GRAND DESIGN AND FLOCCULENT SPIRALS IN THE SPITZER SURVEY OF STELLAR STRUCTURE IN GALAXIES (S⁴G)

In order to highlight disk symmetry, each image was rotated 180° using the IRAF task rotate and subtracted from the original using imarith to get the asymmetric part. Negative values were set equal to 0 in the asymmetric images, using the IRAF task imreplace, and this truncated image was then subtracted from the original to get the symmetric part (Elmegreen et al. 1992, hereafter EEM):

$$\begin{equation} S_2(r,\theta)=I(r,\theta)-[I(r,\theta)-I(r,\theta +\pi)]_T. \end{equation} \tag{ 1 }$$

Here, I is the original image, S₂ is the symmetric image, and T stands for truncation. Sample two-fold symmetric images are shown in the second row of Figures 1–4. The grand design symmetric images look like cleaned-up images of the originals, with prominent two-arm structure. It is difficult to remove stars globally from channel 1 images without also removing star-forming regions. These symmetric images eliminate virtually all the stars and most of the star-forming regions—all but those that happen to be symmetric in the galaxy. Typically 80%–85% of the light in these images is in the S₂ image and 15%–20% is in the asymmetric image; most of this symmetric light is from the azimuthally averaged disk, unrelated to spirals.

Three-arm structure can be highlighted by following a similar procedure but rotating 120° twice (EEM):

$$\begin{eqnarray} S_3(r,\theta)&=&2I(r,\theta)-[I(r,\theta)-I(r,\theta +2/3\pi)]_T\nonumber\\ &&- [I(r,\theta)-I(r,\theta -2/3\pi)]_T. \end{eqnarray} \tag{ 2 }$$

The results are shown in the third row of Figures 1–4. The most prominent three-arm structures appear in multiple arm galaxies. In contrast, grand design galaxies are dominated by two arms. For the flocculent galaxies, which do not have a dominant symmetric component, the S₃ images resemble the S₂ images and both show a lot of arms.

Most of our galaxies have the same spiral Arm Class at all passbands. The optically flocculent galaxies are also flocculent in 3.6 μm because of patches from PAH emission and young supergiants. The exceptions in our sample are the flocculents NGC 5055 and NGC 2841 (Figure 3), which have long and smooth spiral arms at 3.6 μm. These arms were discovered by Thornley (1996) in K_s band and also shown by Buta et al. (2010) and Buta (2011). The long-arm spiral in NGC 2841 is predominantly seen as a dust arm. NGC 7793 may have a subtle two-arm component as well, which shows up weakly in the symmetric image.

To consider a broader sample of underlying long-arm structure in flocculent galaxies, we examined Two Micron All Sky Survey (2MASS) images of 197 galaxies cataloged as optically flocculent by Elmegreen & Elmegreen (1987). We find that only ∼15% have underlying weak two-arm structure in the near-IR 2MASS images; most are still dominated by flocculent structure.

One measure of spiral arm amplitudes is a comparison of arm and interarm surface brightnesses, which were measured on azimuthal scans of deprojected images with sky subtracted. The position angles and inclinations used for this are in Table 1. Position angles were measured from contours of the 3.6 μm images; if the outer and inner disks had different orientations, then the position angle was selected to be appropriate for the disk in the vicinity of the main arms. The values we used are generally within a few degrees of those listed in the RC3 or Hyperleda.²¹ Slight differences like these changed the derived arm amplitudes by less than 10%, because most of the galaxies in our sample have relatively low inclination. Gadotti et al. (2007) discuss general errors for measurements that depend on inclination, pitch angle, and the assumption of circular disks.

Each galaxy image was transformed into polar coordinates, (r, θ), as shown in the lower panels of Figures 1–4. The y-axis is linear steps of radius, while the x-axis is the azimuthal angle from 0 to 360°. From this polar image, pvector was used to make 1 pixel wide azimuthal cuts for radii in steps of 0.05R₂₅ (the values we used for R₂₅, from RC3, are listed in Table 1).

Spiral arms were measured from the azimuthal scans, often out to ∼1.5R₂₅, which is further than in previous near-IR studies because of the greater sensitivity of the S⁴G survey. Star-forming spikes and foreground stars were obvious on the azimuthal scans and avoided in the arm measurements. This avoidance of point sources gives the arm–interarm contrast an advantage over Fourier transform measurements in uncleaned images.

The arm–interarm intensity contrast at radius r was converted to a magnitude using the equation

$$\begin{equation} A(r)=2.5\log \left[\frac{2I_{\rm arm}(r)}{I_{\rm interarm1}(r)+I_{\rm interarm2}(r)}\right]; \end{equation} \tag{ 3 }$$

I_arm is the average arm intensity measured at the peak of the broad component, and the denominator contains the adjacent interarm regions. Figure 5 shows A(r) in the top row and in the third row from the top. Two galaxies from each of Figures 1–4 are included. The average arm–interarm contrast for each galaxy, averaged over the whole disk (beyond the bar, if there is one), is listed in Table 2; the errors are ∼0.1 mag.

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The average arm–interarm contrast in the disk versus Hubble type is shown in Figure 6. Different Arm Classes have different symbols. For a given Hubble type, the average arm–interarm contrast decreases from grand design galaxies to multiple arm to flocculent. This is consistent with the more prominent appearance of arms in grand design spirals. The averages for all Hubble types combined are 1.14 ± 0.44, 0.81 ± 0.28, and 0.75 ± 0.35 in these three Arm Classes, respectively. Within a given Arm Class, the arm–interarm contrast increases slightly for later Hubble types, probably because of more star formation contributing to the arms; the later type galaxies in our sample are all flocculent or multiple arm.

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Sixteen galaxies in our sample were previously measured in optical and near-IR bands. The arm–interarm contrasts at 3.6 μm, and for some of them at 4.5 μm, are listed for the same radius as the optical and near-IR results for these galaxies in Table 3. The contrasts are qualitatively similar between optical, near-IR, and mid-IR bands; the average arm–interarm contrasts are 0.93 ± 0.32, 0.81 ± 0.49, and 0.99 ± 0.46 mag for B, I, and 3.6 μm, respectively, for all galaxies. For multiple arm and grand design galaxies, the averages are 1.04 ± 0.28, 0.99 ± 0.46, and 1.17 ± 0.38, which are all the same to within the errors. The basic reason these spiral arm amplitudes are nearly independent of color is that the arms are intrinsically strong mass perturbations; color variations are relatively minor.

For flocculent galaxies, the contrasts are smaller than the multiple arm and grand design galaxies and significantly larger in the blue than at longer wavelengths: 0.63 ± 0.24, 0.34 ± 0.18, and 0.44 ± 0.13 for for B, I, and 3.6 μm, respectively. This decrease in amplitude from the B to I band is most likely the result of a young age for these spiral arm features. They appear to be star formation superposed on a somewhat uniform old stellar disk. The slight increase from the I band to 3.6 μm could result from PAH emission and red supergiants in the 3.6 μm band. We attempted to avoid small regions of emission in our measurements at 3.6 μm, so if there is contamination from PAHs, it would have to be somewhat extended. S. Meidt & S⁴G team (2011, in preparation) are exploring techniques to remove PAH and point-source emission from the 3.6 μm images to produce mass maps.

Another explanation for decreasing arm–interarm contrast with increasing wavelength in flocculents might be an excess of gas and dust between the arms. The dust would depress the interarm brightness for the B band but not for 3.6 μm. While it is more likely that the gas and dust are associated with star formation, which is what we see as flocculent arms, it is still possible that shells and other debris around the star formation site darken the interarm regions at short wavelengths.

Fourier transforms are an independent method for measuring arm amplitudes besides arm–interarm contrasts, so Fourier components of azimuthal intensity profiles on the 3.6 μm images were also measured. For number of arms m = 2, 3, and 4, the Fourier transform was determined from the equation

$$\begin{equation} F_{m}(r) = \frac{\sqrt{[\Sigma I(r,\theta) \sin (m\theta)]^2 + [\Sigma I(r,\theta) \cos (m\theta)]^2}}{\Sigma I(r,\theta)}. \end{equation} \tag{ 4 }$$

I(r, θ) is the intensity at azimuthal angle θ and radius r in the deprojected, sky-subtracted image. All sums are over azimuthal angles with steps of 1 pixel. Note that according to this definition, the relative amplitude of a spiral arm is twice the value of the Fourier component. For example, a spiral arm with an amplitude profile I(θ) = 1 + Asin (mθ) has a Fourier m-component from Equation (4) equal to A/2.

Figure 5 shows Fourier components in the second and fourth rows, below the corresponding arm–interarm contrasts. (Fourier transforms on the 4.5 μm images were indistinguishable from those at 3.6 μm and are not considered further here.) The Fourier transforms were done on images that were cleaned to remove most foreground stars; masks for this process are part of Pipeline IV in the S⁴G data reduction. Some scans (such as NGC 5457) still contained some foreground stars, and the Fourier components of these stand out as narrow spikes. The star peaks are avoided in the average values used in the discussion below.

Table 2 lists the average values of the m = 2 components. Uncertainties in the m = 2 components range from ∼ ± 0.02 to 0.10 in different galaxies. The m = 4 components are about half the m = 2 components in all cases. The ratio of the m = 3 component to the m = 2 is largest for the flocculent galaxies, as expected for galaxies with irregular structure and lots of arm pieces. Grand design galaxies are dominated by two main arms, so their m = 3 component is weaker. The average values of F₃/F₂ are 0.58 ± 0.11, 0.48 ± 0.24, and 0.33 ± 0.19 for flocculent, multiple arm, and grand design galaxies, respectively.

In Figure 7, the average m = 2 value of the Fourier component in the spiral region is compared with the average arm–interarm contrast. The two measures of arm amplitude are obviously related. The curves in the figure show the expected trends in two cases: the dashed line follows if the arm–interarm contrast is given exclusively by the amplitude of the m = 2 Fourier component, and the solid line follows if the arm–interarm contrast comes from the combined m = 2 and m = 4 components. If only the m = 2 component contributes, then A/I = (1 + 2F₂)/(1 − 2F₂). If the m = 4 component contributes half the two-arm amplitude of the m = 2 component as found above, then A/I = (1 + 3F₂)/(1 − 3F₂). In the figure, the curves are 2.5log (A/I). The solid line traces the data points reasonably well, but neither line is a good fit in a statistical sense because the scatter in the data is large (the reduced chi-squared values comparing the lines with the data are 0.69 for the solid line and 2.25 for the dashed line, with r-values of 0.588 and 0.629, respectively).

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Other studies have also measured peak m = 2 Fourier amplitudes. For comparison, Laurikainen et al. (2004) has eight galaxies in common with ours from the OSUBSGS; their H-band images yield amplitudes that are similar to the 3.6 μm values, taking the factor of two (mentioned above) into account to compare our F₂ values with their amplitudes.

In previous optical studies of symmetry images (Elmegreen et al. 1989, 1992), we saw large-scale amplitude modulations in spiral arms that looked like interference between inward and outward moving waves in a spiral wave mode (Bertin et al. 1989). Evidence for these can be seen in the mid-IR m = 2 symmetric image of NGC 4321 in Figure 1. The symmetric image of NGC 4321 is shown deprojected with circles in Figure 8, along with arm–interarm contrast measurements and Fourier transform measurements discussed in the previous sections, now plotted with radius on a logarithmic scale. Referring to the top left panel of Figure 8, the parts of the arms at the ends of the central oval (inner circle, at ∼0.25R₂₅) are more prominent than the parts along the oval minor axis. Further out in radius the arms get brighter again (middle circle, at ∼0.45R₂₅); then there is another gap and another brightening near the ends of the arms. The outer circle in the figure is at R₂₅. These bright regions trace the prominent trailing arms which should have an inward group velocity (Toomre 1969). If there are also leading spirals from a reflection of the trailing spirals in the central parts, then the group velocity of the leading spirals would be outward. The bright regions could then be regions of constructive interference where these two wave types intersect.

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The lower left panel of Figure 8 shows the arm–interarm contrast for each arm as in Figure 1. The arrows at the bottom are the radii for the arm maxima found before in B- and I-band images (Elmegreen et al. 1989). Both arms follow an alternating pattern of brightness with a logarithmic spacing, although arm 1 is more variable than arm 2. The arm–interarm contrast profile from the deprojected m = 2 symmetric image is shown in the lower right. A modulation with the same log spacing is present. The two inner circles in the top left panel are at the radii of the two peaks in the lower right panel. The strength of the arm modulation in the symmetric image is 50%–100%. Fourier transform amplitudes are shown in the top right panel. The amplitude variations are most pronounced for the m = 4 Fourier component, but they also show up in the m = 2 component.

Close inspection of the 3.6 μm image shows that the inner peak amplitude at ∼0.25R₂₅ is at the end of the inner oval, in a broad region of bright star formation. The next peak at ∼0.45R₂₅ is the broad ridge of star formation in the main spiral arm, presumably from density wave compression in a shock. The outer peak in the arms is another star formation feature. Because of the mixture of old stars, PAH emission, and AGB stars in the 3.6 μm image, it is difficult to tell if these peaks are present in the old stellar component. At the very least, it appears that the amplitude modulations found here are the locations of extended star formation regions. Perhaps such star formation highlights an underlying wave interference or resonance pattern (Knapen et al. 1992).

Non-SB grand design galaxies in our sample often show distinct broad outer spirals beyond and separate from the main inner bright spirals. The inner spirals, as in NGC 1566 (Figure 1), are associated with bright star formation and dustlanes that are probably spiral shocks (e.g., Roberts 1969). Beyond that there are sometimes separate arms that are smoother and without concentrated star formation. In NGC 1566, the outer arms begin on each side of the galaxy at ∼0.5R₂₅, and extend at least as far as the edge of the image at ∼1.4R₂₅, forming a pseudo-ring. Also in NGC 1566, there is a ridge of star formation disjoint from and leading the main outer spiral in the south. This ridge is described as a "plume" by Buta et al. (2007). Plumes usually occur in barred galaxies where they appear as short disjoint arms of enhanced emission at the ends of bars in the leading direction (Buta 1984; Buta et al. 1995).

Other non-SB grand design galaxies in our survey have disjoint smooth outer arms too (Figure 1). NGC 4321 has two strong inner arms with considerable m = 3 symmetry, and outside of these starting at about 0.5R₂₅ is a pair of spirals at higher pitch angle. The outer arms are clear in the top image of NGC 4321 in Figure 1. On the left of the image there is a branch from the inner spiral to the outer spiral, which continues toward the companion galaxy NGC 4322. Just as in NGC 1566, these outer arms are broader and smoother than the inner arms, they contain little concentrated star formation, and they extend far out in the image.

NGC 5194 has an interacting companion galaxy that could have modified any outer spirals, so we do not consider it in this context here. Still, Tully (1974) and others noted that the spiral arms in M51 have a kink in the middle and seem to be composed of two separate arm systems. We see that kink also at the 7 o'clock position of the bright outer arm in Figure 1. Dobbs et al. (2010) reproduced the kink in a companion-interaction model of M51, and noted that it arose at the intersection of an old, inner spiral pattern, generated by the first fly-by of the companion, and a new, outer spiral pattern generated by the second fly-by.

NGC 5248 in Figure 1 has strong and symmetric inner arms and faint, broad, and disjoint outer arms. The outer arms were also present in a deep R-band image in Jogee et al. (2002) and they were first described by Burbidge et al. (1962) in a deep B-band image. In the northwest, there is a second ridge or short arm that is disjoint from the main outer arm, as in the southern plume of NGC 1566.

The grand design SB galaxies in Figure 4 do not have disjoint outer arms. Instead, the inner arms continue smoothly to the outer galaxy. There is also no obvious counterpart to smooth disjoint outer arms in the multiple arm galaxies of Figure 2. Multiple arm galaxies generally have an inner two-arm symmetry, as seen in the figure, and narrow, irregular and asymmetric arms with star formation all the way to the edge of the image.

Presumably the outer arms in non-SB grand design galaxies are stellar features that are excited by the inner arms or inner oval. There is no evidence from physical connections with the inner arms that the outer arms have the same pattern speed. The outer arms, if they are density waves, should extend only to their own outer Lindblad resonance (OLR). For a flat rotation curve, the OLR is at 1.707 times the corotation radius. Considering the m = 2 Fourier transforms in Figure 5, the dip at 0.45R₂₅ could be corotation, because this is also where the bright star formation ridge ends. If the pattern speed for the inner and outer arms were the same, then the OLR would be at 0.77R₂₅. This is in the middle of the outer arm and therefore impossible for a standard spiral density wave. Alternatively, if corotation were at the outer edge of the second peak in the m = 2 Fourier transform, at 0.8R₂₅, then the OLR would be at 1.36R₂₅ for a flat rotation curve. This is close to the outer edge of the third peak in the m = 2 Fourier transform and the end of the outer spiral. This is an acceptable fit, but it implies the unconventional interpretation that corotation is well beyond the end of the star formation ridge.

A second possibility is that the outer arms are driven by manifolds emanating from the Lagrangian points at the ends an inner oval disk. The viewing angles we used to rectify the image were chosen to force the disk to be circular. Different values are found from kinematics. Pence et al. (1990) found a position angle of 41° ± 3° and an inclination of 27° ± 3° from their Hα Fabry–Perot data. Deprojecting with these viewing angles, we find that the inner part stays oval and thus manifold-driven spirals are an alternative. These can extend well beyond the OLR. Also, the shape of the arms with the kinematic deprojection is in agreement with a permissible manifold shape (see, e.g., Figures 4–6 of Athanassoula et al. 2009a and Section 4.3 of Athanassoula et al. 2009b). Multiple arm or flocculent galaxies would not have smooth outer arms in the manifold interpretation because they would not create the strong m = 2 perturbations required.

A third possibility is that the outer spiral is a resonance response to the inner spiral at a different pattern speed (Sellwood 1985; Tagger et al. 1987). For example, the OLR of the inner spiral could be the source of excitation at corotation for the outer spiral. For inner spiral corotation at the edge of the star formation ridge in NGC 1566, at 0.45R₂₅, and an OLR of the inner spiral and corotation of the outer spiral at 0.77R₂₅ (in agreement with the OLR found by Elmegreen & Elmegreen 1990) for a flat rotation curve, then the OLR of the outer spiral would be at 1.3R₂₅. This is also an acceptable fit because that is about the radius where the outer spiral ends.

In NGC 4321, the ridge of star formation in the south ends at about 0.4R₂₅ in the north and beyond that there is a second arm with a small pitch angle extending to the north until ∼0.7R₂₅. A similar ridge and second arm is on the other side of the galaxy. These features make the two peaks in the m = 2 Fourier transform plot of Figure 5. The second arm could lie between corotation and the OLR of the inner spiral. The outer spiral mentioned above is beyond that, from ∼R₂₅ to ∼1.3R₂₅, as shown by a broad ledge in the m = 2 Fourier transform plot. As for NGC 1566, the outer spiral extends too far to end at the OLR of the inner spiral if the inner spiral has corotation at the end of a star formation ridge. The outer spiral in NGC 4321 could be a manifold or resonance phenomenon too. With corotation of the inner spiral at 0.4R₂₅, and corotation of the outer spiral at the OLR of the inner spiral, at 0.7R₂₅, the OLR of the outer spiral would be at 1.2R₂₅ for a flat rotation curve. This is about the extent of the outer spiral.

These examples suggest that smooth outer spirals observed at 3.6 μm in non-SB grand design galaxies are either driven by manifolds surrounding an oval inner disk, or excited by a resonance with the main inner spirals. The resonance may be one where corotation of the outer spiral is at the OLR of the inner spiral. Barred (SB) galaxies may have similar manifolds and resonance excitations but in that case most of the main spiral would be in this form (Sellwood & Sparke 1988; Athanassoula et al. 2009b), and no additional smooth spirals would exist beyond that, unless they are higher-order excitations and too faint to see here. Multiple arm and flocculent galaxies, which do not show smooth outer spirals in this survey, could lack sufficiently strong m = 2 perturbations in the inner disk to drive them.

The outer southern spiral in NGC 4321 has what appears to be a sharp edge in Figure 1; that is, an abrupt transition from the outer arm to the disk or sky. The western arm in NGC 5194 and the inner arms in NGC 1566 also have sharp outer edges. These outer edges do not look as sharp on azimuthal profiles because they are stretched out by a factor equal to the inverse tangent of the pitch angle.

The intensities along strips cutting nearly perpendicular to the spiral arms and going through the galaxy centers are shown for NGC 4321 and NGC 5194 in Figure 9. The strips are 10 pixels wide to reduce noise. Sharp drop-offs occur at the outer parts of several spiral arms. In both galaxies, the arms at ∼0.5R₂₅ show a steep decline on the outer edge. Linear least-square fits to these drop-offs reveal that they have an approximately exponential profile about twice as steep as the local disk measured on ellipse-fit profiles, which averages over azimuth.

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A sharp outer edge corresponds to a strong amplitude spiral superposed on an exponential disk. A typical azimuthal profile of a grand design galaxy (EE84) has a sinusoidal variation in surface brightness, which is a logarithmic intensity scale. Assume the amplitude of this variation is μ₀. Considering also the exponential disk with scale length r_D, this spiral profile means that the intensity varies with radius r and azimuthal angle θ approximately as

$$\begin{equation} I(r,\theta)=I_0\,{\rm exp} ({-}r/r_D+0.4(\ln 10)\mu _0\sin (2[\theta (r)-\theta _0])), \end{equation} \tag{ 5 }$$

where μ₀ is the surface brightness in units of mag arcsec⁻². For a logarithmic spiral,

$$\begin{equation} \theta (r)=\theta _0+\ln (r/r_0){/}\tan (i) \end{equation} \tag{ 6 }$$

with spiral pitch angle i. The inverse of the local scale length at the outer part of the arm, where the radial gradient is largest, is given by

$$\begin{equation} {{dI}\over {Idr}} = -{1\over {r_D}} - {{0.8(\ln 10})\mu _0 \over {r\tan i}}. \end{equation} \tag{ 7 }$$

The first term is from the underlying disk and the second term is from the outer part of the spiral arm. Evaluating the second term, 0.8(ln 10) = 1.84, r ∼ 2r_D for these spirals, and tan i ∼ 0.27 for i = 15°. Thus, the second term is approximately 3μ₀/r_D for μ₀ ∼ 1 in magnitudes. It follows that the total gradient can be ∼4 times the underlying disk gradient for a strong spiral.

Grand design spiral arms can have sharp outer edges that are comparable in scale to the epicyclic radius for stars. This is consistent with theoretical predictions that the arms are nonlinear waves in which a large fraction of the stars have their epicycles in phase in spiral coordinates. Sharp edges also imply strong radial force gradients, softened by the disk thickness, which is comparable to the scale length there. Such radial forcing causes the arm amplitudes to grow by locking in more and more stars to the common epicycle pattern.

Sharp outer edges in tidally induced arms are present in recent simulations by Oh et al. (2010). The sharp edges result from a growing accumulation of stars in a moving wave front at the galaxy edge. They are a caustic in the distribution of particle orbits (Struck-Marcell 1990; Elmegreen et al. 1991). The sharp edge in the southwest of M51 is also in the simulation by Dobbs et al. (2010).

Our sample includes 18 SB, 18 SAB, and 10 SA galaxies. The average value of the Fourier transform for the m = 2 component in the arms is weaker in non-barred galaxies than in barred galaxies; the averages are 0.15 ± 0.042, 0.26 ± 0.12, and 0.34 ± 0.14 for SA, SAB, and SB galaxies, respectively. Similarly, the arm–interarm contrast is weaker in non-barred galaxies, with averages 0.69 ± 0.030, 0.94 ± 0.33, and 0.94 ± 0.44 for SA, SAB, and SB galaxies, respectively. These results indicate that the presence of bars or ovals increases the amplitudes of the arms.

Figure 10 shows bar intensity profiles for the four galaxies in Figure 4. Two of these galaxies are grand design early types (NGC 986 and NGC 1097) and two are flocculent late types (NGC 5068 and NGC 5147). For all barred galaxies in the sample, profiles were measured along and perpendicular to the bar (shown in the top row of the figure for these four galaxies), and from azimuthal averages based on ellipse fits (shown in the bottom row in the figure). Image counts were converted to surface brightness using a formula in the online IRAC Instrument Handbook: μ(AB mag arcsec⁻²) = 20.472 − 2.5 × log (Intensity[MJy sr⁻¹]). The figure shows a difference in the profiles for the early and late types. The bars in early-type galaxies are long and their major axis profiles are flat (that is, slowly declining), as are their azimuthally averaged profiles. The late-type bars are short and have an exponential decline similar to the disk, with just a small flattening on the major axis. Their azimuthal profiles hardly show the bars at all. Evidently, bars in early-type galaxies reorganize the inner disk regions of their galaxies, but bars in late-type galaxies do this to a much lesser extent.

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Flat or exponential bar profiles are listed in Table 2. In our whole sample, the SB galaxies are dominated by flat bars (13 flat, five exponential), while the SAB galaxies have mostly exponential bars (three flat, 15 exponential). Among the early-type galaxies, SAB and SB galaxies are somewhat biased toward flat bars (15 flat, 10 exponential), while the late-type galaxies are dominated by exponential bars (one flat, 10 exponential). In our sample, all of the grand design barred galaxies have flat bars, while all of the flocculent barred galaxies have exponential bars.

Arm–interarm contrasts in the spiral region vary as a function of radius (Figure 5), with either falling, constant, or rising trends. These trends are listed in the last column of Table 3. Three examples of radial contrast variations are shown in Figure 11: falling in NGC 4579, constant in NGC 1566, and rising in NGC 5457. The arrows on the abscissa indicate the locations of the ends of the bars or ovals. The middle and bottom panels of Figure 11 show histograms for the early and late types, coded for falling, flat, or rising arm–interarm contrasts, and subdivided according to Arm Class. The galaxies are also coded by bar types SA, SAB, and SB. The SB grand design and multiple arm galaxies in our sample tend to have falling arm contrasts with radius, while the SA and SAB galaxies for all Arm Classes tend to have rising or flat arm contrasts in our sample. Flocculent and late-type galaxies tend to have rising arm contrasts. We also find that 11 out of 12 galaxies with falling arm–interarm contrasts have flat bars (10 of which are SB and all of which are multiple arm or grand design), while only 3 out of 18 galaxies with rising arm–interarm contrasts have flat bars (and all of these are multiple arm or grand design).

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Falling spiral arm amplitudes beyond SAB and SB bars could be an indication that these bars end near corotation (Contopoulos 1980). Corotation is generally where spiral waves are amplified, in both the WASER theory (Lau et al. 1976) and the swing amplifier theory (Toomre 1981). The arm amplitudes decrease away from corotation and generally approach zero amplitude at the inner and outer Lindblad resonances, where the waves are absorbed. Thus, the decreasing amplitudes can be an indication that the bars are driving the spirals where these two features meet, and that they both have the same pattern speed. The decrease in arm amplitude beyond the bars is also consistent with predictions by the "manifold theory" of spiral structure (Romero-Gómez et al. 2006, 2007; Athanassoula et al. 2009b, 2009c, 2010). For strong bars, the manifolds have the shape of spiral arms and widen with increasing radius, thereby diminishing the arm–interarm contrast. If there is interference with other spiral components, e.g., with a weak leading component, this could lead to bumps on an otherwise smoothly decreasing density profile.

The peak value of F₂ in the bar is a measure of bar amplitude. Table 3 lists this peak value along with the radius (in units of R₂₅) at which it occurs. The radius of the peak is slightly less than, but correlated with, the end of the bar, according to simulations (Athanassoula & Misiriotis 2002). In Figure 12, the peak F₂ amplitude in the bar for SB galaxies is shown as a function of radius at which this peak occurs. Longer bars tend to have higher peak F₂ amplitudes, as found also in optical and near-IR work mentioned previously and by Elmegreen et al. (2007). The Spearman's rank correlation coefficient is 0.68, with a significance at the 99% confidence level. A bivariate least-squares fit to the relation gives a slope of 1.16 ± 0.328 for peak versus relative bar length.

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This result is consistent with ideas of secular bar evolution (e.g., Debattista & Sellwood 2000; Athanassoula & Misiriotis 2002; Athanassoula 2003; Valenzuela & Klypin 2003; Athanassoula et al. 2009a). In simulations of bar evolution (Athanassoula 2003), angular momentum is emitted mainly by near-resonant material in the bar and absorbed mainly by near-resonant material in the outer disk and halo. As bars lose angular momentum, they can become more massive and/or thinner and/or longer. In the two first cases, the peak m = 2 amplitude increases, while in the third one the radius of the peak amplitude increases.

Our Fourier transform results are similar to those found in studies of barred galaxies in other passbands. Two of the galaxies in K_s-band studies by Elmegreen et al. (2007), NGC 986 and NGC 7552, are in our current sample. The m = 2 Fourier component of NGC 986 in the K_s band has a peak of 0.62 at a radius of 0.6R₂₅, and NGC 7552 has a peak value of 0.6 at a normalized radius of 0.6; we find the same peaks and peak radii in 3.6 μm to within ∼10%, as shown in Table 2. EE85 have NGC 3504, NGC 4314, and NGC 7479 in common with our list. Their respective I-band m = 2 bar components are 0.52, 0.8, and 0.8, compared with our 3.6 μm values of 0.53, 0.45, and 0.45; NGC 3504 is about the same in the I band and 3.6 μm, but the other two are stronger in the I band.

Figure 13 shows the F₂ bar peak as a function of Hubble type for SAB and SB galaxies, sorted by Arm Class. There is a steady decrease of bar amplitude with later Hubble types, although the correlation is weak; the Pearson correlation coefficient is 0.403, significant at the 98% confidence level. Early types tend to be grand design (for the highest amplitude bars) or multiple arm (for weaker amplitude bars), while later types tend to be flocculent with weak bars. The early types are predominantly flat bars, while the later types are mostly exponential bars. These results are consistent with optical (EE85) and near-IR (Regan & Elmegreen 1997) studies. The exponential SABs have a mixture of Arm Classes, while the exponential SBs are all flocculent and mostly late type. Conversely, most of the flocculent exponential-barred galaxies are late type, while most of the multiple arm and all of the grand design exponential-barred galaxies are early type. All of the flat bars of either SAB or SB type are early Hubble type with multiple arm or grand design spiral structure. These results reinforce the idea that high amplitude flat bars of either SAB or SB type occur in early Hubble types and drive spirals in the outer disks.

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There are no grand design non-barred galaxies in our sample. Of the 10 SA types, three are flocculent and seven are multiple arm. Among the early-type SA galaxies, two are flocculent and two are multiple arm, while among the late-type SA galaxies, one is flocculent and five are multiple arm. This observation reinforces the idea that bars or oval distortions help drive grand design spirals. Of course, our sample is small; there could be non-barred isolated grand design galaxies that were not studied in this paper. (M81 is an example of a non-barred grand design galaxy, although it has interacting companions.) Kendall et al. (2011) studied a SINGS sample of 31 galaxies; 15 of their 17 barred galaxies (SAB or SB) have multiple arm or grand design structure, while only 7 of their 13 non-barred (SA) galaxies do. When their sample is divided into early- and late-type galaxies, the 11 early-type SAB or SB galaxies include one flocculent, five multiple arm, and five grand design galaxies, i.e., early-type bars are strongly correlated with stellar spirals. Their six late types include one flocculent, five multiple arm, and no grand designs, which means late-type bars are correlated with weak or no stellar spirals.

Figure 14 shows the average F₂ spiral arm amplitude versus the peak F₂ bar amplitude, sorted by SAB and SB types. There is a weak correlation with larger amplitude bars corresponding to larger amplitude arms. The Pearson correlation coefficient is 0.59 for the plotted points, increasing to 0.85 if only SB galaxies are considered, with a significant at the 99% confidence level. A bivariate least-squares fit to the points gives a slope of 0.49 ± 0.08. This correlation has been seen in the near-IR also (Buta et al. 2009; Salo et al. 2010). The result suggests that bars drive stellar waves, and that stronger bars drive stronger waves.

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There is little systematic difference in the bar amplitudes between SAB and SB bar types. In the figure, the SB types are slightly shifted to the lower right, suggesting slightly larger amplitudes for SB compared with SAB bars for a given arm amplitude. However, it is not generally true that SB types are strong bars and SAB types are weak bars. Both types have a range of bar amplitudes that correlate in the same way with Hubble type (Figure 13). The SAB types have lower eccentricities than the SB types, with about the same amplitudes.

Seigar & James (1998a) and Seigar et al. (2003) found no connection between bar and arm strengths in their K_s-band study of 45 spirals, although they did not plot Fourier transform results. Instead, they used a measure of arm strength they called the "equivalent angle," based on measuring the angle subtended by a disk segment containing the same flux as the arm or bar at a given radial range. Their Figure 8 shows a wide variation of arm strength for a given bar strength, but generally smaller bar strengths correspond with smaller arm strengths.