DYNAMICAL EVOLUTION OF AGN HOST GALAXIES—GAS IN/OUT-FLOW RATES IN SEVEN NUGA GALAXIES

For a detailed study of gas inflow rates, we selected seven galaxies from our H i-NUGA sample (Haan et  al. 2008). The selection of the targets is based on the presence of bright H i emission as well as a fairly regular H i morphology (i.e., no highly disturbed gas disks). All the galaxies are spiral galaxies ranging in Hubble type from Sa to Sc and are barred, with the exception of NGC 7217. The distance of our sample targets ranges from 4 to 24 Mpc with a mean distance of ∼13 Mpc. A complete overview is given in Table 1. The galaxies host various kinds of nuclear activity:¹⁰ LINERs (four galaxies), Seyferts (two galaxies), and starburst (one galaxy).

To trace the dynamics of the outer disk we observed the atomic gas distribution and kinematics in the H i emission line at 21 cm using the NRAO VLA. Previous data obtained with the VLA in C and D array configuration (∼20'' resolution) were already presented by Haan et  al. (2008). For the seven galaxies analyzed here we combined these data with newly obtained VLA B-array data resulting in a final resolution of ∼7''. Two galaxies (NGC 3627, NGC 4736) were already observed as part of the THINGS project (Walter et  al. 2008). The data reduction was performed using the Astronomical Image Processing System (AIPS; Greisen 2003) following the data reduction scheme of the previous CD array data (Haan et  al. 2008). Flux calibrator measurements were performed at the beginning and at the end of each observation cycle. The phase calibrator was observed before and after each source cycle with a maximum angular distance between source and phase calibrator of 12°. The data have an average on-source integration time of 8.5 hr (B-array), 2.7 hr (C-array), and 2.8 hr (D-array). The correlator spectral setup used was set to line mode 4 with Hanning smoothing and 64 channels per 1.5625 MHz channel width per frequency band providing a frequency resolution of 24.414 kHz/channel (∼5.2 km s⁻¹). Calibration solutions were first derived for the continuum data set (inner 3/4 of the spectral band width) and then transferred to the line data. The bandpass solutions were determined from the phase calibrator measurements to account for channel-to-channel variations in phase and amplitude. The B and CD data have been combined. The CLEANing parameters are adjusted to the new combined BCD data, as described in the following. The cell size of each grid was set to 13/pixel with a field of view (FOV) of ∼22'. We produced the CLEANed data cubes using robust weighted imaging with a velocity resolution of ∼10.4 km s⁻¹ and an average angular resolution of ∼7''. To find the best compromise between angular resolution and RMS, several robust weighting parameters were tested and a robust parameter of 0.3 was selected. The RMS values and beam sizes are listed in Table 2 with an average achieved RMS value of 0.32  mJy beam⁻¹. The RMS flux sensitivity of 0.4 mJy/beam/channel corresponds to a 3σ detection limit of ∼0.55 × 10¹⁹ cm⁻² column density for the combined BCD array data. To separate real emission from noise, we produced masks by taking only into account those regions which show emission above a set level (3σ) in cubes that have been convolved to 30'' resolution (task CONVL). Using these masks, we blank areas that contain noise in our robust weighted data cubes (task BLANK).

The subsequent analysis has been done with the Groningen Image Processing System (GIPSY; van der Hulst et  al. 1992). The channel maps were combined to produce zeroth (intensity map), first (velocity field), and second (dispersion map) moments of the line profiles using the task MOMENT. The RMS values have been measured in two regions where no line signal was apparent and averaged over all channels of the cubes. A flux cut-off of three times the channel-averaged RMS value was used for the moment maps. The velocity-integrated H i intensity maps are presented in Figure 1.

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The spectroscopic imaging of CO lines provides information about the distribution and kinematics of the molecular gas. The molecular gas has been observed in the transition of the J = 1–0 and J = 2–1 lines of ¹²CO with maximum angular (∼05) and spectral resolution (3–6 km s⁻¹) using the IRAM Plateau de Bure mm-interferometer (PdBI) as part of the NUGA project and the larger NUGA supersample (García-Burillo et  al. 2003). The primary beam size is ∼42'' (∼21'') in all the CO 1–0 (2–1) line observations. Only for NGC 5248 no PdBI data were available. Therefore, we used for NGC 5248 the publicly available BIMA-SONG data (Helfer et  al. 2003) which was observed in the CO (1–0) line with an angular resolution of 61 and spectral resolution of 10 km s⁻¹ with the BIMA mm-interferometer. An overview of the observational parameters for the CO data is presented in Table 3.

Archival NIR images from Spitzer, HST, and ground-based telescopes have been used to estimate the stellar mass distribution in our galaxies. An overview of all NIR images used is presented in Table 4. Foreground stars have been removed using the software tool NFIGI (A. Baillard et  al. 2009, in preparation) and, in case additional cleaning was necessary, by hand. In addition, background is subtracted from the NIR images by calculating the mean value in the outer region of the image outside the galaxy disk to ensure that no light contribution from the galaxy itself is subtracted. Other methods for background subtraction have been tested (e.g., fitting the radial luminosity profile to find constant background value), but could not be applied successfully to all NIR images because of limited FOVs. After background subtraction all values below zero, which might be caused by small overestimation of the background value (case only for outer regions of image), were set to zero.

In preparation for our analysis of radial gas flows, we derived the following disk parameters by fitting tilted rings to the CO and H i velocity fields using the task ROTCUR within GIPSY.

• 1.  
  The dynamical center and its offset from the optical center (taken from Hyperleda).
• 2.  
  The systemic velocity v_sys in km s⁻¹.
• 3.  
  The position angle (PA) in degrees, defined as the angle between the north direction on the sky and the receding half of the major axis of the galaxy in anti-clockwise direction.
• 4.  
  The inclination (i) in degrees.

The kinematic parameters have been derived in an iterative way (Begeman 1989) as described in our analysis for the entire H i-NUGA sample (Haan et  al. 2008). The parameters were assumed to be the same at all radii, except for the circular velocity. We weighted the obtained value in each ring with its standard deviations, in order to derive the mean parameter. For the fit we excluded data points within an angle of 20° of the minor axis. The widths of the radii were set to 04 (6'') which corresponds roughly to the angular resolution of our CO (H i) data. No radial velocity component was fitted as disk parameter. We have applied the same systemic velocity, inclination, and PA derived from our H i velocity field for the CO analysis as we assume that these disk orientation parameters are constant over the entire galaxy disk. We found no evidence for significantly warped disks in our sample using the derived inclination angle as a function of radius (see Figure 2). For the estimation of the dynamical center position we have taken into account only data of the inner third of the H i gas disk to exclude possible shifts of the kinematic center due to spiral arms and disturbances in the outer H i disk. In addition, we derived the center position from the CO kinematics for most galaxies of our sample. However, our derived values for the kinematic center show only small offsets from the photometric center with an average offset of ∼15 which is similar to typical errors expected from photometric center fitting. An overview of all derived parameters for each galaxy is presented in Table 5. Finally, the rotation velocities were obtained for the atomic and molecular gas with fixed inclination, PA, center, and systemic velocity.

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In this study, we attempt to detect noncircular motions and subsequently evidence for radial gas flow directly from the observed gas kinematics. In principle, one would expect that noncircular motions are caused by the nonaxisymmetric part of the gravitational potential. However in practice, other physical effects such as gas viscosity and shocks in the gas may have significant contributions to any observed noncircular motion. Furthermore, interactions with galaxy companions as well as AGN and star formation feedback might work as additional source for noncircular motions. To estimate the contribution from these effects, we compare the measurement of the noncircular motions with the results obtained via our gravitational torque analysis (see Section 5.3). To obtain a fair estimate of the circular and noncircular part of our gas velocity fields, we have used the GIPSY task RESWRI which performs a harmonic expansion of the velocity fields. This expansion is made by first fitting a tilted-ring model to the velocity field of the gas disk and subsequently decomposing the velocity field along each ring into its harmonic terms. Since the data points may not be uniformly distributed in azimuth, RESWRI performs a least-square fitting (singular value decomposition) rather than a direct Fourier expansion. In practice, after the convergence of the ROTCUR part, RESWRI makes a harmonic expansion of the line-of-sight velocity v_los along each ring,

$$\begin{equation} v_{\rm los}(r) = v_{\rm sys}(r)+\sum _{n=1}^k \left[ c_n(r) \cos (n\psi) + s_n(r) \sin (n\psi) \right], \end{equation} \tag{ 1 }$$

where k is the order of the fit, r is the radial distance from the center, ψ is the azimuthal angle, measured from the receding side of the line of nodes, and v_sys is the systemic velocity as zeroth harmonic component. The coefficients c_n and s_n are determined by making a least-squares fit to the data points up to third order, which requires three sine terms (s₁, s₂, s₃) and three cosine terms (c₁, c₂, c₃). Since the line-of-sight velocity is given in the most general case of a velocity field as

$$\begin{equation} v_{\rm los}(r) = v_{\rm sys}(r)\,+\,v_{\theta }(r) \cos (\psi) \sin (i)\, +\, v_R(r) \sin (\psi) \sin (i), \end{equation} \tag{ 2 }$$

where v_θ and v_R are the circular and radial components of the velocity field, respectively. Hencefore, c₁ = v_θ(r)sin(i) reflects the observed circular velocity, whereas all other terms are contributions to noncircular motions (see for a detailed discussion of the harmonic terms Schoenmakers et  al. 1997; Schoenmakers 1999). To estimate the total amount of noncircular motions we calculate the quadratically added amplitude of all noncircular harmonic components v_nc up to the order of N = 3 (see also Trachternach et  al. 2009),

$$\begin{equation} v_{nc}(r) = \sqrt{s_1^2(r) + c_2^2(r) + s_2^2(r) + c_3^2(r) + s_3^2(r)}, \end{equation} \tag{ 3 }$$

which is basically the vector sum of all noncircular velocity contributions. The fraction v_nc/v_rot shows how the contribution of noncircular motions to the total velocity may vary with radius. At the corotation radii (R_CR) of a bar or spiral the s₃-terms become dominant over the s₁-terms (Canzian & Allen 1997).

For the outer disk (probed by our H i gas kinematics), most of the galaxies in our sample exhibit fairly regular velocity fields that are dominated by circular motions (see Figure 1). The ratio of noncircular motions to rotational velocity v_nc/v_rot averaged over the radius for H i is typically in the range of 0.05–0.09, except for NGC 3627 and NGC 4736 which exhibit a slightly larger ratio of 0.12 and 0.14, respectively. For NGC 3627, that can be explained by the fact that it belongs to the Leo Triplet and noncircular motions are very likely induced by the past encounter with NGC 3628 (Zhang et  al. 1993). Although NGC 4736 exhibits a one-arm spiral in the outer H i disk which might be a hint for a previous interaction, no optical companion is obvious. The mean values of v_nc/v_rot are derived by weighting the data points with their standard deviations and are presented together with the standard deviation of the mean value in Table  6. Although the mean ratios v_nc/v_rot are quite small, the ratio of noncircular motions in H i is increasing to 40% as a function of radius, primarily in the inner- and outer-most region of the H i disk. For the center at ≲1 kpc, a larger fraction of noncircular motions is derived from our CO velocity fields with typical values of (7–14)% for NGC 7217, NGC 5248, NGC 3368, NGC 6951 and very large fractions of (22, 33, 83)% for NGC 4736, NGC 4321, and NGC 3627, respectively. In particular, for NGC 3627 noncircular motions seem to dominate the velocity field not only in the center, but also have significant contributions in the outer disk.

In Figure 3, we plot the rotation curve v_rot and the fraction of noncircular motion to circular motion $v_{nc}/v_{c_1}$ as a function of radius for each galaxy. The total power of noncircular motions v_nc is typically larger for the central (CO kinematics) than for the outer gaseous disk (H i) with mean values of 26% and 9%, respectively. A comparison of noncircular motions and radial flow directions obtained from our kinematics and the ones on the basis of gravitational torques is performed in Section 5.3. Further, we compare in Section 5.3 candidate radial gas flow regions derived from our velocity fields with the inflow/outflow rates due to gravity torques. In fact, all studies that have tried to determine an actual inflow/outflow on the basis of the harmonic decomposition method alone were not very successful so far. The main reason for that is that elliptical streaming in a bar or spiral potential seems to be the dominant contributor to noncircular motions (see Wong et  al. 2004, for more details). To derive net inflow/outflow a phase shift between the gas and the gravitational potential has to be present, which is the underlying concept for our gravity torque study in the following section.

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A detailed description of estimating gravitational torques in spiral galaxies is given by García-Burillo et  al. (2005). We used a similar method to derive gravitational torques which is briefly described below and consists of the following steps:

• 1.  
  Evaluation of the stellar potential using high-resolution NIR images from the Spitzer telescope, ground-based telescopes, and HST. To obtain the total gravitational potential we scale the stellar potential with mass-to-light ratios which are estimated by fitting the gaseous rotation curves derived from our H i and CO observations (see Section 3.1). This implicitly assumes that the stellar disks are nearly "maximal."
• 2.  
  Computation of gravitational forces and gravity torques based on the stellar potential.
• 3.  
  Weighting of the torque field with the gas column density in order to link the derived torque field to angular momentum variations.
• 4.  
  Estimation of the gas flows induced by these angular momentum variations using azimuthal averages of the torques at each radius.
• 5.  
  Finally, the time-scales and gas masses associated with inflow/outflow are derived by estimating the average fraction of angular momentum transferred in one rotation.

As the average fraction of gas to dynamical mass for our sample is ∼5%, the total mass budget is expected to be dominated by the stellar contribution, so that we will neglect self-gravity of the gas.

4.1.1. Computation of the Gravitational Potential

The gravitational potential is computed on a Cartesian grid based on the NIR images used as tracers for the stellar potential (following the method of Garcia-Burillo et  al. 1993). First, we deproject the NIR and gas images using the inclination i and PA derived from our kinematic analysis (see Section  2.4). An estimation of systematic errors of our calculations due to possible parameter errors will be presented in Section 4.2. Since a bulge would be artificially elongated by the deprojection, the bulge component has to be excluded from the deprojection. Thus, a bulge model is created first using results from fitting a de Vaucouleurs and exponential function to the radial stellar profile representing the bulge and disk brightness profile, respectively. The bulge model is assumed to be spherical. Then the bulge component is subtracted from the NIR image. After the deprojection of the galaxy (without bulge), the bulge model (with the same scaling as before) is added again to the deprojected galaxy disk. The deprojected NIR and gas images are shown in Figure 4.

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The gravitational potential of the galaxy,

$$\begin{equation} \Phi (\mathbf {x})=-G \int \frac{\rho (\mathbf {x^{\prime }})d^3 \mathbf {x^{\prime }} }{| \mathbf {x} -\mathbf {x^{\prime }} |}, \end{equation} \tag{ 4 }$$

can be written as a convolution of the three-dimensional mass density ρ(x, y, z) and the function g(r) = 1/r (the gravitational potential of a pointlike source),

$$\begin{equation} \Phi (x,y,z) = -G \cdot \rho (x,y,z) \otimes g(r), \end{equation} \tag{ 5 }$$

where G is the gravitational constant. We use a smoothing length , to avoid singularities for distances close to or equal to 0. This results in a slight change of the convolution kernel to

$$\begin{equation} g(r, \epsilon) = \frac{1}{\sqrt{r^2 + \epsilon ^2}}, \end{equation} \tag{ 6 }$$

and we find the normal 1/r function when goes to 0. Further, the thickness of the stellar component of galactic disks is not negligible. Observations have shown that the vertical scale height of stellar disks is roughly constant as a function of radius (van der Kruit & Searle 1982a, 1982b; Wainscoat et  al. 1989; Barnaby & Thronson 1992). To take this into account we assume a noninfinitesimally thin disk and use a model for the vertical distribution h(z), namely an isothermal plane with

$$\begin{equation} h(z)\,=\,^{1}\!\!/_{2}\;h\; h^2(z/a), \end{equation} \tag{ 7 }$$

and a constant scale height a, equal to ∼1/12 of the radial scale length of the galaxy disk. In that case, the mass density distribution is written as

$$\begin{eqnarray} \rho (x,y,z) & = & \Sigma (x,y) \times h(z) \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} \mbox{with} \int _{-\infty }^{\infty } h(z) & = & 1\,. \end{eqnarray} \tag{ 9 }$$

To compute the potential in the equatorial plane, we integrate the contributions from all heights. This results in a convolution kernel function which can be written as

$$\begin{equation} g(r, \epsilon) = \int _{-\infty }^{\infty } \frac{h(z)}{\sqrt{r^2 + \epsilon ^2 + z^2}} \,{d}z\,. \end{equation} \tag{ 10 }$$

Hence, we still can write the potential in a simple form, at least in the plane of the disk,

$$\begin{equation} \Phi (x,y,z=0) = -G \cdot \Sigma (x,y) \otimes g(r, \epsilon)\,. \end{equation} \tag{ 11 }$$

In practice, given the vertical thickening function h(z) and the smoothing length , we tabulate the function g(r, ) numerically at the beginning of our code. The convolution is then performed using fast Fourier transforms. Note that in order to avoid edge effects, we extend the grid to a factor of ∼4 times larger than the initial grid.

After that, the potential has to be factorized with the appropriate mass-to-light ratio which is calculated via comparing the observed rotation curve and the rotation velocities derived from the potential,

$$\begin{equation} f_{\rm scal}(R) \equiv \frac{v_{\rm model}(R)}{v_{\rm rot}(R)}, \end{equation} \tag{ 12 }$$

where v_model represents the circular velocity derived from the stellar potential and v_rot the rotation curve velocity of our combined and interpolated CO and H i rotation curve at the radius R. Then, we employed a fit on f_scal(R) as function of radius using a decomposition into disk, bulge, and dark matter component. The derived scaled circular velocities from the gravitational potential match very well the observed circular velocities within ±5% along the radial axis.

4.1.2. Computation of Gravitational Forces and Torques

At first, the forces per unit mass in x- and y-direction F_x,y for our derived gravitational potential are calculated at each pixel,

$$\begin{equation} F_{x,y}(x,y) = -\nabla _{x,y}\Phi (x,y)\,. \end{equation} \tag{ 13 }$$

The gravity torques (per unit mass) τ are defined as

$$\begin{equation} {\bm \tau} = {\bi r} \, \times \, {\bi F}\,, \end{equation} \tag{ 14 }$$

and calculated in the plane

$$\begin{equation} \tau (x,y)=xF_y - yF_x. \end{equation} \tag{ 15 }$$

The positive or negative sign of the torque τ(x, y) defines whether the gas accelerates or decelerates. An example of the gravitational potential and torque is presented in Figure  5 for NGC 6951. The link between torque field and angular momentum variations is made through the observed distribution of gas (García-Burillo et  al. 2005) as explained in the following. At first we assume that the CO and H i emission lines are good tracers of the total gas column density in the central and outer disk, respectively. The reason for using two different lines is due to the fact that the neutral ISM undergoes a phase transition from atomic to molecular gas toward the center of a galaxy (Young & Scoville 1991). Since the gas distribution is the convolution of the gas density with the orbit path density, the gas distribution is indirectly equivalent to the time spent by the gas clouds along the orbit paths. Thus, we implicitly average over all possible orbits of gaseous particles. To do this link between the derived torque field and angular momentum variations, the torques are weighted with the gas column density N(x, y) and averaged over the azimuth,

$$\begin{equation} \tau (R) = \frac{\int _{\theta } \left[ N(x,y) \cdot (xF_y - yF_x) \right] }{\int _{\theta } N(x,y)}\,. \end{equation} \tag{ 16 }$$

By definition, τ(R) represents the azimuthal-averaged time derivative of the specific angular momentum L of the gas, i.e.,

$$\begin{equation} \tau (R) = \frac{{d}L}{{d}t} \bigg\vert _{\theta }. \end{equation} \tag{ 17 }$$

The positive or negative sign of τ(R) defines whether the gas may gain or loose angular momentum, respectively. The fueling efficiency can be estimated by deriving the average fraction of the gas specific angular momentum transferred in one rotation (T_rot) by the stellar potential, as a function of radius, defined as

$$\begin{equation} \frac{\Delta L}{L} = \frac{{d}L}{{d}t}\bigg\vert _{\theta } \, \cdot \, \frac{1}{L} \bigg\vert _{\theta } \, \cdot \, T_{\rm rot} = \frac{\tau (R)}{L_{\theta }} \, \cdot \, T_{\rm rot}\,, \end{equation} \tag{ 18 }$$

where the azimuthal-averaged angular momentum L_θ is assumed to be well represented by its axisymmetric average, i.e., L_θ = R · v_rot. The inverse of ΔL/L determines how long it will take (in terms of orbital time periods) for the gravitational potential to transfer the equivalent of the total gas angular momentum. The mass inflow/outflow rate of gas per unit length d²M(R)/(dRdt) (in units of M_☉ yr⁻¹ pc⁻¹) is calculated as follows:

$$\begin{equation} \frac{{d}^2 M(R)}{{d}R {d}t} = \frac{{d}L}{{d}t}\bigg\vert _{\theta } \, \cdot \, \frac{1}{L} \bigg\vert _{\theta } \cdot 2\pi R \cdot N(x,y)\vert _{\theta }, \end{equation} \tag{ 19 }$$

where N(x, y)|_θ is the gas column density of the atomic (H i) and molecular gas using the conversion factor from CO to H₂ typical for galaxy nuclei of X = N(H₂)/I(CO) = 2.2 × 10²⁰ cm⁻² K⁻¹ km⁻¹ s (Solomon & Barrett 1991). Then, the integrated inflow/outflow rates dM(R)/dt are derived by integrating from R = 0 out to a certain radius R,

$$\begin{equation} \frac{{d} M(R)}{{d}t} = \sum _0^R \left[ \frac{{d}^2 M(R)}{{d}R {d}t} \cdot \Delta R \right], \end{equation} \tag{ 20 }$$

with ΔR as radial binning size (in units of pc).

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In order to examine the reliability of our results, we estimated how small errors in our assumed fitting parameters will affect the results. One main source for errors arises from uncertainties in the input parameters that were derived from our kinematic study (see the previous section). First, we checked the parameters that describe the geometry of the disk given by inclination i, PA, and position of the center (x, y). We assume that these parameters do not vary with radius. To evaluate the effects of changes in these parameters on the derived gravity torque map, we tested two simple disk models (see Figure 6): (1) an exponential disk which produces no torque pattern because of its axisymmetrical distribution, and (2) an exponential disk plus an additional constant oval distribution mimicking a barred galaxy, which produces a typical torque that changes sign between the four quadrants. The oval distribution has a minor-to-major axis ratio of 0.5 and a constant surface brightness of 20% of the maximum brightness of the exponential disk. For comparison, typical torque patterns without parameter errors are presented in Figure 7 for different disk models. We tested the robustness of the results from our code with uncertainties based on our kinematic analysis (Section 3):

• 1.  
  An inclination error of 3° at a typical inclination of 45°: the result shows an additional torque pattern that changes sign between the four quadrants. This pattern corresponds to one induced by a small oval contribution. Because of its axisymmetry and uniform gas distribution no contribution in the radial profile is seen. In principle, an inclination error will produce a fake bar; or oval but it will be along one of the principal axes of the disk (major, minor) and hence might provide a way to disentangle this false pattern from a true one.
• 2.  
  A PA error of 2°: here, the torque also changes sign between the four quadrants, causing a pattern similar to an oval distribution, but along an axis that is rotated by 45° to one of the principal axes.
• 3.  
  An error of the center position of 1 pixel which corresponds to ∼50 pc at a distance of 10 Mpc and a pixel size of 1'' that results in a bimodal pattern, which increases the torque budget of one side of a torque pattern produced by an oval distribution, and decreases the torque at the opposite side.

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Since this simple test does not take into account the gas distribution of a "real" galaxy which very likely differs from an homogeneous axisymmetric distribution, we also estimated the effects of these uncertainties using the NIR images and gas maps of a typical galaxy from our sample, namely NGC 6951. In order to estimate the errors in the radial torques, we compared the torque results using the best input parameters with those where typical errors of the input parameter were applied. The total error as a function of radius is shown in Figure 8. The results of this test are summarized in Table 7, separately for the central CO based and outer H i-based torque calculation.

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Table 7. Error Estimation

Parameter	dτ (CO)	dτ (H i)
Position Angle (dPA = 2°)	4%	<1%
Inclination (di = 2°)	24%	<1%
Shift of IR Image (02)	10–30%	∼13%
Shift of Gas Image (03)	5–20%	5–20%
Background Subtraction	<5%	<1%
Bulge-Nobulge Subtraction	<2%	3–10%
Rot. Vel. (dv = 15 km s−1)	20%	10%
Sum	60–80%	13–70%
Typical	67%	30%

Notes. Estimation of systematic errors in the radial torques for CO and H i of a typical galaxy from our sample, namely NGC 6951. For this exercise, the torque results using the best input parameters have been compared with those where typical errors of the input parameters were applied. Note that the errors have a radial dependency (see Figure 8)

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Not only geometric uncertainties can have an effect on our results, but also the assumed mass-to-light ratio. Since we scale our potential with the derived (deprojected) rotation curve from the velocity maps, our results do not depend on uncertainties in the mass-to-light ratio, e.g., derived from stellar population models. Instead, our method relies on a scaling factor between the circular velocity derived from the stellar potential and the combined CO and H i rotation curve as given by Equation (12). Since rotation curves are derived from azimuthal-averaged velocities, different mass-to-light contributions of nonaxisymmetric components such as spiral arms (cool supergiants or dust glowing star formation regions) and bars will affect the scaling. Furthermore, possible deviations from a constant inclination of the galaxy disk (i.e., warping disk) might have an impact. However, most spiral galaxies show only small variations in the inclination across the stellar disk and we found no evidence for a significant warped disk in our sample as described in Section 3.1.

Another possible source of error might be that the gas distribution is not 100% recovered at all spatial scales due to missing short spacings of the interferometer observations. As our H i observations are continuous and traces all spatial scales, H i is a good tracer for the atomic gas distribution in our galaxies. For CO, the PdBI observations do not trace all spatial scales, and thus, might miss some flux from short spacings. However, to estimate the effect of short spacing corrections, S. García-Burillo et  al. (2008, in preparation) have compared the gravity torque results for NGC 4579 without (only CO PdBI-data) and with short spacing corrections using additional CO data obtained with the IRAM 30 m telescope. The result shows that the gravitational torques with short spacings are roughly (5–40)% larger than without short spacings.

The accurate determination of the corotation (corotation resonance (CR)) radii of density wave patterns is an important parameter for the characterization of the dynamical state of a galaxy and to understand the relation between morphologies and kinematics of galaxies. We have applied the potential density phase shift method (Zhang & Buta 2007) to derive the CR radii for our galaxies. This technique is based on the calculated radial distribution of an azimuthal phase shift between the stellar potential and a stellar density wave pattern which results in a torque action between the wave pattern and the underlying disk matter. Consequently, the material inside (outside) the CR radius looses (gains) angular momentum which causes an inflow (outflow). The validity of the phase-shift method is based on the global self-consistency requirement of the wave mode (i.e., both the Poisson equation and the equations of motion need to be satisfied at the same time). For a self-sustained global spiral mode, the radial density variation of the modal perturbation density, as well as the pitch angle variation, together determine that the Poisson equation will lead to the zero crossing of the phase-shift curve being exactly at the CR radius of the mode (see for more details Zhang & Buta 2007).

We calculated the CR radius using the method of Zhang & Buta (2007) as follows: the rate of angular momentum exchange between the density wave pattern and the axisymmetric part of the disk can be either expressed as (Zhang 1996)

$$\begin{equation} \frac{dL}{dt}(r)=-\frac{1}{2\pi }\int _0^{2\pi }\Upsilon (r,\phi)\frac{\partial \Psi (r,\phi)}{\partial \phi }d \phi, \end{equation} \tag{ 21 }$$

with the perturbation density waveform ϒ and the perturbation potential waveform Ψ, or as two sinusoidal waveform,

$$\begin{equation} \frac{dL}{dt}(r)=\frac{m}{2}A_{\Upsilon }(r)A_{\Psi }(r)\sin \left[ m \phi _0(r)\right], \end{equation} \tag{ 22 }$$

with the amplitudes of the density wave A_ϒ and potential wave A_Ψ, the nonaxisymmetric mode number m (e.g., for two spiral arms or a bar: m = 2), and the phaseshift ϕ₀ between these two waveforms. Using these two equations, the phaseshift ϕ₀ can be calculated from

$$\begin{equation} \phi _0(r,\phi)=\frac{1}{m}\sin ^{-1}\left[ \frac{1}{m} \frac{\int _0^{2\pi }\Upsilon (r,\phi)\frac{\partial \Psi (r,\phi)}{\partial \phi }d \phi }{\sqrt{\int _0^{2\pi }\Psi ^2 d \phi } \sqrt{\int _0^{2\pi }\Upsilon ^2 d \phi }} \right]. \end{equation} \tag{ 23 }$$

The potential Ψ and the density distribution ϒ are taken from our gravity torque calculation (see Section 4.1). We assume that the phaseshift is positive when the potential lags the density wave in the azimuthal direction in the sense of the galactic rotation. For all of our galaxies we derived the phaseshift as a function of radius and we defined the CR radii as the positive-to-negative crossings of the phaseshift ϕ₀ (see Figure 9). At this location the direction of angular momentum transfer between the disk matter and the density wave changes sign.

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In fact, the phase shift method of Zhang & Buta (2007) is mathematically very similar to our gravity torque study as both rely on a phase shift between the gravitational potential and a density wave pattern. Thus, also the gravity torque is expected to change sign at each resonance with the gas as test particles for measuring the gravity torques. On the other hand, the gravity torque method might differ from the potential density phase shift in the case of a superposition of density patterns or a shift of gas due to viscosity. However, in the case of a strong dominating pattern, both methods are expected to reveal a similar location for the CR radius.

To verify our estimation of the CR radii we derived additionally the CR radii using the bar length and the method of Canzian (1993; see Section 3.2). A comparison of the results derived from these different studies is presented in Table 8 for all our galaxies. The CR radius is assumed to be 1.1–1.6 times the radius of the bar (Rautiainen et  al. 2008) and is visually estimated directly from the deprojected NIR images (Figure 4) using the change in PA and the ellipticity of the bar. All three methods (Canzian, bar length, phase shift) reveal roughly the same radii for the CR of the bar. For our galaxies the phase-shift method appeared to be the most precise method with uncertainties of (5–10)% and the Canzian method the most unprecise one with uncertainties ranging up to 50%. The estimation of the CR using the bar length suffers from the uncertainty in the estimation of the bar length and its relation to the CR radius, which lies in the range of 10%–30%. The comparison between the CR radii determined by our gravity torques study and the phase-shift method of Zhang & Buta (2007) reveals significant differences for the location of the CR radii of the bar for NGC 3368, NGC4321, and NGC 6951 (see Table 8). One possible explanation for these differences is that all three galaxies exhibit a strong stellar bar as well as a spiral pattern (also visible in the NIR images) which might be superimposed.

To derive the location of the inner Lindblad resonance (ILR), the ultra harmonic resonance (UHR), and outer Lindblad resonance (OLR) we used a simple method presented in Figure 10 which is described in the following. The angular velocity Ω is calculated from a fit using a cubic spline interpolation to our measured CO and H i rotation curve. After determining the bar pattern speed at the location of the CR (taken from the phase-shift method), the frequency curves Ω ± κ/2 and Ω − κ/4 are derived with the epicyclic frequency

$$\begin{equation} \kappa = \sqrt{4 \Omega ^2 + R \frac{{d} \Omega ^2}{{d}R}}\,. \end{equation} \tag{ 24 }$$

The derived CR, ILR, UHR, and OLR are also listed in Table  8. Interestingly, the OLR caused by the bar seems to overlap for some galaxies (NGC 3368, NGC 6951) with the CR radius of the spiral determined by the phase-shift method, suggesting a coupling between bar and spiral resonances. However, as our uncertainties in the estimation of the OLR are quite large, this correlation is not very significant.

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We have derived neutral gas inflow rates for seven nearby spiral galaxies as a function of radius as well as location within the disks (see the top panel of Figure 11). By definition, the torque τ(R) represents the azimuthally-averaged time derivative of the specific angular momentum of the gas. The positive or negative sign of τ(R) defines whether the gas may gain or loose angular momentum, respectively. For almost all galaxies of our sample the torque changes sign in different quadrants which corresponds to inflow and outflow in adjacent quadrants. As such a pattern is generally expected to be generated by m = 2 modes, (e.g., bar or oval potential), we conclude that this mode is the dominant mode in our sample. Only NGC 5248 shows a bipolar torque pattern for the center. In addition, the torque pattern is very sensitive to the presence of stellar spiral arms which cause a positive (negative) torque on the side of the spiral arm toward (away from) the galaxy center. Such a torque pattern is present in all galaxies of our sample, except for NGC 7217 and NGC 4736.

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To estimate radial gas flows and time scales we derived the azimuthally averaged torque τ(R) and the angular momentum transferred in one rotation dL(R)/L as a function of radius which are presented in Figure 11. Similarly to the torque maps, a positive or negative sign of τ(R) and dL(R)/L defines whether the gas may gain or loose angular momentum corresponding to gas outflow or inflow, respectively. The transferred angular momentum and gas masses averaged over the entire galaxy disk are listed in Table 9 for each galaxy. We found that the average-transferred angular momentum 〈dL/L〉 over the entire gas disk lies in the range of −0.04 (NGC 6951) to 0.06 (NGC 5248) for H i and −0.5 (NGC 3368) to 0.16 (NGC 3627) for CO. To estimate the mean amplitude of angular momentum exchange within the disk we derive the mean value of the absolute angular momentum transfer, 〈|dL/L|〉. This parameter is used later in Section 5 for a comparison to the noncircular motions identified in our analysis of the gas kinematics. We found values of 〈|dL/L|〉 in the range of 0.03 (NGC 7217) to 0.26 (NGC 3627) for H i and 0.03 (NGC 7217) to 0.55 (NGC 3368) for CO. Clearly, NGC 7217 exhibits the lowest torque strength and angular momentum transfer in our sample due to its nearly axisymmetric potential (Combes et  al. 2004) whereas NGC 3627 and NGC 3368 exhibit the largest amplitude of angular momentum exchange. In general, the amplitude of 〈|dL/L|〉 is larger in the center (CO) than in the outer disk (H i).

To estimate the transferred gas mass at a certain radius, we derived the mass inflow/outflow rates d²M(R)/(dRdt) (in units of M_☉ yr⁻¹ pc⁻¹) as a function of radius (see Figure 12). In addition, we examined the net mass flow dM(R)/(dt) (in units of M_☉ yr⁻¹) outward to a given radius R by integrating over the gravitational torques of the molecular and atomic gas disk, separately. These values represent the net gas mass inflow/outflow within these regions and can be compared to the typical gas masses required by star formation and AGN fueling (see Section 5). A positive (negative) sign indicates gas outflow (inflow). The net gas mass flow for the entire molecular and atomic gas disks (integrated from 0 to R_max) are listed in Table  9 with a range of −0.01 to 0.5 M_☉ yr⁻¹ for the atomic gas (R_max ≃ 20 kpc) and much larger values of −23 to 50 M_☉ yr⁻¹ for the molecular gas in the center (R_max ≃ 1 kpc). However, to study the redistribution of gas within the galaxy disk, e.g., caused by bar and spiral wave patterns, one has to examine the gas mass flow as a function of radius (see Figure 12). For example, the net gas mass flow over the entire disk can be zero while a large redistribution of gas may occur within the disk. This seems to be the case for, e.g., NGC 3627 where we found a large net gas mass inflow in a region of 0 < R < 3 kpc (∑^3 kpc₀M_☉ yr⁻¹ = −1 M_☉ yr⁻¹), and a similar value with the opposite sign (gas mass outflow) from 3 kpc to the outer region of the galaxy (∑^6 kpc_3 kpcM_☉ yr⁻¹ = 1M_☉ yr⁻¹), so that the sum over the entire disk results coincidentally in an almost zero net gas mass flow $(\sum _0^{R_{\max }} M_{\odot }\, {\rm yr}^{-1} = -0.1M_{\odot}\,{\rm yr}^{-1})$. In general, we found for our sample that the amplitude of the gas mass flow within the disk is much larger than the net gas mass flow over the entire gas disk. This indicates that the redistribution of gas within the galaxy exceeds the net gas mass outflow/inflow from the entire galaxy. The significant inflows and outflows within the disk are likely caused by the dynamical action of a stellar bar and/or spiral pattern.

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To evaluate in more detail whether the angular momentum transfer changes sign at characteristic dynamical locations within the disk, we derived the corotation resonance (CR) radii as described in Section 4.3 (see Table 9 for an overview of all CR radii for our galaxies). The CR radius is defined as the radius where the density wave pattern (e.g., bar, spiral) and the differentially rotating disk have the same angular velocity. This leads to a gain of angular momentum outside and loss of angular momentum inside the regions close to the CR radius. Since galaxies can harbor several spiral and/or bar patterns, also multiple CR radii can exist within one galaxy. In Table 9, we listed all CR radii for our galaxies with a significant phaseshift inversion.

Summarizing, almost all galaxies of our sample (except the center of NGC 5248) exhibit torque patterns typical for m = 2 modes (bars, spirals). NGC 3627 seems to have a very strong torque efficiency in the center (dL/L > 0.6) while NGC 7217 shows the lowest torque strength and angular momentum transfer in our sample. In general, we found that the transfer of angular momentum and the gas mass flow is larger in the center (R ≲ 1 kpc) than in the outer disk (R ≳ 1 kpc).

In this section, we describe the gas flow for each of our seven galaxies and search for relations to their dynamical states and stellar and gaseous morphologies. As most of our galaxies are different in their morphology and kinematics, only case studies can provide information about the mechanisms that are acting. A general picture of the gas flow follows from a comparison of these individual studies and is described in Section 5.2.

5.1.1. NGC 3368

5.1.2. NGC 3627

5.1.3. NGC 4321

5.1.4. NGC 4736

5.1.5. NGC 5248

5.1.6. NGC 6951

5.1.7. NGC 7217

We found that the typical torque pattern changes sign in four quadrants. This typical butterfly pattern is characteristic for the action of an oval or barred gravitational potential. The orientation of the torque pattern in the center is coincident with the one of the outer disk (except for NGC 4736). Toward the outer disk, the gas flow is governed by the action of the stellar spiral arms, if present, which cause a positive (negative) torque on the side of the spiral arms toward (away from) the galaxy center. Thus, we conclude that for almost all galaxies in our sample a dominant m = 2 mode is present and has roughly the same orientation in the outer disk and the center. We found nested bars within large-scale bars for NGC 3368, NGC 3627, NGC 4736 whose major axes have an offset of 25°–60° to the large-scale bars. No signs for unidentified patterns (i.e., not showing up as a prominent feature in the NIR images) are found in the torque maps.

The radial profiles of dL/L and d²M(R)/(dRdt) reveal that torques are more efficient in the redistribution of gas in the center of galaxies than in the outer disks, presumably due to larger gravitational torques induced there by the bar component. We found for a majority of our galaxies (5/7) a significant reversal from outflow to inflow in radial direction at radial distances ranging between (0.1–0.9) kpc, leading to an increasing gravitational pressure on the gas at this position due to gravitational torques and in the case of stable orbits to an accumulation of gas at this position. A comparison of our radial gas density profiles and the torque profiles revealed for most of the outer disks (H i density) no correlation while for some central disks (CO density) an overlap or a shift between both profiles is found (see Figure 12 and Section 5.1). A possible explanation is that gravitational torques will redistribute the gas within a few orbital rotation periods. As the transport of angular momentum per rotation period is on average ∼0.3, the expected timescale for the redistribution of the total angular momentum for a given radius is ∼3 rotation periods (≅3 × 10⁷ yr at a radius of 0.5 kpc). Another explanation might be that the gas is consumed in star formation and hence does not accumulate as expected. For the very center (0–0.1) kpc we found no significant gas transport toward the AGN due to gravitational torques, suggesting that other torques such as viscous torques might be become important for fueling the AGN (see also García-Burillo et  al. 2005).

In the dynamics of galaxies the CR radius is expected to separate inflow (between ILR and CR) from outflow (between CR and OLR). However, in all galaxies of our sample the CR radius of the bar does not correlate with any inflow/outflow separation and thus, sets no barrier for gas transport within the disk. This contradiction can be possibly explained by a coupling of dynamical modes, e.g., of a spiral and a bar or an oval potential. Thus, the orbital paths of the gas may change in comparison to the orbits of a single dynamical mode and allow the transport of gas across the CR. A similar case has been evolved for NGC 4579 which shows a transportation of gas inward across the CR of the bar, but in this case, due to (morphological) decoupling of bar and spiral pattern on different spatial scales S. García-Burillo et  al. (2008, in preparation).

With regard to the CR of a spiral pattern, most of our galaxies with spirals (NGC 3627, NGC 4321, NGC 5248) show a correlation between CR radius of the spiral and a reversal of inflow/outflow of gas due to gravitational torques. Only for NGC 6951 no correlation is found. Instead a significant outflow due to gravitational torques occurs at the CR radius of the spiral, suggesting that the bar potential is still acting at this radius and may overlap with the one of the spiral pattern. Further, the OLR of the bar (18.6 kpc) does not overlap with the CR of the spiral and, thus, seems to substantiate the dominance of the bar for NGC 6951. Also for NGC 3368, NGC 4736, and NGC 7217, which have none or only very weak stellar spiral arms, the gravitational torque is dominated by a bar or oval potential.

Summarizing, gravitational torques can be very efficient in transporting the gas from the outer disk to the center at ∼100  pc. Due to a dynamical coupling of bar and spiral arms, the gas can bridge even resonances such as the CR of a bar (which could act as a natural barrier for further gas inflow).

5.3.1. Estimation of Radial Motion

5.3.2. Streaming Motions versus Radial Inflow/Outflow

As an alternative to radial gas motion where the gas looses or gains angular momentum, gas clouds with streaming motion can follow under certain assumptions closed orbits and no angular momentum is transferred when collisional shocks and other nonlinear effects are excluded. For instance, if the principal axes of an elliptical orbit are aligned with those of a stellar bar, the phase shift between the bar and the gas is zero and hence no inflow will occur at all, but still strong streaming motions are present. To produce a net inflow/outflow a phase shift between the gas and the potential is required. Streaming motions are present in m = 2 perturbation such as in a barlike potential where the gas follows elliptical closed orbits or in a two arm spiral density wave where a phase shift occurs that varies as a function of radius. This collisionless orbit approximation can be simple described by first order Fourier terms (Franx et  al. 1994). In a realistic treatment the nonlinear effects will contribute additional Fourier terms and elliptical streaming motions may overlap with radial gas flow.

To estimate a possible dependence of streaming motion on pure inflow/outflow rates, we have conducted the following study. If we assume that the noncircular motions derived from our kinematic harmonic decomposition are dominated by elliptical streaming, it is possible to compare the amplitude of noncircular motions v_nc/v_rot as tracer for elliptical streaming with the amplitude of angular momentum transport caused by gravitational torques 〈|dL/L|〉 as a function of radius (see Figure 13). Note that the absolute scales are not really expected to be similar as they quantify different physical measurements, but indicate the significance of the values obtained with each method.

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On average, we find that the amplitude of angular momentum exchange derived from our torque analysis (v_nc/v_rot = 0.09 for H i and v_nc/v_rot = 0.16 for CO) is similar to the amplitude of noncircular motions obtained from our kinematic analysis (v_nc/v_rot = 0.09 for H i and v_nc/v_rot = 0.26 for CO). In more detail, v_nc/v_rot seems to be larger for the center as 〈|dL/L|〉 for most of our galaxies, except for NGC 3368 and NGC 6951. For the outer disk 〈|dL/L|〉 is larger than v_nc/v_rot, at least for those galaxies with a bar. Galaxies without a bar (NGC 4736 and NGC 7217) exhibit very weak gravitational torques and hence small values of 〈|dL/L|〉. Interestingly, noncircular motions in unbarred galaxies have roughly the same amplitude for the outer disk than barred galaxies.

The peaks in the amplitude of noncircular motions to the amplitude of inflow/outflow seem to be correlated as a function of radius for some galaxies: center and outer disk of NGC 3368, outer disks of NGC 3627, NGC 4321, NGC 4736, and NGC 6951. Hence one might infer that noncircular motions are not pure elliptical streaming motions, but include to some extent radial motion as well. On the other hand, we found also cases with significant streaming motions but negligible inflow/outflow rates (e.g., for NGC 4736 and NGC 7217). Thus, our study demonstrates that signatures for streaming motions alone can not be used as reliable tracers for inflow/outflow which requires a phaseshift between the gas and the gravitational potential.

The fueling of AGN requires transportation of gas from large kpc scales down to the inner pc. To bring the gas to the center the angular momentum of the gas must decrease by orders of magnitude. Several types of gravitational instabilities, such as bars, nested bars, and spiral density waves have been proposed that might combine to transport gas from large kpc scales down to the very center (e.g., Shlosman et  al. 1990; Combes 2003; Wada & Koda 2004). Inflow of gas is expected to occur mainly inside the CR radius, transporting the gas toward the ILR, where the gas might accumulate and start to form clumps of star formation. To test this scenario we have compared the gas inflow/outflow rates to the dynamical behavior of the gas at the resonances. Our results suggest for some galaxies of our sample (NGC 3368, NGC 4321, and NGC 6951) that the CR radius of the bar is overcome by gravitational torques, most likely due to a coupling of several pattern speeds (e.g., spiral arms plus bar wave pattern). Thus, the gas reservoir inside the CR can be refueled by gas from the outer disk. However, a generalization of such a scenario is not possible and detailed case studies are required to disentangle the underlying mechanisms. For example, S. García-Burillo et  al. (2008, in preparation) find for NGC 4579 that a decoupling of bar and spiral patterns on different spatial scales allows the gas to efficiently populate the UHR region inside corotation of a bar and thus explain how the CR barrier can be crossed.

For the central disk, we found that the integrated gravitational torques indicate net outflow (two galaxies) and inflow (five galaxies) in our galaxies. In particular, NGC 3627 (Seyfert 2  type) shows a large gas outflow (dL/L = 0.5) inside the inner 0.4 kpc, suggesting a redistribution of the total angular momentum of the gas in ∼2.5 rotational periods at this radius (⋍2.4 × 10⁷ yrs). Interestingly, the kinematic analysis reveals also a large amplitude of noncircular motions (dv_nc/v_rot = 0.8) at scales of R ⩽ 0.15 kpc which may substantiate this scenario.

These findings are similar to other studies of the gas flow for the molecular gas disk: a gravity torque study for NGC 6574 using CO as tracer (Lindt-Krieg et  al. 2008) has shown that gas is flowing inward down to a radius of 400 pc which overlaps with a high nuclear gas concentration suggesting that the gas has been piling up at this location quite recently, since no starburst has been observed yet. An analysis of the torques exerted by the stellar gravitational potential on the molecular gas in four other (NUGA) galaxies found a mostly positive torque inside r < 200 pc and no inflow on these scales suggesting that viscous torques may act as fueling mechanism as well (García-Burillo et  al. 2005). An explanation of this variety of morphologies might be related to timescales (García-Burillo et  al. 2005). In this scenario the activity in galaxies is related to that of bar instabilities, expecting that the active phases are not necessarily coincident with the phase where the bar has its maximum strength. The resulting implication is that most AGN probably occur between active accretion episodes of the galaxy disk. Evidence for such a scenario was found by Hunt et  al. (2008) for NGC 2782 where molecular gas inside the ILR of the primary bar, transported by a second nuclear bar, suggests that the gas is fueling the central starburst, and in a second step might directly fuel the AGN.

The integrated gas mass transfer rates in the center range from 0.01–50 M_☉ yr⁻¹ based on our assumptions (see Section  4.1) and are sufficient enough to preserve AGN activity which requires typical mass accretion rates of 0.01–0.1 M_☉ yr⁻¹ for LINER and Seyfert activity. However, to bridge the last 100  pc to the AGN other mechanisms than gravity (e.g., viscous torques) have to play a major role, since no significant gravity torques are present at these scales (see also García-Burillo et  al. 2005).