CO Multi-line Imaging of Nearby Galaxies (COMING). VII. Fourier decomposition of molecular gas velocity fields and bar pattern speed

Abstract

The ¹²CO (J = 1→0) velocity fields of a sample of 20 nearby spiral galaxies, selected from the CO Multi-line Imaging of Nearby Galaxies (COMING) legacy project of Nobeyama Radio Observatory, have been analyzed by Fourier decomposition to determine their basic kinematic properties, such as circular and noncircular velocities. On average, the investigated barred (SAB and SB) galaxies exhibit a ratio of noncircular to circular velocities of molecular gas larger by a factor of 1.5–2 than non-barred (SA) spiral galaxies at radii within the bar semimajor axis a_b at 1 kpc resolution, with a maximum at a radius of R/a_b ≈ 0.3. Residual velocity field images, created by subtracting model velocity fields from the data, reveal that this trend is caused by kpc-scale streaming motions of molecular gas in the bar region. Applying a new method based on radial velocity reversal, we estimated the corotation radius R_CR and bar pattern speed Ω_b in seven SAB and SB systems. The ratio of the corotation to bar radius is found to be in a range of |$\mathcal {R}\equiv R_\mathrm{CR}/a_\mathrm{b} \approx 0.8$|–1.6, suggesting that intermediate (SBb–SBc), luminous barred spiral galaxies host fast and slow rotator bars. Tentative negative correlations are found for Ω_b vs. a_b and Ω_b vs. total stellar mass M_*, indicating that bars in massive disks are larger and rotate slower, possibly a consequence of angular momentum transfer. The kinematic properties of SAB and SB galaxies, derived from Fourier decomposition, are compared with recent numerical simulations that incorporate various rotation curve models and galaxy interactions.

1 Introduction

The distribution of cold interstellar gas (mainly molecular gas) is closely related to galaxy evolution via its role in star formation (e.g., Schruba et al. 2011). In a typical disk-shaped galaxy, molecular gas orbits the dynamical center within a thin disk where most of the star formation activity takes place. If the distribution of mass in the disk is non-axisymmetric, in systems such as barred and non-barred spiral galaxies, there are gravitational torques that induce noncircular motions, observable through tracers such as CO and H i.

Molecular gas is an excellent probe of galactic dynamics owing to its low velocity dispersion and highly dissipative nature (Simkin et al. 1980; Schwarz 1981, 1984; Athanassoula 1992). For instance, many barred galaxies exhibit large noncircular motions of molecular gas indicative of gas inflows (e.g., Sakamoto et al. 2000; Sorai et al. 2000; Salak et al. 2016). The phenomenon is understood to be a consequence of shocks in the bar (which propagate as adensity wave through the disk), typically offset from the potential minimum, where the frequency of cloud–cloud interactions may be higher than in the rest of the disk (e.g., Fujimoto et al. 2014). In the bar region, gas clouds lose angular momentum and gradually (on a Gyr timescale) inflow toward the galactic center (Athanassoula 1992). Consequently, the accumulation of gas in the central region can fuel bursts of star formation and the growth of a pseudobulge linked to secular evolution (e.g., Combes & Elmegreen 1993; Kormendy & Kennicutt 2004; Sellwood 2014; Salak et al. 2017). To understand the secular evolution of galaxies, it is essential to examine the large-scale kinematics of molecular gas in spiral galaxies and systematically quantify the differences in gas motions between barred and non-barred spiral systems.

On the other hand, in order to investigate the evolution of bars, as prominent non-axisymmetric structures, and their interaction with other components of the host galaxies, such as the dark matter halo, stellar disk, and cold gas, it is important to measure their basic characteristics. The two perhaps most interesting properties of bars from the aspect of galaxy evolution are (1) the bar length a_b and (2) the pattern speed Ω_b, defined as the angular speed of the bar figure as an elongated “rigid body” rotating within the galactic disk.¹ From a theoretical point of view, bars are expected to grow and slow down their rotation over time (Kormendy 2013; Sellwood 2014).

The bar length has been studied observationally, e.g. by isophotal analysis (elliptical fitting and mode analysis) of surface brightness structures of near-infrared images that trace the distribution of old stars—the main constituents of bars (e.g., Elmegreen & Elmegreen 1985; Herrera-Endoqui et al. 2015; Salo et al. 2015; Díaz-García et al. 2016). On the other hand, the bar pattern speed, as well as the pattern speed of spiral arms (both often regarded as density waves), have been estimated usually from the velocity fields of stars and interstellar gas. Some examples of the often applied methods include the Tremaine–Weinberg method (Tremaine & Weinberg 1984a; Merrifield & Kuijken 1995; Aguerri et al. 2015; Guo et al. 2019), which utilizes a tracer that obeys the continuity equation (e.g., old stars), the potential-density phase-shift method (Zhang & Buta 2007), various measurements using molecular gas data (e.g., Kuno et al. 2000; Koda et al. 2002; Hirota et al. 2014), the Font–Beckman method (Font et al. 2011, 2017), and others. While recent studies have yielded measurements of Ω_b in an increasing number of nearby galaxies, the importance of measuring this quantity and investigating its relation with fundamental galactic parameters, such as the total stellar mass, have motivated us to make new estimates of Ω_b using ¹²CO (J = 1→0) data as a well-established tracer of molecular gas. In this paper we report two major results of an analysis of molecular gas velocity fields of 20 nearby spiral galaxies, including a subsample of seven barred galaxies: (1) new measurements of basic galactic parameters, such as the position angle, inclination, and systemic velocity, and (2) kinematic parameters in terms of circular and noncircular velocity components and bar pattern speed.

The structure of the paper is as follows. In section 2, we describe the analysis and the sample data. The results, including the derived basic galactic parameters, circular and noncircular velocities, are presented in section 3, followed by a discussion of the bar pattern speed from the aspect of observations and numerical simulations (section 4) and a summary (section 5).

2 Data analysis

We begin by describing the basic principles behind the mode analysis (Fourier decomposition) of the CO (1–0) velocity fields used in this work, and the tools that were used to perform the computation.

2.1 Velocity field as Fourier series

The velocity field of a rotating disk galaxy (barred or non-barred) can be represented as a set of elliptical rings centered at the position of the dynamical center.² If a is the semimajor axis of an ellipse and ϕ is the azimuthal angle in polar coordinates measured from the semimajor axis, then the velocity field v = v(a, ϕ) sampled on any such elliptical ring is a periodic function of ϕ and can be expressed as a Fourier series,

$$\begin{equation} v(a,\varphi ) = c_0+\sum _{m=1}^\infty c_m\cos (m\varphi )+\sum _{m=1}^\infty s_m\sin (m\varphi ), \end{equation}$$

(1)

and the coordinates of the ellipse are given by x_e = a cos ϕ and y_e = qa sin ϕ, where q is the ratio of the minor to major axis, also called flattening (see figure 1). Note that, in the case that m = 1 is the only mode, equation (1) becomes v = c₀ + c₁ cos ϕ + s₁ sin ϕ. If we define c₀ ≡ v_sys, c₁ ≡ v_ϕ sin i, and s₁ ≡ v_r sin i, the above equation can be written as v = v_sys + v_ϕ sin i cos ϕ + v_r sin i sin ϕ, where v_sys is the systemic velocity, i is the inclination angle of the plane of the galactic disk, v_r is the radial component, and v_ϕ is the circular component of the velocity vector in the disk, as defined in figure 1b. This equation describes the planar motion of gas in a thin disk, and the coefficients s₁ and c₁, respectively, correspond to the axisymmetric radial and circular velocity components in the disk observed at an inclination angle i. The sign of s₁ can be used as an indicator of axisymmetric inward or outward motions if the orientation of the disk (near/far side) is known. In a galaxy with purely circular rotation, the velocity field is given by v = v_sys + c₁ cos ϕ, and c₁/sin i is equivalent to the rotation curve. If noncircular motions are present, their axisymmetric component is described by s₁, while nonaxisymmetric components are included in higher-order coefficients m ≥ 2 (Schoenmakers et al. 1997). The higher coefficients, in general, provide indications of local irregular motions in the disk, gravitational perturbations of order higher than 2 (Meidt et al. 2008), and may arise from errors in the galactic center position and geometric parameters (Krajnović et al. 2006). For rotating disk galaxies, it is reasonable to expect that c₁ and s₁ are the dominant terms at most radii (section 3). Note that all coefficients are functions of the semimajor axis only.

In the present analysis, by “Fourier decomposition” we refer to a method of calculating the coefficients of equation (1) up to m = 3 from a given velocity field data set.

Fig. 1.

(a) Illustration of the geometry of an elliptical ring at a position angle α₀ with respect to the sky coordinates (RA, Dec). The semimajor and semiminor axes are a and qa, respectively. The color gradient indicates the change in velocity from redshift to blueshift. The standard position angle, defined as the angle of the redshifted side measured anticlockwise from north, in this case is PA = 270° + α₀. The angle ϕ (eccentric anomaly) in equation (1) is measured anticlockwise from the x-axis. In the case of a barred galaxy, r_b is the projected bar radius and α_b is its position angle (subsection 3.3). The bar is illustrated with a dot-dashed gray ellipse. (b) Definitions of radial velocity v_r and azimuthal velocity v_ϕ at point P in the reference frame of a disk. (Color online)

2.2 The data

The data used in the analysis consist of ¹²CO (1–0) images acquired from CO Multi-line Imaging of Nearby Galaxies (COMING), which is a legacy project of Nobeyama Radio Observatory (Sorai et al. 2019). On-the-fly mapping observations of 147 nearby galaxies in the ¹²CO, ¹³CO, and C¹⁸O (1–0) lines were carried out using the FOur beam REceiver System on the 45 m Telescope (FOREST: Minamidani et al. 2016). The acquired ¹²CO (1–0) data have an effective spatial resolution of 17″, image grid spacing of d = 6″, and a velocity resolution of Δv_ch = 10 km s⁻¹. Although the distances to the galaxies in the sample range from 4 to 39 Mpc (table 1), the average distance yields a physical resolution of ∼1.6 kpc; thus, the results of the present study hold on a kiloparsec scale. A detailed description of the observations and data reduction, as well as a gallery of CO (1–0) images and star formation properties of all sample galaxies, are given in the forthcoming COMING overview by Sorai et al. (2019) and the papers by Muraoka et al. (2019) and Yajima et al. (2019). Throughout this paper, CO refers to the ¹²CO isotopologue.

Table 1.

Basic parameters of the sample galaxies.*

	αJ2000.0	δJ2000.0	Morphology	Distance	〈α0〉	〈PA〉	〈q〉	〈i〉	v  sys
Galaxy	(h : m : s)	(° : ′ : ″)		(Mpc)	(°)	(°)		(°)	(km s−1)
NGC (Messier)	(1)	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
157	00:34:46.76	−08:23:47.2	SAB(rs)bc	12.1	−43.5 ± 1.2	226.5 ± 1.2	0.536 ± 0.083	57.6 ± 5.6	1641.0 ± 1.8
613	01:34:18.170	−29:25:06.10	SB(rs)bc	26.4	27.0 ± 0.4	297.0 ± 0.4	0.405 ± 0.025	66.1 ± 1.6	1477.7 ± 1.3
628 (M 74)	01:36:41.747	+15:47:01.18	SA(s)c	9.02	−69.1 ± 3.4	20.9 ± 3.4	0.934 ± 0.057	20.9 ± 9.1	651.1 ± 1.3
1084	02:45:59.908	−07:34:42.48	SA(s)c	20.9	−58.0 ± 0.5	212.0 ± 0.5	0.703 ± 0.089	45.3 ± 7.2	1402.2 ± 3.6
2903	09:32:10.11	+21:30:03.0	SAB(rs)bc	9.46	−64.9 ± 0.8	205.1 ± 0.8	0.258 ± 0.038	75.0 ± 2.3	540.7 ± 0.9
2967	09:42:03.295	+00:20:11.18	SA(s)c	22.2	35.3 ± 1.5	305.3 ± 1.5	0.62 ± 0.15	52 ± 11	1875.0 ± 5.2
3147	10:16:53.65	+73:24:02.695	SA(rs)bc	39.3	56.1 ± 0.7	146.1 ± 0.7	0.908 ± 0.028	24.8 ± 3.8	2773.0 ± 2.7
3627 (M 66)	11:20:14.964	+12:59:29.54	SAB(s)b	9.04	76.4 ± 1.5	166.4 ± 1.5	0.535 ± 0.044	57.7 ± 3.0	704.6 ± 1.8
3810	11:40:58.760	+11:28:16.10	SA(rs)c	16.4	−58.9 ± 1.2	211.1 ± 1.2	0.891 ± 0.065	27.0 ± 8.2	992.9 ± 2.7
3893	11:48:38.19	+48:42:39.0	SAB(rs)c	15.7	75.6 ± 3.6	345.6 ± 3.6	0.789 ± 0.032	37.9 ± 3.0	980.7 ± 1.3
3938	11:52:49.45	+44:07:14.6	SA(s)c	17.9	−67.7 ± 1.1	202.3 ± 1.1	0.951 ± 0.042	18.0 ± 7.8	812.2 ± 0.6
4030	12:00:23.627	−01:06:00.34	SA(s)bc	29.9	−53.8 ± 0.5	36.2 ± 0.5	0.933 ± 0.045	21.1 ± 7.2	1458.7 ± 2.5
4303 (M 61)	12:21:54.895	+04:28:25.13	SAB(rs)bc	16.5	45.3 ± 0.4	315.3 ± 0.4	0.942 ± 0.031	19.6 ± 5.3	1562.6 ± 1.3
4579 (M 58)	12:37:43.522	+11:49:05.498	SAB(rs)b	16.5	0.3 ± 0.7	90.3 ± 0.7	0.780 ± 0.040	38.7 ± 3.7	1507.6 ± 1.8
5055 (M 63)	13:15:49.33	+42:01:45.4	SA(rs)bc	9.04	8.8 ± 0.3	98.8 ± 0.3	0.506 ± 0.017	59.6 ± 1.1	498.5 ± 2.3
5248	13:37:32.024	+08:53:06.64	SAB(rs)bc	13.0	12.7 ± 1.3	102.7 ± 1.3	0.874 ± 0.044	29.1 ± 5.2	1151.7 ± 1.4
5676	14:32:46.846	+49:27:28.45	SA(rs)bc	34.7	−47.1 ± 0.6	222.9 ± 0.6	0.735 ± 0.015	42.7 ± 1.3	2116.8 ± 8.2
5678	14:32:05.610	+57:55:17.20	SAB(rs)b	35.7	−83.2 ± 0.7	186.8 ± 0.7	0.731 ± 0.093	43.0 ± 7.8	1899.0 ± 7.5
6643	18:19:46.41	+74:34:06.1	SA(rs)c	21.3	−50.0 ± 0.7	30.0 ± 0.7	0.629 ± 0.033	51.0 ± 2.4	1496.2 ± 2.1
7479	23:04:56.65	+12:19:22.4	SB(s)c	36.8	−62.7 ± 1.8	207.3 ± 1.8	0.495 ± 0.061	60.3 ± 4.0	2346.8 ± 5.5

*(1) Adopted from the NASA/IPAC Extragalactic Database. (2) de Vaucouleurs et al. (1991). (3) NGC 613: Tully et al. (2009), NGC 628: Dhungana et al. (2016), NGC 2967: Bottinelli et al. (1984), NGC 3893: Springob et al. (2009), NGC 3938: Poznanski et al. (2009), NGC 4303 and NGC 4579: Mei et al. (2007), others: Tully et al. (2013). (4) The derived kinematic position angle 〈α₀〉 (measured anticlockwise from west) is also expressed as (5) the position angle of the receding side of the galaxy 〈PA〉, which is measured anticlockwise from north. (6) Ellipse flattening q = cos i. (7) Galactic disk inclination. (8) The velocities are in radio definition with respect to the local standard of rest (LSR).

In the data analysis, we used moment 1 images (two-dimensional data), defined as intensity-weighted velocity 〈v〉 ≡ ∫VT_mbdV/∫T_mbdV, where T_mb is the main beam brightness temperature and V is the velocity with respect to the local standard of rest (LSR) in radio definition. The data cubes were clipped at 4 σ in T_mb prior to creating moment 1 images, where the sensitivity was typically better than 1 σ(T_mb) ≈ 80 mK.

The criteria for selecting galaxies for the present analysis were the following: (1) ¹²CO (1–0) emission detected at ≥4 σ in T_mb within a field of radius of at least R ≈ 50″ (∼3 × effective beam size), (2) within a radius larger than the projected bar radius (in barred galaxies), and (3) not edge-on (inclination i ≤ 75°). These conditions allowed us to perform Fourier decomposition of velocity field images, as described in the next section, and derive basic galactic properties, such as position angle, inclination, systemic velocity, and circular/noncircular velocities. The sample of 20 galaxies, listed in table 1, includes all major types of spiral galaxies classified by the presence or absence of a bar according to de Vaucouleurs et al. (1991): SA (non-barred), SAB (weakly barred), and SB (strongly barred) systems.

2.3 Fourier decomposition of CO (1–0) velocity fields

To calculate the coefficients in equation (1) from CO (1–0) moment 1 images, we used Kinemetry (Krajnović et al. 2006), a program that fits elliptical rings on an observed velocity field 〈v〉 by minimizing the noncircular velocity component expressed as |$v_\mathrm{nc} = \sqrt{s_1^2+s_2^2+c_2^2+s_3^2+c_3^2}$| (up to m = 3). This is done by searching for a minimum value of 〈v〉 − c₁ cos ϕ (residuals from circular rotation models) on a grid of flattenings 0.2 ≤ q ≤ 1 and kinematic position angles −90° ≤ α₀ ≤ 90°. The flattening is related to the galactic inclination (assuming a thin disk) as q = cos i. For each ring, the pair (q, α₀) with the minimum v_nc yields the parameters of the best-fit ellipse, whose coordinates are (x_e, y_e). The data points in incomplete rings are retrieved by interpolation and the calculation is performed until a ring is achieved which is filled with less than 75% of the data points.

The program was first set to calculate the initial values of α₀ and q for each ring. The galactic center coordinates (table 1) were fixed, and the semimajor axes of the rings were determined by the program’s default sampling function a_n = (n + 1.1ⁿ)d, where n = 0, 1, 2, …, N is the ring number, and d = 6″ is the grid spacing on images. The initial values of α₀ and q were derived for each radius (figure 2). The mean values of the two parameters across the disk of each galaxy were calculated as |$\langle \alpha _0\rangle = \sum _n^N w_n \alpha _{0n}/\sum _n^N w_n$| and |$\langle q\rangle = \sum _n^N w_n q_n/\sum _n^N w_n$|⁠, where w = w(a) is a weighting function equal to the number of ellipses used to determine the best fit within each ring n, and N is the total number of rings for which fitting was successful. Applying this function ensures that rings with a larger number of data points, which are found in the outer parts of images, are given more weight in calculating the average values. Here, we assume that the CO gas disks are not significantly warped within the investigated regions. The innermost ring and the outermost ring were excluded from calculations because the number of pixels and the signal-to-noise ratio were minimal, respectively. In addition, we found that the initial values of q were notably different in the inner R < 50″ region compared to the rest of the disk in NGC 628 (figure 2), and within R < 40″ in NGC 3893. For these two galaxies, only rings beyond these radii were used in calculating 〈q〉 and 〈α₀〉. For barred galaxies, the weighted averages were calculated using only the values of the rings outside the bar region, defined by the bar length in table 2. This is justified by the expectation that noncircular motions are dominant in the bar region, and may induce an “S”-shaped disturbance of the velocity field that resembles a warp of the disk. Therefore, taking the average over all rings in the disk would yield an average kinematic position angle of that of the inner and outer regions of the disk. We assume that the outer region is less affected by the tangential forces induced by the bar, so that the kinematics of molecular gas there is well represented by circular rotation. As shown in subsection 3.1, this assumption is reasonable given the excellent agreement between the position angles derived by Kinemetry and those by photometric methods for both barred and non-barred spirals. The calculated weighted average values (table 1) were fixed for all radii in the next round of calculations that yielded the final results. The output parameters are the Fourier coefficients c_m and s_m (m = 1, 2, 3), kinematic position angle 〈α₀〉, flattening 〈q〉, and systemic velocity v_sys (≡ c₀). The systemic velocity was calculated as the mean of the values determined in rings that were used to calculate 〈α₀〉 and 〈q〉.

Fig. 2.

(a) Position angle (PA), and (b) flattening q calculated for NGC 628. The horizontal full lines show the weighted average values calculated for rings beyond R > 50″ (table 1). The horizontal dashed line in panel (b) marks q = 1. (c) Moment 1 image of NGC 628. The contours are plotted at 〈v〉 = 631, 641, 651, 661, 671 km s⁻¹; the straight line indicates the derived kinematic major axis, i.e., position angle 〈PA〉 (table 1). Also shown are the innermost (a = |${12^{\prime\prime}_{.}6}$|⁠) and the outermost (a = |${132^{\prime\prime}_{.}3}$|⁠) elliptical rings calculated using 〈PA〉 and 〈q〉.

3 Results

In this section we describe the main results of the analysis using Kinemetry, presented for each galaxy in terms of geometric and kinematic parameters, circular and radial velocity components, and the fraction of noncircular motions as a function of galactocentric radius.

3.1 Basic geometric and kinematic parameters

As described in the previous section, the program fitted elliptical rings on sampled moment 1 images. The solutions yielded several basic kinematic parameters, such as (1) the weighted average of the kinematic position angle 〈α₀〉 of the galactic disk (ellipse), measured within (−90°, +90°) anticlockwise from west, (2) the flattening 〈q〉, and (3) the systemic velocity v_sys, which is the zeroth-term Fourier coefficient c₀. The obtained parameters are listed in table 1; a comparison with the values of position angle (PA) and q, compiled from the literature in Sorai et al. (2019) and derived mainly from photometry, is shown in figure 3. In the case of NGC 2967, the photometric value of PA = |${64{^{\circ}_{.}}0}$|±|${62{^{\circ}_{.}}5}$| (Salo et al. 2015) was largely offset from our value of 〈PA〉 = |${305{^{\circ}_{.}}3}$|±|${1{^{\circ}_{.}}5}$| determined from kinematics and so was excluded from the plots in figure 3. The photometric PA was obtained from analysis of the Survey of Stellar Structure in Galaxies (S⁴G) Spitzer 3.6 μm images. A comparison of the photometric PA as a function of radius in Salo et al. (2015) with 〈PA〉 in table 1 suggests that the two values are consistent in the inner R ≲ 40″ of this galaxy, whereas the photometric PA exhibits a wide scatter at larger radii. The linear relation between photometric and kinematic PAs for the galaxies in figure 3a is very tight, which is important given that the sample includes both barred and non-barred spirals; the consistency for the barred systems supports our assumption made in subsection 2.3 that the kinematic PA in such systems is more reliably determined if only rings outside the bar region are considered.

Fig. 3.

Comparison of (a) position angle PA and (b) flattening q derived from kinematics in this work with the literature values compiled in Sorai et al. (2019), denoted by (*). The uncertainties are approximately the size of the data markers in panel (a).

Note that the relation between the kinematic and photometric estimates exhibits a much larger scatter in the case of q, as evident from figure 3b. The cause of this is not clear, but could be an effect of disk warps at large radii that affect photometric measurements. Many galactic disks are warped in the outermost regions; q determined in the outer disk is thus not necessarily the same as in the inner disk, where we determined the kinematic q from CO velocity fields. For example, our estimate of 〈q〉 = 0.934 ± 0.057 for NGC 628 gives 〈i〉 = 21° ± 9°, compared to some previous works that suggest an inclination as small as 7° in the inner disk (Kamphuis & Briggs 1992). On the other hand, our measurement for this galaxy is in good agreement with i ≈ 20° derived from optical data in HyperLeda³ (Paturel et al. 2003). Compared to the mean inclinations derived from The H i Nearby Galaxy Survey (THINGS) data in de Blok et al. (2008) and Trachternach et al. (2008), the differences between H i and our CO measurements for four galaxies that were analyzed in both projects are: ∼1° in NGC 5055, ∼3° in NGC 3627, ∼6° in NGC 628 (within uncertainties), and ∼9° in NGC 2903. The inclination of NGC 2903, derived from H i as a function of radius in these works, is closer to our weighted average (∼75°) in the inner R ≲ 100″, but decreases at large radii, yielding an average of ∼66°. Given the differences in the angular resolution and properties of H i and CO gases, the results in the two studies are in reasonably good agreement.

3.2 Circular and radial velocities

The deprojected azimuthal velocity, defined as v_ϕ = c₁/sin i, where c₁ is the amplitude of the first-order cosine term and i is the inclination of the galactic disk, was calculated for a number of radii determined by the size of the region in the moment 1 map where CO (1–0) emission was detected at ≥4 σ. The derived v_ϕ was used to obtain the angular velocity, Ω(R) = v_ϕ(R)/R, and these two quantities describe the circular motion of molecular gas in a galactic disk.

The program also calculated the deprojected radial velocity component, v_r = s₁/sin i, where s₁ is the amplitude of the first-order sine term. This velocity corresponds to axisymmetric inward or outward motion corrected for galactic inclination. The term is expected to be nonzero, for example, in a system where noncircular motions of gas arise from the potential of a bar. The ratio of overall noncircular to circular motions in a galactic disk is quantified by

$$\begin{equation} \Delta (R)\equiv \frac{v_\mathrm{nc}}{c_1}, \end{equation}$$

(2)

where |$v_\mathrm{nc} = \sqrt{s_1^2+s_2^2+c_2^2+s_3^2+c_3^2}$| is a measure of noncircular motions as a function of radius (subsection 2.3).

Examples of the results of calculations of v_ϕ, Ω, v_r, and Δ for two representative galaxies of SA (NGC 5055) and SB (NGC 2903) types are shown in figure 4, while the rest of the sample is plotted in the Appendix. Figure 4 shows several important features of these systems:

• The kinematics is clearly dominated by circular rotation at all radii (v_r ≪ v_ϕ), especially outside the bar region, and the derived v_ϕ of both galaxies are consistent with the shapes and amplitudes of the rotation curves derived from H i data (de Blok et al. 2008).

• The fraction of noncircular motions is somewhat larger in the inner region of the barred galaxy; within R < 50″, we find Δ ≈ 0.15 in NGC 5055 and Δ ≈ 0.20 in NGC 2903. A comparison with v_r/v_ϕ suggests that Δ is dominated by the s₁ term, i.e., axisymmetric radial flow. As shown below, this behavior is a general property of molecular gas motion in all investigated barred galaxies. While noncircular motions are not entirely absent in NGC 5055, they are a factor of 1.5–2 smaller than in NGC 2903; noncircular motions in this SA galaxy are possibly caused by gravitational torques due to the spiral arms.

• Panel (c) shows a nearly linear rise of v_ϕ in NGC 2903 up to the radius R ≈ 4 kpc, resembling the rotation of a rigid body. This is a consequence of the fact that the bar position angle of this galaxy is nearly the same as the position angle of the host galaxy (the difference is θ ≈ 3°; see table 2). As discussed in sub-subsection 4.1.3, the observed circular velocity in this case traces the rotation of the bar itself and reveals its angular velocity, which appears to be nearly constant within 30″ < R < 90″.

Fig. 4.

Examples of results for one SA and one SB galaxy. Panels (a) and (c) show circular velocity v_ϕ (blue circles) and angular velocity Ω (black squares). Panels (b) and (d) show radial velocity v_r (red diamonds) and noncircular to circular velocity ratio Δ (green stars). In panel (c), a_b and R_r are the bar radius and reversal radius, respectively, and the purple horizontal line in panel (c) marks the angular velocity at R = R_r. The reversal radius marks the location where v_r changes sign (sub-subsection 4.1.1). (Color online)

Having calculated the circular velocity, position angle, and inclination of the disk, it is possible to generate rotation models from the c₁ term. After subtracting the models from the moment 1 images, we can obtain residuals that reveal the location of noncircular motions in the disk at kpc scale. Some examples of such investigation are shown in figure 5. The residual velocity images of two SB galaxies, NGC 2903 and NGC 7479, in panels (c) and (f), respectively, exhibit amplitudes of the order |v| ≈ 20 km s⁻¹ in the bar region NW and SE of the galactic center (indicated by arrows). Assuming that the spiral arms are trailing in both galaxies, as appears to be the case in the majority of barred spiral galaxies (e.g., Athanassoula et al. 2009), it is possible from the velocity field to determine which side of the disk is closer to the observer, and distinguish inward from outward velocities in the disk. Since the near and far sides of the disk are opposite in NGC 2903 and NGC 7479, the radial velocity corrected for the orientation of the disk is v_r < 0 in the central 1′ of both galaxies (see figure 8). In subsection 3.4 we will see that not all galaxies exhibit negative v_r.

Fig. 5.

(a) Moment 1 (mean velocity) image 〈v〉 of NGC 2903, (b) rotation model v_mod, and (c) residuals calculated as v_res = 〈v〉 − v_mod. Panels (d)–(f): same, but for NGC 7479. The arrows in panels (c) and (f) indicate noncircular motions in the bar region. The contours are 3.6 μm Spitzer maps acquired from NASA/IPAC Extragalactic Database. The beam size (17″) is shown at the bottom left corner.

3.3 Analysis of barred galaxies

In order to analyze the molecular gas kinematics in SAB and SB galaxies, it is important to estimate the bar radius (= semimajor axis) and its position angle with respect to the position angle of the host galaxy. The orientation of bars with respect to galactic disks is random, so the apparent length of a bar can vary depending on this angle difference and inclination. In this work, the projected bar radius r_b, defined as the distance from the galactic center to the bar end (as in figure 1) and its position angle α_b were adopted from Herrera-Endoqui et al. (2015), who analyzed the S⁴G Spitzer 3.6 μm data by visual inspection and searching for the radius of ellipticity maximum. The differences between the two values are usually of the order of a few percent, and we adopt a conservative uncertainty of 10% in a_b. The projected radius r_b was converted to a deprojected bar radius a_b by correcting for inclination and position angle using the equation |$a_\mathrm{b} = r_\mathrm{b}q^{-1}\sqrt{q^2\cos ^2{\theta }+\sin ^2{\theta }}$|⁠, where q is the ellipse flattening (see subsection 2.2), and θ ≡ α_b − α₀ is the difference between the position angle of the bar and that of the galactic disk.⁴ The position angles and deprojected bar radii are listed in table 2. Note that some recent works (Buta et al. 2007, 2015; Herrera-Endoqui et al. 2015) classify some galaxies listed in table 1 as SA, in contrast to the SAB designation in de Vaucouleurs et al. (1991). Since there was no reliable data for the bars in these systems, they were excluded from the sample of barred galaxies, and the final list consists of three SAB (NGC 3627, NGC 4303, NGC 5248) and four SB (NGC 613, NGC 2903, NGC 4579, NGC 7479) objects. For comparison, we include more recent classifications of galaxies in table 2.

Table 2.

Bar parameters.*

		r  b	αb	θ	a  b	a  b	R  r	|$\mathcal {R}$|	Ωb
Galaxy	Morphology	(′)	(°)	(°)	(′)	(kpc)	(′)		(km s−1 kpc−1)
NGC	(1)	(1)	(1)	(2)	(3)	(3)	(4)	(5)	(6)
157	SA(s)bc	—	—	—	—	—	—	—	—
613	SB(rs)b	77.1	35	8.0	80.8	10.3 ± 1.0	61.0 ± 6.0	0.75 ± 0.11	21.5 ± 2.2
2903	(R’)SB(rs)b	68.1	−62	2.9	69.3	3.18 ± 0.32	75.6 ± 6.0	1.09 ± 0.14	48.7 ± 4.2
2967	(R’)SAB(rs)c	8.1	—	—	—	—	—	—	—
3147	SAB(r’l)b	14.9	—	—	—	—	49.1 ± 6.0	—	—
3627	SAB(s)b pec	66.4	70	−6.4	67.4	2.95 ± 0.29	104.0 ± 6.0	1.54 ± 0.18	44.3 ± 3.0
3893	SA(s)c	—	—	—	—	—	30.6 ± 6.0	—	—
4303	SAB(rs)bc	36.1	88	42.7	37.1	2.97 ± 0.30	45.1 ± 6.0	1.22 ± 0.20	52 ± 15
							(60.8 ± 6.0)	(1.64 ± 0.23)	(40 ± 11)
4579	(RL,R’)SB(rs)a	40.7	−37	−37.3	45.2	3.62 ± 0.36	61.9 ± 6.0	1.37 ± 0.19	52.7 ± 6.6
							(76.5 ± 6.0)	(1.69 ± 0.22)	(43.9 ± 5.0)
5248	(R’)SAB(s)bc	27.4	38	25.3	28.2	1.78 ± 0.18	45.5 ± 6.0	1.61 ± 0.27	73 ± 16
5678	(R’L)SA(rs)b pec	—	—	—	—	—	25.4 ± 6.0	—	—
7479	(R’)SB(rs)b	49.3	−82	−19.3	57.0	10.2 ± 1.0	>68.1	>1.09	≲14

*(1) Adopted from Herrera-Endoqui et al. (2015). The bar position angle (α_b) is measured anticlockwise from west. (2) θ ≡ α_b − α₀. (3) Deprojected bar semimajor axis. The uncertainty of a_b is adopted to be 10%. (4) Reversal radius, where v_r changes sign (secondary reversal radius is given in brackets where applicable). (5) |$\mathcal {R}\equiv R_\mathrm{r}/a_\mathrm{b}$|⁠. (6) Ω_b ≡ v_ϕ(R_r)/R_r.

Figure 6 shows the relations between the deprojected bar radius a_b, total molecular gas mass M_mol, and total stellar mass M_*. Here, the molecular gas mass was calculated as M_mol = |$1.36\:{\rm m}_\mathrm{H_2}AX_\mathrm{CO}\sum _i(I_\mathrm{CO})_i$|⁠, where |$m_\mathrm{H_2}$| is the mass of a single hydrogen molecule, A is the area of a map pixel, I_CO ≡ ∫T_mbdV is the integrated intensity, X_CO = 2 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ is a standard conversion factor (Bolatto et al. 2013), and summation is performed over all pixels. The factor 1.36 is the correction for the abundance of helium. The uncertainty of I_CO was evaluated for each pixel as |$\Delta I_\mathrm{CO} = \Delta T_\mathrm{mb}\sqrt{\Delta v_\mathrm{int}\Delta v_\mathrm{ch}}$|⁠, where ΔT_mb is the r.m.s. of the emission-free channels, Δv_int is the velocity range of the signal, and Δv_ch = 10 km s⁻¹ is the channel width. Since M_mol ∝ I_CO, the uncertainty ΔI_CO was translated into the uncertainty of M_mol ± ΔM_mol (Sorai et al. 2019).

Fig. 6.

(a) Total stellar mass M_* plotted against the bar radius a_b; distance uncertainties are not included in the uncertainties of a_b. (b) Total molecular gas mass M_mol and M_* for all 80 barred galaxies in the COMING sample. Open circles are SAB and filled stars are SB galaxies based on the morphological classification in de Vaucouleurs et al. (1991); the barred galaxies studied in this paper are shown as red circles (table 3). The error bars are typically the size of the data markers. The full line shows a power-law fit for all objects. (Color online)

The total stellar mass M_* was calculated from the Wide-field Infrared Survey Explorer (WISE) 3.4 μm data products by Sorai et al. (2019), who applied the light-to-mass conversion method described by Wen et al. (2013). The masses of the barred galaxies are listed in table 3; the ratio M_mol/M_* is typically of the order of 10% in the studied objects. The plots indicate that the galaxies with massive stellar disks in our sample generally tend to have longer bars and larger total molecular gas reservoirs. The relation between M_* and a_b also holds when a larger sample is studied, as shown by Díaz-García et al. (2016). Note that the plot in figure 6b includes all 44 SAB and 36 SB galaxies (without SA) in the COMING sample. The relation in panel (b) can be fitted with a power law:

$$\begin{equation} \log {M_\mathrm{mol}} = (-3.10\pm 0.48)+(1.193\pm 0.046)\log {M_\ast }. \end{equation}$$

(3)

Figure 6b shows that the seven barred galaxies analyzed in this work are among the luminous ones (M_* > 2 × 10¹⁰ M_⊙) compared to the entire sample.

Table 3.

Stellar mass and molecular gas mass in the selected barred galaxies.*

Galaxy NGC	M  mol (M⊙)	M  * (M⊙)	M  mol/M*
613	(1.075 ± 0.009) × 1010	(1.22 ± 0.03) × 1011	0.09
2903	(4.46 ± 0.02) × 109	(4.06 ± 0.09) × 1010	0.11
3627	(4.65 ± 0.02) × 109	(4.02 ± 0.09) × 1010	0.12
4303	(9.33 ± 0.06) × 109	(5.93 ± 0.14) × 1010	0.16
4579	(2.93 ± 0.03) × 109	(6.90 ± 0.16) × 1010	0.04
5248	(3.93 ± 0.02) × 109	(2.34 ± 0.05) × 1010	0.17
7479	(1.157 ± 0.010) × 1010	(1.07 ± 0.03) × 1011	0.11

*The molecular gas masses were derived from CO (1–0) integrated intensity, using a single conversion factor of X_CO = 2 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ and correcting for the abundance of He. The total stellar mass was calculated from WISE 3.4 μm images (Sorai et al. 2019).

3.4 Gas streaming motions

One of the main results of the mode analysis of the velocity fields is presented in figure 7, where the noncircular velocity parameter Δ = v_nc/c₁ and the deprojected radial velocity v_r = s₁/sin i of 12 SA galaxies (on the left) and 7 SAB + SB galaxies (on the right) are plotted against galactocentric radius. The radial velocity is corrected for inclination and for the near/far side of the disk of the barred galaxies, assuming that these galaxies all have trailing spiral arms. The sign of i depends on disk orientation, such that 0° < i < 90° when the near side is at −180° < ϕ < 0°, and −90° < i < 0° when the near side is at 0° < ϕ < 180° (figure 8). Radial velocities uncorrected for disk orientation, i.e., as derived by the program, are shown in figure 11 in the Appendix. For SA galaxies, we plot the results in their uncorrected form in figure 7 to avoid uncertainty in the choice of orientation of their disks.

Fig. 7.

Upper panels: Noncircular velocity parameter Δ = v_nc/c₁ is shown for (a) non-barred and (b) barred galaxies. Middle panels: Average values of the plots in panels (a) and (b) with 1 σ uncertainty. The data were smoothed in bins of δR = 1 kpc in panel (c) and δ(R/a_b) = 0.2 in panel (d). The innermost data point of NGC 4579, which has Δ = 2.9, is regarded as an outlier and excluded from the smoothed curve in panel (d)—see text. Lower panels: Radial velocity v_r is plotted for (e) non-barred and (f) barred galaxies. The horizontal axis in panels (b), (d), and (f) is normalized as the galactocentric radius divided by the bar radius. The color tick marks in panel (f) indicate the locations of R_r, and the star (e.g., N7479*) indicates objects with −90° < i < 0° for which v_r has been corrected.

Fig. 8.

Moment 1 (mean velocity in km s⁻¹) images of the barred galaxies. Shown are the calculated kinematic position angle (gray line), bar orientation (black ellipse; semimajor axis =r_b), and the adopted near side of the disk with respect to the observer, assuming that the spiral arms are trailing. The near/far orientation of the disk was used to correct the sign of v_r = s₁/sin i in figure 7d. When the near side is at −180° < ϕ < 0°, where ϕ is measured anticlockwise from the galactic major axis (west side), then 0° < i < 90°. Five contours are plotted on each image, with equal separations from minimum to maximum values. The illustration in the bottom right shows two possible orientations of the disk from the perspective of an observer.

Figure 7 shows a clear difference in the Δ parameter between the barred and non-barred systems at 1 kpc resolution: inspecting the regions within the bar radius a_b in SAB and SB galaxies, we find a maximum of Δ ≈ 0.26 at R/a_b ≈ 0.3, which is a factor of 1.5–2 larger than in SA systems, where no significant variation with radius is observed. On the other hand, the values appear to be similar (Δ ≈ 0.10–0.15) in both barred and non-barred objects at large radii.

The radial velocities v_r, plotted in figure 7f, indicate that the rise in Δ can be attributed to some extent to the s₁ coefficient in v_nc. Molecular gas is affected by the presence of the bar and exhibits noncircular (streaming) motions with radial velocity amplitudes typically |v_r| ≈ 20 km s⁻¹ at 1 kpc resolution. Note that both v_r < 0 and v_r > 0 velocities are observed. The reason for this is that there can be both inward and outward motions of gas in the bar region as there are negative and positive gravitational torques (tangential forces) in the vicinity of the bar that give rise to tangential acceleration |$|\ddot{\varphi }|>0$| and |v_r| > 0; gas is accelerated as it approaches the bar on the upstream side and decelerated on the downstream side. The streaming motions are clearly visible in figure 8, where isovelocity contours tilt with respect to kinematic minor axis in the same sense as the bar. If the gas orbits were ellipses, the isovelocities in the bar would remain without tilt only if θ = 0 or θ = π/2 (Kalnajs 1978). However, we find substantially tilted isovelocity contours even in NGC 2903, where θ ≈ 0. This behavior can be explained in terms of gas inflow in the bar region.

Among the barred galaxies, NGC 4579 exhibits notably peculiar CO (1–0) velocities with a large value of Δ in the innermost ring. The high fraction of noncircular motions appears to be related to a blueshifted velocity component in the galactic center, as evident in the moment 1 image in figure 8. Similar peculiar motions of gas can also be seen in the velocity field of ionized gas traced by [O iii] in the central R < 20″ (Dumas et al. 2007), as well as within R < 30″ of the interferometric CO (1–0) data in Sofue et al. (2003).

NGC 3147 is not included in the plots because it is very weakly barred (classified as |$\mathrm{S\underline{A}B}$| or SA). The bar length in the catalogue of Herrera-Endoqui et al. (2015) suggests a nuclear bar (⁠|$r_\mathrm{b} \approx {14^{\prime\prime}_{.}9}$|⁠); similarly, Casasola et al. (2008) reported a radius of |$r_\mathrm{b} \approx {7^{\prime\prime}_{.}5}$|⁠, which is too small to be resolved in our images. Nevertheless, the result of Fourier decomposition of this galaxy (the Appendix) exhibits noncircular motions that peak at the innermost radii, in agreement with other barred systems.

Similarly, NGC 5678 is classified as SAB in de Vaucouleurs et al. (1991) and as SA in Herrera-Endoqui et al. (2015). There is no clear indication of a bar in this galaxy, so we place it in the SA plot in figure 7e. However, we note that the behavior of v_r exhibits a change in sign at a radius of R ≈ 4 kpc, indicating that the velocity field is perturbed. The optical/near-infrared morphology suggests asymmetric morphology with an outer pseudo-ring.

On the other hand, although NGC 2967 is classified as SA in de Vaucouleurs et al. (1991), ellipticity measurements reported in Herrera-Endoqui et al. (2015) and Salo et al. (2015) indicate an SAB type; the bar radius, however, is only ∼8″, which is smaller than the resolution of our images, and the influence of this nuclear bar can be neglected. This is supported by the fact that v_r behaves as expected from an SA (figure 11).

4 Discussion

4.1 Corotation resonance and bar pattern speed

Based on analytical models of weak bar potentials,⁵ a bar is expected to extend between the radius of an inner Lindblad resonance (if there is any) and the corotation radius R_CR, where the disk material and bar rotate with the same angular velocity (⁠|$\Omega _{R = R_\mathrm{CR}} = \Omega _\mathrm{b}$|⁠; Contopoulos 1980; Elmegreen 1996; Binney & Tremaine 2008). It is therefore expected that the corotation radius in the majority of weakly barred galaxies is at least as large as the bar radius. Numerical simulations also suggest that the ratio of the corotation radius to the bar radius, defined as the parameter |$\mathcal {R}\equiv R_\mathrm{CR}/a_\mathrm{b}$|⁠, is typically in the range |$\mathcal {R}$| = 1.0–1.4 (Athanassoula 1992); these are referred to as the fast bars, in contrast to the slow bars, which are loosely defined as having |$\mathcal {R}\gtrsim 1.4$|⁠. On the other hand, since the gravitational torques generated by the bar are generally smaller at radii R > R_CR compared to the bar region (Díaz-García et al. 2016), the velocity dispersion of gas is expected to decrease at the corotation radius. In accordance with the density wave theory, the velocity field may even reverse phase at R ≈ R_CR, in the sense that radial streaming motions of gas associated with spiral arms change sign at resonance radii, such as corotation and outer Lindblad resonance (OLR), in systems where the spiral pattern is forced by a bar (Kalnajs 1978). Therefore, identifying the radii where the velocity reversals occur can yield the locations of resonances.

4.1.1 Radial velocity reversal

The argument of Kalnajs (1978) was recently applied by Font et al. (2011) to high-resolution Hα data to develop a new technique of estimating the pattern speed (resonance radii) in spiral galaxies. In Font et al. (2017), Ω_b is found indirectly by first measuring R_CR, and since the angular velocity of the disk is equal to that of the bar at R_CR, the value of Ω at that radius becomes the bar pattern speed. Here, R_CR is determined at the radial range where the velocity residuals, obtained by subtracting a model velocity field image from the Hα velocity field image, are minimal and reversals are identified. In principle, the method can also be applied to estimate the pattern speed of spiral arms if they are regarded as density waves (Font et al. 2014).

The possibility that the gas velocity field images (e.g., CO, H i, Hα) can be used to determine the corotation radius R_CR of spiral arms, based on the assumption that they are density waves, was also discussed by Canzian (1993), García-Burillo, Sempere, and Combes (1994), Canzian and Allen (1997), Colombo et al. (2014), and others. For example, high-resolution (∼1″) images of the grand-design spiral galaxy M 51, analyzed by Colombo et al. (2014) and Querejeta et al. (2016), reveal streaming motions in the spiral arms that can be associated with different wave modes and corotation, although possibly amplified by tidal interaction with M 51b.

Compared to spiral arms, the bar instability is believed to be a genuine, long-lasting density wave in the disk and the response of gas in terms of streaming motions is more obvious, so that the location of its corotation can be estimated by applying the following simplified method. Since gas streaming motions are present in the bar region, we will observe |v_r| > 0 inside R_CR and v_r ≈ 0 near it, and the spatial extent of the m = 1 mode is determined by the reversal radius R_r in the axisymmetric radial velocity v_r (table 2). The existence of v_r ≈ 0 regions near the corotation can be understood from the simulations of Athanassoula (1992). For example, figure 2b of her paper shows that velocity vectors have magnitudes close to zero in the corotation region in the reference frame where the bar is at rest. Since the material here then must rotate about the galactic center at the same rate as the bar, the rotation is nearly circular and both radial velocity v_r projected along the minor galactic axis and radial velocity dispersion |$\sigma _{v_{\rm r}}$| are expected to be minimal. Note, however, that these models consider isolated (non-interacting) barred galaxies.

In order to examine the velocity reversal method for barred galaxies, we first discuss the behavior of v_r determined by mode analysis. Although OLR is also an important resonance, often related to the formation of outer rings in galaxies (Buta & Combes 1996; Buta et al. 2007), it may lie outside the region where CO (1–0) is firmly detected in our images, so in this section we discuss corotation only. The location of OLR and its relation with the observables is briefly addressed in subsection 4.3.

As evident in figures 4 and 7, the calculated v_r exhibits a reversal at a specific radius R_r in the case of barred galaxies, in contrast to the majority of non-barred spiral galaxies, where v_r ≈ 0 at most radii. In all but one galaxy, the reversal radius R_r is somewhat larger than the bar radius (table 2), which is comparable to the expected value of R_CR based on analytical analyses and simulations of weakly barred galaxies. Assuming that R_r ∼ R_CR, we get |${\bar R}=1.24\pm 0.12$| (1 σ of the mean), with a wide range |$\mathcal {R}$| = 0.8–1.6, in general agreement with previous measurements that apply different methods (e.g., Aguerri et al. 2003, 2015; Rautiainen et al. 2008; Corsini 2011; Font et al. 2017). For example, the simulations in Rautiainen, Salo, and Laurikainen (2008) yield a similar value of |${\bar R} = 1.44\pm 0.29$| for intermediate (SBb) barred galaxies, comparable to the ones in our sample. Although the uncertainty is large, arising mainly from the uncertainty of R_r, the values suggest the presence of both fast and slow rotator bars in intermediate barred spirals. We also note that the minimum of the smoothed Δ in SAB and SB systems in figure 7d lies at R/a_b ≈ 1.3.

The corotation of a trailing spiral pattern is expected to be marked by a reversal from negative (inward) to positive (outward) v_r (Kalnajs 1978). Figure 7f indicates that NGC 4303 and NGC 4579 may have two reversals, one just beyond the bar end and the other farther out in the spiral arms. Since both galaxies exhibit v_r < 0 between the first and second reversal radii, this region may be inside the corotation of the spiral pattern. For instance, similar gas dynamics with streaming motions and reversals in the spiral arms is observed in the SB galaxy NGC 1365 (Lindblad et al. 1996; Elmegreen et al. 2009).

In the discussion that follows, we associate the v_r-reversal radius R_r (where v_r = 0; radial velocity decreases to zero or changes sign) with R_CR as an approximate location of the corotation resonance, adopting an uncertainty given by the resolution of the image grid (ΔR_r = 6″). The error of q, discussed in subsection 3.1 (figure 3), is the dominant contributor to the uncertainty of the deprojected radius, and this is reflected on R_r. In galaxies with large θ, the projection effects nearly cancel out for R/a_b, since both R ∝ q⁻¹ and a_b ∝ q⁻¹, but can be large in objects with small θ. In the extreme case of θ = 0°, the estimated bar radius is a_b = r_b, but we still have R_r ∝ q⁻¹ and the largest errors arise for small-q (close to edge-on) systems. If we take NGC 2903 as an example, figure 3 (see also table 2) shows that the photometric q is larger than the kinematic one. If we Fourier decomposed this velocity field by using the photometric q, we would get somewhat smaller R_r. From table 2, the uncertainty of 6″ corresponds to ∼10%–20% of the bar radius in all objects. The total uncertainty of the bar pattern speed is estimated to be of the order of ∼20%.

4.1.2 Comparison with other studies

Although our measurements of Ω_b are not the first for most of the selected galaxies, the applied v_r-reversal method is new, relatively simple to perform, and applicable to 1 kpc-resolution data. In this section, we demonstrate that there are no significant discrepancies compared to previous works.

The derived value of the pattern speed in NGC 3627 is compatible with |$\Omega _\mathrm{b} = 50^{+3}_{-8}\, \mathrm{km\, s^{-1} \, kpc^{-1}}$| found by Helfer et al. (2003), who applied the Tremaine–Weinberg method. Similarly, the parameter |$\mathcal {R}$|⁠, which is distance independent, is comparable to |$\mathcal {R}\approx 1.6$| in Hirota et al. (2009). We also find a marginal agreement for the value of |$\mathcal {R}\approx 1$| for NGC 613 in Elmegreen, Elmegreen, and Montenegro (1992), who estimated the location of R_CR from an analysis of optical morphology. The values of |$\mathcal {R}$| for both NGC 4303 and NGC 4579 are consistent with the values of |$\mathcal {R} = 1.70\pm 0.45$| and |$\mathcal {R} = 1.46\pm 0.30$|⁠, respectively, in Rautiainen, Salo, and Laurikainen (2008), who simulated model galaxies. We also find agreement with |$\mathcal {R} = 1.07\pm 0.09$| and |$\Omega _\mathrm{b} = 49.6^{+3.2}_{-2.9}\, \mathrm{km\, s^{-1} \, kpc^{-1}}$| (Font et al. 2017), as well as Ω_b = 53 km s⁻¹ kpc⁻¹ for NGC 4303 (Koda & Sofue 2006), who adopted a similar distance to the galaxy.

The low derived bar pattern speed of NGC 7479 (its upper limit) is comparable to the value of Ω_b ≈ 18 km s⁻¹ kpc⁻¹ found by Fathi et al. (2009), who applied the Tremaine–Weinberg method on Hα data and adopted a similar distance to the galaxy. Font et al. (2017) obtained the same value of Ω_b and |$\mathcal {R} = 1.14\pm 0.05$|⁠, as did Sempere, Combes, and Casoli (1995), both consistent with our lower limit.

The case of NGC 5248 is less clear because there are opposite views on the bar length in this system. From inspection of an R-band image, Jogee et al. (2002) claimed the presence of a much larger bar (a_b ≈ 95″) with |$\mathcal {R}\approx 1.2$| that resembles an oval disk in which a spiral pattern is embedded. However, Herrera-Endoqui et al. (2015) and Salo et al. (2015) found an ellipticity maximum at a much smaller radius, compatible with a small bar. The v_r derived from the CO (1–0) velocity field exhibits a reversal at |$\mathcal {R}\approx 1.6$|⁠, and figures 7f and 8 suggest that the velocity field in this galaxy seems to be only mildly disturbed over most radii.

4.1.3 Bar pattern speed of NGC 2903

Among the examined barred galaxies, NGC 2903 is an SB with the position angle of the bar (α_b) found to be nearly equal to that of the disk (α₀) of the host galaxy as determined in our analysis (table 2). This condition allows us to investigate the bar pattern speed by using an alternative approach, as demonstrated by Salak et al. (2016) for NGC 1808 (see also Hirota et al. 2009).

As mentioned in subsection 2.2, the measured mean velocity can be expressed as v = v_sys + v_ϕ sin i cos ϕ + v_r sin i sin ϕ. Assuming that molecular clouds in the bar region propagate in a nearly radial direction (in the non-inertial reference frame of the bar) in the “offset ridges” where prominent dust lanes are often observed, and that the dust lanes are the density maxima in gas flow, the projected azimuthal velocity (in the inertial frame of the observer) in the bar region is much larger than the radial velocity component along the line of sight, hence the equation simplifies to v ≈ v_sys + v_ϕ sin i. Under this condition, the observed azimuthal velocity is related to the rotation of the bar via v_ϕ = RΩ_b cos (α_b − α₀) ≈ RΩ_b.

The expected linear relation v_ϕ ∝ R can be seen clearly in the plot of v_ϕ and Ω in figure 4c, where the angular velocity of the disk appears nearly constant within a wide range of 30″ < R < 80″ that marks the extent of the bar. The flat portion of the angular velocity curve here can be interpreted as the pattern speed of the bar in NGC 2903, and is consistent with the value of Ω_b = 48.7 ± 4.2 km s⁻¹ kpc⁻¹ determined from the v_r reversal method in sub-subsection 4.1.1, where we have assumed that R_r ≃ R_CR (table 2). In fact, figure 4c shows that the angular velocity Ω is lower than the derived value of |$\Omega _{R = R_\mathrm{r}}$| at some radii inside corotation. Since the bar is rotating slower than the disk at R < R_CR, the bar pattern speed is possibly somewhat lower, in better agreement with Hirota et al. (2009).

4.2 Bar radius, total stellar mass, and bar pattern speed

Using the estimated bar pattern speed Ω_b, we now examine its relation with two fundamental parameters of the host galaxies: bar radius a_b and total stellar mass M_* (figure 9). Although the sample is small, Ω_b is found to be relatively low in systems with large bars and massive stellar disks. The first result, shown in panel (a), is not entirely surprising: if the rotational velocities of the galaxies are similar and approximately flat or decreasing at large radii, then the angular velocities are decreasing with R; if R_r is large, as is found in systems with long bars, Ω_b is expected to be relatively low. This trend is generally expected from simulations (Debattista & Sellwood 2000) and analytical analyses (Lynden-Bell 1979).

Fig. 9.

Angular velocity Ω_b at the reversal radius R_r plotted against (a) bar radius a_b, (b) total stellar mass M_*, and (c) dynamical mass (baryonic + dark matter) within the bar radius, M_tot(R < a_b).

Figure 9b tentatively suggests a negative correlation between the bar pattern speed and the total stellar mass M_* in these systems. If large-M_* galaxies have massive dark matter halos, the observed trend may indicate that dynamical friction on bars by the dark matter halos effectively slows them down (Chandrasekhar 1943; Tremaine & Weinberg 1984b). From the point of view of numerical simulations, the following scenario is expected: as galaxies evolve, their masses increase and the rotational speeds of bars decay due to frictional torques exerted by dark matter halos and angular momentum transfer to the outer disk and bulge (Sellwood 1980; Weinberg 1985; Debattista & Sellwood 2000; Athanassoula 2003; Minchev et al. 2012; Saha et al. 2012). In the sample, we have two SB galaxies with massive stellar disks and slowly rotating bars: NGC 613 and NGC 7479. Curiously, we find |$\mathcal {R}\approx 1$| in NGC 613, which classifies it as a fast bar. This would be expected if the galaxy hosts a “maximum disk” with a minimal fractional contribution of the dark matter halo in the bar region (R ≲ R_CR; Debattista & Sellwood 2000). Some studies suggest that maximum disks are indeed more common in luminous spiral galaxies, such as these in our sample, compared to low-luminosity ones (Kranz et al. 2003).

If the baryonic mass in the studied galaxies is proportional to the dark matter mass, the negative relation in figure 9b would be observed for both M_* and the total galactic mass (baryonic |$+$| dark matter) M_tot. The rotation curves of the seven barred galaxies, which can be used as proxies of the total (dynamical) mass within a certain radius (assuming symmetrical distribution of mass) via |$M_\mathrm{tot}(<R) = v^2_\varphi R/G$|⁠, attain maximal measured values in a range between 160 km s⁻¹ ≲ v_ϕ ≲ 270 km s⁻¹, with no clear correlation with M_*. This is possibly because M_* was derived within a larger spatial extent than M_tot. On the other hand, we find a negative correlation between Ω_b and the dynamical mass within the bar radius, M_tot(R < a_b), as shown in figure 9c. To further investigate the extent of coupling between M_* and M_tot with Ω_b, it is important to study the shapes of rotation curves of these galaxies at larger radii.

4.3 The effects of galactic mass distribution and tidal interactions on resonance radii

In the simplistic discussion above, the barred galaxies are regarded as isolated and similar in terms of mass distribution and underlying rotation curves. However, recent numerical simulations give insight into possible complications in systems undergoing tidal interactions during fly-bys, as well as subtleties of gas kinematics and the location of corotation that arise from differences in rotation curves (mass models) of the host galaxies (Pettitt & Wadsley 2018).

4.3.1 Gas kinematics in simulated barred galaxies

In figure 10, we show an example of gas kinematics calculated for simulated galaxies in Pettitt and Wadsley (2018) for different rotation curve models (referred to as the Fall, Rise, and Mid models in figure 1 of their paper; the distribution of gas is shown in their figure 26) and interaction strength: S00 for isolated systems and S05 for weakly interacting ones. In panel (a), we show the azimuthally averaged radial velocity dispersion |$\sigma _{V_{\rm r}}$| as a function of galactocentric radius, the bar semimajor axis a_b, and the corotation radius R_CR at time t = 2.4 Gyr.⁶ Note that the bar length is determined by tracing the linearity of the bar figure such that a deviation of a Gaussian fit by 10° from the one in the inner 2 kpc is regarded as the beginning of a spiral arm/bar end at that radius (Pettitt & Wadsley 2018). This definition is different from the one adopted here for the observed bars (Herrera-Endoqui et al. 2015).

Fig. 10.

(a) Azimuthally averaged radial velocity dispersion |$\sigma _{V_{\rm r}}$| of gas in isolated bars (upper panels) and weakly interacting systems (lower panels) calculated for simulated galaxies in Pettitt and Wadsley (2018) at time t = 2.4 Gyr. The column names refer to the rotation curve models (Fall, Rise, Mid) as defined in their paper. The bar radius a_b and corotation radius R_CR are indicated by vertical blue and red dashed lines, respectively. Dotted lines show the radii of Lindblad resonances. (b) Azimuthally averaged radial velocity V_r.

The results show that, in general, the corotation radius is often found to be in the vicinity of a local minimum in |$\sigma _{V_{\rm r}}$| (and just outside the region of high |$\sigma _{V_{\rm r}}$|⁠), as discussed in sub-subsection 4.1.1, but may be as much as ∼30% offset, especially in galaxies with rising rotation curves. Note that the region of high |$\sigma _{V_{\rm r}}$| is comparable to Δ in subsection 3.4 and figure 7, where it was found that an increase in the ratio of noncircular to circular motions in SAB and SB systems, quantified by Δ, is located within R/a_b ≲ 1. Figure 10a suggests that the global maximum of |$\sigma _{V_{\rm r}}$| is generally found in the vicinity of the inner Lindblad resonance, at a radius of R/a_b ≈ 0.3, which is notably similar to the behavior of Δ in figure 7d. In some models (FallS00 and MidS00), another small jump is observed in the vicinity of the outer Lindblad resonance.

In figure 10b, we show azimuthally averaged radial velocity V_r for the same galaxy models as in panel (a). The gas flow is generally negative (inward) with V_r ≲ 0 within R < R_CR and V_r ≈ 0 beyond R > R_CR in all isolated galaxy types, in agreement with the assumptions made in sub-subsection 4.1.1. In fact, R_CR is found to be at least at the reversal radius and in some cases farther out (R_CR ≳ R_r). Although the offsets seem to be rather small (∼20%), especially in FallS00 and RiseS00 models, corotation does not always mark a clear reversal in V_r. Moreover, note that the situation may dramatically change in systems undergoing weak interactions. In all S05 plots in figure 10, the radius R_r seems to be uncorrelated with R_CR, though we still find that R_CR > R_r in all cases. The plots also show that V_r appears to attain its minimum in the vicinity of the inner Lindblad resonance in all models.

4.3.2 Comparison with observed molecular gas kinematics

The parameters of individual galaxies presented in section 3 and the Appendix can be compared with the simulations. The input information is the shape of the rotation curve and the presence or absence of a tidally interacting neighbor. In addition to relatively isolated galaxies (NGC 613, NGC 2903, NGC 5248, and NGC 7479), the sample presented here includes one member of a compact group (NGC 3627; Leo Triplet), as well as cluster members (NGC 4303 and NGC 4579; Virgo cluster); we also find a variety of rotation curve shapes, which are mainly Fall or Mid types, as judged from published H i data, such as THINGS (de Blok et al. 2008), that trace the dynamics to larger radial extent. These two models correspond to disk-dominated systems with |$\mathcal {R}<1.4$|⁠, unlike Rise which is dark matter dominated and generates slow rotators (⁠|$\mathcal {R}>1.4$|⁠) in the simulations.

Comparing v_r, derived from Fourier decomposition of CO velocity fields, with V_r in figure 10, we find that the models for isolated bars resemble the behavior of radial velocity in the galaxy sample, i.e., generally V_r ≈ 0 at R = R_CR. As an example, we compare the results of simulations with the SB system NGC 2903 discussed above, which has θ ≈ 0 (bar is aligned with the galactic major axis) so that we can compare v_r directly with V_r. The panel in figure 4c suggests a rising rotation curve; however, H i measurements beyond R ≈ 7 kpc clearly indicate a Fall type (de Blok et al. 2008). A comparison of the measured values with the simulation results for FallS00 reveals the following similar features: (1) the corotation radius is close to the bar radius, (2) this radius is in the vicinity of a steep rise in |$\sigma _{V_{\rm r}}$| and Δ, and (3) the radial velocity is v_r ≲ 0 within R ≲ R_CR. Similar behavior is observed in other isolated galaxies too, such as NGC 613 and NGC 7479.

Another example is NGC 3627 (M 66), a member of the Leo Triplet galaxy group that includes NGC 3623 (M 65) and NGC 3628. The rotation curve of NGC 3627 can be regarded as a Mid type based on H i kinematics (de Blok et al. 2008), and due to its interaction past (Zhang et al. 1993), we may consider both MidS00 and MidS05 panels of figure 10. Compared to figure 11, we recognize that R_r lies at a radius larger than a_b and that |$\sigma _{V_{\rm r}}$| and Δ are at minimal values. Note that NGC 3627 exhibits v_r > 0 at some radii, as evident from figure 7f, even though it also has a relatively small θ. This behavior is comparable to V_r in the MidS05 model.

Fig. 11.

Left panels: Circular velocity v_ϕ (blue circles) and angular velocity Ω (black squares). Right panels: Radial velocity v_r (red diamonds) and parameter Δ (green stars). Bar radius a_b and v_r-reversal radius are indicated by vertical dashed and dotted lines, respectively. The radial velocity v_r = s₁/sin |i| is not corrected for the near/far side orientation of the galactic disk. (Color online)

Similarly, figure 7f shows that NGC 4303 and NGC 4579 also exhibit some outward motion of molecular gas at R < R_r. This can be understood if the clouds are accelerated inside corotation due to positive torques exerted by the bar on the upstream side. Curiously, these two objects are members of the Virgo cluster, which makes them good candidates for subtle tidal interactions, as in all S05 models in simulations. For instance, NGC 4303 (M 61) may be a MidS05 type, judging from the large-scale rotation curve in Sofue et al. (1999), and its optical morphology (dust lanes in high-resolution images) resembles the distribution of gas in figure 26 (MidS05) of Pettitt and Wadsley (2018). Figure 10b shows that weak interactions may notably elevate the azimuthally averaged radial velocity V_r at some radii inside corotation.

4.4 Concluding remarks

The above comparisons lead us to two main conclusions of the discussion on bar pattern speed. (1) The observed properties of relatively isolated SAB and SB galaxies are generally consistent with the simulations of non-interacting galaxies, thereby providing a justification for the v_r-reversal method described in sub-subsection 4.1.1 that was motivated by the density wave theory and early numerical works. From its simplicity and good agreement with previous studies, we suggest that the method can be applied to modest-resolution (1 kpc) wide-field velocity field data that are often more easily acquired in large galaxy surveys (such as COMING) compared to high-resolution images. In order to apply the v_r-reversal method, it is essential to detect CO or another tracer of cold gas (e.g., H i) to radii outside the bar at resolution high enough to resolve the basic galactic structure. (2) The effects of rotation curve (mass distribution) differences and tidal interactions on the location of corotation radius, as shown by simulations, should be taken into account when methods based on velocity (phase) reversals are applied in such systems. Our work encourages new studies that can be conducted using facilities such as the Atacama Large Millimeter/submillimeter Array to further investigate molecular gas kinematics (e.g., radial velocity v_r) in a variety of morphological and tidal interaction stages and its relation with the bar pattern speed. Establishing a reliable method of determining Ω_b at 1 kpc resolution for a variety of galactic morphologies is especially important when such measurements are attempted for high-redshift objects to understand the evolution of barred galaxies in the early Universe.

5 Summary

We have presented a mode analysis of the 1 kpc-resolution ¹²CO (1–0) velocity fields of 20 nearby galaxies selected from the COMING legacy project of Nobeyama Radio Observatory. The main results are summarized below.

1. The velocity fields (moment 1 images) of 13 non-barred (SA) and 7 barred (SAB and SB) spiral galaxies were Fourier decomposed into circular (v_ϕ) and radial (v_r) velocity components by ellipse fitting using the program Kinemetry. The fitting results yielded new measurements of the kinematic position angle, disk inclination, and systemic velocity.

2. In barred galaxies, the ratio of noncircular to circular velocity components is of the order of ∼0.25 at inner radii in the bar region, in contrast to an approximately radius-independent value of ∼0.15 in non-barred (SA) spirals. On average, the ratio attains a maximum at R/a_b ≈ 0.3. The enhanced noncircular motions are found to be caused by substantial radial velocities, i.e., streaming motion in the bar region with amplitudes up to |v_r| ≈ 40 km s⁻¹ at 1 kpc resolution.

3. The radial velocity in SAB and SB galaxies generally decreases to v_r ≈ 0 at a reversal radius R_r, which is found to be |$\mathcal {R}$| = 0.8–1.6 the size of the bar in a sample of seven galaxies. Associating R_r with the corotation radius of the bar, we estimated the bar pattern speed of the sample galaxies. The results are consistent with previous studies and suggest that intermediate (SBb–SBc), luminous barred spiral galaxies host fast and slow rotator bars.

4. The bar pattern speed is found to decrease with a_b and total galactic stellar mass M_*. Although more data are needed to improve the significance of the analysis, the result is in agreement with theoretical predictions that bars evolve by growing and slowing down their rotation as a consequence of angular momentum transfer.

5. Recent numerical simulations of barred galaxies, that include the effects of different galactic mass distributions and tidal interactions, suggest that corotation radius R_CR is often located in the vicinity of the v_r-reversal radius R_r (where v_r ≈ 0) in relatively isolated galaxies, as assumed in the method to derive Ω_b. However, we also find that R_CR can be largely offset from R_r in galaxies undergoing tidal interactions, suggesting that the velocity reversal method of determining the bar pattern speed in such systems may suffer from large uncertainties.

Acknowledgments

The authors thank the referee for sending valuable comments and suggestions. We are grateful to the staff of Nobeyama Radio Observatory for providing generous help in our observations with the 45 m telescope. The Nobeyama 45 m radio telescope is operated by Nobeyama Radio Observatory, a branch of the National Astronomical Observatory of Japan. This research has made use of the NASA/IPAC Extragalactic Database, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

Appendix.

Circular and noncircular velocity components of individual galaxies

The velocities v_ϕ, Ω, v_r, and Δ derived by mode analysis for all galaxies in the sample (table 1) are plotted in figure 11, except for NGC 2903 and NGC 5055, which are shown in figure 4. For the definitions of the parameters, see section 3.

Footnotes

1

The non-axisymmetric potential of a bar has the ability to trap stellar orbits of different radii into a common precession rate (pattern speed), that otherwise tend to precess at different rates if the potential were axisymmetric (Lynden-Bell & Kalnajs 1972; Lynden-Bell 1979).

2

A ring is assumed to be circular but inclined with respect to the observer so that it appears as an ellipse.

3

〈http://leda.univ-lyon1.fr/〉.

4

The formula is obtained from the equation of an ellipse, x²/a² + y²/(qa)² = 1, expressed in polar form, where a = a_b, x = r_b cos θ, and y = r_b sin θ; see figure 1.

5

A commonly used definition of a weak bar is the condition |Φ_b/Φ₀| ≪ 1, where Φ₀ is the axisymmetric component and Φ_b is the non-axisymmetric component of the potential.

6

The azimuthally averaged radial velocity in simulated galaxies is written as V_r to distinguish it from the axisymmetric radial velocity derived from Fourier decomposition, which we write as v_r. The velocity dispersion is |$\sigma _{V_{\rm r}} = \sqrt{\langle V_{r,\varphi }^2\rangle -\langle V_{r,\varphi }\rangle ^2}$|⁠, where 〈〉 means averaging in azimuth ϕ.