MEASUREMENT OF THE BLACK HOLE MASS IN NGC 1332 FROM ALMA OBSERVATIONS AT 0.044 ARCSECOND RESOLUTION

The Atacama Large Millimeter/submillimeter Array (ALMA) has a revolutionary capability to resolve cold molecular gas kinematics on angular scales well below 1''. This opens the possibility of using cold molecular gas disks as dynamical tracers to measure the masses of supermassive black holes (BHs) in galaxy centers (Davis et al. 2013b; Onishi et al. 2015). Accurate measurement of a BH mass (${M}_{\mathrm{BH}}$) generally requires observations that resolve kinematics within the BH's gravitational radius of influence ${r}_{{\rm{g}}}$, the radius within which the BH dominates the gravitational potential, and ALMA is the first millimeter/submillimeter interferometer having the capability to resolve ${r}_{{\rm{g}}}$ in large numbers of galaxies.

NGC 1332 is a nearby S0 or E galaxy that contains a highly inclined circumnuclear dust disk. Its BH mass has been measured previously via stellar dynamics (Rusli et al. 2011) and by modeling the hydrostatic equilibrium of its X-ray emitting halo (Humphrey et al. 2009), making it an excellent laboratory for direct comparison of independent measurement techniques. In Paper I (Barth et al. 2016), we described 03 resolution Cycle 2 ALMA observations designed to test for the presence of CO emission from within ${r}_{{\rm{g}}}$. The data showed very clean disklike rotation with a slight warp and central emission extending to ±500 km s⁻¹. Our simplest dynamical model fits converged on ${M}_{\mathrm{BH}}=6.0\times {10}^{8}$ ${M}_{\odot }$, but beam smearing in the highly inclined disk caused a severe degeneracy between rotation and turbulent motion at small radii, precluding the derivation of stringent confidence limits on ${M}_{\mathrm{BH}}$. Here, we present new ALMA Cycle 3 observations at 0044 resolution and dynamical modeling of the data. We adopt a distance of 22.3 Mpc to NGC 1332 for consistency with Paper I and with Rusli et al. (2011); at this distance, 1'' corresponds to 108.6 pc.

NGC 1332 was observed in ALMA Band 6 as part of project 2015.1.00896.S. Three separate observations on 2015 September 16 and 23 were carried out, for a total on-source integration time of 101 minutes. The observations were obtained in a frequency band centered at 229.354 GHz, corresponding to the redshifted ¹²CO(2–1) line in NGC 1332. We processed the data using the Common Astronomy Software Applications (McMullin et al. 2007) package to produce an emission-line data cube with $0\_\_AMP\_\_farcs;01$ pixels, a field of view of $6\_\_AMP\_\_farcs;4\times 6\_\_AMP\_\_farcs;4$, and 75 frequency channels with a channel spacing of 15.4 MHz, corresponding to a velocity width of 20.1 km s⁻¹ per channel for the CO(2–1) line. The synthesized beam has major- and minor-axis FWHM sizes of 0052 and 0037 with major-axis position angle (PA) 64°, giving a geometric mean resolution of 0044.

We created kinematic maps by fitting Gaussian line profiles to the data. In order to increase the signal-to-noise ratio (S/N) of the profile fits, we first binned the data cube spatially by block averaging in 4 × 4 pixel blocks, producing 004 pixels. The CO intensity, line of sight mean velocity (${v}_{\mathrm{LOS}}$), and width (${\sigma }_{\mathrm{LOS}}$) are shown in Figure 1. The disk kinematics are illustrated over an elliptical region with major and minor axes of 37 and 05, corresponding to the largest region over which we were able to obtain successful profile fits at each binned pixel. At the 004 binned pixel scale, the rms noise level is 0.142 mJy beam⁻¹ per channel, and the the median S/N per 004 spatial pixel is 7.2 over this elliptical region. The ${v}_{\mathrm{LOS}}$ map illustrates remarkably regular disklike rotation, with subtle signs of the mild kinematic twist or warp we found in the Cycle 2 data. In the ${\sigma }_{\mathrm{LOS}}$ map, the "X" shaped feature is characteristic of rotational broadening in an inclined disk. The major-axis position–velocity diagram (PVD) exhibits a central velocity upturn to $\approx 500$ km s⁻¹ (Figure 2), similar to that seen in the Cycle 2 data.

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Our dynamical modeling follows the methods described in Paper I. We model a thin disk in circular rotation in a gravitational potential containing a central point mass and an extended mass distribution from stars. The enclosed stellar mass ${M}_{\star }(r)$ is based on a deprojected luminosity profile ${L}_{\star }(r)$ multiplied by a mass-to-light ratio ϒ. We neglect the contributions of dark matter and of molecular gas, both of which are ≲10⁸ ${M}_{\odot }$ within r = 250 pc; in contrast, ${M}_{\star }(250\;\mathrm{pc})\approx {10}^{10}$ ${M}_{\odot }$ (Paper I). Models are computed on a spatial grid with 001 pixels. At each grid element, we determine the line of sight component of the disk's rotation velocity (relative to the systemic velocity ${v}_{\mathrm{sys}}$), observed at inclination i and major-axis PA Γ. The emission line at each grid point is assumed to have a Gaussian velocity profile, with turbulent velocity dispersion ${\sigma }_{\mathrm{turb}}$.

This procedure produces a simulated cube with the same velocity channel spacing as the data cube. The total line flux at each spatial element is scaled to match a CO surface brightness map measured from the ALMA data. Then, each velocity channel image in the model is convolved with the ALMA synthesized beam, and the model is rebinned to match the 004 spatial scale of the binned ALMA data cube. The goodness of fit (quantified by ${\chi }^{2}$) is computed by direct comparison of the model cube with the binned ALMA data. Calculating ${\chi }^{2}$ at a spatial scale of approximately one pixel per synthesized beam substantially mitigates the problem of spatially correlated noise in the data, which would be a significant issue if ${\chi }^{2}$ were computed for 001 pixels (see Paper I). Free parameters include ${M}_{\mathrm{BH}}$, ϒ, ${\sigma }_{\mathrm{turb}}$, i, Γ, ${v}_{\mathrm{sys}}$, the dynamical center position $({x}_{0},{y}_{0})$, and an overall normalization factor f₀ applied to the CO surface brightness model. We assume a uniform value of ${\sigma }_{\mathrm{turb}}$ over the disk surface and discuss this assumption in Section 4.3. We optimize the model fits using the amoeba downhill simplex method in IDL.

To model ${M}_{\star }(r)$, we use the deprojected stellar luminosity profile from Rusli et al. (2011), based on a combination of K-band VLT adaptive-optics data at small radii ($r\lt 4\_\_AMP\_\_farcs;5$) and R-band imaging at larger radii. Their best-fitting stellar-dynamical model implies an R-band mass-to-light ratio of ${{\rm{\Upsilon }}}_{R}=7.35$ (J. Thomas 2016, private communication). Rusli et al. (2011) give full details of the ${L}_{\star }(r)$ measurement, and Paper I presents a plot of ${M}_{\star }(r)$.

4.1. Fits to the Full Disk

We first describe models fitted to the entire disk, over the elliptical region illustrated in Figure 1. This region contains 840 binned (004) spatial pixels. We compute ${\chi }^{2}$ over 56 frequency channels, for a total of 47,040 data points. With all model parameters free, the best fit is found at ${M}_{\mathrm{BH}}=6.86\times {10}^{8}$ ${M}_{\odot }$, with ${\chi }_{\mathrm{min}}^{2}=55206.9$ or ${\chi }_{\nu }^{2}=1.17$. Other parameters for the best-fitting model are ${{\rm{\Upsilon }}}_{R}=7.53$, $i=84\_\_AMP\_\_fdg;1$, ${\rm{\Gamma }}=117\_\_AMP\_\_fdg;2$, and ${\sigma }_{\mathrm{turb}}=22.2$ km s⁻¹; these are similar to the best-fitting values found in Paper I. Figure 3 shows ${\rm{\Delta }}{\chi }^{2}={\chi }^{2}-{\chi }_{\mathrm{min}}^{2}$ as a function of ${M}_{\mathrm{BH}}$. The 68.3% confidence region (corresponding to ${\rm{\Delta }}{\chi }^{2}=1$) is $(6.86\pm 0.14)\times {10}^{8}$ ${M}_{\odot }$. Kinematic maps measured from the best-fitting model (Figure 1) match the structure in the data closely. While the model is generally successful at reproducing the disk kinematics, a result of ${\chi }_{\nu }^{2}=1.17$ for 47,032 degrees of freedom is formally unacceptable. We attribute this to systematic inadequacies of the model, such as neglect of the disk's slight warp, the assumption of a spatially constant value of ${\sigma }_{\mathrm{turb}}$, and noise in the CO surface brightness model.

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The apparently high precision of this result is typical of gas-dynamical models fitted to a large number of pixels (Gould 2013). However, the model-fitting (statistical) uncertainties only represent a portion of the true error budget for the full-disk fits. In this case, the major source of systematic uncertainty is likely to be the stellar mass profile. Pixels at $r\gg {r}_{{\rm{g}}}$ in the ALMA data dominate the full-disk fit, leading to very tight constraints on ϒ and, consequently, on ${M}_{\mathrm{BH}}$ as well. If the assumed stellar mass profile shape is systematically wrong, model fits will force ${M}_{\mathrm{BH}}$ to an incorrect (but ostensibly precise) value.

4.2. Fits to Smaller Regions

4.3. Turbulent Velocity Dispersion

This work presents the first BH mass derived from ALMA observations that resolve the BH's gravitational radius of influence, demonstrating ALMA's powerful capability to improve our understanding of local BH demographics. The primary challenges for gas-dynamical measurements of BH masses are identifying disks that exhibit suitably clean rotation and obtaining observations of sufficient angular resolution and S/N to map the kinematics on the required scales. While a great deal of effort has been invested in measurement of ionized gas kinematics, primarily with Hubble Space Telescope spectroscopic observations, the cold molecular gas in circumnuclear disks is known to be less turbulent than the ionized component, making molecular kinematics a more accurate tracer of a galaxy's gravitational potential than optical emission-line kinematics (Young et al. 2008; Davis et al. 2013a). Resolving ${r}_{{\rm{g}}}$ in CO observations of nearby galaxies was at the limiting edge of the capabilities of earlier millimeter-wave interferometers (Davis et al. 2013b), but ALMA will be able to carry out such observations for a large number of galaxies.

In Paper I, we stressed the importance of resolving angular scales corresponding to ${r}_{{\rm{g}}}\mathrm{cos}i$ (the BH radius of influence projected along the disk's minor axis) in order to fully resolve disk kinematics within the BH's sphere of influence. This Cycle 3 observation of NGC 1332 fully resolves ${r}_{{\rm{g}}}$, but ${r}_{{\rm{g}}}\mathrm{cos}i$ is unresolved by a factor of two. Thus, the observed kinematics are still blurred by beam smearing in the minor-axis direction, as seen in the ${\sigma }_{\mathrm{LOS}}$ map and the PVD, but this effect is far less severe than in our Cycle 2 observation. Higher-resolution observations would be required in order to determine the disk's central ${\sigma }_{\mathrm{turb}}(r)$ profile accurately and reduce the remaining degeneracy between rotation and turbulence in the inner disk. This would permit even tighter constraints on ${M}_{\mathrm{BH}}$ and provide unique information on the physical properties of molecular gas deep within the gravitational potential well of a supermassive BH.

The ideal situation for a gas-dynamical BH mass measurement is achieved when observations have sufficiently high S/N and angular resolution that ${M}_{\mathrm{BH}}$ can be well constrained by models fitted only to points within $r\lt {r}_{{\rm{g}}}$, so that stellar mass represents a small contribution to the total enclosed mass and systematic uncertainties in the stellar mass profile have a negligible impact on the measurement. This has previously been accomplished for a few objects using ionized gas kinematics (e.g., M87; Macchetto et al. 1997; Walsh et al. 2013) as well as for H₂O maser disk galaxies such as NGC 4258 (Miyoshi et al. 1995; Kuo et al. 2011). ALMA is certainly capable of achieving this goal for observations of CO or other molecular tracers. With an optimal combination of high angular resolution and S/N, ALMA observations of other targets could in principle provide BH mass precision at the level of a few percent in future measurements. All else being equal, the most precise BH masses will be determined from observations of disks at intermediate inclination angles satisfying the competing requirements that ${r}_{{\rm{g}}}\mathrm{cos}i$ is resolved and $v\mathrm{sin}i$ is large enough to permit accurate kinematic mapping. NGC 1332 is a challenging target in that the high disk inclination requires observations at very high angular resolution. At high resolution, the S/N per spatial pixel is low, and for this measurement, S/N is the primary limiting factor for determining ${M}_{\mathrm{BH}}$. Significant improvement in S/N would require a very large investment of ALMA time; however, far more than the ∼5 hr (including calibrations and overhead time) used for this observation.

The CO kinematics are highly inconsistent with a central mass as high as ${M}_{\mathrm{BH}}=(1.45\pm 0.20)\times {10}^{9}$ ${M}_{\odot }$, the value found by Rusli et al. (2011) based on stellar-dynamical modeling of VLT adaptive-optics spectroscopic data. For our constrained fits with ${r}_{\mathrm{fit}}=50$ pc, we find ${\rm{\Delta }}{\chi }^{2}=101$ at ${M}_{\mathrm{BH}}=1.45\times {10}^{9}$ ${M}_{\odot }$ and ${\rm{\Delta }}{\chi }^{2}=67$ for ${M}_{\mathrm{BH}}=1.25\times {10}^{9}$ ${M}_{\odot }$. This disagreement is particularly stark in that we are using the same stellar luminosity profile measured by Rusli et al. (2011) in our dynamical models, and our fits find a stellar mass-to-light ratio very similar to their best-fit value. It would be worthwhile to revisit the stellar-dynamical models for NGC 1332 to determine whether this disagreement can be resolved.

Our best-fitting BH mass is consistent with the value ${5.2}_{-2.8}^{+4.1}\times {10}^{8}$ ${M}_{\odot }$ (90% confidence) found by Humphrey et al. (2009) from modeling the hydrostatic equilibrium of the diffuse X-ray-emitting hot gas halo of NGC 1332. This agreement provides new motivation to explore the hydrostatic equilibrium method as a promising approach for measurement of the central mass profiles in elliptical galaxies and to carry out additional direct comparisons with other mass measurement techniques.

The scatter in the ${M}_{\mathrm{BH}}\mbox{--}{\sigma }_{\star }$ correlation at low redshift provides a fossil record of the cosmic history of BH evolution through accretion and mergers (Robertson et al. 2006; Peng 2007). Determination of this scatter requires a large sample of ${M}_{\mathrm{BH}}$ measurements with reliable measurement uncertainties. The factor of two disagreement between our gas-dynamical ${M}_{\mathrm{BH}}$ value and the stellar-dynamical result from Rusli et al. (2011) serves as an important cautionary note regarding the measurement and interpretation of the scatter in the ${M}_{\mathrm{BH}}\mbox{--}{\sigma }_{\star }$ relation, which is primarily determined from stellar-dynamical measurements (Gültekin et al. 2009; Kormendy & Ho 2013; McConnell & Ma 2013). Our results suggest that much more effort will be required in order to arrive at a definitive empirical understanding of the scatter in the local BH–bulge correlations. ALMA is poised to make a major contribution to our understanding of local BH demographics, if significant numbers of cleanly rotating molecular disks similar to the NGC 1332 disk can be identified.

This Letter makes use of data from ALMA program 2015.1.00896.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. We thank Jens Thomas for providing the stellar mass profile from Rusli et al. (2011).

L.C.H. acknowledges support from the Chinese Academy of Science through grant XDB09030102 (Emergence of Cosmological Structures) from the Strategic Priority Research Program and from the National Natural Science Foundation of China through grant 11473002.

Facility: ALMA.