TESTING THEORIES IN BARRED-SPIRAL GALAXIES

Spiral arms in barred galaxies have been explained in the past as density waves (e.g., Korchagin & Marochnik 1975) or spiral waves that result from the crowding of gas orbits (Huntley et al. 1978). Kaufmann & Contopoulos (1996) invoked for the first time the need for chaotic orbits as building blocks of spirals near the end of the bar. In the Kaufmann & Contopoulos (1996) models, regular orbits dominate the main structure of the bar and the outermost portions of spiral arms. The inner portions of spiral arms are supported by chaotic orbits. Recently it has been proposed that chaotic motion can support the spirals in barred-spiral systems. The new theory proposes that unstable Lagrangian points (L₁ or L₂) near the end of the bar are the sites where chaotic orbits are guided by invariant "manifolds" and are the origin of spirals and (inner and outer) rings (Voglis & Stavropoulos 2006; Patsis 2006; Romero-Gómez et al. 2006, 2007; Voglis et al. 2006a, 2006b; Tsoutsis et al. 2008, 2009; Athanassoula et al. 2009a, 2009b, 2010; Harsoula & Kalapotharakos 2009; Contopoulos & Harsoula 2011). In this scenario the spiral dynamics are coupled to the bar, and are driven by the manifolds. This approach has been studied by two different groups of people.

One of those groups (Romero-Gómez et al. 2006, 2007; Athanassoula et al. 2009a, 2009b, 2010) considers a continuous flow of orbits along the manifolds emanating from L₁ or L₂. When spirals form, stars move away from the corotation in a radial movement (Athanassoula et al. 2010), and material is needed to replenish the manifolds. One prediction of this "manifold theory" (or "Lyapunov tube model"), not accounted for in the density wave scenario, is that stronger bars should have more open spirals compared to weaker bars, i.e., the spiral arms pitch angle¹ should increase with bar strength (Athanassoula et al. 2009a). This kind of correlation was previously predicted by Schwarz (1984), though for gas arms driven by a bar perturbation.

Another view of the "invariant manifold theory" (Voglis et al. 2006a, 2006b; Tsoutsis et al. 2008, 2009) considers the locus of all points with initial conditions at the unstable manifolds that reach a local apocentric (or pericentric, see Harsoula et al. 2011) passage, i.e., the apsidal sections of the manifolds. In this scenario, there is no need for the replenishment of material to obtain long-lived spirals (see, e.g., Efthymiopoulos 2010). Both views of the "invariant manifold theory" predict a trailing spiral pattern for strong perturbations and similar pattern speeds for the bar and spiral, i.e., Ω^bar_p = Ω_p^spiral. However, in the view of Voglis et al. (2006a, 2006b) and Tsoutsis et al. (2008, 2009), the "azimuthal tilt" of the spiral response (Tsoutsis et al. 2009), i.e., the difference between the bar's major axis and the Lagrangian points L₁ or L₂ at the moment of the onset of the spiral, determines how open the spiral arms will be. In this case, the pitch angles are smaller than the ones predicted by Athanassoula et al. (2009a) and become even smaller for pure bar models when the "azimuthal tilt" is not taken into account (C. Efthymiopoulos 2011, private communication).

Patsis et al. (2010) describe one more dynamical mechanism that supports spiral arms through stars in chaotic motion. They propose this mechanism by describing the spiral arms of the barred-spiral NGC 1300. Together with the bar, these spiral arms are inside the corotation and are not related to the presence of unstable Lagrangian points and the associated families of periodic orbits. This alternative mechanism may be linked to some range of pitch angles of spiral arms encountered in barred-spiral systems.

Do manifolds drive spiral dynamics in barred galaxies? Or are the dynamics driven by the bar? The bar may drive the dynamics, affecting the spiral amplitude locally, as reported by Salo et al. (2010; see also Block et al. 2004) and previously discarded (or weakly corroborated) by other authors comparing bar strength to spiral arm strength (Buta et al. 2009; Durbala et al. 2009; Seigar & James 1998). Bars driving the dynamics would imply an accordance with (linear) density wave theory. These spirals may be a continuation of the bar mode or an independent mode coupled to the bar (e.g., Tagger et al. 1987; Masset & Tagger 1997). In the "Lyapunov tube model," the strength of the bar affects the pitch angle of the spirals, but not its amplitude. The amplitude of the spirals depends on how much material is trapped by the manifolds, although the amplitude of the spirals should in general decrease outward (Athanassoula et al. 2010). Grosbøl et al. (2004) investigated the relation between the amplitude of the spirals with the pitch angle in non-barred and weakly barred galaxies.

One prediction of the density wave theory (Hozumi 2003; see Section 6.1) entails that different pitch angles are expected for spirals when observed in different bands (e.g., optical versus near-infrared (NIR)). According to Athanassoula et al. (2010), the "invariant manifold theory" predicts that stars of different ages will be guided by the same manifold, and no difference between the winding of the spirals is expected.

In this paper, we will investigate whether the predictions of pitch angles are observed for real galaxies or not. Two methods were applied for this purpose: the "slope method" (Section 4.1), which is especially good for determining how long the logarithmic shape is maintained for spiral arms, and the "Fourier method" (Section 4.2), which was used to determine the "dominant" pitch angle inside a given annulus for each object.

The initial galaxy sample consists of 104 galaxies classified as Fourier bars in Laurikainen et al. (2004). The data were acquired from the Ohio State University Bright Galaxy Survey (OSUBGS; Eskridge et al. 2002). From this initial sample, it was found that only 84 objects present spiral-like features. Nevertheless, not all the objects are suitable for this kind of study due to asymmetries, e.g., short, faint, or ragged spiral arms, or prominent rings. The following criteria were established in order to obtain a sample, including objects with a morphology candidate to be explained by "chaotic" spirals.

• 1.  
  The spiral arms must remain logarithmic, i.e., with a constant pitch angle (i), at least for 50° in the azimuthal range, α.² This was verified with the "slope method" (see Section 4.1). The lower limit value of α was chosen according to Figure 4 in Athanassoula et al. (2009a), where the manifold loci remain logarithmic (for the adopted model parameters) and maintain a "nearly" logarithmic geometry up to ∼100°. We consider that the manifold loci and the density maximum along the spirals coincide. According to Patsis (2006), spirals supported by chaotic particles may extend up to π/2 radians. Variations of α toward larger angles will be discussed in Section 6.
• 2.  
  The object presents two spiral arms visually connected to the bar.
• 3.  
  No prominent inner rings (near the bar's end) are present.³ Ring structures are connected to the bar on both sides. The pitch angle definition as applied in this investigation only refers to spiral arms. A dependence of the inner ring shape on bar strength has been investigated by Grouchy et al. (2010).

After applying these selection criteria, the final sample consists of 27 barred spirals (see, e.g., Table 1).

In order to use the bar strength values of Laurikainen et al. (2004, see Section 3), we adopt the same deprojection parameters of those authors, i.e., the same values for position angle (ϕ) and minor-to-major axial ratios (q = b/a).⁴ To determine these parameters, Laurikainen et al. (2004) fit ellipses to the outer isophotes on the disk. They were based on the OSUBGS B-band images that are deeper than H-band images.

To test the Athanassoula et al. (2009a, 2010) predictions regarding spiral arms pitch angles, we use the NIR H-band since we are interested in "long"-lived structures rather than young stars, H ii regions, or gas that would be present in optical data.

The predicted trend in Athanassoula et al.'s (2009a) "manifold models" requires the strength of the bar at the radius of the Lagrangian points L₁ or L₂. It should be mentioned that for these models the self-gravity of the spirals was not taken into account. On the other hand, the addition of the spiral potential in Tsoutsis et al.'s (2009) models shifts the positions of the Lagrangian points L₁ or L₂ both in the radial and azimuthal directions.

The strength of the bar can be obtained from the Laurikainen et al. (2004) radial profiles of the perturbation strength. Laurikainen et al. (2004) used the gravitational torque method (Combes & Sanders 1981; Buta & Block 2001; Block et al. 2002), taking care of the artificial bulge stretch (see also Speltincx et al. 2008). The perturbation strength is calculated as

$$\begin{equation} Q_{t}(r) = \frac{ ({\partial {\Phi (r,\theta)}}/{ \partial {\theta }})_\mathrm{max} }{r {d{\Phi _{0}(r)}}/{d{r}}}, \end{equation} \tag{ 1 }$$

which represents the ratio between the maximum amplitude (over azimuth) of the tangential force and the mean axisymmetric radial force derived from the m = 0 component of the gravitational potential. The potential is inferred from the luminous mass and can be represented as (see Combes & Sanders 1981; Quillen et al. 1994)

$$\begin{equation} \Phi (r,\theta) \approx \Phi _{0}(r) + \sum _{m=2,4,6} \Phi _{m}(r) \cos {[m\theta ]}. \end{equation} \tag{ 2 }$$

The angle θ is given in the deprojected image, and θ = 0 along the bar major axis. For this investigation we assume that L₁ = L₂ = L. For real galaxies, L₁ may differ from L₂ due to odd terms in the gravitational potential.

We analyzed three cases in which the bar's strength is estimated in three different ways.

• 1.  
  In the first case, the bar's strength is estimated at r = r_L. The Lagrangian point or corotation radius (Sellwood & Wilkinson 1993), r_L, was obtained from Buta & Zhang (2009), who applied the "potential-density phase shift method" to the OSUBGS sample. There have been significant discussions on the validity of this method. This is partly because Zhang & Buta (2007) found some cases (e.g., NGC 4665) where r_L/r_bar < 1, i.e., corotation before the end of the bar. According to Contopoulos (1980), self-consistent bars are not possible to be modeled in this regime. One important difference between "manifold" models (Romero-Gómez et al. 2006; Voglis et al. 2006a, 2006b; Athanassoula et al. 2009b) and the "potential-density phase shift method" is that Zhang & Buta (2007) and Buta & Zhang (2009) considered potentials varying considerably with time. The time-independent (rigid) potentials of the "manifold" models generate "passive" chaotic orbit responses. Although the Zhang & Buta (2007) models involve chaos in the individual stars' trajectories, "collective dissipation" makes possible the existence of coherent structures (e.g., spiral arms). We define "r_BZ09" as the corotation radius obtained from Buta & Zhang (2009). Table 1 shows the Q_t(r = r_BZ09) values for the 27 OSUBGS barred galaxies.
• 2.  
  In the second case, we estimate the bar's strength at a distance r_L = 1.2r_bar. According to various studies (Athanassoula 1992; Elmegreen 1996; Aguerri et al. 2003), the expected range for the bar length lies between r_bar = r_L/1.0 and r_bar = r_L/1.4. Elmegreen (1996) and Aguerri et al. (2003) also discuss objects where r_bar = r_L/1.7. A mean value of r_bar = r_L/1.2 is expected for large samples of galaxies. For bar strengths, the effect of having r_L = 1.0r_bar or r_L = 1.4r_bar, instead of r_L = 1.2r_bar, could be much larger than deprojecting a galaxy within a 10% error in the projection angles. For this study the bar length, r_bar, was taken from Laurikainen et al. (2004). In Table 1 we show the Q_t(r = 1.2r_bar) values adopted for this investigation.
• 3.  
  The third case involves the maximum of the radial Q_t(r) profiles or Q_g. These were tabulated in Laurikainen et al. (2004).

The adopted Q_t(r) values from Laurikainen et al. (2004) were computed assuming a constant M/L ratio throughout the disk and an empirical correlation for the vertical scale height (h_z). Also, it is assumed that dark matter has little impact on the bar strength. For the Q_t(r) error calculation shown in Table 1, the Q_g error of Laurikainen et al. (2004) was summed in quadrature with the error inherent to digitization⁵ of the Q_t(r) plots and the r_L (Buta & Zhang 2009) errors for the Q_t(r = r_BZ09) values.

A technique for separating the gravitational torques of bars and spirals was developed by Buta et al. (2003, 2005). This technique separates the bar + disk image to obtain the bar strength Q_b (at the respective maximum of Q_t(r)) unaffected by the spiral gravitational influence. Nevertheless, for the majority of barred galaxies in the OSUBGS sample, the bar strength, Q_b, dominates over the spiral arm strength, Q_s (Durbala et al. 2009). Also, the correction of the spiral arms does not affect the tendencies for Q_g in the Hubble sequence (Laurikainen et al. 2007). In either case, for this investigation it is assumed that Q_g ∼ Q_b, and that the Q_t(r) values are affected by the spirals within the errors.

Spiral arms pitch angles have been measured in the literature with different methods. Danver (1942) measured the spiral arms on photographic plates. Kennicutt (1981) measured the spiral shapes using the intensity and H ii region distributions. Ma et al. (1999) fit the shapes of spiral arms directly on the images. Fourier decomposition methods had also been used (e.g., Considere & Athanassoula 1988; Puerari & Dottori 1992; Saraiva Schroeder et al. 1994; Seigar et al. 2006), yielding similar results to other methods (Considere & Athanassoula 1988; Puerari & Dottori 1992).

4.1. "Slope Method"

This method is similar to the one used in Seigar & James (1998). It is assumed that the arms can be represented by logarithmic spirals, which implies a constant pitch angle. However, variable pitch angles may be a better and more adequate representation for some objects (see, e.g., the case of NGC 1365 in Ringermacher & Mead 2009).

Before deprojection, the spiral regions were isolated by masking the bar, foreground stars, strong star-forming regions (visually selected), bad pixels, and other structures not associated with the corresponding arm region. After deprojection (H-band data), the centers of the objects were determined by fitting ellipses to the central isophotes close to the bar region.⁶ Afterward, the spiral arms were "unwrapped" by plotting them in a ln r versus θ map (e.g., Iye et al. 1982; Elmegreen et al. 1992; Grosbøl et al. 2004). Under this geometric transformation, logarithmic spirals appear as straight lines. The pitch angle, i, is related to the slope of the line, s, as

$$\begin{equation} \cot {i} = k |s|, \end{equation} \tag{ 3 }$$

wherein k is a constant⁷ due to "pixelation" and unit conversion.

Two arm segments were selected closest to the bar's end with the condition that the slope, s, was maintained nearly constant along them (see Figures 1–4). Due to this "slope restriction," in many cases the critical segment, including the part of the arms attached to the bar, was not able to be considered. The slope (s) is determined by first selecting for each column in the arm segment the pixels with a maximum in intensity (see as an example Figure 5, for the case of NGC 1832). A least-squares fit is then obtained for the resulting pixels. As already mentioned, these fits were done in the H-band aiming to trace Population II stars. Young stars and clusters can contribute locally up to 20%–30% of the observed radiation in the NIR (e.g., Rix & Rieke 1993; Rhoads 1998; Patsis et al. 2001; Grosbøl & Dottori 2008). How these young objects affect the pitch angles' measurements depends on the star formation conditions and young stars kinematics. For this investigation, it is assumed that young stars and clusters affect the spiral arms pitch angles within the errors involved in the methods applied.

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As previously mentioned, all the objects with inner rings, asymmetries, unclear, or "logarithmically short" arms were discarded from the analysis. For the remaining 27 objects, the arm segments (I or II) best determined and with the clearest spiral structure were also identified. These are marked with an asterisk (*) in Table 2, together with the adopted radial ranges, Δr, tabulated from innermost (r₀) to outermost radius (r), and azimuthal ranges, α. Azimuthal ranges are obtained by the equation

$$\begin{equation} \alpha = \cot {i_{H}} \ln {\left(\frac{r}{r_{0}} \right)} \end{equation} \tag{ 4 }$$

and are displayed graphically in Figures 1–4 with regions delimited by solid lines. These values do not indicate the end of the spirals, since spiral arms may extend further with a variable pitch angle (Ringermacher & Mead 2009). However, if the extensions of the spirals have reduced amplitudes with respect to the logarithmic part, their phases will be difficult to determine, as will their pitch angles. The estimations are done independently of the amplitude (strength) of the spiral itself. Grosbøl et al. (2004, their Figure 8) found a tendency between the amplitude of the m = 2 spiral and pitch angles in SA and SAB galaxies.

Table 2. "Slope Method" Derived Parameters

Galaxy and Segment	iH	Δr	α	Segment	iH	Δr	α
	(deg)	(arcsec)	(deg)		(deg)	(arcsec)	(deg)
NGC 150 Arm I*	24.6+5.3− 1.5	(34.7–55.4)	58 ± 8	Arm II	33.2+7.6− 10.3	(41.8–51.4)	18 ± 5
NGC 210 Arm I*	18.4+0.7− 0.9	(64.8–107.6)	88 ± 4	Arm II	29.5+1.8− 1.8	(63.6–105.6)	51 ± 4
NGC 289 Arm I	16.5+2.1− 1.2	(22.8–28.2)	41 ± 4	Arm II*	19.3+4.8− 0.9	(22.8–30.8)	50 ± 7
NGC 578 Arm I*	23.8+0.4− 1.3	(20.8–50.3)	115 ± 4	Arm II	24.8+1.4− 1.3	(20.8–50.3)	110 ± 7
NGC 864 Arm I*	24.3+0.7− 0.6	(33.0–68.5)	93 ± 3	Arm II	28.8+4.1− 1.2	(34.8–53.1)	44 ± 4
NGC 1073 Arm I*	29.0+7.8− 6.7	(37.8–55.3)	39 ± 10	Arm II	12.6+4.4− 1.8	(48.7–70.0)	93 ± 18
NGC 1187 Arm I	35.7+3.8− 4.1	(35.7–48.8)	25 ± 3	Arm II*	29.1+3.2− 0.5	(25.2–55.6)	82 ± 6
NGC 1300 Arm I*	21.5+7.4− 7.8	(84.8–116.9)	47 ± 14	Arm II	18.8+6.1− 5.5	(74.3–100.5)	51 ± 13
NGC 1703 Arm I*	16.7+0.4− 1.9	(15.6–28.1)	112 ± 7	Arm II	25.0+0.7− 0.7	(16.8–28.1)	64 ± 2
NGC 1832 Arm I*	20.7+1.3− 1.2	(20.0–45.8)	126 ± 8	Arm II	27.0+6.1− 2.9	(20.0–27.8)	37 ± 6
NGC 3059 Arm I*	31.5+1.9− 1.6	(26.8–54.0)	66 ± 4	Arm II	33.7+7.5− 2.1	(23.6–33.9)	31 ± 5
NGC 3261 Arm I*	17.1+15.6− 2.5	(27.4–34.2)	41 ± 14	Arm II	11.8+3.3− 0.8	(30.0–36.1)	50 ± 7
NGC 3513 Arm I	32.9+1.3− 0.9	(26.1–72.6)	91 ± 4	Arm II*	40.7+1.4− 1.7	(16.3–72.6)	100 ± 5
NGC 3583 Arm I	20.4+0.5− 3.3	(34.7–59.3)	83 ± 7	Arm II*	24.1+2.0− 0.7	(34.7–68.8)	87 ± 5
NGC 3686 Arm I*	24.6+0.9− 0.5	(20.2–46.2)	104 ± 3	Arm II	67.0+6.6− 2.2	(20.2–43.7)	19 ± 4
NGC 4145 Arm I*	17.0+2.4− 0.4	(50.5–87.9)	104 ± 8	Arm II	17.6+3.3− 1.4	(50.5–87.9)	100 ± 12
NGC 4303 Arm I*	38.4+1.0− 1.2	(29.2–60.5)	53 ± 2	Arm II	44.1+1.5− 1.9	(29.2–57.3)	40 ± 2
NGC 4902 Arm I*	11.2+6.6− 1.8	(21.4–30.0)	98 ± 26	Arm II	23.1+5.3− 0.6	(19.5–34.7)	77 ± 9
NGC 4930 Arm I*	18.7+0.6− 2.7	(38.9–58.2)	68 ± 6	Arm II	74.6+5.5− 16.1	(36.2–53.1)	6 ± 4
NGC 4995 Arm I	47.8+3.7− 4.9	(22.6–34.3)	22 ± 3	Arm II*	38.5+6.3− 2.1	(17.3–37.7)	56 ± 8
NGC 5483 Arm I*	29.2+0.5− 4.4	(14.8–39.4)	100 ± 9	Arm II	31.8+1.4− 3.0	(14.8–34.7)	79 ± 6
NGC 5921 Arm I*	16.1+0.5− 0.6	(61.0–95.3)	89 ± 3	Arm II	16.4+0.9− 1.1	(69.5–91.9)	54 ± 3
NGC 6221 Arm I*	19.5+1.3− 0.7	(39.0–57.6)	63 ± 3	Arm II	20.0+2.5− 1.0	(41.2–56.5)	50 ± 4
NGC 6300 Arm I*	17.3+0.6− 0.4	(42.2–78.9)	116 ± 3	Arm II	18.3+0.4− 0.4	(42.2–78.9)	108 ± 2
NGC 6384 Arm I*	24.5+0.4− 1.9	(31.4–84.9)	125 ± 6	Arm II	18.0+0.7− 1.6	(40.1–57.2)	63 ± 4
NGC 7479 Arm I*	36.6+2.3− 1.4	(56.1–95.8)	41 ± 3	Arm II	23.0+2.8− 0.7	(60.4–95.8)	62 ± 5
IC 5325 Arm I	49.2+2.3− 9.7	(11.4–23.0)	35 ± 7	Arm II*	20.3+3.0− 0.4	(14.3–27.4)	100 ± 8

Notes. Column (1): object and spiral arm segment, see Figures 1–4. Columns (2) and (6): H-band pitch angles, i_H, in degrees. Columns (3) and (7): radial ranges, r₀ to r, in arcsec. Columns (4) and (8): azimuthal ranges, $\alpha = \cot {i_{H}} \ln {({r}/{r_{0}})}$, in degrees. Column (5): spiral arm segment (see Figures 1–4) for the same object as Column (1).

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Figure 6 shows a histogram of the maximum azimuthal range distribution (either arm segment I or II) for each object presented in Table 2.

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4.1.1. Error Determination

Errors introduced by deprojection parameters (ϕ and q) translate into different slopes or deviations of a straight line in the ln r versus θ plots. For each object, five deprojected frames were obtained to better account for these errors. The images were deprojected with the parameters ϕ, q; ϕ + sd, q; ϕ − sd, q; ϕ, q + sd; and ϕ, q − sd, where sd is the respective standard deviation. Pitch angle values were measured and compared to the case when ϕ and q were used as the deprojection parameters (i.e., when sd = 0). The cases with the highest (positive) or lowest (negative) discrepancies were adopted to account for the +σ and −σ errors, respectively (see Table 2).

4.2. "Fourier Method"

Figure 7(a) plots the pitch angles in arm segment I versus arm segment II for each object as obtained with the "slope method." Figure 7(b) shows a histogram of the absolute value difference between arm segments I and II. As shown in the figures, some scatter is present when analyzing spiral arm segments within the same galaxies. Since we are interested in comparing single values of pitch angles for each object, we need a method that provides the "dominant mode" for the pitch angle measurement. The "Fourier method" is perfectly adequate for this purpose.

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In this method, it is again assumed that the arms can be represented by logarithmic spirals.⁸ The Fourier amplitudes for each component are given by

$$\begin{equation} A(m,p) = \frac{\sum _{i=1}^{I}\sum _{j=1}^{J} I_{ij}(\ln {r},\theta) \mathrm{exp}[-i(m\theta +p\ln {r})]}{\sum _{i=1}^{I}\sum _{j=1}^{J} I_{ij}(\ln {r},\theta)}, \end{equation} \tag{ 5 }$$

where r and θ are the polar coordinates, I_ij is the intensity at coordinates ln r, θ, m is the number of spiral arms (or modes), and p is related to the spiral arms pitch angle (i_H) by

$$\begin{equation} \tan {i_{H}} = -m/p_{\mathrm{max}}, \end{equation} \tag{ 6 }$$

where p_max corresponds to the maximum of A(m,p) and m = 0, 1, 2, 3, ..., i.e., the maximum of the Fourier spectrum (see, e.g., Puerari & Dottori 1992; Saraiva Schroeder et al. 1994) for mode m. Most of the analyzed objects present m = 2 as the dominant mode for the spiral arms in the H-band (see Table 3), so it was adopted for this investigation. The exceptions are NGC 3261 and NGC 4930 in which m = 1 dominates and was used instead. For NGC 1300 and NGC 7479, other Fourier modes (m) compete with the m = 2 mode because of the spiral arm segments with variable pitch angles. The pitch angles corresponding to the m = 2 Fourier mode were adopted for these objects in the subsequent analysis.

Table 3. "Fourier Method" Derived Parameters

Galaxy	$\frac{m_{1}}{m_{2}}$	$\frac{m_{2}}{m_{2}}$	$\frac{m_{3}}{m_{2}}$	$\frac{m_{4}}{m_{2}}$	$\frac{m_{5}}{m_{2}}$	$\frac{m_{6}}{m_{2}}$	iH	iB	Δr
							(deg)	(deg)	(arcsec)
NGC 150	0.416	1.000	0.344	0.213	0.395	0.167	27.9+0.9− 1.9	17.6+1.3− 2.0	(34.7–55.4)
NGC 210	0.525	1.000	0.189	0.478	0.255	0.220	16.7+0.5− 0.5	15.7+0.5− 0.5	(63.6–107.6)
NGC 289	0.103	1.000	0.309	0.213	0.091	0.109	19.7+0.8− 0.5	17.2+1.3− 1.8	(22.8–30.8)
NGC 578	0.656	1.000	0.289	0.255	0.080	0.116	24.2+1.6− 1.5	23.0+0.9− 1.3	(20.8–50.3)
NGC 864	0.721	1.000	0.462	0.361	0.307	0.325	20.7+1.2− 1.5	18.0+0.9− 0.9	(33.0–68.5)
NGC 1073	0.778	1.000	0.515	0.506	0.307	0.349	34.7+1.4− 5.9	12.4+1.1− 0.5	(37.8–70.0)
NGC 1187	0.668	1.000	0.460	0.436	0.655	0.239	21.4+1.3− 1.2	19.4+1.1− 1.0	(29.0–59.4)
NGC 1300	0.617	1.000	1.001	0.854	0.399	0.330	11.2+15.3− 0.3	13.1+7.7− 0.3	(74.3–116.9)
NGC 1703	0.245	1.000	0.255	0.371	0.161	0.196	18.8+1.5− 2.0	17.2+1.6− 0.6	(15.6–28.1)
NGC 1832	0.242	1.000	0.492	0.399	0.100	0.128	25.1+1.8− 1.1	24.4+1.1− 1.6	(20.0–45.8)
NGC 3059	0.728	1.000	0.371	0.475	0.424	0.694	27.0+3.0− 2.6	8.9+0.5− 0.9	(23.6–54.0)
NGC 3261	1.043	1.000	0.246	0.226	0.259	0.322	9.1+13.3− 0.3	10.2+20.1− 0.4	(27.4–36.1)
NGC 3513	0.481	1.000	0.229	0.472	0.141	0.220	25.7+1.2− 1.7	24.2+1.7− 0.7	(27.8–72.6)
NGC 3583	0.446	1.000	0.470	0.225	0.300	0.146	24.5+2.4− 2.1	22.6+2.0− 1.3	(34.7–68.8)
NGC 3686	0.450	1.000	0.886	0.271	0.276	0.493	14.4+0.4− 0.6	14.4+0.6− 0.8	(20.2–46.2)
NGC 4145	0.448	1.000	0.250	0.348	0.240	0.275	23.9+1.0− 0.7	17.3+0.4− 1.3	(50.5–87.9)
NGC 4303	0.260	1.000	0.174	0.294	0.149	0.186	42.8+2.8− 1.9	37.7+1.6− 2.2	(29.2–60.5)
NGC 4902	0.467	1.000	0.210	0.665	0.277	0.401	20.8+2.5− 1.6	25.3+3.5− 4.0	(19.5–34.7)
NGC 4930	1.081	1.000	0.550	0.272	0.181	0.223	30.1+2.1− 2.1	13.9+0.9− 2.0	(36.2–58.2)
NGC 4995	0.898	1.000	0.356	0.595	0.234	0.204	90.0+5.6− 8.8	78.3+5.4− 8.3	(22.5–37.7)
NGC 5483	0.163	1.000	0.134	0.439	0.158	0.280	26.3+1.9− 1.8	21.6+1.4− 0.9	(14.8–39.4)
NGC 5921	0.188	1.000	0.454	0.524	0.152	0.313	30.6+4.9− 3.1	22.5+2.0− 2.2	(61.0–95.3)
NGC 6221	0.602	1.000	0.315	0.413	0.115	0.235	35.4+4.7− 1.4	19.9+1.6− 0.7	(39.0–57.6)
NGC 6300	0.199	1.000	0.201	0.330	0.157	0.283	23.1+0.7− 0.9	20.3+0.6− 1.1	(42.2–78.9)
NGC 6384	0.501	1.000	0.574	0.312	0.385	0.273	26.3+1.2− 1.2	28.0+0.9− 2.0	(31.4–84.9)
NGC 7479	1.073	1.000	0.656	1.200	0.610	0.473	26.8+3.8− 1.8	31.5+1.6− 3.2	(56.1–95.8)
IC 5325	0.367	1.000	0.426	0.176	0.200	0.090	69.4+3.8− 5.2	81.5+4.2− 12.7	(11.4–27.4)

Notes. Column (1): galaxy name. Column (2): ratio between the maximum amplitudes of Fourier modes m = 1 and m = 2, in the H-band. Columns (3), (4), (5), (6), and (7): ratio between the maximum amplitudes of the respective Fourier modes, in the H-band. Column (8): H-band (see Section 4.2) pitch angles, in degrees. Column (9): B-band (see Section 6.1) pitch angles, in degrees. Column (10): radial ranges, r₀ to r, in arcseconds.

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For the galaxies of the sample, it has been realized that the presence of foreground stars does not affect the value of the pitch angle in general. Nevertheless, caution must be taken when foreground stars (or objects) compete in extension with spiral arms (see, e.g., annulus for NGC 864 in Figure 1). In these cases the need for masks is required.

Objects were deprojected as explained in Section 4.1. Radial ranges were selected to cover the spiral segments previously analyzed with the "slope method." The azimuthal coverage is 2π radians. The analyzed annuli are shown graphically in Figures 1–4 (dashed lines). These are the regions where the Fourier analysis was performed.

Table 3 shows the results for the Fourier pitch angle values, which agree with the "slope method" within a ∼16° difference (this corresponds to 1σ in Figure 7) in the majority of the objects. NGC 5921 and NGC 6221 present the largest differences (∼15°). For two objects, NGC 4995 and IC 5325, the computed pitch angles are close to ∼90°. This is due to the fact that the spiral arms have a low surface brightness (as compared to the disk) and the bar component is difficult to isolate in the analyzed annulus. The "slope method," for the "best-defined arm," was used instead for these two objects in the subsequent analysis.

4.2.1. Error Determination

Errors were determined in the same way as in the "slope method." These were added in quadrature with the error intrinsic to the method. A program was built that computes the two-dimensional fast Fourier transform in Equation (5). The output of this program is a 128 × 2048 (m, p) matrix. The two closest values near p_max were used to approximate the error of the method.

NGC 210. "Skinny" spiral arms compared with the bar.

NGC 289. The outer spiral arms have a greater pitch angle (i_H ∼ 40°) compared to the inner ones (i_H ∼ 25°; "slope method").

NGC 578. Two symmetric spiral arms near the bar.

NGC 1073. Spiral arms difficult to trace (low signal-to-noise ratio).

NGC 1187. The two arm features analyzed are visually attached to the bar. A third arm feature, not visually attached to the bar, is present. The radial ranges for the "Fourier method" were modified with respect to the "slope method" to allow a better signal-to-noise ratio in the ln r versus θ map.

NGC 1300. Two well-defined logarithmic spiral arms, although short in azimuthal range. The adopted deprojection parameters were changed compared to the ones of Laurikainen et al. (2004). This was done because the values provided in Laurikainen et al. (2004) do not agree with the outer isophotes of the OSUBGS images. An average between Hyperleda (Paturel et al. 2003), RC3 (de Vaucouleurs et al. 1991), and a visual determination of the outer isophotes was used.

The deprojection parameters from Lindblad et al. (1997) were also tried for the pitch angle measurements. These parameters, ϕ = 87° ± 2° and q = 0.82 ± 0.05, are based on H i data and are independent of kinematical or dynamical criteria (see also Kalapotharakos et al. 2010). Using these parameters, spiral arms are difficult to follow in a ln r versus θ map (assuming a logarithmic geometry). For arm region I, a pitch angle of 212 ± 6° was obtained. Arm region II was not possible to measure via the "slope" method. The pitch angles obtained by applying the "Fourier" method led to values with a contrary sign to the one expected, i.e., an inverse sense of winding for the spiral arms.

NGC 1703. Difficult to analyze the spiral arms in the inner regions due to few pixels in a ln r versus θ map.

NGC 1832. The bar region is distorted (not straight).

NGC 3059. "Hard to follow" logarithmic shape for the spiral arms.

NGC 3583. Two symmetric spiral arms can be appreciated in the outer disk. The region close to the bar presents a structure similar to a ring or a tight spiral arm.

NGC 4145. Double bar system?

NGC 4303. This object presents three main spiral arms.

NGC 4902. Three spiral regions are present in this object.

NGC 5921. This object presents an inner ring and spiral features.

NGC 6300. This object presents spiral features and apparently a ring feature.

NGC 6384. Spiral arms with bifurcations.

NGC 7479. In general the spiral arms for this object do not present a clear logarithmic geometry.

IC 5325. This object presents four well-defined segments of spiral arms. Only the ones near the bar's end were analyzed.

Figure 8 shows the results for the pitch angle, i_H (Fourier method, except for NGC 4995 and IC 5325; see Section 4.2), versus perturbation strengths, Q_t(r = r_BZ09). A first inspection of the data, where the "azimuthal range"⁹ is α > 50°, shows considerable scatter around the predicted correlation for models A (Ferrers 1877, bar potential) and D (Dehnen 2000, bar potential) in Athanassoula et al. (2009a).¹⁰ However, if the α criterion is changed to logarithmic spiral segments that extend up to α > 70°, α > 90°, and α > 110°, the scatter is reduced. The reduced Pearson's chi-square, χ²/n, obtained as

$$\begin{equation} \chi ^2 = \sum _{k=1}^{n} \frac{(i_k - i_p)^2}{i_p}, \end{equation} \tag{ 7 }$$

where i_k is the kth Fourier-measured pitch angle and i_p is the predicted pitch angle value for models A and D in Athanassoula et al. (2009a), gives the results 3.10, 1.55, 1.83, and 2.00 for α > 50° (n = 27), α > 70° (n = 17), α > 90° (n = 13), and α > 110° (n = 5), respectively.

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Figures 9 and 10 show the results for the cases Q_t(r = 1.2r_bar) and Q_g, respectively. For α > 70°, reduced Pearson's chi-square values obtained as

$$\begin{equation} \chi ^2 = \sum _{k=1}^{n} \frac{(Q_k - Q_p)^2}{Q_p}, \end{equation} \tag{ 8 }$$

where Q_k is the kth bar strength value corresponding to the kth Fourier-measured pitch angle and Q_p is the predicted bar strength value for models A and D in Athanassoula et al. (2009a), yield the results 0.049, 0.075, and 0.084 for the Q_t(r = r_BZ09), Q_t(r = 1.2r_bar), and Q_g plots, respectively.

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According to this result, the best concordance with the Athanassoula et al. (2009a) model is obtained by comparing the pitch angles with Q_t(r) given at Buta & Zhang (2009) bar corotation radii (r = r_BZ09). This last point is not discussed in Athanassoula et al. (2009a).

One important aspect in the Athanassoula et al. (2009a) prediction is that the self-gravity of the spirals was not taken into account. The potential created by the "confined" chaotic orbits is neglected. Conversely Tsoutsis et al. (2009) emphasize the contribution of the spiral part for studying the dynamics of the "chaotic" spirals. Also, realistic bar potentials are hard to model. If many different realistic potentials are used, the predicted correlations may become broader (Athanassoula et al. 2009a, 2010). This may explain in Figure 8 the tendency of the points (squares and circles) to be above the predicted correlation for Q_t(r = r_BZ09) < 0.2.

6.1. Density Wave Theory Prediction

The modal approach explains the density wave phenomena as generated by intrinsic mechanisms in the disk (Bertin et al. 1989a, 1989b; Bertin & Lin 1996). Normal modes of oscillation generate spontaneously and evolve according to the physical and dynamic properties of the system. Three physical properties determine the morphology in disk galaxies: the disk mass, the gas content, and the stellar velocity dispersion. When the disk mass is "high," bar structures are generated as oscillating modes of the system. The modal theory considers bars and spirals equally, i.e., as normal modes of oscillation in the disk.

Based on the dispersion relation, linear density wave theory predicts (Hozumi 2003) that the pitch angle should increase with increasing velocity dispersion, or that

$$\begin{equation} \tan {i} \propto c^{2}_{r}\big/\Sigma, \end{equation} \tag{ 9 }$$

where i is the arms pitch angle, c_r is the radial velocity dispersion, and Σ is the surface density of the disk. Spiral structure shows different morphologies when observed in optical versus NIR bands (Block & Wainscoat 1991; Grosbøl & Patsis 1998). NIR bands can trace both the old populations of bar and spiral arms, assuming that red young stars do not contribute globally to the observed radiation (Rix & Rieke 1993). Also, older populations have a higher velocity dispersion compared to younger ones (Barbanis & Woltjer 1967; Wielen 1977; Nordström et al. 2004; Binney & Tremaine 2008, Section 8.4). For most galaxies at the arm location, we have that ∼98% (by mass) of the stars belong to evolved populations (see Gonzalez & Graham 1996 and references therein). Nevertheless, young stars contribute to most of the light in optical wavelengths. According to this, NIR images of spiral perturbations should present higher pitch angles compared to optical ones. Azimuthal age (color) gradients (e.g., Gonzalez & Graham 1996; Martínez-García et al. 2009a, 2009b; Martínez-García & González-Lópezlira 2011) may also affect the pitch angles observed in the optical versus NIR bands, but these are very difficult to trace by just comparing the light distributions in two bands (Gonzalez & Graham 1996; Seigar & James 1998). Besides, azimuthal gradients are not located continuously along the spiral arms but in specific regions (Gonzalez & Graham 1996; Martínez-García et al. 2009a; Martínez-García & González-Lópezlira 2011).

From Equation (9), taking into account that young and old stars are similarly affected by the gravitational potential of the disk (which depends on the surface density), we obtain

$$\begin{equation} i_{B} = \arctan { \left\lbrace \left(\frac{c_{r_{B}}}{c_{r_{H}}} \right)^{2} \tan {i_{H}} \right\rbrace }, \end{equation} \tag{ 10 }$$

where i_B is the B-band pitch angle, i_H is the H-band pitch angle, $c_{r_{B}}$ is the radial velocity dispersion of young stars, and $c_{r_{H}}$ is the radial velocity dispersion of old stars.

In the case of the invariant manifold theory, where chaotic orbits are "confined" in the spiral locus, no difference between pitch angles of spiral arms traced in different wavelengths is predicted (Athanassoula et al. 2010).

Seigar et al. (2006) found a nearly 1:1 correlation between pitch angle measurements in the B and H bands, for 57 galaxies in the OSUBGS (Eskridge et al. 2002) sample. Nevertheless, based on the sample of five non-barred and weakly barred spirals, Grosbøl & Patsis (1998) notice that the main two-armed spiral is tighter when measured in bluer colors. For the barred-spirals data presented in this investigation, we measured the pitch angles in the B-band images for the same objects analyzed in the H-band from the OSUBGS sample, applying the "Fourier" method. The B-band images were registered to the H-band images, so the high-resolution data (B-band) were degraded to the low-resolution data (H-band in this case). Annulus regions were selected in the same positions as the H-band, and the pitch angles were measured identically¹¹ with the method described in Section 4.2. The results are shown¹² in Figure 11 (and Table 3) where a tendency of ∼30% of the points toward higher H-band pitch angles is observed. However, if we apply the same azimuthal range (α) criteria as in Figures 8, 9, and 10, we can notice that ∼80% of the α > 70° data lie very close to the 1:1 relation as expected (independently of α) from Athanassoula et al. (2010).

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6.2. Invariant Manifolds as Apsidal Sections

The results of this investigation show the following.

• 1.  
  Although the adopted deprojection parameters may introduce some biases (see, e.g., Barnes & Sellwood 2003), a trend can be observed where some strong barred spirals have more open spiral arms when compared to galaxies with weaker bars. This kind of trend was also discussed in Block et al. (2004), where a similar behavior was found. The correlation predicted by the manifold models of Romero-Gómez et al. (2006, 2007) and Athanassoula et al. (2009a, 2009b, 2010) is better reproduced by observations on two conditions.

  • (a)  
    The corotation values obtained with the "potential-density phase shift method" (Buta & Zhang 2009) are adopted.
  • (b)  
    The spirals logarithmic geometry is maintained for large azimuthal ranges, α > 70°.
• 2.  
  ∼60% of the 27 galaxies in the analyzed sample seem to reproduce the investigated correlation.
• 3.  
  The pitch angles calculated via the "Fourier method" in the B (young stars) and the H (mostly old stars) bands yield similar values for ∼80% of the objects where the azimuthal range, α, is greater than 70°. This kind of behavior is expected in the "Lyapunov tube model" (Athanassoula et al. 2010), although no restriction on the azimuthal range was given by the authors.
• 4.  
  Other possible mechanisms to generate spiral features in barred galaxies, such as bar-driven spirals (e.g., Salo et al. 2010), models where the Lagrangian points of the system are specified by both bar and spirals (e.g., Tsoutsis et al. 2009), or chaotic spirals inside corotation (thus not related with the presence of unstable Lagrangian points; Patsis et al. 2010), cannot be excluded by the present investigation.

I am grateful to the anonymous referee for many important remarks and helpful comments that have improved this paper. I acknowledge postdoctoral financial support from UNAM (DGAPA), México. I thank Christos Efthymiopoulos for clarifying my inquiries about spiral arms driven by "manifolds." This work made use of data from the Ohio State University Bright Spiral Galaxy Survey, which was funded by grants AST-9217716 and AST-9617006 from the United States National Science Foundation, with additional support from the Ohio State University.