THE MEGAMASER COSMOLOGY PROJECT. III. ACCURATE MASSES OF SEVEN SUPERMASSIVE BLACK HOLES IN ACTIVE GALAXIES WITH CIRCUMNUCLEAR MEGAMASER DISKS

The primary goal of the Megamaser Cosmology Project (MCP; Braatz et al. 2009; Reid et al. 2009a; Braatz et al. 2010) is to determine the Hubble constant H₀ to ∼3% accuracy in order to constrain the equation-of-state parameter w of dark energy. The key to achieving this goal is to measure accurate distances to galaxies well into the Hubble flow (50–200 Mpc). A proven method to measure accurate angular-diameter distances involves sub-milliarcsecond resolution imaging of H₂O maser emission from sub-parsec circumnuclear disks at the center of active galaxies, a technique established by the study of NGC 4258 with Very Long Baseline Interferometry (VLBI; Herrnstein et al. 1999). Throughout the paper, we will refer to such circumnuclear disks as "megamaser disks" and galaxies that contain such megamaser disks as "megamaser galaxies." As the MCP discovers and images megamaser disks, an important result, in addition to the distance determination, is the accurate measurement of the enclosed masses of the massive dark objects (MDOs; Kormendy & Richstone 1995; Richstone et al. 1998) at the centers of these megamaser galaxies. We assume that the MDOs are black holes (BHs) and justify this assumption in Section 4.

The VLBI plays a crucial role in measuring BH masses with high precision especially in galaxies with lighter supermassive BHs (i.e., M_• ∼ 10⁶–10⁷ M_☉). The critical advantage provided by VLBI is an angular resolution two orders of magnitude higher than the best optical resolution. For any given galaxy with a nearly constant central mass density of stars, M_BH is proportional to R³_inf, where R_inf is the radius of the gravitational sphere of influence of BH (Barth et al. 2004). So, a factor of 100 increase in resolution permits measurements of masses up to 10⁶ times smaller. Similarly, the central density limits that can be set are up to 10⁶ times higher, high enough to rule out extremely dense star clusters as the MDOs based on dynamical argument (see Section 4) in most megamaser disks presented here.

Megamaser disks, such as the archetypal one in NGC 4258, are usually found in the centers of Seyfert 2 galaxies. Because maser emission is beamed and long path lengths are required for strong maser amplification, the megamaser disks are observable only if the disk is close to edge-on. In NGC 4258 the disk is inclined ∼82° and its rotation curve is Keplerian to better than 1% accuracy, which makes the BH mass determination robust with very few assumptions. The megamaser disks are typically smaller (r∼0.2 pc in NGC 4258) than the gravitational sphere of influence of their supermassive BHs (r ∼ 1 pc in NGC 4258); this guarantees that the gravitational potential is dominated by the central mass. Applying dynamical arguments to the extremely high mass density ρ₀ ∼ 10¹¹ M_☉ pc⁻³ within the NGC 4258 maser disk can rule out a dense cluster of stars or stellar remnants for most of the central dark mass (Maoz 1995), implying that the measured central mass is dominated by a BH.

The megamaser disk method for estimating BH masses has some practical limitations. First, finding megamaser disks is difficult, partly because detectable disks need to be nearly edge-on. Only eight BH masses have been published based on measurements of megamasers: NGC 1068 (Greenhill et al. 1996), NGC 2960 (Mrk 1419; Henkel et al. 2002; not based on VLBI), NGC 3079 (Kondratko et al. 2005), NGC 3393 (Kondratko et al. 2008), UGC 3789 (Reid et al. 2009a; MCP Paper I), NGC 4258 (Herrnstein et al. 1999), NGC 4945 (Greenhill et al. 1997), and Circinus (Greenhill et al. 2003). Second, some rotation curves are significantly flatter than Keplerian, e.g., NGC 1068, NGC 3079, and IC 1481 (Mamyoda et al. 2009). The origin of the flatter rotation curves in these galaxies is unclear. It could be caused by self-gravity of a massive disk or the presence of a nuclear cluster, in which case the enclosed mass would be larger than the BH mass, or it could be caused by radiation pressure, in which case the enclosed mass would be smaller than the BH mass (Lodato & Bertin 2003). Without fully understanding the causes of the flatter rotation curves and making correct modeling for these megamaser disks, accurate BH mass measurement would be difficult. Therefore, the key to measuring reliable BH masses is to find well-defined, edge-on megamaser disks with Keplerian rotation curves.

Over the past two decades, there has been substantial progress in detecting BHs and constraining their masses, especially in quiescent or mildly active nearby galaxies (Kormendy & Richstone 1995; Kormendy & Gebhardt 2001; Kormendy 2004; Ferrarese & Ford 2005). This rapid progress was mainly facilitated by the high angular resolution provided by the Hubble Space Telescope (HST) and by gradually maturing techniques for modeling stellar dynamics of galaxies. The number of BH detections has increased to the degree that the field has shifted from confirming the existence of BHs to comparing their properties with those of the host galaxies. One of the most discussed relationships found between BHs and host galaxies is the correlation between BH mass and the velocity dispersion of stars in the bulge (i.e., the M_BH–σ_⋆ relation; Ferrarese & Merritt 2000; Gebhardt et al. 2000; Gültekin et al. 2009, and references therein). It is often suggested that this relation is a manifestation of a causal connection between the formation and evolution of the BH and the bulge. As pointed out by Gebhardt et al. (2000), it is natural to assume that bulges, BHs, and quasars formed, grew, or "turned on" as parts of the same process, in part because the collapse or merger of bulges might provide a rich fuel supply to a centrally located BH. However, the nature of this connection is still not well understood.

Clearly, accurate BH mass measurements in megamaser galaxies can help to improve our understanding of the M_BH–σ_⋆ correlation, especially at the low-mass end. We present high-resolution images and rotation curves of megamaser disks in six new galaxies and their BH masses. In addition, we re-analyze the BH mass in UGC 3789 using the data from Reid et al. (2009a) and present the new measurement here. In Section 2, we present our sample of galaxies, VLBI observations, and data reduction. Section 3 shows the VLBI images and rotation curves of the megamaser disks, followed by our analysis of BH masses. In Section 4, we rule out compact clusters of stars or stellar remnants as the central dominant masses for the majority of the megamaser disks. In Section 5, we compare results from other mass measuring techniques to our maser masses. The discussion of our main results is presented in Section 6, and we summarize the results in Section 7. A more in-depth discussion of the M_BH–σ_⋆ relation, including the new maser BH masses, is presented in a companion paper by Greene et al. (2010).

2.1. The Megamaser Disk Sample

2.2. Observations

2.3. Data Reduction

2.4. Relativistic Velocity Assignment

The sub-parsec megamaser disks presented in this paper are in a deep gravitational potential and the majority have recession velocities over 1% of the speed of light c. For this reason, we made both special and general relativistic corrections to the maser velocities in the data before we used the data to analyze the BH masses. We made the relativistic corrections in the following way.

The observed local standard of rest (LSR) velocities V_op listed in Table 3 are based on the "optical" velocity convention

$$\begin{equation} {V_{\rm op} \over c} = \biggl ({\nu _0 - \nu \over \nu }\biggr), \end{equation} \tag{ 1 }$$

where ν is the observed frequency and ν₀ = 22.23508 GHz, the rest frequency of the H₂O 6₁₆ − 5₂₃ transition.

Because of gravitational time dilation, the emitting frequency ν₀ of a maser at distance r from a compact object of mass M and the actual observed frequency ν_∞ for an observer at r = ∞ differ by a factor

$$\begin{equation} {\nu _0 \over \nu _\infty } = \biggl (1 + {{\it GM} \over r c^2}\biggr). \end{equation} \tag{ 2 }$$

For a maser in a circular orbit moving at speed β_mc, balancing the gravitational and centripetal accelerations gives GM/r² = (β_mc)²/r, so

$$\begin{equation} {\nu _0 \over \nu _\infty } = 1 + \beta _{\rm m}^2. \end{equation} \tag{ 3 }$$

We multiplied the observed frequency ν of each maser line by (1 + β²_m) to correct for gravitational time dilation.

The megamaser lines at the systemic velocities were also corrected for a transverse Doppler shift

$$\begin{equation} {\nu _0 \over \nu } = 1 + \beta _{\rm m}^2/2. \end{equation} \tag{ 4 }$$

The relativistically correct Doppler equation for a source moving radially away with velocity v = βc is

$$\begin{equation} {\nu \over \nu _0} = \biggl ({1 - \beta \over 1 + \beta }\biggr)^{1/2}. \end{equation} \tag{ 5 }$$

After the observed frequency had been corrected for the gravitational time dilation and transverse Doppler shift, we used Equation (5) to convert the corrected frequency to its relativistically correct radial velocity v = βc. For the megamaser disks, typically β_m < 0.003. So, for galaxies in the Hubble flow (z>0.01), the general relativistic corrections are smaller than the special relativistic corrections. For example, for masers in UGC 3789, which has an optical-LSR recession velocity of 3262 km s⁻¹, the special relativistic corrections range from 10 to 26 km s⁻¹ whereas the general relativistic corrections range from 0 to ∼2 km s⁻¹.

Finally, when fitting the rotation curves, we used the relativistic formula for the addition of velocities to decompose the observed β value of each maser spot into a common β_g associated with the radial velocity of the center of each megamaser disk and an individual β_m associated with orbital motion of a specific maser spot. The redshifted maser spots have β_m>0 and blueshifted maser spots have β_m < 0.

$$\begin{equation} \beta = {\beta _{\rm g} + \beta _{\rm m} \over 1 + \beta _{\rm g} \beta _{\rm m}}. \end{equation} \tag{ 6 }$$

3.1. VLBI Images, Rotation Curves, and BH Masses

Figure 1 shows the GBT single-dish spectra for all megamaser galaxies presented here except for UGC 3789, which can be found in Reid et al. (2009a). Figures 2, 3, and 4 show the VLBI maps and the position–velocity (P–V) diagrams along with the fitted rotation curves of the maser spectral components (spots) in UGC 3789, NGC 1194, NGC 2273, NGC 2960 (Mrk 1419), NGC 4388, NGC 6264, and NGC 6323. Reid et al. (2009a) published the UGC 3789 VLBI map and the rotation curve, and we performed a new analysis of the BH mass for this galaxy based on those data. We show the VLBI map and rotation curve for this galaxy again for direct comparison with the other six megamaser disks. The data points in the VLBI maps and rotation curves are color-coded to indicate redshifted, blueshifted, and systemic masers, where the "systemic" masers refer to the maser spectral components having velocities close to the systemic velocity of the galaxy. Except for NGC 4388, the maser spot distributions are plotted relative to the average position of the systemic masers. Systemic masers are not detected in NGC 4388, so we plotted its maser distribution relative to the dynamical center determined from fitting the data in the P–V diagram with a Keplerian rotation curve.

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To estimate the inclination and dynamical center (i.e., the position of the BH) of each disk, we rotated the coordinate system to make the disk horizontal and used the fitted horizontal line that passes through the high-velocity masers as the zero point of the y-coordinate of the dynamical center (see Figures 3 and 4). The zero point of the x-coordinate is defined to be the unweighted average θ_x of the systemic masers.

We assumed that the systemic masers have about the same orbital radii as the high-velocity masers to estimate the maser disk inclination cos⁻¹(〈θ^(sys)_y〉/θ_r), where 〈θ^(sys)_y〉 is the average θ_y position of the systemic masers and θ_r is the orbital radius of the systemic masers. In principle, one can determine θ_r exactly only when good rotation curves for both systemic and high-velocity masers can be obtained, and precise acceleration measurements for the systemic masers are available. Among our data we only have such information for UGC 3789 at this point (Braatz et al. 2010), and so we use 〈θ_x〉 of high-velocity masers as an estimate of θ_r. We note that all our megamaser disks with systemic masers detected have inclinations larger than 80°, so assuming the disk is exactly edge-on will only cause errors less than 1% in the derived BH masses. In NGC 4388 we could not measure a disk inclination, but even if the disk were 20° from edge-on, the contribution to the error in the BH mass would be only 12%.

We determined the rotation curve for each megamaser disk as a function of the "impact parameter" defined as the projected radial offset θ = (θ²_x + θ²_y)^1/2 of the maser spots so that we can account for the warped structures in some megamaser disks. We then performed a nonlinear least-squares fit of a Keplerian rotation curve to the P–V diagram with the assumption that the high-velocity masers lie exactly on the mid-line of the disk. In addition, the systemic velocity of each galaxy was fitted as a free parameter, and we report the best fits of the systemic velocities of our megamaser galaxies in Table 1.

The fitted Keplerian rotation curves can be expressed with the following form:

$$\begin{equation} \vert v_{\rm K} \vert = v_1 \biggl ({\theta \over 1 {\rm mas}}\biggr)^{-1/2}, \end{equation} \tag{ 7 }$$

where |v_K| is the orbital velocity (after relativistic corrections) of the high-velocity masers and v₁ is the orbital velocity at a radius 1 mas from the dynamical center. The BH mass is

$$\begin{equation} M_\bullet = \biggl ({\vert v_{\rm K} \vert ^2 \theta \over \rm G}\biggr) D_{\rm A} = \biggl ({ \pi v_{1}^2 \over 6.48 \times 10^8\, \rm G}\biggr) D_{\rm A}, \end{equation} \tag{ 8 }$$

where D_A is the angular diameter distance to the galaxy. We show all the measured BH masses in Table 4.

Table 4. The BH Masses and Basic Properties of the Maser Disks

Name	Dist.	BH mass	Disk Size	P.A.	Incl.	rc	ρ0	mmax	$\emph {T}_{\rm age}$	Rinf
	(Mpc)	(107M☉)	(pc)	(°)	(°)	(pc)	(M☉ pc−3)	(M☉)	(yr)	(arcsec)
NGC 1194	53.2	6.5 ± 0.3	0.54-1.33	157	85	0.260	1.2 × 109	13	>1.0 × 1010	0.033
NGC 2273	25.7	0.75 ± 0.04	0.028-0.084	153	84	0.015	6.1 × 1011	0.05	<2.2 × 106	0.010
UGC 3789	46.4	1.04 ± 0.05	0.084-0.30	41	>88	0.022	2.3 × 1011	0.1	<1.8 × 107	0.010
NGC 2960	72.2	1.16 ± 0.05	0.13-0.37	−131	89	0.056	1.7 × 1010	0.48	<1.0 × 109	0.005
NGC 4388	19.0	0.84 ± 0.02	0.24-0.29	107	–	0.090	3.3 × 109	0.9	<6.6 × 109	0.034
NGC 6264	139.4	2.91 ± 0.04	0.24-0.80	−85	90	0.085	1.2 × 1010	1.4	>1.0 × 1010	0.012
NGC 6323	106.0	0.94 ± 0.01	0.13-0.30	10	89	0.046	2.3 × 1010	0.3	<4.7 × 108	0.003

Notes. Column 1: galaxy name. Column 2: the Hubble flow distances we adopt from NED. Column 3: BH masses measured in our project. The mass uncertainty here only includes errors caused by source position uncertainty and from fitting a Keplerian rotation curve for a given distance. Except for NGC 4388, the actual BH mass uncertainty is dominated by the error of the latest H₀ measurement (∼6%). NGC 4388 has a larger BH mass error (11%) which is limited by the uncertainty of the Tully–Fisher distance determination(see the explanation in Section 3.2). Column 4: sizes of the inner and outer edge of the maser disks. Column 5: position angle (P.A.) of the disk plane measured east of north. P.A. equals zero when the blueshifted side of the disk plane has zero east offset and positive north offset. Column 6: inclination of the maser disk. Note that the inclination of NGC 4388 is unconstrained because we did not detect systemic masers. Column 7: the core radius of the Plummer cluster in parsecs. For NGC 1194, NGC 2273, NGC 2960, and NGC 4388, the core radii are derived from Equation (12). For UGC 3789, NGC 6264, and NGC 6323, the radii are derived from the Plummer rotation curve fitting. Column 8: the central mass density of the Plummer cluster ρ₀ = 3M_∞/4πr³_c. Here, M_∞ is obtained from the Plummer rotation curve fitting. Column 9: the maximum stellar mass of the Plummer cluster below which the cluster will not evaporate in less than 10 Gyr. In all cases except NGC 1194, a cluster of neutron stars can be directly ruled out because m_max is less than ≈1.4 M_☉. Column 10: lifetime ($\emph {T}_{\rm age}$) of a cluster. The values shown here are limited by the collision timescale, which is the maximum lifetime of the cluster composed of either main-sequence stars, brown dwarfs, white dwarfs, or neutrons stars. Column 11: the radius of the gravitational sphere of influence for the maser BHs in arcsec. We calculate R_inf using Equation (1) of Barth et al. (2004) with the bulge velocity dispersion measurements from Greene et al. (2010).

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3.2. The Error Budget for the BH Mass

3.3. Notes on Individual Galaxies

3.4. Search for Continuum Emission

The near-Keplerian shapes of the rotation curves of many circumnuclear megamaser disks allows one to determine the enclosed (spherical) mass within the innermost radius of the disk. While the enclosed masses of the megamaser disks are very likely dominated by supermassive BHs, as we assumed in the previous sections, it is still important to find ways to justify this assumption. The question of whether the enclosed mass can be attributed to a point mass or a compact cluster of stars or stellar remnants has been addressed by Maoz (1995) & Maoz (1998). The main argument is that if the lifetime of a central cluster, limited by evaporation or collision timescales, is significantly shorter than the age of its host galaxy, then it is unlikely to persist, and a central supermassive BH would be required to account for the enclosed mass. Here, the evaporation timescale t_evap ≈ 136t_relax (t_relax is the half-mass relaxation timescale; Binney & Tremaine 2008) is considered to be the upper limit of the lifetime of any bound stellar system, whereas the collision timescale is the characteristic timescale that a star suffers a physical collision (i.e., an inelastic encounter; Binney & Tremaine 2008).

To estimate the lifetime of the central cluster, we follow Maoz (1995) and assume that the central cluster has the Plummer density distribution (Plummer 1915):

$$\begin{equation} \rho (r) = \rho _0 \biggl (1 + {r^2\over r_{\rm c}^2}\biggr)^{-5/2}, \end{equation} \tag{ 9 }$$

where ρ₀ is the central density and r_c is the core radius. (The reason for choosing the Plummer distribution is described in Section 2.1 of Maoz 1998.)

We constrain ρ₀ and r_c by fitting the P–V diagram of the megamaser disk with the rotation curve of a Plummer cluster:

$$\begin{equation} v_{\rm P} = \biggl [ {{\it GM}_\infty r^2 \over \big(r_{\rm c}^2 + r^2\big)^{3/2}}\biggr ]^{1/2}, \end{equation} \tag{ 10 }$$

where M_∞ = 4πρ₀r³_c/3 is the total mass of the cluster. Here, M_∞ and r_c are fitted as free parameters. From these two parameters, we calculated ρ₀ = 3M_∞/4πr³_c. In all cases M_∞ is very close to the "enclosed" mass measured from the Keplerian rotation curve fit in Section 3, and the differences are less than 4%–18%. In Figures 3 and 4, we show the fitted Plummer rotation curves along with the fitted Keplerian rotation curves.

Rather than using the method described above to constrain the core radius r_c, in some cases we could apply another approach that places even tighter constraints. We note that the Plummer rotation curve does not decrease monotonically with radius; instead the rotation curve turns over at a maximum rotation speed v_max:

$$\begin{equation} v_{\rm max} = \biggl ({2 {\it GM}_\infty \over 3^{3/2} r_{\rm c}}\biggr)^{1/2}. \end{equation} \tag{ 11 }$$

Having a maximum rotation velocity is not unique to a Plummer cluster. It is a general feature for clusters having the same form of density profile with the exponent smaller than −3/2.

The core radius r_c of a Plummer sphere having maximum rotation speed v_max is

$$\begin{equation} r_{\rm c}^{(\rm max)} = {2 {\it GM}_\infty \over 3^{3/2} v_{\rm max}^2}. \end{equation} \tag{ 12 }$$

We used Equation (12) to estimate r_c for the megamaser disks in NGC 1194, NGC 2273, NGC 2960, and NGC 4388. In these cases, we do not have very well-sampled or high quality rotation curves (see Figures 3 and 4), so using r^(max)_c from Equation (12) actually sets a tighter constraint on the core radius of the cluster than using rotation-curve fitting. For these four cases, we use the highest observed velocity in the PV diagram as an estimate (lower limit) of v_max and use it to calculate r^(max)_c with M_∞ from the Plummer rotation curve fitting from Equation (10). For UGC 3789, NGC 6323, and NGC 6264, we used r_c from rotation curve fitting because the qualities of the rotation curves are good, and they give tighter constraints on r_c. In Table 4, we give the Plummer model parameters for all of our megamaser disks.

We constrained the lifetime of the Plummer cluster $\emph {T}_{\rm age}$ by first requiring the cluster not evaporate in a timescale less than the age of its host galaxy (⩾10 Gyr if the galaxy has formed before z = 2). This requirement sets an upper limit to the mass of the constituent stars of the cluster because evaporation is unimportant so long as the mass of its stars satisfies the following equation:

$$\begin{equation} \biggl ({ m_\star \over \,M_\odot }\biggr) \lesssim [\ln (\lambda M_\infty /m_\star)]^{-1} \biggl ({r_{\rm h} \over 0.01\,{\rm pc}}\biggr)^{3/2} \biggl ({M_\infty \over 10^8 \,M_\odot }\biggr)^{1/2}, \end{equation} \tag{ 13 }$$

where m_⋆ is the mass of each star, λ ≈ 0.1 (Binney & Tremaine 2008), and r_h is the radius of half total mass (r_h ≈ 1.305r_c). We call the maximum m_⋆ that satisfies the above equation m_max (Table 4) and used it to calculate the collision timescale of the Plummer cluster with the Plummer model parameters:

$$\begin{equation} t_{\rm coll} = \biggl [16 \sqrt{\pi } n_\star \sigma _\star r_\star ^2 \biggl (1 + {G m_{{\rm max}} \over 2 \sigma _\star ^2 r_\star }\biggr)\biggr ]^{-1}, \end{equation} \tag{ 14 }$$

where n_⋆ = ρ₀/m_max is the number density of stars, σ_⋆ is the rms velocity dispersion of the stars, and r_⋆ is the stellar radius (Maoz 1998; Binney & Tremaine 2008). If t_coll < 10 Gyr, then $\emph {T}_{\rm age}$ is constrained by t_coll and we can rule out the Plummer cluster as an alternative to the BH. If t_coll ⩾10 Gyr, we cannot rule out a cluster whether $\emph {T}_{\rm age}$ is dominated by evaporation or collision.

For UGC 3789 we obtained an upper limit m_max ≈ 0.10 M_☉ to the mass of individual stars and a lower limit N ≳ 1.0 × 10⁸ to the number of stars in the cluster. The mass limit directly rules out neutron stars as the constituents of the cluster, and only brown dwarfs, very low mass stars, or white dwarfs are possible. If the constituents of the cluster are brown dwarfs, the collision timescale for our Plummer model in UGC 3789 is t_coll < 3.9 × 10⁶ years, much less than the age of a galaxy. The timescale is even shorter if the constituent stars are main-sequence stars. However, if the cluster is composed of white dwarfs, the collision timescale can be as long as 1.8 × 10⁷ years, but this is still much shorter than the lifetime of a galaxy. Therefore, we conclude that a compact cluster is not likely to survive long, and the dominant mass at the center of UGC 3789 is a supermassive BH. For all other megamaser galaxies except NGC 1194 and NGC 6264, the lifetimes of the central clusters are also shorter than the age of a galaxy (Table 4). The constraints on lifetime weakly rule out massive clusters in NGC 4388 ($\emph {T}_{\rm age} \approx 7 \times 10^{9}$ yr) but strongly rule out clusters in the others ($\emph {T}_{\rm age} < 1.1 \times 10^{9}$ yr).

In summary, by setting tight constraints on the sizes and central mass densities of possible Plummer clusters, we have been able to strongly rule out clusters as the dominant central masses in UGC 3079, NGC 2273, NGC 6323, and NGC 2960 and weakly rule out the clusters in NGC 4388. We argue that supermassive BHs are the dominant masses in these megamasers. Together with the Milky Way Galaxy, NGC 4258, and M31 (Kormendy 2004), the number of galaxies with strong evidence to rule out a massive star cluster as the dominant central mass increases from three to seven.

It is important to compare results from different methods for measuring BH masses, because such comparisons can provide insight into potential systematic errors for each method (Siopis et al. 2009; Kormendy 2004; Humphrey et al. 2009; Greene et al. 2010). Comparing optically measured dynamical BH masses with those from the H₂O megamaser method is especially valuable, since the megamaser galaxies with Keplerian rotation curves provide the most direct and accurate BH mass measurements for external galaxies, These maser BH masses can be used to test the more commonly used BH mass measuring techniques in the optical, such as the stellar or gas dynamical methods.

One cannot meaningfully compare the stellar or gas dynamical BH masses with our maser BH masses unless the gravitational spheres of influence can be resolved. Otherwise, even if the optically determined dynamical masses agree with the maser BH masses within the errors, the uncertainty can be too large to tell the real accuracy of the stellar- or gas-dynamical method. The angular diameters of the spheres of influence of our maser BH masses, 2R_inf (Table 4), range from 0006 to 006. Among these megamasers, only the spheres of influence in NGC 4388 and NGC 1194 can be barely resolved by the HST (resolution ≈007 at λ ≈ 6500 Å) or the Very Large Telescope (VLT) assisted with adaptive optics (resolution ≈01 at λ = 2 μm). Therefore, of the galaxies here, stellar or gas dynamical measurements are feasible only for these two galaxies. We have obtained VLT time to measure the BH mass in NGC 4388, and we will apply the stellar dynamical method to this galaxy and compare the BH mass to the maser BH mass in the future.

Another commonly used BH mass measuring technique is the virial estimation method (Greene & Ho 2006; Kim et al. 2008; Vestergaard & Osmer 2009). We are able to compare this technique with the megamaser disk method in four galaxies, and this is the first time that the virial estimation method is directly tested. In this method, one estimates the BH mass as $M_{\rm BH} = \emph {f} R_{\rm BLR}\sigma _{{\rm line}}^2/G$, where f is an unknown factor that depends on the structure, kinematics, and orientation of the broad-line region (BLR), $\emph {R}_{\rm BLR}$ is the radius of the BLR, σ_line is the gas velocity dispersion observed in the BLR, and G is the gravitational constant. Here, we adopt the latest empirically determined 〈f〉 = 5.2 ±  1.3 from Woo et al. (2010). Vestergaard (2009) suggests that the virial method is accurate to a factor of ≈4. Since we cannot directly detect the BLRs in megamaser galaxies that have Seyfert 2 nuclei, we estimated σ_line using the scattered "polarized broad lines" (PBL) from the hidden BLRs in those four megamaser galaxies with detected PBLs: NGC 1068, NGC 4388, NGC 2273, and Circinus. We estimated $\emph {R}_{\rm BLR}$ via the $\emph {L}_{(2\hbox{--}10\, \rm kev)}\hbox{--}R_{\rm BLR}$ correlation (Kaspi et al. 2005). Table 6 shows the resulting virial BH masses for these four galaxies.

Under the assumption that the observed line width of broad H_α or H_β emission approximates the "intrinsic line width",¹⁴ we find that for NGC 1068, NGC 2273, NGC 4388, and Circinus, the BH masses at the 1σ level measured by the virial estimation method agree within a factor of five with the megamaser BH masses. So, while our comparison is so far limited to these four galaxies, the BH mass measured from the virial method matches the megamaser BH mass to about the expected accuracy.

The application of the virial method to megamaser galaxies is described in detail in the Appendix.

Table 4 shows the BH masses for all seven megamaser galaxies. Along with the previously published BH masses from megamaser observations, it is interesting to see that the maser BH masses, except in Circinus, are within a factor of three of 2.2 × 10⁷ M_☉, with the uncertainty of each BH mass ⩽12%. This small range of masses could be a selection bias. Disk megamasers are preferentially detected in Seyfert 2 spiral galaxies, and since the local active BH mass function for Seyfert 2 galaxies peaks at M_BH ≈ 3 × 10⁷ M_☉ (Heckman et al. 2004), we may be limited to BHs in this range of masses by analyzing disk megamasers.

Our new maser BH masses more than double the number of galaxies having dynamical BH masses M_BH ∼ 10⁷ M_☉. These measurements play a particularly important role in constraining the M–σ_⋆ relation at the low-mass end of known nuclear BH masses. In a companion paper, Greene et al. (2010) find that the maser galaxies as a group fall significantly below the M–σ_⋆ relation defined by more massive elliptical galaxies. As a result, the M–σ_⋆ relation that fits later-type and lower-mass galaxies has both a larger scatter and lower zero point than the relation for elliptical galaxies alone. However, there is a potential caveat that measuring robust (well-defined) σ_⋆ in late-type galaxies is challenging (Greene et al. 2010), and the contribution of systematic biases in σ_⋆ to the deviation from the M–σ_⋆ relation still needs to be explored in the future. With this caveat in mind, the observed deviations from the M–σ_⋆ relation at low mass imply that the relation may not be a single, low-scatter power law as originally proposed, and our BH mass distribution is consistent with the best simulations of galaxy growth regulated by "radio" feedback (see Figure 4 of Croton et al. 2006).

In addition to our seven galaxies with maser BH masses, we currently have VLBI data sets for another four disk megamaser candidates, and we are monitoring the spectra of more than six disk megamaser candidates that are currently too faint to be observed with VLBI, but may flare up in the future. Along with these galaxies and more megamasers we may discover in the future with the GBT, we expect to obtain a larger sample of maser BH masses which could help to clarify the M_BH–σ_⋆ relation.

Our main conclusions are the following.

• 1.  
  The maser distributions in all seven megamaser galaxies are consistent with edge-on circumnuclear disks surrounding central massive objects in the AGNs. The inner radii of the disks are between 0.09 and 0.5 pc, similar to all previously published megamaser disks. The rotation curves of all seven megamaser disks are consistent with Keplerian rotation. Four of the megamaser disks reveal evidence for warps.
• 2.  
  VLBI observations of circumnuclear megamaser disks are the only means to measure directly the enclosed mass and the mass density well within the radius of the gravitational sphere of influence of the central mass in these galaxies. The high central mass densities ((0.12–60) × 10¹⁰ M_☉ pc⁻³) (within the central 0.3 pc) of the seven megamaser disks indicate that in all, except one, disks, the central mass is dominated by a supermassive BH rather than an extremely dense cluster of stars or stellar remnants.
• 3.  
  The BH masses measured are all within a factor of three of 2.2 × 10⁷ M_☉ and the accuracy of each BH mass is primarily limited by the accuracy of the Hubble constant. The narrow range of BH mass distribution may reflect selection from the local active-galaxy mass function.
• 4.  
  Stellar dynamics cannot be applied to the majority of the megamasers presented here to measure BH masses with high precision. The gravitational spheres of influence in all cases except NGC 1194 and NGC 4388 are too small to be resolved by current optical telescopes. Observations with the VLT for measuring the BH mass in NGC 4388 with stellar-dynamical modeling are in progress.
• 5.  
  Under the assumption that the broad emission line widths can be estimated from polarized scattered light, we have calibrated for the first time the BH mass determination by the virial estimation method based on optical observations. With the latest empirically determined 〈f〉 = 5.2 ± 1.3, the virial estimated BH mass is within a factor of five at the 1σ level of the accurate BH mass based on megamaser disks in NGC 1068, NGC 2273, NGC 4388, and Circinus. This is comparable to the factor of four accuracy expected for the virial estimation method.
• 6.  
  The accurate BH masses in the seven megamaser galaxies contribute to the observational basis of the M–σ_⋆ relation at the low-mass end. The deviation from the mean relation of the several accurate maser BH masses suggests that the relation may not be a single, low-scatter power law as originally proposed, which has interesting implications for the universality of the M–σ_⋆ relation Greene et al. (2010).

The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. We thank Ed Fomalont for his kind help with our VLBI data reduction and Lincoln Greenhill for his substantial assistance for this project. C.Y.K. thanks Mark Whittle for his numerous insightful comments on our project. This research has made use of NASA's Astrophysics Data System Bibliographic Services, and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

The virial estimation method for measuring BH masses in AGNs (e.g., Greene & Ho (2006), Kim et al. (2008), Vestergaard & Osmer (2009)) uses the BLR gas as a dynamical tracer. It is usually only applied to Type 1 AGNs, where the BLRs can be directly observed. In this method, one estimates the BH mass with the following equation:

$$\begin{equation} {M_\bullet } = {f R_{\rm BLR}\sigma _{\rm line}^2 \over \rm G}, \end{equation} \tag{ A1 }$$

where f, $\emph {R}_{\rm BLR}$, and σ_line have been defined in Section 5. Since one can directly detect light from the BLRs in Type 1 AGNs, σ_line can be measured from the broad line spectra and one can use the continuum luminosity λL_λ(5100 Å) to estimate $\emph {R}_{\rm BLR}$ via the $\lambda L_{\lambda }(5100\,\AA)\hbox{--}\emph {R}_{\rm BLR}$ correlation (e.g., Kaspi et al. 2000, 2005).

In a Type 2 AGN, including the megamaser galaxies we study here, our line-of-sight to the BLR is blocked by heavy dust extinction, so one cannot directly measure λL_λ(5100 Å) and σ_line. Instead, one can probe the BLRs in megamaser galaxies with polarized scattered light and hard X-rays. Among all the megamaser galaxies with measured BH masses, polarized scattered light from the BLRs has been detected in NGC 1068, NGC 4388, NGC 2273, and Circinus (see Table 6), and X-ray measurements are also available for these four galaxies.

We can estimate σ_line from the relation σ_line/V_FWHM = 2.09 ±  0.45 from Woo et al. (2010), where V_FWHM is the FWHM width of polarized scattered light in the broad H_α line (NGC 2273, NGC 4388, and Circinus) or H_β line (NGC 1068). The major concern with these line widths is that the observed values may not be the same as the line width one would measure if the BLRs could be observed directly. We identify two effects that can induce such a difference from the well-studied case NGC 1068 (Miller et al. 1991). First, the broad H_β lines in the polarized flux spectra can contain a contribution from the narrow H_β lines. Without removing the contribution from narrow lines, the BLR line width may be underestimated by ∼20%–30% in NGC 1068. Second, the polarized emission from AGNs may originate from light being scattered by electrons with a temperature a few times 10⁵ K, which results in significant thermal broadening (∼50%) of the spectral lines. Because of these two effects, there will be systematic errors in σ_line if one directly uses the observed line width of the polarized lines to estimate σ_line. Among the four megamaser galaxies we consider here, only NGC 1068 has been studied in enough detail to remove these two effects. Luckily, the two effects change the line width in opposite directions, and hence could offset each other to a certain extent. As in NGC 1068, if no correction is made for these two effects, the systematic error will be only ∼10%, just slightly larger than the measurement error. Without knowing the actual contributions of these two effects for the other three galaxies, we assume that the two effects cancel each other to the same extent as in NGC 1068 and use the observed line widths as the approximations for the intrinsic widths. The reader should be aware of this caveat when interpreting the comparisons made.

In this work, $\emph {R}_{\rm BLR}$ was estimated from the "intrinsic" $\emph {L}_{\rm (2-10 \,kev)}$ of the nuclear region via the $\emph {L}_{\rm (2-10 \,kev)}\hbox{--}\emph {R}_{\rm BLR}$ correlation (Kaspi et al. 2005). Since three (NGC 1068, NGC 2273, and Circinus) of the four megamaser galaxies considered here are Compton-thick (i.e., the X-ray absorbing column density is >10²⁴ cm⁻²), we paid particular attention to how the $\emph {L}_{\rm (2-10 \,kev)}$ were measured. The Compton thick nature of these AGNs is a problem because the intrinsic radiation is mostly suppressed and the X-ray spectrum is dominated by the reflected or scattered components. It is difficult to measure the actual absorbing column density and it is very likely that the intrinsic hard X-ray luminosity is severely underestimated (e.g., Levenson et al. 2006; Bassani et al. 1999). Therefore, we excluded those measurements that did not consider the Compton-thick nature of these sources and failed to give the absorbing column density in the expected range. We mainly considered those measurements from either data with appropriate modeling or from observations with instruments capable of directly measuring the transmission components of X-ray above 10 keV. We took at least two different measurements for each galaxy from the literature and used the average value to calculate R_BLR from the correlation in Kaspi et al. (2005).

Given R_BLR and σ_line, we estimated the BH masses using Equation (A1) with the empirically determined 〈f〉 = 5.2^+1.3_−1.3 from Woo et al. (2010). The resultant BH masses were compared with the maser BH masses in Section 5.