THE MEGAMASER COSMOLOGY PROJECT. II. THE ANGULAR-DIAMETER DISTANCE TO UGC 3789

As a complement to observations of the cosmic microwave background (CMB), an independent measurement of the Hubble constant, H₀, within 3% rms would place a valuable constraint on the dark energy equation-of-state parameter, w (e.g., Hu 2005; Olling 2007). The most widely accepted value for H₀ is 72 ± 7 km s⁻¹ Mpc⁻¹ (Freedman et al. 2001), based on the "extragalactic distance ladder" and using the Large Magellanic Cloud (LMC) to calibrate Cepheid variables which are treated as standard candles. The Cepheid metallicity correction has been controversial, as highlighted by Sandage et al. (2006), who used similar methods but a different metallicity correction to obtain H₀ = 62 ± 6 km s⁻¹ Mpc⁻¹. If measured at the ≲3% level, a value of H₀ = 72 km s⁻¹ Mpc⁻¹ would be consistent with the cosmological constant as an explanation of dark energy (w = −1), whereas H₀ = 62 km s⁻¹ Mpc⁻¹ would challenge that model (e.g., Figure 14 of Riess et al. 2009).

Macri et al. (2006) used NGC 4258 rather than the LMC to calibrate Cepheids for the extragalactic distance scale and measured H₀ = 74 ± 7 km s⁻¹ Mpc⁻¹. Riess et al. (2009) subsequently refined the measurement, also using Cepheids in NGC 4258 to calibrate those in selected SN Ia host galaxies, and they determined H₀ = 74.2 ± 3.6 km s⁻¹ Mpc⁻¹ (5%). Riess et al. were able to reduce the systematic uncertainty in H₀ compared to the LMC-based results because the Cepheids in NGC 4258 are more similar in metallicity and period to those in the galaxies being calibrated. NGC 4258 is a good foundation for the extragalactic distance scale because its distance (7.2 Mpc) is known to high accuracy from observations of circumnuclear water megamasers in its active galactic nucleus accretion disk (Herrnstein et al. 1999). NGC 4258 is too close, though, for determining H₀ directly because its recession velocity may be highly influenced by its peculiar motion. Lo (2005) gives a review of megamasers, including a discussion of the maser distance technique used in NGC 4258.

The Megamaser Cosmology Project (MCP) aims to determine H₀ by measuring angular-diameter distances to galaxies in the Hubble flow, ∼50–200 Mpc distant, using the maser technique. Being independent of Cepheids and the CMB, the MCP will provide a valuable check of these other methods for measuring H₀. The project is a multi-year effort that begins with surveys for new megamaser disks. Measuring a distance to each galaxy detected with a megamaser disk requires high-fidelity very long baseline interferometry (VLBI) imaging, which we pursue with the High Sensitivity Array (VLBA+GBT+VLA+Effelsberg), and spectral-line monitoring to measure the centripetal accelerations of individual lines in the maser spectrum. The MCP aims to measure maser distances with ∼10% or better accuracy to each of about 10 galaxies, thereby determining H₀ to ∼3%.

In a survey with the Green Bank Telescope (GBT), Braatz & Gugliucci (2008, hereafter BG08) discovered water maser emission from the Seyfert 2 galaxy UGC 3789. The maser spectrum has three sets of Doppler components spaced nearly symmetrically at and around the galaxy's systemic recession velocity (Figure 1). This profile is characteristic of megamasers in an edge-on accretion disk. The megamaser in UGC 3789 is similar to that in NGC 4258, where "high-velocity" blueshifted and redshifted maser components arise from the tangent points of the disk rotating toward and away from the observer, and components near the systemic recession velocity originate on the near side of the disk, along the line of sight to the central black hole.

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Rotation velocities indicated by the high-velocity maser lines in UGC 3789 are ∼400–800 km s⁻¹. By monitoring the maser spectrum roughly monthly over six months, BG08 measured secular, redward drifts in radial velocity of 2–8 km s⁻¹ yr⁻¹ among the systemic group of maser components, while the redshifted and blueshifted features remained relatively fixed in velocity. The radial velocity drift is interpreted as centripetal acceleration in the maser gas as it orbits the central supermassive black hole. The spectral profile and the measured velocity drifts demonstrate that the maser disk in UGC 3789 is suitable for measuring an angular-diameter distance to the galaxy, using the method pioneered by Herrnstein et al. (1999).

In Paper I from the MCP (Reid et al. 2009), we presented a VLBI image of the maser disk in UGC 3789. The masers reveal nuclear gas in an edge-on disk ranging in radii from 0.08 to 0.30 pc. The high-velocity lines trace a Keplerian rotation curve with high precision. While the high-velocity lines indicate relatively little disk warping, the systemic features show some structure that suggests that the front side of the disk may be warped, tilted, or geometrically thick.

Here, we present single-dish observations of the UGC 3789 maser made with the GBT over a period of 3.2 years. Using these observations, we refine the acceleration measurements made by BG08 and present a model to estimate the distance to UGC 3789 using the maser observations.

Between 2006 January 12 and 2009 March 31, we observed the 22 GHz water maser emission from UGC 3789 using the GBT of the National Radio Astronomy Observatory. We observed the source 36 times during this period, with the observations spaced roughly monthly throughout, except for the summer months when the humidity makes observations at 22 GHz inefficient. Table 1 shows the observing date and sensitivity for each observation. The first nine epochs are the same as reported in BG08.

We configured the spectrometer with two 200 MHz spectral windows, each with 8192 channels, giving a channel spacing of 24 kHz, corresponding to 0.33 km s⁻¹ at 22 GHz. The first spectral window was centered on the systemic recession velocity of the galaxy and the second was offset by 180 MHz to the red, giving a total bandwidth coverage of 380 MHz (5100 km s⁻¹). We observed both circular polarizations simultaneously using a total-power nodding mode in which the galaxy was placed alternately in one of the two beams of the 18–22 GHz K-band receiver, observing for 2.5 minutes at each position and then repeating the cycle.

We reduced and analyzed the data using GBTIDL. In order to enhance weak signals, we smoothed the blank sky reference spectra using a 16-channel boxcar function prior to calibration. We corrected the spectra for atmospheric attenuation using opacities derived from weather data and applied an elevation-dependent gain curve. Our flux density scale is uncertain by about 15%, limited by noise-tube calibration and pointing errors. To remove residual baseline shapes from each 200 MHz spectrum, we subtracted a polynomial (typically third order) fit to the line-free channels of the calibrated spectrum. Finally, we Hanning smoothed the spectra. Integration times were typically about 2 hr for each epoch prior to 2007 May and about 4 hr after.

Unless stated otherwise, velocities quoted in this paper use the optical convention (v_opt = cz) and are given in the frame of the local standard of rest (LSR). For UGC 3789, the LSR frame relates to the heliocentric frame by v_hel = v_LSR − 0.3 km s⁻¹.

In Figure 2, we plot the radial velocities of UGC 3789 maser peaks as a function of the epoch observed. The local maxima in the maser profiles were determined by eye and are accurate to about one channel width (0.3 km s⁻¹). The trends are evident: high-velocity spectral lines are roughly invariant in velocity throughout the monitoring period, while the radial velocities of the systemic features show a clear drift redward. The maser spectral lines in the systemic range remain within a window of velocities such that new lines may appear near the low-velocity limit and disappear beyond the high-velocity limit. Some lines can be tracked throughout the entire monitoring period, while others appear or disappear over the course of the monitoring. If sufficiently long-lived, it would take individual maser spectral components at least 10 years, and in some cases much longer, to drift through the entire velocity window.

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The drift pattern evident in Figure 2 is consistent with the picture of high-velocity maser emission originating near the midline of the edge-on megamaser disk and maser emission near the systemic velocity originating on the near side of the disk. Here, we define the midline as the diameter of the disk perpendicular to the line of sight. Maser features along the line of sight to the central black hole but on the far side of the disk would show a drift to lower velocities, but there is no evidence in our data for such blueward drifts in UGC 3789. The absence of detected masers from the far side of the disk may be understood if systemic masers amplify radio continuum emission originating interior to the maser disk. In that case, masers on the far side of the disk would be beamed away from us. Alternatively, maser emission from the far side of the disk might be free–free absorbed near the central black hole.

To measure accelerations for each of the maser lines in UGC 3789, we applied a global, nonlinear, least-squares χ² minimization routine that decomposes the spectra into multiple Gaussians at each epoch and solves simultaneously for the amplitude, width, velocity, and constant drift (acceleration) for each maser component. The method is described by Humphreys et al. (2008). We allowed the amplitude, width, and velocity to vary for each spectral component from epoch to epoch.

Finding consistent and stable solutions from the fitting algorithm required experimentation with the initial parameters used by the fitting algorithm, and many trials were required to determine the final accelerations. We found it convenient to divide the spectra into 15–30 km s⁻¹ segments, selected by choosing edges near local minima in the profile. We then solved for accelerations in each segment separately. We began the fitting process using initial parameters describing equally spaced spectral components across the velocity range of interest. In the initial trials, we spaced spectral components by about 4 km s⁻¹. We then added components in later trials until the solution was satisfactory.

After each trial, we judged the results by examining the fits and residuals at each epoch. Figure 3 gives an example of the Gaussian components and residuals determined from a good fit to a section of the data among the systemic components. We also compared the fitted accelerations to the trends apparent in plots showing velocities of maser peaks over time, as in Figure 2. The line segments plotted in the central panel of Figure 2 correspond to accelerations measured from the global fit, while the plus symbols represent the velocities of maser peaks determined by eye from each spectrum. Where the fit accelerations were clearly in disagreement with the trends in velocities of the maser peaks, we rejected the solution and tried a new set of initial parameters. When the residuals were poor over the entire velocity range being fitted, we added components and maintained equal velocity spacing. When the residuals were poor only in a narrow range of the velocities being fit, we kept a base set of equally spaced spectral components and inserted additional spectral components at the specific velocities where the residuals were unsatisfactory. A good fit to masers in the systemic velocity range typically required that we use one Gaussian component for every 2–3 km s⁻¹ of velocity space being fit. We confirmed that the solutions remained stable and consistent near the edges of each segment by running test cases with overlap on the segment boundaries.

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It is apparent in Figure 2 that the fitting algorithm is able to find stable solutions even in parts of the spectrum where there are no distinct peaks in the spectral profile. In these velocity ranges, the emission lines are typically weak or heavily blended. We note, however, that the data used in the final distance analysis come only from velocity ranges where there are both distinct peaks detected by eye and stable solutions determined from the fitting algorithm.

For high-velocity lines, we used data from all 36 epochs to fit for acceleration. The small accelerations among high-velocity lines make it possible to track lines for long durations and obtain stable solutions using data for all 3.2 years of monitoring. For the systemic features, however, we found that using all 36 epochs made the global solution unstable. Fitting systemic lines using all epochs may be complicated by the appearance and disappearance of many fast-drifting spectral components, making it difficult to account for all components over long durations. Our aim is to determine accelerations appropriate for the VLBI observing date of 2006 December 10. We therefore fit accelerations of systemic components using a subset of the epochs roughly centered on this date and including at least the epochs between 2006 September 9 and 2007 April 14. In Figure 2, the lengths of the line segments showing the acceleration fits represent the number of epochs used in each fit.

In Figure 4, we show the best-fit accelerations for each maser line as measured by the global least-squares fit. The redshifted lines have a mean value of −0.13 ± 0.04 km s⁻¹ yr⁻¹ and the blueshifted lines have a mean of −0.16 ± 0.04 km s⁻¹ yr⁻¹. Here, we quote unweighted, standard errors of the mean. Accelerations of the systemic features are consistent with the preliminary work of BG08 and range from 0.5 to 7.3 km s⁻¹ yr⁻¹. Tables 2–4 list the accelerations measured for each line. The uncertainties listed in these tables are formal values from the global fit and may underestimate the true uncertainties.

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In this section, we discuss the "megamaser disk method" for measuring the angular-diameter distance to a galaxy based on a straightforward model of the maser emission arising from an edge-on Keplerian disk. This model gives a good approximation to the maser emission from NGC 4258. To measure the distance to UGC 3789, we make a simple extension of this model, as presented in Section 5.

First, we consider a model in which maser emission originates in a thin, flat, edge-on disk and the dynamics are dominated by a central massive object, with all maser clouds in circular orbits. In this model, high-velocity masers trace gas near the disk midline. Systemic masers occupy part of a ring orbiting at a single radius and covering a small range of azimuthal angles on the near side of the disk. When plotted on a position–velocity (P–V) diagram, the high-velocity masers trace a Keplerian rotation curve and the systemic masers fall on a line segment. If the line segment is extended to the rotation curve traced by the high-velocity features (as in the dotted lines shown in Figure 3 of Paper I), the intersecting points (θ₁, v₁) and (θ₂, v₂) determine both the angular radius (θ) and magnitude of the rotation velocity (v_r) of the narrow ring traced by systemic masers, according to θ = |θ₁ − θ₂|/2 and v_r = |v₁ − v₂|/2. In this model, the angular-diameter distance to the galaxy is

$$\begin{equation} D_a = r/\theta = v_r^2/(a \theta), \end{equation} \tag{ 1 }$$

where r is the radius of the ring traced by the systemic maser components (r = v²_r/a) and a is the centripetal acceleration of systemic components, measured from the change in Doppler velocity with time.

In practice, we do not measure θ and v_r directly. Instead we measure the Keplerian rotation constant k, which we determine by fitting the high-velocity lines to the rotation curve v = k(θ − θ₀)^−1/2 + v_sys. In addition, we measure a curvature parameter Ω, which is the slope of the line traced by systemic features on the P–V diagram (Ω ≃ dv/dθ for the systemic features). With this parameterization, we can write the angular-diameter distance to the galaxy as

$$\begin{equation} D_a = a^{-1} k^{2/3} \Omega ^{4/3}. \end{equation} \tag{ 2 }$$

The fractional uncertainty in D_a is

$$\begin{equation} \frac{\sigma _{Da}}{D_a} \simeq \sqrt{\left(\frac{\sigma _a}{a} \right)^2 + \frac{4}{9} \left(\frac{\sigma _k}{k} \right)^2 + \frac{16}{9} \left(\frac{\sigma _\Omega }{\Omega } \right)^2}. \end{equation} \tag{ 3 }$$

Here, we use a notation in which the uncertainty in a measured or derived value, x, is represented as σ_x, so the fractional uncertainty in x is $\frac{\sigma _x}{x}$.

The straightforward method for measuring the angular-diameter distance described above works well for NGC 4258, where systemic accelerations fall in the range 7.7–8.9 km s⁻¹ yr⁻¹ (Humphreys et al. 2008), indicating that they are seen within a narrow range of radii, r, from the central black hole (r∝ 1/$\sqrt{a}$). The tightly confined range of radii in NGC 4258 is a consequence of the masers forming in the bottom of a bowl-shaped region of the warped accretion disk (Humphreys et al. 2008). While it may be fortuitous that the systemic masers prefer a single radius, we note that the high-velocity features in NGC 4258 also seem to prefer specific radii and even show periodicity suggestive of spiral structure (Maoz 1995; Humphreys et al. 2008).

The masers in UGC 3789 also appear to cluster around discrete radii from the black hole. In Figure 5, we reproduce the VLBI map presented in Paper I. The "red" features in particular are seen mainly at three distinct radii from the central black hole. Evidence that the systemic features are also clustered in radius comes from the acceleration profile (Figure 4), as we describe in the next section.

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Except for the westernmost "red" maser, all the high-velocity maser lines in UGC 3789 fall on a straight line, within the measurement uncertainties (Figure 5). The line has a position angle of 411, east of north. The maser distribution is consistent with emission from a thin, edge-on disk. The systemic maser spots show systematic deviations from the line defined by the high-velocity masers, with the easternmost systemic masers appearing north of the line and the westernmost systemic masers south of it. One way to interpret this arrangement is that the near side of the disk, traced by the systemic masers, is warped while the high-velocity lines trace a part of the disk that appears relatively unwarped from our point of view.

We define an "impact parameter" for each maser spot as the projection of its angular offset onto the line defined by the high-velocity masers, with an impact parameter θ₀ identifying the location of the dynamical center. To determine the impact parameters, we rotated the map on the sky by 489 (counterclockwise, when viewed as in Figure 5) and measured the impact parameter for each maser component by its abscissa on the rotated axes.

When modeling the data, we corrected the recorded "optical" velocities of all maser spots for relativistic effects, as in Argon et al. (2007). First, we recalculated each velocity using the relativistic Doppler formula. We estimated the black hole mass using these velocities, then corrected for the gravitational time dilation caused by the central black hole. Finally, we corrected velocities of the systemic features for the relativistic transverse Doppler effect. The effect of the relativistic corrections is to reduce the measured optical velocities by 12–27 km s⁻¹, the blueshifted features getting the smaller corrections. We then used these corrected velocities in all of the analysis, including the determination of the final black hole mass reported in this paper.

5.1. The Keplerian Rotation Curve

5.2. Accelerations as Evidence for a Two-ring Model

5.3. Curvature Parameter Ω

Figure 6 shows the distribution of systemic maser features on a P–V diagram. Rather than tracing a single line, as would be expected if all masers were at a common radial distance from the central black hole, there is a kink in the distribution with the masers on each side of the kink roughly defining a line. The distribution is consistent with the masers being detected primarily in two rings. In Figure 6, we use unique symbols to differentiate the masers associated with each of the rings. We can measure the slope Ω separately for each ring: Ω = 695 ± 65 km s⁻¹ mas⁻¹ for ring 1 and Ω = 1654 ± 440 km s⁻¹ mas⁻¹ for ring 2. The poorer fit for ring 2 results, in part, from that ring covering a narrower velocity range than ring 1, so it has fewer spectral channels to fit from the VLBI map.

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5.4. The Angular-diameter Distance to UGC 3789

We can determine the distance to UGC 3789 separately for each of the two rings using Equation (2). For ring 1, we find D_a = 50.2 ± 7.7 Mpc and for ring 2 D_a = 48.1 ± 17.4 Mpc. The weighted mean of these two values gives our best determination of the maser distance to UGC 3789: D_a = 49.9 ± 7.0 Mpc (14%).

The rotation curve determined from high-velocity lines provides a central black hole mass

$$\begin{equation*} M_{\rm bh} = 1.13 \times \left(\frac{k}{\rm km\,s^{-1}\,mas^{0.5}}\right)^2 \times \left(\frac{D_a}{\rm Mpc}\right)\ \hbox{$M_{\odot }$}. \end{equation*}$$

At 49.9 Mpc and k = 440 km s⁻¹ mas^0.5, the mass of the nuclear black hole is 1.09 × 10⁷ M_☉ ± 14%, with nearly all of the mass uncertainty being caused by the distance uncertainty.

The peculiar radial velocity of UGC 3789 relative to the CMB is −151 ± 163 km s⁻¹ (K. L. Masters 2009, private communication), using a flow model based on the SFI++ sample (Masters et al. 2006; Springob et al. 2007). Here, peculiar velocity is defined as v_pec = v_rec − H₀D, where v_rec is the galaxy's observed recession velocity. Writing the UGC 3789 maser radial velocity in the CMB frame (v_CMB ≃ v_LSR + 60 km s⁻¹) and correcting for the peculiar velocity, we obtain a relativistic, recessional flow velocity of 3481 ± 163 km s⁻¹. Applying a standard ΛCDM model with Ω_m = 0.26, we therefore determine H₀ = 69 ± 11 km s⁻¹ Mpc⁻¹.

The main source of uncertainty in the distance to UGC 3789 comes from the measurement of the orbital curvature parameter Ω. The uncertainty in the acceleration is also a significant contributor, while the contribution from the Keplerian rotation constant is negligible. Therefore, we can best reduce the total uncertainty in the distance to UGC 3789 by obtaining a more sensitive VLBI map. We are currently processing additional VLBI observations of UGC 3789, which we can average to increase the map sensitivity. Improving the measurement of the acceleration profile requires more sensitive monitoring observations. Recognizing this need, we increased the duration of GBT observations after 2007 May. These efforts should reduce the overall distance uncertainty to <10%, which would determine H₀ with accuracy comparable to the Hubble Key Project, but with entirely independent data and methods.

Ultimately, to reach an uncertainty in H₀ of ≲3% from the MCP, we require distances to at least 10 galaxies comparable to UGC 3789. We are currently pursuing observations of the megamaser galaxies Mrk 1419, NGC 6264, and NGC 6323, which could eventually lead to comparable distance measurements. There are another ∼6 maser disk galaxies known that may also be suitable for distance measurements, but the quality of these (i.e., brightness of the masers, richness of the spectrum, and velocity coverage) is not as good. Surveys for additional maser galaxies therefore remain an important part of the MCP.

As part of the MCP, we have observed the water megamaser disk in the nucleus of the galaxy UGC 3789. Paper I (Reid et al. 2009) presents a VLBI map of the maser emission and here we present measurements of orbital accelerations from GBT monitoring observations spanning 3.2 years. The high-velocity maser emission traces a thin, edge-on disk 0.08–0.30 pc in radius and demonstrates Keplerian rotation about a 1.09 × 10⁷ M_☉ black hole. The dynamical center corresponds to a (relativistic) LSR recession velocity of 3270 ± 4 km s⁻¹. We model the emission from the systemic masers using two narrow rings and apply the "maser distance method" to each. Combining the results, we determine a distance to UGC 3789 of 49.9 ± 7.0 Mpc, corresponding to H₀ = 69 ± 11 km s⁻¹ Mpc⁻¹. Additional observations of the maser in UGC 3789 will improve the signal-to-noise ratio and can reduce the distance uncertainty to about 10%. We have demonstrated that angular-diameter distance determination to galaxies with observed megamaser disks is a promising approach to determining H₀, independent of the traditional distance ladder approach based on Cepheid variables. The MCP aims to measure the distances to additional megamaser disk galaxies in order to reduce the overall uncertainty in H₀ and eventually place meaningful constraints on the dark energy equation-of-state parameter.

We thank the referee for helpful comments, and we are grateful to the NRAO staff in Green Bank for their many contributions to this research program. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.