the gravitational sphere of influence of the BH, which is not required for AGN. Recent studies have suggested that this criterion may bias the quiescent galaxy sample, and accounting for this bias substantially reduces the discrepancy between the fits (W13; Shankar et al. 2016). Expanded samples of AGN and quiescent galaxies, at both the high and low M_BH ends, are key to further investigating this difference and determining if it is physically meaningful, or simply the result of selection effects.

We estimate f by fixing the slope in Equation (4) to that for quiescent galaxies and taking the difference between the intercepts for the two samples:

$$\begin{eqnarray}&&\mathrm{log}\ f={\alpha }_{q}-{\alpha }_{\mathrm{AGN}}.\end{eqnarray} \tag{ 6 }$$

where α_q is the intercept for the quiescent galaxy sample and α_AGN is the intercept for the AGN sample.

We use an adapted version of LINMIX_ERR that allows for fixing the slope and use the two quiescent galaxy parameterizations from Section 4.1, as well as that of W13 (β = 5.31, α = 8.37), to provide a comparison with the literature. W13 is chosen because the sample of σ_⋆ that they use contains some rotation-corrected values, so it is more consistent with our sample than others available in the literature.

The results are summarized in Table 3. The lowest value of f is found with the parameterization of W13, while the highest comes from Equation (3). They are all consistent with previous estimates (e.g., Graham et al. 2011; Grier et al. 2013; Woo et al. 2015; W13), though our highest value is notably higher than most quoted in the literature. Our determinations are also consistent with the results of dynamical modeling of the BLR by Pancoast et al. (2014), who modeled five active galaxies (including NGC 5548 and NGC 6814) and determined f separately for each, finding a mean of $\langle \mathrm{log}\ f\rangle =0.68\pm 0.40$.

For comparison we perform the same fitting using the alternative definition of σ_⋆, which are listed in Table 3. They are generally steeper, but consistent within the errors.

As can be seen, f varies significantly depending on the chosen parameterization of the quiescent M_BH–σ_⋆ relation, and this is driven by the different measurements of σ_⋆ for the quiescent sample. Given the previously discussed issues with the r_e determinations for the quiescent sample, it is clear that the best value to use for f is still not settled.

IFS provides more information about the galaxy kinematics, so IFS-based σ_⋆ measurements are an improvement over previous estimates. The quiescent galaxy parameterizations presented in this work rely on such estimates and provide a more consistent basis for comparison with our AGN sample, so f values determined from these parameterizations are preferable. Equation (3) should provide the best parameterization to use with the AGN sample, since the σ_⋆ measurements do not include contributions from the rotational velocity. However, the bias that arises from poorly constrained r_e measurements for the quiescent sample does not affect the AGN sample, so they are not completely consistent. Following the discussion in Section 4.1, it is reasonable to consider that the f value obtained from Equation (3) is too high, and the f value obtained from the parameterization of W13 is too low, so we recommend f = 4.82 ± 1.67. This is close to the median f value of all those listed in Table 3, though the scatter among those values is 2.6, which is higher than the quoted uncertainty.

Figure 2 shows the M_BH–σ_⋆ relation, with the lines of best fit, for both samples. While the slopes are different, the samples clearly overlap, and the absence of AGN with high M_BH and high σ_⋆ may well be responsible for the difference in slopes. The AGN sample is split into barred and unbarred galaxies (red and black points, respectively), since previous studies have suggested morphological dependencies in the M_BH–σ_⋆ relation. We see no obvious difference between these subsamples, though we caution that this sample is too small to draw any definite conclusions.

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