REVISITING THE SCALING RELATIONS OF BLACK HOLE MASSES AND HOST GALAXY PROPERTIES

Our power-law fit to a given sample is defined in log space by an intercept α and slope β:

$$\begin{equation} \log _{10} M_\bullet = \alpha + \beta \, \log _{10} X , \end{equation} \tag{ 2 }$$

where M_• is in units of  M_☉, and X = σ/200 km s⁻¹, L/10¹¹L_☉, or M_bulge/10¹¹M_☉ for the three scaling relations. We have also tested a log-quadratic fit for the M_•–σ relation:

$$\begin{eqnarray} &&\log _{10} M_\bullet = \alpha + \beta \, \log _{10} X + \beta _2 \, [ \log _{10} X]^2 , \end{eqnarray} \tag{ 3 }$$

where X = σ/200 km s⁻¹. Results for the quadratic fit are discussed separately in Section 3.2.6 below.

For the power-law scaling relations, we have compared three linear regression estimators: MPFITEXY, LINMIX_ERR, and BIVAR EM. MPFITEXY is a least-squares estimator by Williams et al. (2010). LINMIX_ERR is a Bayesian estimator by Kelly (2007). Both MPFITEXY and LINMIX_ERR consider measurement errors in two variables and include an intrinsic scatter term, ₀, in log(M_•). LINMIX_ERR can be applied to galaxy samples with upper limits for M_•. For the M_•–σ sample with upper limits, we also use the BIVAR EM algorithm in the ASURV software package by Lavalley et al. (1992), which implements the methods presented in Isobe et al. (1986). The ASURV procedures do not consider measurement errors, and we use this method primarily for comparison with B12. All three algorithms are publicly available.³

For each of the global scaling relations and galaxy subsamples, we obtain consistent fits from MPFITEXY and LINMIX_ERR, although LINMIX_ERR usually returns a slightly higher value of ₀. Table 2 includes the global fitting results from both methods. In Table 2 we also include results from LINMIX_ERR in cases where ₀ is poorly constrained by MPFITEXY. For the M_•–σ relation including upper limits, the BIVAR EM procedure returns a lower intercept than LINMIX_ERR, but the slopes from the two methods are consistent within errors. Recently, Park et al. (2012) investigated the M_•–σ relation using four linear regression estimators, including MPFITEXY and LINMIX_ERR. All four estimators yielded consistent fits to empirical data, and MPFITEXY and LINMIX_ERR behaved robustly for simulated data with large measurement errors in σ.

3.2. M_•–σ Relation

3.3. M_•–L and M_•–M_bulge Relations

In Table 3, we present V-band luminosities for 44 galaxies and dynamically measured bulge masses for 35 galaxies. For several of the late-type galaxies in our sample, the literature contains one or more estimates of the bulge-to-total light ratio. Rather than judging between the various estimates, we present results for the early types only. Our best fit values are β = 1.11 ± 0.13 and α = 9.23 ± 0.10 for the M_•–L relation, and β = 1.05 ± 0.11 and α = 8.46 ± 0.08 for the M_•–M_bulge relation.

Additional fits to subsamples of these galaxies are listed in Table 2. The M_•–L and M_•–M_bulge relations do not show statistically significant differences between core and power-law galaxies (see also Figures 4(b) and (c)).

Figures 2 and 3 show that our M_•–L and M_•–M_bulge samples both appear to have a central knot, where black holes with 10⁸ M_☉ < M_• < 10⁹ M_☉ exhibit relatively weak correlation with L or M_bulge. This feature makes it difficult to interpret the fits to high-L and low-L (or high-M_bulge and low-M_bulge) subsamples. We find tentative evidence that the most luminous and massive galaxies (L > 10^10.8 L_☉; M_bulge > 10^11.5 M_☉) have steeper slopes in M_•(L) and M_•(M_bulge), as exemplified in Table 2. Both samples are sparsely populated at the low-M_• end.

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3.4. Comparison to Previous Studies

For a given black hole scaling relation, the differences between the measured values of M_• and the mean power-law relation are conventionally interpreted as a combination of measurement errors and intrinsic scatter. We assume the scatter in M_• to be lognormal, and define the intrinsic scatter term ₀ such that

$$\begin{equation} \chi ^2 = \sum _i \, \frac{[ \log _{10} (M_{\bullet ,i}) - \alpha - \beta x_i ]^2}{\epsilon _0^2 + \epsilon _{M,i}^2 + \beta ^2 \epsilon _{x,i}^2} \;\;, \end{equation} \tag{ 4 }$$

where x = log₁₀(σ/200 km s⁻¹) for the M_•–σ relation, x = log₁₀(L/10¹¹ L_☉) for the M_•–L relation, and x = log₁₀(M_bulge/10¹¹ M_☉) for the M_•–M_bulge relation. Here, _M is the 1σ error in log₁₀(M_•), and _x is the 1σ error in x. For a given sample and power-law fit, we adopt the value of ₀ for which χ²_ν = 1 (χ² = N_dof). G09 tested several forms of intrinsic scatter in M_• and found lognormal scatter to be an appropriate description.

Fitting the full galaxy sample for each scaling relation, we find the intrinsic scatter in log₁₀(M_•) to be ₀ = 0.38 for the M_•–σ relation, ₀ = 0.49 for the M_•–L relation, and ₀ = 0.34 for the M_•–M_bulge relation (or ₀ = 0.17 for M_• versus stellar M_bulge). While it is tempting to conclude that M_bulge is the superior predictor of M_•, the relative errors in M_bulge, σ, and L demand a more cautious interpretation. As noted in Section 2, we have assumed that all M_bulge values have an error of at least 0.24 dex. We have repeated our fits to M_•(M_bulge) with a minimum error of only 0.09 dex. Fitting the full M_bulge sample with this reduced error in M_bulge, we obtain a larger intrinsic scatter (₀ = 0.39) as expected from Equation (4), while the slope and intercept of the fit do not change significantly. Similarly, our measurements of ₀ for the M_•–σ and M_•–L relations depend in part upon the assumed errors in σ (⩾5%, following G09) and L (typically <0.05 dex). In addition to evaluating ₀, Novak et al. (2006) used earlier data sets to assess which correlation yielded the lowest predictive uncertainty in M_•, given a set of host galaxy properties with measurement errors. They found the M_•–σ relation to be marginally favorable for predicting black holes with M_• ∼ 10⁸ M_☉, but noted that uncertainties in the relations' slopes complicated predictions near the extrema of the relations. Our global M_•–σ relation has a steeper slope (β = 5.64) than the samples evaluated by Novak et al. (2006), with β from 3.69 to 4.59.

Beyond the intrinsic scatter ₀ in black hole mass for the full sample, the dependence of ₀ on σ, L, and M_bulge is a useful quantity for constraining theoretical models of black hole assembly. Successive mergers are predicted to drive galaxies toward the mean M_•–M_bulge relation, especially when black hole growth is dominated by black hole–black hole mergers (e.g., Peng 2007; Jahnke & Macciò 2011). Understanding intrinsic scatter in M_• is also crucial for estimating the mass function of black holes, starting from the luminosity function or velocity dispersion function of galaxies (e.g., Lauer et al. 2007c; Tundo et al. 2007; G09).

Figure 5 illustrates how ₀ varies across each of the M_•–σ, M_•–L, and M_•–M_bulge relations for our updated sample of measurements. For each relation, we construct three or four bins containing equal numbers of galaxies and perform multiple estimates of the scatter in each bin. One estimate, represented by blue stars in Figure 5, is to perform an independent Bayesian fit (with LINMIX_ERR) for the scaling relation in each bin, and assess the posterior distribution of ₀. This method provides uncertainties for ₀ in each bin, but the fits to narrow data intervals typically yield poor estimates for the slope and intercept. A second estimate, represented by black squares in Figure 5, is to compute ₀ as in Equation (4), using the global fit to the scaling relation to define the same values of α and β for all bins. The third and simplest estimate, represented by red diamonds in Figure 5, is to evaluate the root-mean-squared (rms) residual between log(M_•) and the global scaling relation. The ₀ term in Equation (4) provides a reliable assessment of intrinsic scatter only if random measurement errors are small: measurements with large uncertainties can yield χ²_ν ⩽ 1 with no intrinsic scatter term. In comparison, the rms estimate has no explicit dependence on measurement errors.

Figures 5(a) and (b) illustrate the scatter in M_• as a function of σ. Two definitions of σ for the 12 galaxies in Table 1 are shown for comparison, where σ is computed from data between radii  r_inf and  r_eff, or between 0 and  r_eff. Considering the large error bars in ₀ from the Bayesian fits, we find no significant variation in M_• with respect to σ. The high- and low-mass ends of the M_•–σ relation both exhibit ∼0.3–0.4 dex of scatter, regardless of how σ is defined or how scatter is estimated.

For the M_•–L and M_•–M_bulge relations, we find possible evidence that galaxies with low spheroid luminosities (L < 10^10.3 L_☉) and small stellar masses (M_bulge < 10¹¹ M_☉) exhibit increased intrinsic scatter in M_•. However, the scatter appears constant for galaxies above this range, spanning 10^10.3 L_☉ < L < 10^11.5 L_☉ and 10¹¹ M_☉ < M_bulge < 10^12.3 M_☉. More measurements in the range M_bulge ∼ 10⁸–10¹⁰ M_☉ are needed to reveal whether intrinsic scatter in M_• varies systematically across an extended range of bulge luminosities or masses. The Bayesian estimates of ₀ in each bin have large uncertainties; adopting this method, we do not detect a significant change in scatter for any interval in σ, L, or M_bulge.

The Local Group galaxy M32 is separated from the other early-type galaxies in our sample by almost an order of magnitude in L, and more than an order of magnitude in M_bulge. We have excluded its contribution in Figures 5(c) and (d), so that the sizes of the leftmost bins better reflect the sampled distributions of L and M_bulge. Including M32 does not substantially change the amount of scatter in the lowest-L and lowest-M_bulge bins.

Although our three estimates of scatter do not yield the same absolute values, their qualitative trends as a function of σ, L, or M_bulge are very similar. The similar behavior of ₀ and rms indicates that variations in measurement errors are not responsible for the apparent trends in intrinsic scatter.

We have compiled an updated sample of 72 black hole masses and host galaxy properties in Table 3; a more detailed version of Table 3 is available at http://blackhole.berkeley.edu. Compared with the 49 objects in G09, 27 black holes in our sample of 72 are new measurements and 18 masses are updates of previous values from improved data and/or modeling. Our present sample includes updated distances to 44 galaxies.

We have presented revised fits for the M_•–σ, M_•–L, and M_•–M_bulge relations of our updated sample (Table 2 and Figures 1–3). Each relation is fit as a power law: log₁₀(M_•) = α + β log₁₀(X). Our best fit to the full sample of 72 galaxies with velocity dispersion measurements (X ≡ σ/200 km s⁻¹) is α = 8.32  ±  0.05 and β = 5.64  ±  0.31. A quadratic fit to the M_•–σ relation with an additional term β₂ [log₁₀(X)]² gives β₂ = 1.68  ±  1.82 and does not decrease the intrinsic scatter in M_•. Including 92 additional upper limits for M_• decreases the intercept but does not change the slope: α = 8.15  ±  0.05 and β = 5.58  ±  0.30.

For the 44 early-type galaxies with reliable V-band luminosity measurements (X ≡ L_V/10¹¹ L_☉), we find α = 9.23  ±  0.10 and β = 1.11  ±  0.13. For the 35 early-type galaxies with dynamical measurements of the bulge stellar mass (X ≡ M_bulge/10¹¹ M_☉), we find α = 8.46  ±  0.08 and β = 1.05  ±  0.11.

We have also examined the black hole scaling relations for different subsamples of galaxies. When the galaxies are separated into early and late types and fit individually for the M_•–σ relation, we find similar slopes of β = 5.20  ±  0.36 (early types) and 5.06  ±  1.16 (late types). The intercepts, however, differ significantly: α = 8.39  ±  0.06 for the early types, a factor of ∼2 higher than α = 8.07  ±  0.21 for the late types. The steep global slope of 5.64 is therefore largely an effect of combining different galaxy types, each of which obeys a shallower M_•–σ relation and different intercepts.

When the early-type galaxies are further divided into two subsamples based on their inner surface brightness profiles, the resulting M_•–σ relation has a significantly larger intercept for the core galaxies than the power-law galaxies (Table 2 and Figure 4(a)). The slopes of the M_•–L and M_•–M_bulge relations do not show statistically significant differences between core and power-law galaxies, but M_• follows L and M_bulge more tightly in core galaxies than power-law galaxies (Table 2).

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In the literature, the exact value of the M_•–σ slope has been much debated by observers and regarded by theorists as a key discriminator for models of the assembly and growth of supermassive black holes and their host galaxies. We suggest that the individual observed M_•–σ relations for the early- and late-type galaxies provide more meaningful constraints on theoretical models than the global relation. After all, these two types of galaxies are formed via different processes. Superficially, our measurement of β = 5.64  ±  0.32 for the global M_•–σ relation favors thermally driven wind models that predict β ∼ 5 (e.g., Silk & Rees 1998) over momentum-driven wind models with β ∼ 4 (e.g., Fabian 1999). However, the intercept of the empirical M_•–σ relation is substantially higher than intercepts derived from thermally driven wind models (e.g., King 2010a, 2010b).

For the subsamples, the early-type galaxies give β > 4.0 with 99.96% confidence (Δχ² = 11.3 for ₀ = 0.34) and β > 4.5 with 97% confidence (Δχ² = 3.8), whereas the late-type and power-law galaxy subsamples are each consistent with β = 4.0 (Δχ² < 1). Core galaxies exceed β = 4.0 with marginal significance (Δχ² = 1.1) and are consistent with β = 4.5. Including central kinematics in the definition of σ further erodes the significance of high β for the early-type and core galaxy subsamples. More robust black hole measurements and more sophisticated theoretical models taking into account of galaxy types and environment (e.g., Zubovas & King 2012) are needed before stronger constraints can be obtained.

The intrinsic scatter in M_• plotted in Figure 5 for different intervals of σ, L, and M_bulge serves as an independent test for theoretical models of black hole and galaxy growth. Our data set shows decreasing scatter in M_• with increasing σ. However, there are currently insufficient data to probe the 30–100 km s⁻¹ range, where scatter in M_• could identify the initial formation mechanism for massive black holes (Volonteri et al. 2008; Volonteri & Natarajan 2009). Theoretical models of hierarchical mergers in Λ cold dark matter cosmology predict that scatter in M_• should decline steadily with increasing stellar mass (M_⋆), even when M_• and M_⋆ are initially uncorrelated (Malbon et al. 2007; Peng 2007; Hirschmann et al. 2010; Jahnke & Macciò 2011). The semi-analytic models by Malbon et al. (2007), for instance, predict that black holes with present-day masses >10⁸ M_☉ have gained most of their mass via black hole–black hole mergers, yielding extremely low scatter (₀ ∼ 0.1) at the upper end of the M_•–M_bulge relation. More recent models by Jahnke & Macciò (2011) use fully decoupled prescriptions for star formation and black hole growth, and attain a more realistic amount of scatter on average, yet these models still exhibit decreasing scatter as M_bulge increases from ∼10⁹ M_☉ to ∼10^11.5 M_☉. In comparison, we observe nearly constant scatter from M_bulge ∼ 10¹¹ M_☉ to 10¹² M_☉, beyond the highest bulge masses produced in the Jahnke & Macciò (2011) models.

Our final comment is that investigations using the M_•–σ correlation should consider the definition of σ, i.e., whether it is measured from an inner radius of zero or  r_inf. We find that both definitions yield similar amounts of scatter in the M_•–σ relation (Table 2), so neither has a clear advantage for predicting M_•. Excluding data within  r_inf corresponds more closely to cases where  r_inf is unresolved, such as seeing-limited galaxy surveys, high-redshift observations, or numerical simulations with limited spatial resolution. From a theoretical perspective, the evolutionary origin of an M_•–σ relation and the immediate effects of gravity may warrant separate consideration. On the other hand, the total gravitational potential of a galaxy includes its black hole. Our test of how redefining σ alters the M_•–σ relation has only considered 12 galaxies for which data within  r_inf contribute prominently to the spatially integrated velocity dispersion. At present, the full sample of M_• and σ measurements comprises a heterogeneous selection of kinematic data. Rather than advocating a particular definition, we wish to call attention to the nuances of interpreting the M_•–σ relation and encourage future investigators to consider their options carefully.

As this manuscript was being finalized, van den Bosch et al. (2012) reported a measurement of M_• = 1.7  ±  0.3 × 10¹⁰ M_☉ in NGC 1277. They reported σ = 333 km s⁻¹ for data between  r_inf and  r_eff, and M_bulge = 1.2  ±  0.4 × 10¹¹ M_☉; the latter measurement suggests that NGC 1277 lies two orders of magnitude above the mean M_•–M_bulge relation. Adding NGC 1277 to our 72-galaxy sample changes our global power-law fit to the M_•–σ relation only slightly: α = 8.33  ±  0.05, β = 5.73  ±  0.32, and ₀ = 0.39 (from MPFITEXY). Our global fit to M_•(M_bulge) for 36 galaxies including NGC 1277 yields α = 8.51  ±  0.09, β = 1.05  ±  0.13, and ₀ = 0.44.

Dynamical measurements of M_• require substantial observational resources and careful analysis, and are often published individually. Nonetheless, recent and ongoing efforts are rapidly expanding the available M_• measurements and revising the empirical black hole scaling relations. Our online database⁴ aims to provide all researchers easy access to frequently updated compilation of supermassive black holes with direct dynamical mass measurements and their host galaxy properties. Updated scaling relations can be used to estimate M_• more accurately in individual galaxies. This can improve our knowledge of Eddington rates and spectral energy distributions for accreting black holes, as well as time and distance scales for tidal disruption events. Moreover, the M_•–σ relation for quiescent black holes has been used to normalize the black hole masses obtained from reverberation mapping studies of active galaxies (Onken et al. 2004; Woo et al. 2010; Park et al. 2012). This important calibration could be improved by addressing morphology biases in the reverberation mapping samples and the M_•–σ relations for different galaxy types.