A MEASUREMENT OF THE BLACK HOLE MASS IN NGC 1097 USING ALMA

NGC 1097 is a nearby Type-1 Seyfert galaxy at a distance of 14.5 Mpc (Tully 1988) (∼70 pc arcsec⁻¹). The position of the nucleus is determined by the peak position of the 6 cm continuum emission (Hummel et al. 1987): ${\rm R}.{\rm A}.({\rm J}2000.0)\;={{02}^{{\rm h}}}{{46}^{{\rm m}}}18\buildrel{\rm{s}}\over{.}96$, ${\rm decl}.({\rm J}2000.0)=-30{}^\circ 16^{\prime} 28\buildrel{\prime\prime}\over{.} 9$. The peak position of the 860 μm continuum emission coincides with the 6 cm peak (Izumi et al. 2013). Properties of NGC 1097 are summarized in Table 1.

Table 1.  Properties of NGC 1097

Parameter	Value	Reference
Morphology	SB(s)b	1
Nuclear activity	Type 1 Seyfert	2
Position of nucleus	⋯	3
R.A.(J2000.0)	${{02}^{{\rm h}}}{{46}^{{\rm m}}}18\buildrel{\rm{s}}\over{.}96$	⋯
Decl.(J2000.0)	$-30{}^\circ 16^{\prime} 28\buildrel{\prime\prime}\over{.} 9$	⋯
Systemic velocity (km s−1)	1253a	4
Position angle (°)	130	1, 4
Inclination angle (°)	46 ± 5	5
Distance (Mpc)	14.5	6
Linear scale (pc arcsec−1)	70	6
I-band luminosity (mag)	8.09	7

Note. ^aSystemic velocity here is a heliocentric velocity determined with molecular lines. Koribalski et al. (2004) shows the heliocentric velocity to be 1271 km s⁻¹ determined with HIPASS observation.

References. (1) de Vaucouleurs et al. (1991), (2) Storchi-Bergmann et al. (1993), (3) Hummel et al. (1987), (4) this work, (5) Ondrechen et al. (1989), (6) Tully (1988), (7) Springob et al. (2007).

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The SMBH mass in NGC 1097 is estimated to be $(1.2\pm 0.2)\times {{10}^{8}}{{M}_{\odot }}$ by Lewis & Eracleous (2006) using the empirical ${{M}_{{\rm BH}}}-\sigma$ relation from Tremaine et al. (2002) with an observed $\sigma =196\pm 5$ km s⁻¹. The uncertainty in this estimate is large, depending on the assumed ${{M}_{{\rm BH}}}-\sigma$ relation. The latest ${{M}_{{\rm BH}}}-\sigma$ relation [${{{\rm log} }_{10}}({{M}_{{\rm BH}}}/{{M}_{\odot }})=8.32+5.64{{{\rm log} }_{10}}(\sigma /200$ km s⁻¹), McConnell & Ma (2013)] would yield SMBH mass of $(1.9\pm 0.3)\times {{10}^{8\,}}{{M}_{\odot }}$. Note that this relation is a fit to both late-type and early-type galaxies. When selecting only the late-type galaxies, the ${{M}_{{\rm BH}}}-\sigma$ relation becomes ${{{\rm log} }_{10}}({{M}_{{\rm BH}}}/{{M}_{\odot }})=8.07+5.06{{{\rm log} }_{10}}(\sigma /200$ km s⁻¹) (McConnell & Ma 2013) and the estimated SMBH mass becomes $(1.1\pm 0.3)\times {{10}^{8}}{{M}_{\odot }}$.

The enclosed mass in 40 pc radius has been studied by (Izumi et al. 2013) to be $2.8\times {{10}^{8\,}}{{M}_{\odot }}$, using the dynamics from ${\rm HCN}(J=4-3)$ emission line. In contrast, Fathi et al. (2013) report a dynamical mass in $40{\rm pc}$ radius as $8.0\times {{10}^{6\,}}{{M}_{\odot }}$ from the same data of Izumi et al. (2013). The difference occurs because Fathi et al. (2013) assume a thin disk and extracts the non-circular motions of the gas while Izumi et al. (2013) assume a simple Keplerian rotation. Note but the dynamical mass of Izumi et al. (2013) includes all the mass within that radius, not showing the intrinsic SMBH mass. A more detailed study of NGC 1097 is thus necessary to precisely measure the SMBH mass.

We express the mass distribution as a summation of the SMBH mass and the stellar mass profile, expressed as the stellar luminosity distribution multiplied by a constant I-band mass-to-light ratio (M/L). Assuming that the galaxy is axisymmetric, the stellar luminosity distribution along the galaxy major axis is modeled as a superposition of Gaussian components, using the idea of MGE (Emsellem et al. 1994). The major axis defined here is at a position angle of 130° from de Vaucouleurs et al. (1991). We use an I-band image observed with the Advanced Camera for Surveys Wide Field Channel F814W filter on Hubble Space Telescope (HST) to obtain the luminosity distribution at the major axis (black line in Figure 2). We subtract the contribution from the Active Galactic Nucleus (AGN) and flatten the starburst dominated region (both are shaded in Figure 2) in order to estimate the underlying stellar luminosity profile. The AGN profile is calculated by the convolution of a delta function with the Point Spread Function (PSF), measured to be $0\buildrel{\prime\prime}\over{.} 2$ (full width half maximum, FWHM) from five unresolved stars in the same image. The PSF is also checked by using "Tiny Tim" package (version 6.3) developed by Krist et al. (2011). The luminosity value of the delta function is determined to obtain the residual luminosity distribution larger than 0 at any radius. Note that the AGN subtraction does not gravely affect the SMBH mass estimation. Even the peak before subtraction gives a mass of less than $6.00\times {{10}^{6\,}}{{M}_{\odot }}$ for a M/L ratio 5.14, and it is at least one digit smaller than the SMBH mass we put in the model. The starburst ring region is determined to be in the range of radii from $-11^{\prime\prime}$ to $-7^{\prime\prime}$ and from 8'' to 13'' along the major axis. The ring includes younger stars than inside or outside of the ring, and is bluer in color. We model the stellar luminosity profile by calculating the least-square value with the luminosity distribution without the ring and the AGN. The model (blue line in Figure 2), therefore, consists mainly of old stars, but does not include younger stars on the ring. See also Table 3 for MGE parameters we give for the data. The mass profile is simply modeled by multiplying the constant I-band M/L ratio to the modeled stellar luminosity profile and adding the assumed SMBH point mass in the center. Note again that we model older stars, by which means we are assuming that the radial difference of the M/L ratio is negligible.

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Table 3.  MGE Parameters for the Stellar Luminosity Obtained from WFC F814W Surface Brightness of NGC 1097

j	Ij(${{L}_{\odot ,I}}/$pc−2)	${{\sigma }_{j}}$(arcseconds)	qj
1..........	7451.23	0.500	0.900
2..........	5122.72	3.50	0.700
3..........	3725.62	1.25	0.900
4..........	1862.81	16.0	0.700

Note. ^aSee the text for how we deal with the AGN and the starburst region.

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With a given mass profile from Section 3.1, we calculate the velocity field from a gravitational potential field derived with equations from Cappellari et al. (2002). We use MGE_circular_velocity code, which is in the JAM modeling package of Cappellari (2008). The inclination angle is set to be 46° (Ondrechen et al. 1989). See Section 4.2 for more details regarding the inclination angle.

We calculate a PVD model along the galaxy major axis by assuming that the observed molecular gas follows the velocity field obtained by our calculation. The observational effects are taken into account by utilizing KinMS (Davis et al. 2013). We convolve the model cloud distribution with our beamsize ($1\buildrel{\prime\prime}\over{.} 60\times 2\buildrel{\prime\prime}\over{.} 20$) to express the beam-smearing, and assume the molecular gas disk to be an axisymmetric thin disk. The position angle of the galaxy major axis is set to be $130{}^\circ$ (de Vaucouleurs et al. 1991), which is consistent with the kinematical position angle estimated from our observational data. The position angle is also consistent with the one calculated by Spitzer Survey of Stellar Structure in Galaxies (S4G, Sheth et al. 2010), $-52{}^\circ$, and global properties calculated via their pipeline 3 by Muñoz-Mateos et al. (2015). Note that we can mostly avoid the region with non-circular motion pointed out by Fathi et al. (2013) by using a PVD along the galaxy major axis. We assume that the streaming motion remaining in the PVD is negligible. We can also comment that when considering the error propagation for a simple equation of ${{v}^{2}}/2=GM/r$, 10 percent error in the velocity could result in 20 percent error for the SMBH mass, which is consistent the error bar we derive from Figure 5.

The observed PVD of ${\rm HCN}(J=1-0)$ is shown in the upper panel of Figure 3 with color filled contours. We set the kinematical position angle to be $130{}^\circ \pm 5{}^\circ$. The center of the galaxy is assumed to be the peak of 6 cm observation (Hummel et al. 1987), 860 $\mu {\rm m}$, and ${\rm HCN}(J=4-3)$ observation (Izumi et al. 2013). We fit the observed PVD with PVD models calculated with 2 free parameters, the I-band M/L ratio and the SMBH mass. We find the two parameters to be around M/L ∼ 5.0 and ${{M}_{{\rm BH}}}\sim 1.0\times {{10}^{8\,}}{{M}_{\odot }}$ by initial robust-grid calculation. Then the finer grid of model parameters is set to be from M/L = 4.80 to 5.35 in steps of 0.01 and ${{M}_{{\rm BH}}}=0.5\times {{10}^{8}}{{M}_{\odot }}$ to $2.5\times {{10}^{8}}{{M}_{\odot }}$ in steps of $0.1\times {{10}^{8}}{{M}_{\odot }}$. We calculate an optimal rotation curve from the observed PVD above $3\sigma \;\sim$ 8 mJy as follows: we make two cuts—one in the horizontal and one in the vertical direction. A cut in the vertical direction gives us a spectrum at the pixel whereas a cut in the horizontal direction gives us an intensity profile at that velocity. Peak positions for both are determined with the Gaussian fit. We use the points when the two are consistent, but abandon them when they do not match. In the latter case the spectrum is not well characterized by a Gaussian because the asymmetric distribution of the molecular gas and the beam smearing effect is skewing the profile. 106 points are extracted from the observed PVD (see Figure 3). The error bar along the velocity axis for each representative point of the PVD is defined to be a sum in quadrature of the channel width (3.284 km s⁻¹) and the error from Gaussian fitting. After fitting a Gaussian, to the spectrum at each position, we determine the error budget to be the range of all the channels which has an observed value within the difference of half of the noise level from the maximum value of the fitted Gaussian.

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Chi-square values are calculated for each model for the 106 points in the observed PVD. Note that the degree of freedom becomes 104, because we have two free parameters, the SMBH mass and the I-band M/L ratio. Figure 4 shows the chi-square value distribution in the parameter space. The contour level is defined to be (2, 3, 4, 5, 6, 8, 12) × $(\chi _{{\rm min} }^{2})$. The smallest chi-square value of 113 (reduced chi-square value is 1.09 when divided with the degree of freedom) is realized with parameters of ${{M}_{{\rm BH}}}=1.40\times {{10}^{8}}{{M}_{\odot }}$ and the I-band M/L ratio = 5.14. See Figure 3 to compare three PVD models in black contours and lines calculated with different values of parameters (${{M}_{{\rm BH}}}=0$, I-band M/L ratio = 5.14 for the left column, ${{M}_{{\rm BH}}}=1.40\times {{10}^{8\,}}{{M}_{\odot }}$, I-band M/L ratio = 5.14 for the middle column, ${{M}_{{\rm BH}}}=7.00\times {{10}^{8\,}}{{M}_{\odot }}$, I-band M/L ratio = 5.05 for the right column). The red contour in the upper panel shows the observed PVD by ${\rm HCN}(J=1-0)$. Red dots in the middle panel represent the extracted points from the observed PVD. Residuals are plotted in the lower panels. The chi-square values are 244, 113, and 1090 (reduced chi-square values are 2.35, 1.09, and 10.5) for each.

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We determine the error bar for each parameter, M/L ratio and SMBH mass, by taking the parameter value within 99.73% confidence level (${\Delta }{{\chi }^{2}}\leqslant 9$, where ${\Delta }{{\chi }^{2}}\equiv {{\chi }^{2}}-\chi _{{\rm min} }^{2}$). Figure 5 show the polynomial fit to the ${\Delta }{{\chi }^{2}}$ distribution for each parameter. In the left panel, the SMBH mass is thus estimated to be $1.40_{-0.32}^{+0.27}\times {{10}^{8}}{{M}_{\odot }}$ by considering all the values below the black straight line. The M/L ratio is estimated to be $5.14_{-0.04}^{+0.03}$ by also considering values below the black straight line in the right panel. Note here that the derived SMBH mass is consistent with the one presumed from the ${{M}_{{\rm BH}}}-\sigma$ relation reported in (McConnell & Ma 2013) and the central velocity dispersion (196 ± 5 km s⁻¹, Lewis & Eracleous 2006) (see also Section 1.1).

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We estimate the expected stellar mass profile by excluding the bright AGN profile and luminosity enhancement by the starburst ring (Section 3.1), but we could not avoid the dust extinction effects, which could be important for this starburst galaxy with its prominent dust lane around the starburst ring. One way to mitigate the dust extinction effects is to observe at longer wavelengths such as the near infrared. NICMOS on HST has a narrow band filter F190N which observes at 1.9 microns. We calculate a velocity field from the assumed SMBH mass and the stellar mass profile derived from the luminosity profile of 1.9 microns, and obtain a PVD by following the method described in Section 3.3. We then compare the two PVDs calculated from the F190N luminosity profile and the F814W luminosity profile. Both luminosity profiles need to have the same PSF for a proper comparison. For the NICMOS data, which have a small field of view and no feasible stars available for a measurement of the PSF, we refer "Tiny Tim" package (version 6.3 Krist et al. 2011) and assume the shape as a Gaussian with its FWHM of $0\buildrel{\prime\prime}\over{.} 4$. We convolve the I-band luminosity distribution with this PSF. Figure 6 shows the luminosity distribution of the PSF-convolved I-band and 1.9 microns normalized by each maximum value (upper panel) and the difference of two profiles (lower panel). This comparison already shows that the difference between the two luminosity profiles is negligible. We also examine the S4G data at 3.6 microns to compare the stellar profile with the HST data but the Spitzer Infrared Array Camera PSF is too large ($\sim 1\buildrel{\prime\prime}\over{.} 8$) to do any detailed comparison.

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We normalize the peak luminosity at 1.9 microns to the I-band peak, and then subtract the luminosity enhancement of the AGN, as described in Section 3.1. We also ignore the starburst region and use the same M/L ratio to calculate the velocity field for the two stellar mass profiles. We find that a ${{M}_{{\rm BH}}}=1.40\times {{10}^{8\,}}{{M}_{\odot }}$ and M/L = 5.14 gives the best-fit value for an inclination of 46°. The PVDs calculated as such are shown in Figure 7. Black and gray contours in Figure 7 are the PVD calculated from two stellar mass profile models—these do not differ much between the two luminosity profiles. We therefore conclude that the dust extinction effect with the F814W filter is not too serious for the measurement of the SMBH mass.

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We discuss briefly on the difference coming from how we set the inclination angle. The accuracy of the inclination angle is critical for calculating the velocity and therefore crucial for the SMBH mass estimation. It is however not straightforward to determine the inclination angle when comparing observations at different field of view. Previous studies of NGC 1097 have determined a dynamical inclination angle of 46 ± 5° (HI observations at 3 kiloparsec scales, Ondrechen et al. 1989), or 34° (Hα line profile study, Storchi-Bergmann et al. 2003). Hsieh et al. (2011) reported that the inclination angle of NGC 1097 is 41.7 ± 06 using the kinematic parameters of $^{12}{\rm CO}(J=2-1)$ observed with Submillimeter Array. They argue that the circumnuclear ring is nearly circular for the inclination of ∼42°, by which means the ring has an intrinsic elliptical shape in the galactic plane, of which case is not symmetric to the axis. Though the suggested asymmetry is interesting to note, we would like to leave it as a further discussion, since in this work we assume an axisymmetric distribution for stars and molecular gas when calculating the circular velocity field. This time we assume the galaxy inclination angle to be 46–51° by referring to Ondrechen et al. (1989) and the axis ratio of the HST I-band observation. We evaluate the SMBH mass to be $1.40_{-0.32}^{+0.27}\times {{10}^{8}}{{M}_{\odot }}$ and the the I-band M/L ratio to be $5.14_{-0.04}^{+0.03}$ at the inclination angle of 46°, with the chi-square value of 113 (1.09 when divided with the degree of freedom 104). We also follow the same process in Section 3 with the inclination angle of 51° and evaluate the SMBH mass to be $1.20_{-0.34}^{+0.35}\times {{10}^{8}}{{M}_{\odot }}$ and the I-band M/L ratio to be 5.11 ± 0.03 with the chi-square value of 117 (1.13 when reduced with the degree of freedom). See also dashed curves in Figure 5 for the chi-square distribution, used to determine the error bar for each parameter.

We can also consider the case of the inclination angle is 34–41° by multiplying a factor of ${\rm sin} ({{i}_{{\rm intrinsic}}})/{\rm sin} (46{}^\circ )$ ∼0.78–0.91 to the velocity where we write ${{i}_{{\rm intrinsic}}}$ as an inclination angle of the observed component. Under the simplified assumption that the SMBH mass is proportional to the square of the velocity, we can estimate the change of the SMBH mass to be smaller than $0.31\times {{10}^{8\,}}{{M}_{\odot }}$, which is mostly included in the error bar of our result $1.40_{-0.32}^{+0.27}\times {{10}^{8\,}}{{M}_{\odot }}$. We therefore consider that it is not crucial to count this error into our error budget. Note that, however, this galaxy could have a warped or a misaligned structure, which could be interesting to investigate but requires a calculation for an asymmetric potential field.

Our main result is obtained using the HCN line because it had the highest signal-to-noise ratio. It is important to measure the SMBH mass from other molecular species as well for consistency. We therefore repeat our method using the ${\rm HC}{{{\rm O}}^{+}}(J=1-0)$ emission line.

We apply the fitting procedure described in Section 3.4 to the PVD for ${\rm HC}{{{\rm O}}^{+}}(J=1-0)$, and estimate the SMBH mass to be $(1.40\pm 0.30)\times {{10}^{8}}{{M}_{\odot }}$ and the I-band M/L ratio to be 5.15 ± 0.03 with a galaxy inclination of $46{}^\circ$. These derived values are consistent with the measurement using ${\rm HCN}(J=1-0)$. From Figure 8, we see that the observed PVDs of two molecular gases are in good agreement, indicating that the fitting parameters will be consistent between the two.

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Reaching the velocity structure from multiple molecular species is one of the particular benefit of the SMBH mass measurement with millimeter/submillimeter wavelength observation, which enable one to observe more than two molecular species at the same frequency band.