PATTERN SPEEDS OF BARS AND SPIRAL ARMS FROM Hα VELOCITY FIELDS^*

When applying the Tremaine–Weinberg method, the slits must be placed along the line of nodes, which can be approximated by the kinematic and/or photometric major axes.

The orientation of the kinematic major axis derived from the quantification scheme described in Section 3 is the result of the second iteration in the procedure for deriving V_rot, i.e., after determining and fixing the position of the dynamical center (x₀, y₀) and V_sys. Similar to the standard tilted-ring method, the algorithm minimizes the expression χ² = V²_los − (c²₁ + s²₁), which can be seen ideally in cases when V_los is the result of a circular cosinusoidal term (c₁) and a radial sinusoidal term (s₁). In the presence of weak non-circular motions caused by weak perturbations or large beam smearing effects as in radio observations, this approximation is reasonable. However, for our data, where local velocities from strongly star-forming regions as well as significant flows along the bars in the objects could significantly skew the isovelocity contours toward the bar axis (e.g., Duval 1977; Huntley 1978; Buta 1987), we are bound to derive misplacements between the kinematic line of nodes position angle ϕ_k and the photometric counterpart ϕ_p. One step forward in treating this problem is the expansion of the minimization routine to the expression χ² = V²_los − Σ³_m=1(c²_m + s²_m). However, patchiness of an observed field, the presence of strong dust content internal to the observed galaxy, and asymmetric drift cause considerable computational complications which are beyond the scope of this paper (e.g., Weijmans et al. 2008).

We derive the outer disk line of nodes position angle (ϕ_p) from the near-infrared images presented in Section 5.1 by using the average position angle for the outer isophotes over a 2–4 kpc range in radius. We compare the values with those obtained from the RC3 catalogue as well as previous studies of our sample galaxies, and find that in most cases, the photometrically derived values are in fair agreement (see Table 3); however, for IC 342, NGC 5921, and NGC 6946, the ϕ_p values from the literature are based on comprehensive analysis; we use these angles for deriving the pattern speeds.

Furthermore, we compare the χ² values for the linear fits to the 〈V〉, 〈X〉 pairs, and as it is expected that the Tremaine–Weinberg method delivers the least scatter along the correct line of nodes, we use the χ² values to assess the ϕ_p we have decided to use. Figure 3 illustrates the reduced χ² values for a range of position angles, where the minimum χ² values correspond to position angles almost identical to or to within 10° from ϕ_p. This figure also shows that extracting (〈V〉, 〈X〉) pairs along offset position angles of ±10° still delivers reasonably linear fits to the 〈V〉, 〈X〉 diagrams such as those shown in Figure 1.

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We use these uncertainties to measure differences in the Ω_p by changing the orientation of the nominal slits. The 〈V〉/〈X〉/sin(i) values illustrated in Figure 4 show that for ϕ_p uncertainties of ±10°, we can bracket the Ω_p values to within the limits listed in Table 4. Allowing the slit widths to vary between 1 and 10 pixels implies additional differences in the derived Ω_p of the order of 10%.

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Previous analysis of the Ω_p uncertainties due to incorrect slit orientations has been carried out by Debattista (2003) and Rand & Wallin (2004), who found that ≈5° P.A. offset is sufficient to cause Ω_p values those differ by up to 30%. In Figure 5, illustrated as error bars, we show the difference in the measured Ω_p between an estimate with the correct line of nodes, and an estimate using an incorrect line of nodes, displaced by 10°, as a function of the number of slits used in these estimates. The figure shows that this difference grows quickly as the number of slits falls. When the number of slits has fallen to around 15, matching this condition, the differences have grown to 30%, and thus consistent with the findings of Rand & Wallin (2004). However, we note that in our derivations of the Ω_p we use over 100 slits per galaxy.

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The ideal form of the Tremaine–Weinberg method uses the luminosity-weighted velocities and the distances in the full radial range of the galaxy disk, i.e., the integrals range from −∞ to ∞. Observations, on the other hand, cover only a limited radial range, and ours reach typically the r₂₅ radius of the galaxy disk. It is thus important to see whether our observations are deep enough so that they do not yield incorrect Ω_p.

We note that the Tremaine–Weinberg method gives pattern speeds strictly when the limits of the integrals reach well beyond the bar. However, in this test, we restrict the radial range of integration in progressive steps, starting from an innermost radius of 10 pixels and increasing in steps of 10 pixels until the entire observed disk is covered (see Figure 6). Each derivation yields a different 〈V〉/〈X〉 value, treated as if it were a value for Ω_p (see Figure 7). As shown in the detailed analysis by Zimmer et al. (2004), these values decrease asymptotically as the disk coverage increases. In particular, the differences between the outermost values of these trials and the values obtained using the complete disks are much smaller than those between the values obtained limiting the field to a few tens of pixels and those found when using the full disk region. Furthermore, we confirm that by using all the pixels inside the slit (i.e., without the radial truncation as shown in Figure 6), the derived Ω_ps are much more stable. This is in support of the result by Zimmer et al. (2004) who found that most of the variation in the derived pattern speed occurs within an inner radius which they set to 100'' for their observations.

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A pattern speed derived over a radial range significantly shorter than the bar length will not give a meaningful value since the integrals do not converge, so we do not wish to justify here any variations (or indeed the absence of variations) calculated within these ranges. Moreover, as found by Shetty et al. (2007) for M 51, quasi-steady-state continuity is not satisfied by a single-valued pattern speed, so it is important to analyze the derived Ω_p variations to make sure they converge on a unique value, which can then be taken as the main value for a single and dominant m = 2 perturbation (Henry et al. 2003). In our work, we have used integrations over the full observed field for all galaxies to derive the pattern speeds presented in Table 4 and Figure 1.

In an ongoing study of the Tremaine–Weinberg method based on N-body + SPH numerical simulations carried out by Isabel Pérez (see, e.g. Beckman et al. 2008) we find that in the case of one constant pattern speed in a disk, scans of untruncated pseudo-slits in the inner regions as well as outer regions, all recover the input Ω_p of the simulations. When truncating the slits to cut the outer sections of the disk (compare with Figure 6), the Tremaine–Weinberg integrals applied to inner truncated slits result in different 〈V〉/〈X〉 values when compared with using the full simulated galaxy. This is a clear demonstration of the necessity of using full slit coverage and deep observations (even in the inner regions) due to the non-converging integrals. However, as expected, we have found that as soon as the radial disk section is comparable with or larger than the size of the bar, the Ω_p input into the simulations can be recovered. This feature is shared by real data as well as by numerical simulations.

Furthermore, we have applied the Tremaine–Weinberg method using the simulated velocity field weighted by the simulated density maps as well as the simulated star formation maps and found that although using the density maps provide more accurate results, the star formation maps also deliver comparable (to within the errors) pattern speeds, implying that the Hα velocities are reliable when applying the Tremaine–Weinberg method. For the galaxies presented here, Table 4 shows that the Hα and continuum deliver comparable Ω_p (note that the continuum maps used for this exercise also come from our data cubes). The exception is NGC 4519, where the continuum image assigns higher weights to the velocities inside the bar, and thereby introduces significant difference between the Ω_p derived using the continuum and Hα maps. The higher pattern speed of the bar using the continuum image delivers a higher total Ω_p.

In the presence of multiple structures with different pattern speeds, when deriving the 〈V〉/〈X〉 for an inner component, such as an inner bar inside a large bar, by applying the notional slits on the inner parts of a two-dimensional map, the majority of the pixels will belong to the disk region outside the bar. These pixels will therefore affect the derived pattern speed of the inner structure. This effect was tested by, e.g., Zimmer et al. (2004, their Figure 7) who found that once the slit coverage is comparable with the size of the main bar, the Tremaine–Weinberg method delivers reliable Ω_ps. Moreover, cutting the disk in radial bins seems to reveal the presence of multiple pattern speeds when the disk section is comparable to the structure with a correspondingly distinct pattern speed. In Figure 1, we illustrate the 〈V〉, 〈X〉 points from the inner regions of two galaxies where this effect can be seen. However, as the derivation of secondary pattern speeds is more complicated (see, e.g., Meidt et al. 2008), this test is to be used with caution in those cases.

Finally, we confirm that the Ω_p differences using the continuum images instead of the Hα surface brightness maps do not significantly change the location of the resonances throughout our analysis.

Empirical studies by Erwin (2005) and simulations by Martinez-Valpuesta et al. (2006) and Gadotti et al. (2007) have shown that in systems with single bars, careful examination of ellipticity ( = 1 − b/a) profiles can deliver a reliable m = 2 bar radius. For our sample galaxies, we analyze near-infrared K_s-band images from the Two Micron All Sky Survey (2MASS) catalogue (Jarrett et al. 2000) and deep 3.6 μm images from the Spitzer archives (PI: Fazio, PI: Kennicutt, PI: Kenney) in order to derive the bar length. For the galaxies where Spitzer data were not available, we retrieved deep H-band images from the 2.2 m Calar Alto Telescope (NGC 4519; PI: Boselli) and deep K-band images from the 4.2 m William Herschel Telescope (NGC 5921 and NGC 5964; Knapen et al. 2003).

We detect the presence of a bar within a galaxy disk from the ellipticity and line of nodes position angle ϕ profiles traced by the galaxy isophotes (Jedrzejewski 1987). The transition from the (round) bulge-dominated center to the disk is typically characterized by an increase in ellipticity. Following Martinez-Valpuesta et al. (2006), we define the semimajor axis length of the bar (r(bar)) as the radius where the ellipticity value has fallen by 15% (empirically derived from the numerical simulations by Martinez-Valpuesta et al. 2006) from the radially outermost local maximum accompanied by no significant change in ϕ (see Figure 8). It is important that no position angle variation is detected around this radius, as twists could imply erroneous ellipticity diagnostics.

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We compare the r(CR) derived from kinematic analysis with the bar radius derived from the images (see Table 5), and find that the ratio r(CR)/r(bar) agrees with the predictions from numerical simulations (Athanassoula et al. 1990; O'Neill & Dubinski 2003). This result further suggests that peak ellipticities could underestimate the radial extent of the bar, though not by a large factor, since the ellipticity profiles shown in Figure 8 illustrate that the radius where the ellipticity value has fallen by 15% is usually not located far from the radius of peak ellipticity. Furthermore, for almost all galaxies, we derived comparable ellipticity profiles using the 2MASS K_s-band images. However, these images are not deep enough to deliver good-quality ellipticity profiles at large radii.

Table 5. Corotation Radius r(CR) and Bar Lengths Derived form Ellipse Fitting (r(bar)), and Fourier Analysis of the Near-infrared Images with Uncertainties Calculated by Allowing for the Ω_p to Vary within the Limits Listed in Table 4

Object	r(CR) ('')	r(CR) (kpc)	r(bar) (kpc)	r(CR)/r(bar)	r(bar)F (kpc)	r(CR)/r(bar)F
IC 342	344+26−79	6.5+0.5−1.5	5.8	1.1+0.1−0.3	7.2	0.9+0.1−0.2
NGC 2403	390+?−32	6.2+?−0.5	7.8	0.8+?−0.1	6.6	0.9+?−0.1
NGC 4294	29+4−6	1.9+0.3−0.4	1.7	1.1+0.2−0.2	2.5	0.8+0.1−0.2
NGC 4519	...	...	4.7	...	4.7	...
NGC 5371	...	...	13.8	...	13.0	...
NGC 5921	65+4−8	8.0+0.5−1.0	6.3	1.3+0.1−0.2	11.0	0.7+0.1−0.1
NGC 5964	55+38−13	7.0+4.5−1.5	7.6	0.9+0.6−0.2	4.3	1.6+1.0−0.3
NGC 6946	254+?−63	8.0+?−2.0	7.6	1.1+?−0.3	7.0	1.1+?−0.3
NGC 7479	94+6−25	13.0+1.0−4.0	11.8	1.1+0.1−0.3	10.0	1.3+0.1−0.4
NGC 7741	109+?−17	5.5+?−1.0	5.6	1.0+?−0.2	7.5	0.7+?−0.2

Notes. In NGC 4519, r(CR) falls outside our observed field, and for NGC 5371, we have measured the r(CR) for the spiral arms, hence no ratios calculated.

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As an alternative to analyzing the ellipticity profiles, the isophotal shapes can be described by means of the Fourier decomposition method (Elmegreen & Elmegreen 1985b; Buta 1986a, 1986b; Aguerri et al. 1998). Certain preference can be attributed to this method as it avoids confusion caused by the morphological changes that accompany the onset of spiral structure in the disk, which could artificially lengthen the bar. Simulations predict that Fourier decomposition of the azimuthal luminosity profile is dominated by the (m = 2) at the bar and falls sharply when the bar ends (Ohta et al. 1990); however, in reality, matters are more complicated, and a combination of the Fourier components or the phase shift of the second component can be used to derive the length of a bar (Athanassoula & Misiriotis 2002; Aguerri et al. 2000, 2003; Zhang & Buta 2007).

We apply the Fourier decomposition to the deprojected Spitzer, William Herschel, and Calar Alto near-infrared images described in Section 5.1, and present the second and forth components (I₂, I₄) in Figure 8 (see Elmegreen & Elmegreen 1985b, for a detailed presentation of the method). In this paper, since we are mainly interested in the lengths of the bars in our sample, we do not analyze the Fourier decomposition extensively, but only use it to ensure that these profiles indicate the presence of a bar in agreement with those derived from the ellipse fitting analysis. We use a combination of the following criteria to determine the bar lengths (r(bar)_F), and where these three criteria agree, we can specify the end of the bar quite well.

• 1.  
  The bar ends after the m = 2 component has been dominant, and starts decreasing accompanied by an increase of the odd components. This indicates that the twofold symmetry of the bar component has decreased.
• 2.  
  The contrast parameter defined by Ohta et al. (1990), I_b/I_ib = (1 + I₂ + I₄ + I₆) /(1 − I₂ + I₄ − I₆) is a measure of the intensity of the bar divided by the intensity of the interbar. As these authors have demonstrated, the contrast parameter is smaller in the region of galaxies where the bulge component dominates, then increases and reaches its peak at the middle of the bar, and falls at the end of the bar.
• 3.  
  The phase shift of the second Fourier component (m₂) changes sign (Zhang & Buta 2007).

We note in Figure 8 that not all these three criteria are fulfilled in our profiles, most likely due to the presence of strong star formation knots, foreground stars, and dust lanes. However, we note that in some cases, in the proximity of r(bar), the first two criteria could hold, and in all cases, the phase of the second Fourier component crosses the zero line. We thus define the phase-shift crossing radius as the r(bar)_F. Table 5 shows that this method yields results similar to the ellipticity profile analysis outlined in Section 5.1, and overall we derive comparable r(CR)/r(bar) ratios using either of the photometric analysis methods. One prominent exception is the galaxy, NGC 4294, where the Fourier decomposition method unveils an extended 4–6 kpc oval structure which is not seen in ellipticity profile analysis or directly in the images. This oval structure could cause the severe underestimation of our r(CR)/r(bar)_F for this galaxy.

Numerical simulations of barred galaxies predict the slowdown of bars due to the exchange of angular momentum between the dark matter halo and the galactic disk (e.g., Debattista & Sellwood 1998; Athanassoula 2003; Martinez-Valpuesta et al. 2006; Sellwood & Debattista 2006; Weinberg & Katz 2007). Bars in the fast rotator regime are expected to have r(CR)/r(bar) between 1.0 and 1.4, except for simulations of Valenzuela & Klypin (2003) and Tiret & Combes (2007, 2008), which obtained slow rotators with r(CR)/r(bar)>1.4. This ratio strongly relies on the measurement of the bar radius, which is far from trivial to derive homogeneously from observations; however, the consistency check presented in Section 5 demonstrates that our morphologically derived r(CR)/r(bar) ratios are consistent with those predicted by numerical simulations. The average value for our measurements is r(CR)/r(bar) = 1.1. As our main goal is to derive the Ω_p, the r(CR)/r(bar) can serve as a useful inspection since its implication is that the bars in our sample late-type spiral galaxies are fast bars, as predicted by models (e.g., Laine et al. 1998; Debattista & Sellwood 2000; Rautiainen et al. 2008).

The galactocentric σ profiles, illustrated in Figure 1, show that for most galaxies σ is almost constant throughout the observed field with some general hint of decline toward the 25th magnitude radius. NGC 5371 and NGC 7479 both show smaller Hα σ values in the central ≈10'' with the outer parts following the same trend as the rest of the sample. NGC 5921 has a deficit of ionized gas in the bar region, outside which it displays a very flat σ profile. Consistent with the statistical results of Erwin & Sparke (2002) for early-type bars and assuming that the epicyclic approximation can be used here, we find that three of our 10 late-type spirals (NGC 4519, NGC 6946, NGC 7741) show evidence for an inner Lindblad resonance (ILR; see Section 6.2 and Figure 1). In all three cases, the circular component of the observed V_los (see Section 3) shows a clear presence of a more rapidly rotating component within the ILR. Inside the ILR, the σ values are generally higher than outside this region, with a particularly steep rise toward the nucleus, and as displayed in the rotation curves, the higher σ values appear to accompany more rapid rotation. This might be linked to the scenario where pseudo-bulges are caused by bar dissolution (see the detailed review by Kormendy & Kennicutt 2004).

The results for our sample compare well with two-dimensional kinematic maps from Falcón-Barroso et al. (2006), Peletier et al. (2007), and Böker et al. (2008). To first order, our spatial resolution allows us to distinguish between local effects such as winds from star-forming regions and supernovae, and global secular evolution effects. We find that the σ values are higher at the location of the star-forming H ii regions (cf. Allard et al. 2006), as compared with regions where no strong star formation occurs. This result is similar to that which we found in Paper I, i.e., that locally outflows from H ii regions lead to a higher measured velocity dispersion close to the regions, while globally there is very little radial dependence of sigma, implying large-scale uniformity throughout the disk in the summed inputs of the energy sources to the gas.

IC 342 has angular size of more than 20'' and has a weak primary bar with r(CR) ≈ 47, in the approximately north–south direction (Crosthwaite 2002). IC 342 contains extremely strong molecular gas in the innermost parts of the bar, where the presence of an ILR (≈2'') has been suggested (Sheth et al. 2002; Turner & Hurt 1992; Böker et al. 1997). At this radius a star-forming ring-like structure has been observed to include a ≈15 Myr old supergiant-dominated central stellar cluster (Böker et al. 1997). The near-infrared spectra show no evidence for multiple stellar content within the bar, in support of this system being young. Although the exact molecular content in the bar has not been established, the CO observations by Turner & Hurt (1992) and Sakamoto et al. (1999) suggest gas inflow due to the bar feeding the circumnuclear starburst. Inside the ILR ring, a deficit of molecular gas content has been observed, and our observations show a very weak Hα signal. We find an overall patchy ionized gas distribution with a steeply rising rotation curve, fully consistent with both CO and H i rotation curves which stay around 200 kms⁻¹ out to 15' (Rogstad et al. 1973; Young & Scoville 1982; Sofue 1996; Crosthwaite et al. 2000).

The bar Ω_p has been estimated to 40 kms⁻¹kpc⁻¹ by Turner & Hurt (1992). Radial streaming motions above 5 kms⁻¹ along one arm were detected by Newton (1980) who also found r(CR) between 16 and 21 kpc, which they found consistent with a spiral density wave with Ω_p = 10 kms⁻¹kpc⁻¹. Crosthwaite et al. (2000) derived the bar Ω_p = 21 kms⁻¹kpc⁻¹ based on morphology of the spiral structure and r(CR) at around 400'' with no sign of an ILR. Applying the Tremaine–Weinberg method on the Hα emitting ionized gas, we derive Ω_p = 31⁺⁵₋₅ kms⁻¹kpc⁻¹ and r(CR) ≈ 6.5 kpc. We cannot verify the presence of the location of the ILR due to the large spatial bins which we use in order to increase the signal in the central regions (see Figure 1). Finally, we confirm that the GHαFaS observations deliver a 〈V〉/〈X〉 value comparable to that when using the inner ≈2' section of the FaNTOmM data.

NGC 2403 is a high surface brightness member of the M 81 group with flocculent arms and bright H ii regions and a total star formation rate ≈0.1 M_☉yr⁻¹ (Kennicutt 1989). In the inner kpc, this galaxy contains about half as much molecular gas as its atomic hydrogen content (e.g., Thornley & Wilson 1995), suggesting that the formation of molecular hydrogen is constrained by the low surface density of the gas (Helfer et al. 2003). The central gas surface densities are below the threshold star formation density of Kennicutt (1989). Fraternali et al. (2001) have detected anomalous neutral hydrogen extending beyond the disk plane of the galaxy interpreted as galactic fountains powered by stellar winds from massive stars and supernova explosions, which in turn could be triggered by the bar. Confirmed by a large sample of H ii regions, there is a gradient in O/H across NGC 2403 with a slope of 0.1 dex kpc⁻¹ (Garnett et al. 1997), supporting the study by Beckman et al. (1987) who found that non-circular motions are not very efficient in mixing the ISM in this galaxy. Although Schoenmakers et al. (1997) found, using H i kinematics, that NGC 2403 is axisymmetric to a high degree, we find that the ellipse fits and Fourier analysis of the near-infrared images reveal the presence of a ≈6 kpc bar (see Figure 8).

Our observed Hα rotation curve shows a moderate increase to ≈60 kms⁻¹ in the central kpc followed by almost a solid-body-type increase out to around 10 kpc radius. The curve is consistent with the H i curve presented in Shostak (1973); however, due to strong feedback from the numerous H ii regions in the central ≈200 pc radius (confirmed by the large σ values), we cannot constrain the rotation curve within this region. Shostak (1973) derived the spiral pattern speed 10.5 kms⁻¹kpc⁻¹, whereas, using the Tremaine–Weinberg method, we derive the bar pattern speed Ω_p = 22⁺⁶₋₁ kms⁻¹kpc⁻¹, and r(CR) at 6 kpc, with no sign of Lindblad resonances (Figure 1).

NGC 4294 appears to interact with NGC 4299 with a projected separation of 27 kpc. Both galaxies are similar in morphology, mass, luminosity, gas content, and both exhibit a high global star formation rate (e.g., Koopmann & Kenney 2004). Moreover, Chung et al. (2007) found that due to its position in the Virgo Cluster (0.7 Mpc from M 87) interaction with the intracluster medium could be plausible. NGC 4294 hosts many H ii regions, a quiescent nucleus, and two rather faint flocculent spiral arms. The outer regions of the spiral structure are consistent with enhanced star formation at the radius at which the gas surface density in the disk becomes supercritical for star formation (Thornley & Wilson 1995; Thornley 1996).

The observed Hα velocities show streaming motions along the spiral arms (Chemin et al. 2006, and this work). Although the receding part of the disk appears slightly perturbed, the overall appearance of the velocity field is quite regular. The Hα rotation curve is consistent with that presented by Rubin et al. (1999). Using the Tremaine–Weinberg method, we derive Ω_p = 43⁺³₋₁₂ kms⁻¹kpc⁻¹, with r(CR) ≈ 1.9 kpc in agreement with the location where the spiral arms emerge from the ends of the bar. Within the r(CR), the velocity dispersion follows a plateau, whereas outside this radius it decreases steadily.

NGC 4519 hosts many H ii regions in the bar as well as the two partly broken spiral arms. The central regions exhibit significant molecular gas content (Böker et al. 2002), although the star formation efficiency is constant in the inner two effective radii (Rownd & Young 1999). Our velocity field displays an "S"-shaped zero-velocity curve over the central 4 kpc radius region of the galaxy. The rotation curve rises steeply in the central few arcseconds, which indicates a rapidly rotating component within ≈30'', and it continues to increase to 130 kms⁻¹ at 6 kpc radius. This is consistent with the optical rotation curve from Rubin et al. (1999) and the H i position–velocity diagram of Helou et al. (1984).

We derive Ω_p = 20⁺⁶₋₈ kms⁻¹kpc⁻¹, which we assign to outer spiral arms. We use this Ω_p to locate an outer ILR radius at about 1.5 kpc (20'') and an inner ILR radius at around 200 pc. The outer ILR radius is, to within the errors, consistent with the Hα disk scale length derived by Koopmann et al. (2006).

Similar to Fathi et al. (2007b), we apply the Tremaine–Weinberg method on the pixels interior to the ILR, and find that the inner bar is decoupled from the outer oval, as its pattern speed is Ω_p = 45 kms⁻¹kpc⁻¹. This Ω_p yields that the corotation of the inner bar is located at the ILR of the outer oval. The method we have applied here enables us to determine the secondary Ω_p only with an uncertainty of 50%, since to zeroth order, the Tremaine–Weinberg method applied to the central region has to assume that the outer bars or spiral arms are stationary. This is not a realistic scenario, so we use this higher pattern speed for the inner bar only as an indication that we can detect the phenomenon, and that the pattern speed is higher than that of the outer bar.

NGC 5371 is a grand design, possibly post-starburst, spiral galaxy with a LINER nucleus (Rush et al. 1993; Koornneef 1993; Elfhag et al. 1996). It hosts a prominent 2 kpc stellar bar ≈45° from the orientation of the outer disk major axis with no significant change when comparing B- and H-band images (Eskridge et al. 2002), and displays a star formation rate of 0.9 M_☉yr⁻¹ (Gonzalez Delgado et al. 1997). The rotation curve was measured by Zasov & Sil'chenko (1987) from which they derived an exponential main disk scale length of ≈8.7 kpc, and a massive bulge (M_bulge ⩾ M_disk). Although the outer parts of their rotation curve are not well determined, the inner regions agree well with our Hα rotation curve.

Our observations display a deficiency of Hα emission across the bar, in agreement with the Hα images from Gonzalez Delgado & Perez (1997). The velocity field displays clear disk-like rotation, with various "wiggles" in the zero-velocity curve, often interpreted as streaming motions along spiral arms (Emsellem et al. 2006). The rotation curve rises steadily in the central 30'', and reaches 250 kms⁻¹ at 15 kpc radius. We derive Ω_p = 14⁺⁵₋₁ kms⁻¹kpc⁻¹ and the r(CR) ≈ 17 kpc, i.e., 93''. The deficiency of the Hα emission from the stellar bar, combined with its non-optimal orientation compared with the outer disk major axis, indicates that we are probing the Ω_p of the spiral arms, and not the well-known 2 kpc bar.

We note that by placing corotation at 17 kpc, we bring the inner 4:1 resonance close to the end of the symmetric part of the spiral structure. At this region, in agreement with the predictions by Patsis et al. (1997), a bifurcation of the northern arm is seen in Figure 1. Furthermore, the association of the end of the symmetric part of the spirals with the inner 4:1 resonance is supported by the radial variation of the ratio of the amplitudes of the m = 4 to the m = 2 components (Grosbol 1985; Patsis 1991).

NGC 5921 has a very poor molecular and neutral gas content (Verter 1985) and relatively poor ionized gas in a ≈8–12 kpc prominent stellar bar which is enclosed by a sharp inner ring-like structure. The bar contains two straight dust lanes, and the spiral arms host many H ii regions (Gonzalez Delgado & Perez 1997; Aguerri et al. 1998; Rautiainen et al. 2005) and form the ring at the bar ends, just before they become very open, broad, and diffuse. At the north end of the bar, the arm becomes brighter as it leaves the ring (Eskridge et al. 2002). Martin & Friedli (1997) found that the stellar bar has an axis ratio of 0.5 and is oriented 18° in the north–east direction. These authors also derived an asymmetric star formation efficiency of <0.2 M_☉yr⁻¹ in the bar, and seven times higher value in the circumnuclear region (Gonzalez Delgado et al. 1997).

Like in NGC 5371, the Hα emission is so weak in the bar of this galaxy that we have to make use of large spatial bins to measure any kinematic information. The rotation curve cannot be constrained within the bar. The Tremaine–Weinberg method thus delivers the pattern speed of the prominent spiral structure (Ω_p = 13⁺²₋₂ kms⁻¹kpc⁻¹), with the spiral pattern r(CR) ≈ 8 kpc (or 65'', in good agreement with the r(CR) = 71'' of Rautiainen et al. 2008), which is just outside the 58'' radius of the strong stellar bar (Buta & Zhang 2009). This suggests that the bar shares the same pattern speed as the spiral arms (e.g., Tagger et al. 1987).

NGC 5964 hosts a very prominent and elongated bar surrounded by diffuse, extended, and fragmented spiral arms with no regular pattern (Elmegreen & Elmegreen 1987). This flocculent galaxy is rich in neutral Hydrogen content (M_HI/M_Tot = 82%) and hosts a patchy circumnuclear structure (Giovanelli & Haynes 1983; Böker et al. 2002). The Hα rotation curve displays almost a solid-body rotation in the inner 10 kpc. Our analysis shows that Ω_p = 25⁺¹₋₅ kms⁻¹kpc⁻¹ is well outlined in Figure 1 with the location of the r(CR) at 6.5 kpc.

NGC 5964 displays the largest misalignment (36°) between the ϕ_p and ϕ_k, which could be caused by the projection effect since this galaxy is near face-on. Such a misplacement can also be seen as an indication of strong streaming along the spiral arms and prominent bar. In Figure 1, we show that the r(CR) agrees with the radius where the patchy spiral structure starts, indicating that the bar and spiral arms rotate at the same rate (see Tagger et al. 1987).

Theoretical work such as that by Elmegreen & Thomasson (1993) have shown that flocculent spirals can form via swing amplification of star formation patches, and thus should have no pattern speed. Although their proposed mechanism explains the patchiness of these galaxies, it does not exclude the presence of density waves in the galaxies where swing amplification builds the patchy spiral structure. Support for such a scenario comes from simulations by Wada & Norman (2001) as well as observational studies by Thornley & Mundy (1997), Grosbol & Patsis (1998), Elmegreen et al. (2003). Notably, in NGC 5055, Thornley & Mundy (1997) could outline the spiral arms by a spline fit to the K-band image and show that the non-circular motions in their residual H i velocity field follow these arms. These authors combined the photometry with kinematics to derive the pattern speed for the underlying density wave at 35–40 kms⁻¹kpc⁻¹, and conclude that in some flocculent galaxies the underlying density wave is difficult to observe due to their overall lower gas surface density (e.g., that in NGC 5055 is a factor of 6 lower than the gas density in M 51). To summarize, our results for NGC 5964 imply an underlying density wave with Ω_p = 25⁺¹₋₅ kms⁻¹kpc⁻¹.

NGC 6946 is a grand-design-barred spiral galaxy with three main gravitational distortions: a large oval with radius R ≈ 45, a R ≈ 1' primary stellar bar with ellipticity 0.15, and a R ≈ 8'' nuclear bar with ellipticity 0.4, which is almost perpendicular to the main bar. We have presented a detailed analysis of the resonance structure in this galaxy in Fathi et al. (2007b), and confirm that the maps presented in this paper are fully consistent with our previous work.

NGC 7479 is an extensively studied starburst galaxy (e.g., Hawarden et al. 1986; Aguerri et al. 2000; Laine 2001) with a massive, 2.2 × 10¹⁰ M_☉ stellar bar with an axis ratio of 0.25–0.4 (Burbidge et al. 1960; Martin 1995). Although there is no evidence of recent interaction, Laine & Heller (1999) have found many kinematic and morphological features consistent with a minor merger event for this galaxy, proposing the hypothesis that this galaxy is on its way to evolve toward an earlier Hubble type. Several studies have found the r(CR) to be between 45 and 106'' (Elmegreen & Elmegreen 1985a; Quillen et al. 1995; Beckman & Cepa 1990) and Ω_p to be between 27 and 100 kms⁻¹kpc⁻¹ (Laine et al. 1998; del Rio & Cepa 1998). The mean star formation efficiency in the bar, although asymmetric, is 0.4 M_☉yr⁻¹ and increases to 0.63 M_☉yr⁻¹ in the circumnuclear regions (Martin & Friedli 1997), indicating inflow along the bar.

Our Hα rotation curve is consistent with that derived by Garrido et al. (2005) with a prominent and rapidly rotating component in the central 30''. This curve, as well as the angular frequency curve, cannot be determined in the central kpc region due to the weak Hα emission, and hence we cannot confirm the presence of the ILR reported by Laine et al. (1998). We find Ω_p = 18⁺³₋₃ kms⁻¹kpc⁻¹and r(CR) ≈ 13 kpc. Our r(CR) is almost twice the 55'' found by Puerari & Dottori (1997), the r(CR) suggested by numerical simulations of Laine et al. (1998), and the 58'' found from potential-density phase-shift analysis carried out by Buta & Zhang (2009). Using the r(CR) found by these authors, our angular frequency curve implies Ω_p ≈ 23 km s⁻¹ kpc⁻¹, i.e., slightly above the value presented here. This would also make the bar with r(CR)/r(bar)_F ≈ 0.9, to within the errors, comparable with the value we present in Table 5.

NGC 7741 has a prominent bar with an axis ratio of 0.36, two short and flocculent spiral arms including a significant amount of diffuse ionized gas and many H ii regions (e.g., Duval et al. 1991; Block et al. 2004; Laurikainen et al. 2004). This bulge-less galaxy has a mean star formation efficiency of 0.1 M_☉yr⁻¹ (Martin & Friedli 1997), with very weak CO content in the bar (Braine et al. 1993). Duval et al. (1991) found strong ionized gas flows in the bar region and a ratio of 30% between the bar mass and that of the inner disk inside the bar radius.

Our rotation curve agrees with the low-resolution curve of Duval & Monnet (1985) and Garrido et al. (2002). The Tremaine–Weinberg method results in Ω_p = 19⁺⁸₋₆ kms⁻¹kpc⁻¹ for the main bar with r(CR) ≈ 6.5 kpc, cf. Ω_p = 31 kms⁻¹kpc⁻¹ from Duval & Monnet (1985). We find the location of an outer ILR at ≈1.3 kpc, inside which the rotation curve suggests the presence of a dynamically cold rapidly rotating component. We find also an inner ILR at ≈300 pc, and no hint of a higher pattern speed inside the outer or the inner ILR. In NGC 7741, like in NGC 7479, the ILR radii given here are based on the assumption that the epicyclic approximation can be applied, and it should be noted that in strong bars, such as in NGC 7741, where the epicyclic approximation breaks down, the actual resonances may be in different locations (e.g., Regan & Teuben 2004).