4 Results

In the following subsections we show, for each galaxy of the sample, the |A(p,m)| spectra, with and without noise from field stars, and the |S_m(r)| functions for all the galaxies in our sample. The spectra for $1 \le m \le 6$ are shown in Fig. 3 and the |S_m(r)| functions for $1 \le m \le 5$ in Fig. 1. In all the galaxies in our sample the dominant component is m=2, except in the case of NGC 6814, which shows a dominant four-armed pattern.

  

Figure 1: Sm(r) functions. From left to right and from top to bottom: NGC 157, NGC 753, NGC 895, NGC 4321, NGC 6764, NGC 6814, NGC 6951, NGC 7479 and NGC 7723

4.1 NGC 157

The first impression of this galaxy is that its entire surface is covered with luminous spiral arm segments, intermingled with dust lanes. The most prominent dust lane is on the concave side of the arm that extends from the bulge (very small and without structure) to the north. This lane crosses the arm at only once. Burbidge et al. (1961) found a very flat, almost linear, relative density curve i.e. there is not much mass concentration in the central region of this galaxy. It has been identified as a starburst galaxy (Devereux 1989), with strong H$_\alpha$ and [N II] lines. Zasov & Kyazumov (1981) found very symmetric arms, which suggests there is no substantial large-scale non-circular motion in the spiral zone; but at $R\gt 55\hbox{$^{\prime\prime}$}$ on the northern side, there is an abrupt fall-off in rotational velocity, which Burbidge et al. (1961) did not find in their rotation curve, calculated from H$_\alpha$ and [N II] $\lambda 6583$ emission lines. This effect cannot be due to a companion, because the nearest galaxy, NGC 255, is somewhat more than three degrees away ($\sim \! 1.16$ Mpc). Another important feature of the rotation curve is the long, almost linear, branch, that extend about 8 or 9 kpc

  

Table 1: Relative weight (in %) of the different symmetric components obtained by Fourier transforms and Eq. (7). It is noticeable that in three galaxies the percentage changes after corotation (in NGC 753, NGC 6814 and NGC 7479), in other three galaxies, after or before corotation, two modes have similar weights (NGC 895, NGC 4321 and NGC 6764), and the final three maintain the same and lonely dominant component

from the nucleus. Within the errors it remains constant over this interval. Although other galaxies have this kind of behaviour (like NGC 7479 in our sample), it is not a frequently occurring case. This feature of the rotation curve is incompatible with an exponential decrease in galaxian mass surface (see Zasov & Kyazumov 1981).

With the azimuthal method (Paper III) we found a change in the sign of the difference in the B- and I-band mean position, as well as in the sign of the profile skewness in the B band at $R\sim 40\hbox{$^{\prime\prime}$}$. Also, a dust lane crosses the northern arm at approximately this distance. All this evidence leads us to the conclusion that CR is found at $R\sim 40\hbox{$^{\prime\prime}$}$. In addition, and in spite of the fact that due to the high bi-symmetry of this galaxy the m=2 component is dominant at all radii, we found a local minimum at S₂(r) near $R\sim 40\hbox{$^{\prime\prime}$}$, coinciding with an increase in the other (secondary and odd) components, S₃(r) and S₅(r).

This highly symmetric galaxy shows a spectrum with a two-armed component dominating to the outer part of the disc (Figs. 2 and 4). There is a small radial range ,where secondary components, as m=1 and m=5 dominate slightly over m=2 (Fig. 1), but the OLR is located at this radius (Paper III). In the same figure, near corotation, $R\sim 40\hbox{$^{\prime\prime}$}$, we find a local minimum at S₂(r) and local maxima of the other modes. The Fourier coefficients show a narrow peak at m=2 centred on $p^2_{_{\rm max}}=3$, which corresponds to a two-armed spiral with pitch angle $i\sim 34^\circ$, slightly higher than found the from azimuthal profiles (Paper III). If we fit a Gaussian to the main peak, the maximum is shifted to $p^{\rm g}_{_{\rm max}}=4.5$ which leads to a pitch angle ($i^{\rm g}\sim 24^\circ$) much more similar to the previous one. The use of Gaussians is justified because the p values are integers, so the only pitch angles possible are $90^\circ,\, 63^\circ,\,45^\circ$, etc., while, as we mentioned before, |A(m,p)| represents the "weight'' that a spiral with pitch angle i has in the m-order symmetry. In this case, with the Gaussian fit we found a reasonable average of such values. A secondary maximum at m=1, does not correspond to a one-armed spiral pattern, but reflects the star formation burst with a strong asymmetric component (Figs. 2 and 4).

Using the EEM method, an elongated bulge can be seen from which emerges a small "bar'' of $40\hbox{$^{\prime\prime}$}$, from which, in turn, two vigorous arms originate. But this "bar'' is probably just an artifact of the symmetrizing method, just like the dust lane in the southern arm, which is not present in the original image. Dust is present in the S2 image on the concave side of the arms, until $R\sim 40\hbox{$^{\prime\prime}$}$, beyond which the dust lanes cross to the convex side much more inconspicuously. Finally, the arms end at $R\sim 70\hbox{$^{\prime\prime}$}$, where the OLR could be located (Paper III). The ratio between the radius of both resonances, OLR/CR $\sim 1.75$ is in perfect agreement with that found by EEM. Higher-symmetry images, as in the method of Fourier decomposition, do not show very important structure, despite that found by EEM.

  

Table 2: Data marked with $^\dagger$ are from de Vaucouleurs, those marked with $^\ddagger$ are from Grosbøl. For NGC 4321 whe have used $i=31^\circ$ and PA $= 151^\circ$, from Cepa et al. Data marked with ^* have been calculated in del Río & Cepa (1998b). Finally, those marked with ^** and $^\S$ have been calculated in this work, the former correspond to Considère & Athanassoula method, and the last to Puerari & Dottori method. Angles for NGC 6814 marked with ^* have been calculated with m=4 as dominant mode

4.2 NGC 753

In the image of this apparently small galaxy we can distinguish a brilliant and elongated nucleus. From this nucleus emerge two main arms, with very patchy star formation, and of very irregular brightness, becoming almost invisible beyond $180^\circ$ from their were origin. Around these two arms several others are wound; they are more extended but much weaker and give the impression of a light pinwheel expanding from the nuclear zone to the outest parts of the optical disc. This open structure may be due to a relative close companion, NGC 759, at 0.44 Mpc and $\Delta v \sim 22 \,\rm km\,s^{-1}$(with $H_0 = 75 {\rm \,km\,s^{-1}\,Mpc^{-1}}$).

The skewness and mean position difference in this galaxy were really difficult to measure (Paper III) and none of the results was conclusive, all instead indicating a special situation around $R\sim 30\hbox{$^{\prime\prime}$}$. The pitch angle found by exponential fitting to mean positions in the arms has an abrupt change (from $13^\circ$ to 24$^\circ$) at this position; the skewness and mean position difference behave in a distinct manner before and after this point. Finally, the Fourier transform analysis supports the idea that CR is located at $R\sim 30\hbox{$^{\prime\prime}$}$; at this distance ($\sim 28\hbox{$^{\prime\prime}$}$) we found a local minimum at S₂(r), which is the endpoint of the stronger and more symmetric arms, and close to this radius ($R\sim 38\hbox{$^{\prime\prime}$}$) S₃(r) becomes the main component until the end of the visible disc.

Fourier decomposition is not expected to provide much clarification; in fact, the spectra of this galaxy show a noisy two-armed component (with $S/N \sim 17$), and a moderately strong three-armed component, while the rest of components are not at all significant. The maximum in the S₂(r) mode is located at $p^2_{_{\rm max}} =-5$, which leads to a pitch angle $\approx 22^\circ$, somewhat less than that found with a logarithmic fit ($\sim\!28^\circ$). We must take into account that the signal is quite noisy. If we estimate the maximum position with a Gaussian fit, we find $p^{\rm g}_{_{\rm max}} = -3.88$, or a pitch angle $\approx 27^\circ$, in much better agreement with the logarithmic fit. In both cases, this relatively good agreement with a poor fit is due to the mixing of the inner (more closed) and outer (much more open) arm shapes. The Fourier transform also treats the arm as there were no breaking; in fact the first pitch angle (which is provided by a maximum in the m=2 spectrum) is closer to the value calculated only for the inner arm (where the image m=2 is more intense) using a logarithmic fit, than to the value obtained from the outer arm.

The bi-dimensional distribution of S_m(r) confirms that found from the spectra. In spite of the low intensity of all components, the bi-dimensional distribution S₂ stands out from the others, showing two short tightly wound arms around the nucleus that split at $R\sim 30\hbox{$^{\prime\prime}$}$, one part following the same path part, and the other opening out, as shown in the logarithmic fit (Paper III) and S2 image. In the second place, m=1 reveals irregular star formation in zones close to the nucleus, with a shape very similar to that of A2. The wide fringe around $R\approx 50\hbox{$^{\prime\prime}$}$ is consequence of increasing intensity at the end of the eastern arm. The first relative minimum in S₂(r) should be produced by internal dust located symmetrically with respect to the centre and perpendicular to the major axis, because all components are affected. The next minimum, at $R\sim 28\hbox{$^{\prime\prime}$}$, matches reasonably well with the sign change in $S^{\rm B}$ and $\overline{\theta^{\rm B}} - \overline{\theta^{\rm I}}$ and with a sharp change in pitch angle (see Paper III). This is another indication that supports the idea that CR is near $R\sim 30\hbox{$^{\prime\prime}$}$. The next relative minimum, at $R\sim 38\hbox{$^{\prime\prime}$}$, marks the end of the more intense arms, and a change in dominant mode from the m=2 dominant pattern (due to the strong arms) to m=3, until $R\sim 60\hbox{$^{\prime\prime}$}$ (due to the eastern arm and the external splitting of the western arm). However, the emission in this zone is too weak for these features in the S3 image to be appreciated.

4.3 NGC 895

This galaxy has two long and symmetric arms, as can be see in Figs. 2 and 4, and, external to these, several others that are fainter and rather patchy, as would correspond to a prototypical class 9 galaxy. The m=2 component shows a high, narrow peak, with $S/N\sim 61$, clearly dominating over the other components. The high values of the m=4 and m=6 components, in addition to harmonics of m=2, probably reflect the structure of these secondary arms. However, the m=1 component represents the asymmetric star formation produced in the western arm, which is wider than the eastern one. The maximum of |A(2,p)| is $p^2_{_{\rm max}}=-4$ (-3.55 after a Gaussian fit), which leads to a pitch angle of $26^\circ (29^\circ)$, in very good agreement with the value found using a simple least-squares fit of a logarithmic function to the mean positions, in only one arm (Paper III). This is obviously due to the high degree of symmetry in both arms. Decomposing the image following the EEM method reveals that 79% of the total emission comes from S2, and that S3 and S4 have only remains of axisymmetric and bi-symmetric images, respectively. The arms are relatively short and end in the most intense star formation just beyond corotation, although the weakest parts reach the OLR. At $R\sim 75\hbox{$^{\prime\prime}$}$,where corotation is located (Paper III), the m-components show a small change in the dominant mode over $\sim 10\hbox{$^{\prime\prime}$}$, where the main component, S₂(r), drops to a local minimum, while S₄(r) exceeds slightly this value.

The logarithmic fit of mean positions (Paper III) is exceptionally good in this highly bi-symmetric galaxy, whose arms seem to end slightly beyond CR. It might be that the high degree of bi-symmetry occurs because the arms almost coincide with the main part of the pattern. As previously mentioned, this is an observational confirmation that, before corotation, short and long waves are found, whereas, beyond corotation, only short trailing waves, which only just reach the OLR, are seen. In Fig. 1c it can be seen that the S₂(r) component strongly dominates the others, and that at CR ($\sim 75\hbox{$^{\prime\prime}$}$) there is a soft local minimum of S₂(r) as well as slight local maxima of S₄(r) (which even overshadows S₂(r), S₅(r) and S₆(r), which are not present in the figure). The bi-dimensional S_m(r) distribution shows a galaxy with a complex structure, with two long arms that originate in the very inner part of the small bulge, an S₁ component that reproduce the different star formation rates of both main arms, and S₃ and S₄ components that reproduce the outer and fainter arms.

  

Figure 2: Fourier anti-transforms. Top: NGC 157. Bottom: NGC 753

 

Figure 2: (continued) Fourier anti-transforms. Top: NGC 895. Bottom: NGC 6764

 

Figure 2: (continued) Fourier anti-transforms. Top: NGC 4321. Bottom: NGC 6814

 

Figure 2: (continued) Fourier anti-transforms. Top: NGC 6951. Bottom: NGC 7479

 

Figure 2: (continued) Fourier anti-transforms. NGC 7723

The first pronounced local minimum in S₂(r) is at $R\sim 20\hbox{$^{\prime\prime}$}$, but inside this radius a small spiral is enclosed, which is not visible either in the flux image (Paper II) or in S2. A second important minimum occurs at $R\sim 32\hbox{$^{\prime\prime}$}$, close to the theoretical 3:1 ILR position. The last S₂(r) minimum, at $R\sim 75\hbox{$^{\prime\prime}$}$,coincides with the end of the brightest arms, with the sign change in $\overline{\theta^{\rm B}} - \overline{\theta^{\rm I}}$, with the tendency change in B skewness (Paper III) and with a change in dominant mode, which passes from S₂(r) to S₄(r) over a small interval of $10\hbox{$^{\prime\prime}$}$.

Again we have found serious anomalies near corotation.

  

Figure 3: Amplitudes of |A(m,p)| coefficients. Dotted lines, with field stars, filled lines, without field stars. From left to right and from top to bottom: NGC 157, NGC 753, NGC 895, NGC 4321, NGC 6764 and NGC 6814

 

Figure 3: (continued) NGC 6951 (left), NGC 7479 (right) and NGC 7723 (bottom)

4.4 NGC 4321

This impressive galaxy gives the impression of a fantastic whirlpool of light, with two long arms and a large brilliant bulge. Around this can be distinguished four small arms, in principle completely independent of the two main ones. This is the largest galaxy, in relative terms, in our sample, because it is the second closest (beyond NGC 6951) and the second largest (beyond NGC 753). In their H$_\alpha$ study, Arsenault et al. (1988) found an unusual double-lobe structure in the circumnuclear region. They located CR at $\sim\! 70\hbox{$^{\prime\prime}$}$ and the OLR a little further than $128\hbox{$^{\prime\prime}$}$, while for the ILR they found found two values, the IILR ($\sim\! 5\hbox{$^{\prime\prime}$}$) and the OILR ($\sim\!16\hbox{$^{\prime\prime}$}$). Cepa & Beckman (1990b), with higher spatial resolution, determined that this structure is actually an inner spiral which persists after continuum subtraction and that it emits, like the nucleus, in H$_\alpha$ and N II. The corotation radius, predicted by Cepa & Beckman (1990b) from the ILR positions, delimited by the inner spiral arms (the two most prominent of these continue in the main spiral) is $R\sim 80\hbox{$^{\prime\prime}$}$, in other words, approximately where dust lanes on concave side of the arms cross over to the convex side (Cepa et al. 1992). From $R\sim 100-110 \hbox{$^{\prime\prime}$}$ the brightness in both arms decreases suddenly, and the number of H II regions too (Cepa & de Pablos 1998).

In the EEM symmetrized images it can be seen that, in general, the best part of the star formation and the B-flux distribution is in the S2 image. In the inner region there is a small spiral, wound in the same sense as that of the two main arms. This structure was observed for first time by Hubble (1934), and is present in all bands. It is a structure with two main arms, which also appear in Fourier transform, and two secondary arms. In the S2 image it seems that two arms extend radially from the nucleus and, from $R\sim 60\hbox{$^{\prime\prime}$}$, are "broken'' and then turn around galaxian centre. This strange behaviour can be explained by the presence of a very wide bar, which can be discerned in Fig. 4d and which ends at $R\sim 60\hbox{$^{\prime\prime}$}$;arms extend from this point. Also we can see that two spiral arms emerge from each tip of this bar. The outermost arms rapidly merge with the bright disc; this suggests that fourth-symmetric components could be very important or even dominant radially, but this cannot be appreciated in the S4 image. The asymmetry in the star formation is very rare. This is shown in Fig. 4d, where, in A2, it is the different sizes of these formation regions that matter, rather than their different locations.

It is not surprising that Fourier transform of this galaxy show a single high maximum in m=2 with $S/N\sim 53$. The other components show a lower S/N ratio and much lower peaks. An exception is m=4, whose peak value is a half that of the m=2 one, and whose $S/N\sim 32$.

The maximum in the m=2 coefficient is centred on $p^2_{_{\rm max}}=3$, which corresponds to a dominant two-armed spiral with pitch angle $i\sim 34^\circ$. Again, a Gaussian fit changes this result slightly: $p^{\rm g}_{_{\rm max}}=3.9$ and $i^{\rm g} \sim 27^\circ$. This value is in rather good agreement with that found from a logarithmic fit to the northern arm (Paper III). Other authors find even lower values; Kennicutt (1981), for example, gives $i = 15^\circ \pm 3^\circ$. Due to the enormous difference between RC3's and Grosbøl's values for the position angle and our value (Table 2), we have used the method of Considère & Athanassoula to calculate the best pair (PA, $\omega$), centred on ($30^\circ, 32^\circ$), with an interval of $\pm \,33^\circ$ between them, so Grosbøl's values are also included. The result was that all 568 pairs have $p^2_{_{\rm max}} = 0$.

Figure 1d shows the amplitudes of the inverse Fourier transforms for different m values. For $R<20\hbox{$^{\prime\prime}$}$ the components diverge. Over the interval all the components are rather similar, and none predominates over the others. That is due to the wide bar, where every component has its own representation (even those that lack importance). In a small fringe () S₂(r) has a jump, followed by S₄(r), due to bursts of star formation at the end of the bar. From this point outwards to corotation, S₄(r) is clearly dominant, while S₂(r) decreases even more than S₃(r) and S₅(r).

4.5 NGC 6764

This barred galaxy is the first of two Class 5 (Elmegreen & Elmegreen 1987) galaxies that we analyse in this sample. The emission is mainly concentrated in the brilliant nucleus and in two starbursts at the end of the bar. Due to strong optical emission lines in the nucleus, Rubin et al. (1975) classified this galaxy as a Seyfert 2. Later on, Osterbrock & Cohen (1982) obtained a low-excitation spectrum, so they classified it as a LINER galaxy. These authors also found typical emission from Wolf-Rayet stars in the nucleus, which is direct evidence for recent massive star formation. Eckart et al. (1991) measured CO and ¹³CO emission, as well as obtaining near-infrared (J, H and K) photometry, and calculated an almost linear rotation curve, with a velocity shift between the extremes of the bar of 250 $\rm km\,s^{-1}$. The arms emerge from the edge of the bar, and are very short and faint. Before eliminating field stars, all the Fourier transform components are purely noise, with no distinction among them. That is because the signal, when it exists, is below the noise level. Without these stars the coefficients are consistent with those expected in a barred galaxy; the main m=2 component followed by secondary m=4 component. In this case star formation is so rare outside the nucleus and bar extremes that the m=1 component is not relevant at all. The maximum at m=2 is centred on $p^2_{_{\rm max}}=1$, which corresponds to a pitch angle of $i\sim 63^\circ$. A Gaussian fit shifts this result to $i^{\rm g}\sim 53^\circ$. Even this value is too large for those tiny and almost circular arms. The same problem is present in all the barred galaxies in our sample; the m=2 coefficient is hardly affected by bar emission, which has a pitch angle of $90^\circ$, so the pitch angle derived from $p^2_{_{\rm max}}$ gives an idea of the relative importance of the bar versus the arms, more than the real arm shapes. Looking at bi-dimensional distribution of the Fourier coefficients, it is evident that S₂ is the only important component, and that S₄ contains information on both the arms and the bar, but not on the four-armed structure. The most striking feature of Fig. 2e is that the winding sense of S₁ is contrary to that of the main modes, S₂ and S₄. If, as Pasha (1985) asserts, the galaxy is trailing, then this means that it contains a leading component. The phenomenon is not new, even though it is rather infrequent; Considère & Athanassoula (1982) found both trailing and leading arms, represented by different modes, in a certain galaxy. For ,where there is a local minimum in S₂(r), emission of all the components drop (except for m=1, which falls at $R\sim 70\hbox{$^{\prime\prime}$}$, but this is due to a field star that we could not remove because it is too close to the end of one of the arms). The radius at $25 \,\mu$ is also $R\sim 65\hbox{$^{\prime\prime}$}$. If this is the OLR, as claimed by Elmegreen & Elmegreen (1995), and the rotation curve is locally flat (Eckart et al. 1991), then corotation should be at $R\sim 30-31\hbox{$^{\prime\prime}$}$,coinciding with the deepest local minimum of S₂(r). But this radius is inside the bar (which reaches $R\sim 37-40\hbox{$^{\prime\prime}$}$), and we have not studied this range in Paper III. Instead, we have found outlines of change in skewness and different mean position signs at . This range corresponds to a local minimum in S₂(r) and a change in dominant mode (from m=2 to m=4). If corotation falls between these values, the OLR should be at , too far away, taking into account the galaxian dimensions.

We cannot reach any conclusions concerning the resonances in this galaxy by any method. The arms are much too faint in our image to distinguish different features away from the bar; they are also intrinsically faint, as the very low amplitude in the I band demonstrates (Paper III).

  

Table 3: Summary of correlations between change of modes and minima in dominant modes at or around corotation for the galaxies of the sample. NGC 6814 has been excluded since its arms end at corotation

4.6 NGC 6814

This is a clear exception in our sample, it is a grand design spiral with a four-armed dominant pattern. The visual appearance of multiple arms is confirmed in the Fourier decomposition. Beyond $R\sim 18\hbox{$^{\prime\prime}$}$, where its bright nucleus is dominant (it is a Seyfert 1 galaxy, Ulrich 1971), the S₄(r) component is stronger than the others, included the S₂(r) component. Perhaps due to this anomaly the arms die out at corotation, as derived from spiral density wave theory (first because with m=4 the range of existence is shorter, and secondly, because leading waves die at CR).

This is the galaxy with least inclination in our sample, only $7^\circ$(Grosbøl 1985; Paper II) although other authors give higher values ($21^\circ$, de Vaucouleurs et al. 1991; hereafter RC3). This Seyfert 1 galaxy has strong H$_\alpha$ emission (Knapen et al. 1993). The nuclear region dominates the total emission. The low inclination of this galaxy permits us to appreciate more easily that the arms trace the disc. As they go further from the nucleus, the two arms that emerge from it begin to bifurcate several times, that gives a multi-armed aspect to the galaxy. The brightest arms complete approximately one revolution up to $R\sim 49\hbox{$^{\prime\prime}$}$. From this radius, their luminosity decays abruptly, ending at $R\sim 65\hbox{$^{\prime\prime}$}$. Dust seems to be present in a weak but constant manner, mainly on the convex sides of the arms, out to $R\sim 21-27\hbox{$^{\prime\prime}$}$, where it crosses them for the first

  

Figure 4: Symmetric and asymmetric images from the EEM method. From left to right and from top to bottom: NGC 157, NGC 753, NGC 895, NGC 4321, NGC 6814, NGC 6951, NGC 7479 and NGC 7723

time. Several more crossings are seen, but dust is not conspicuous on the concave sides of the arms.

Fourier decomposition with field stars is very noisy (as in the case of NGC 6764). When cleaned, the signal is very low, in fact the maximum is at |A(4,p)|, with an intensity of only 0.065. This is the only case in our sample where |A(2,p)| is not the most important coefficient. It is worthwhile noting that the intense peak in Fig. 1f is due to a field star over the southern arm, so all bi-dimensional distribution images are affected by it.

Except for the very inner regions () where S₂(r) is the principal component, the image is dominated by a four-armed pattern, with an important S₃(r) contribution (see Table 1).

Corotation is presumably at $R\sim 45\hbox{$^{\prime\prime}$}$ (Paper III) close to an S₄(r) local minimum. As we have already mentioned,

  

Figure 5: Comparison between S2 (left) and S2-S4 (right) images. From left to right and from top to bottom: NGC 157, NGC 753, NGC 895, NGC 4321, NGC 6814, NGC 6951, NGC 7479 and NGC 7723

this is the only case where we do not find a two-armed pattern dominating the morphology of a galaxy; it is also the only case where, in our previous work (Paper III), we found a leading dynamical pattern. NGC 6814 has no companion which can induce leading arms, as Pasha (1985) found in every leading galaxy in his study. These peculiarities could be just a coincidence, but it is more probably that the phenomenon which inhibits leading behaviour in spiral galaxies is related with a mechanism that encourages the formation of two main arms. If this is true, the absence of such a mechanism permits the presence of four-armed leading patterns.

This is an exception in our sample, it is the only case where a grand design spiral has a dominant four-armed pattern. The visual appearance of multiple arms is confirmed in subsequent Fourier decomposition. Although

  

Figure 6: Convergence of angles with Considère & Athanassoula method

this is not a conclusive proof, it confirms the spiral density wave theory prediction that leading waves, both short and long, end at CR, even when the four-armed pattern could extend up to the 4:1 OLR, theoretically at $R\sim 70\hbox{$^{\prime\prime}$}$.

4.7 NGC 6951

This is a galaxy whose major emission is concentrated in a bar-like structure; in fact it has been classified as both S and SB in different catalogues (RC3; Sandage & Tammann 1987). This structure, together with two relatively weak arms, make the Fourier method of finding optimal values of PA and i difficult because $p^2_{_{\rm max}}$ values are always centred on 0.

  

Figure 7: Convergence of angles with Puerari & Dottori method

  

Figure 8: Comparison of |A(2,p)| coefficients found with the Considère & Athanassoula method for the best pair of angles (PA, $\omega$, thick line) and our values (dotted line)

  

Figure 9: Comparison of |A(2,p)| coefficients found with the Puerari & Dottori method for best pair of angles (PA, $\omega$, thick line) and our values (dotted line)

This galaxy has the characteristics of barred and normal spirals. It is a clear example of transition between to "pure'' types of spirals. Several prominent dust lanes are visible, one of these originating in the central region. The bright bulge is elongated in the direction of this dust lane, almost perpendicular to the major axis. The arms are relatively short (around $180^\circ$), the most symmetric part ending approximately at $R\sim 100\hbox{$^{\prime\prime}$}$and extending somewhat beyond some starbursts. The arms are considerably fainter than in the other grand design galaxies in the sample.

The |A(2,p)| coefficient is, without doubt, the most important in the Fourier decomposition, with a unique and very pronounced peak ($S/N\!\sim \!32$), as is expected in the case of barred spirals. Second in order of importance is |A(1,p)|, ($S/N\!\sim\! 16$) and finally, |A(4,p)| ($S/N\!\sim\! 15$). The other components ($\vert A(3,p)\vert,\,\vert A(5,p)\vert,\, {\rm and\, especially} \,\vert A(6,p)\vert$)are essentially noise, without significant peaks. A substantial improvement is appreciated in all the coefficients when field stars are eliminated.

As with NGC 6814, the |A(4,p)| coefficient presents a series of peaks which are symmetric with respect to p=0; this is the reason why the signal-to-noise ratio is much lower than would be expected from Fig. 3g. The maximum of |A(2,p)| is at $p^2_{_{\rm max}} = 0$,with and without field stars, which corresponds to arms that extend radially from the nucleus. This could be due to the deprojection not being completely correct, or, more probably, to the influence of the bar, which dominates the Fourier transforms in the same way as the images obtained by the EEM method.

A Gaussian fit to |A(2,p)| does not vary the results substantially ($i^{\rm g} \sim 77^\circ$). The lack of a symmetric arm in the one studied provokes this behaviour in transforms. Obviously, the dominant component is m=2, but it does not account for the arms in the same way that the EEM method fails to do. There is an arm that is clearly dominant over the other one, and this is shown in Fig. 2g, where we see the bar and the incipient arms S2, as in S2; S₁ show the strongest arm, which is more active in terms of star formation. Finally, in S₄ a feature common to all spirals with a strong bar can be seen: the "arms'' are spurious and are really a consequence of the flux between the wide bar and the main arms. The maximum of the |A(1,p)| coefficient is at $p^1_{_{\rm max}}=-1$, which leads to a pitch angle $i\sim 45^\circ$ ($i^{\rm g} \sim 43^\circ$), in reasonable agreement with that found for logarithmic fits to the main arm (Paper III). This confirms that our initial deprojection was correct. The result is not improved either with Considère & Athanassoula or the Puerari & Dottori method. The latter yields similar PA and $\omega$ values (although somewhat higher for $\omega$), and the same value for i, but the convergence (see Fig. 7) is not very good. The first gives a bad convergence (Fig. 6), low signal (Fig. 8) and high dispersion between the observed and fitted pitch angles. The value found in Paper III falls between these.

The corotation radius is not clearly defined, although all the evidence points to $R\sim 70\hbox{$^{\prime\prime}$}$ (Paper III). Nothing special occurs at this radius in S₂(r), but it seems to be the end point of S₁(r).

In the bi-symmetric image, the arms dilute in the disc at $R\sim 75\hbox{$^{\prime\prime}$}$;nevertheless, in A2 we can follow their faint trace until the limit of the frame, i.e. $R\sim 150\hbox{$^{\prime\prime}$}$, which could correspond to the OLR. This value is in reasonably good agreement with that found theoretically ($\approx 152\hbox{$^{\prime\prime}$}$)assuming a locally flat rotation curve and a corotation radius of $R\sim 70\hbox{$^{\prime\prime}$}$. In A2 there is also present a ring around the nucleus, slightly shifted forward by $90^\circ$ in the image ($\sim 257^\circ$ in the original frame), with a radius $R\sim 9\hbox{$^{\prime\prime}$}$. This singularity corresponds to the (theoretical) ILR, with the same assumptions as before.

4.8 NGC 7479

This is another important barred galaxy in the sample, but the main difference from the former galaxy is that in this case there is a strong blue arm, which has a pitch angle rather similar to that found with a simple least-squares fit.

This galaxy has been identified as a starburst type with moderately wide CO lines (Young & Devereux 1991). This barred spiral shows a weak continuum in its nucleus, with faint H$_\alpha$ and [N II] emission, whereas in the bar it is stronger. The H$_\alpha$ line shape implies the presence of large and brilliant H II regions along the bar (Hua et al. 1980). At the end of the bar, which is $100\hbox{$^{\prime\prime}$}$ long, two arms originate; one of these is bifurcated even before it leaves the bar. This effect, probably an optical one, may be caused by a dust lane accompanying the east-north arm to its end. The west-south arm, much more regular, is also accompanied by a dust lane, but which is shorter than the other one, because it finishes near the supernova remnant of SN 1990V, at $R\sim 60\hbox{$^{\prime\prime}$}$.

The coefficient |A(2,p)| shows the highest maximum in our galaxy sample, $\sim\! \!40\%$ larger than the same coefficient for NGC 6951, the other "intense'' barred spiral in the sample. Although the noise in all the coefficients diminishes notably when field stars are removed, (|A(2,p)| goes from $S/N\sim 18$ to $S/N\sim 71$, and |A(1,p)| from 8 to 48), only the maxima of the secondary components (|A(3,p)|, |A(5,p)| and |A(6,p)|) vary. The maximum of |A(2,p)| is at $p^2_{_{\rm max}}=-1$, which corresponds to $i\sim 63^\circ$, much higher that found in Paper III. A Gaussian fit gives $i^{\rm g}\approx 49^\circ$, which is closer to the value calculated for the first part of the arm (before corotation). Calculation with the Considère & Athanassoula and Puerari & Dottori methods does not improve the results. From Fig. 2g it is not difficult to understand why the pitch angles are so high; the bulk of the image is in the bar, whose pitch angle is $90^\circ$.

A second important maximum occurs in |A(1,p)| (S₁ represents the outer part of the arms, in particular the western arm), with $p^1_{_{\rm max}}=-6$,i.e. a pitch angle $i\!\!\sim\!\! 10^\circ$, somewhat lower than calculated by Paper III for the outer arm (after corotation). This is because arms become circular ($i \sim 0)$) at $R\sim 90\hbox{$^{\prime\prime}$}$, so that azimuthal profiles cannot be computed and the fit must be performed over a range smaller than that of the Fourier transform. In addition to the bar, two small arms, with different pitch angle, can be seen in S₂ (Fig. 2g). Their origin (which marks the tip of an intense emission) matches with an S₂(r) minimum and a change in dominant mode (from S₂(r) to S₁(r)). In fact, all the components exceed S₂(r) in an interval around $R\sim 85\hbox{$^{\prime\prime}$}$, but S₁(r) is dominant until the end of signal.

With the EEM method we see also that all symmetry is in the bar; not even the bulge is symmetric (a part appears in A2). The unusual symmetry of this galaxy is reflected in S2, where there is only 69% of the total emission, much less than in other grand design galaxies, despite the bright bulge and the star formation zones in the bar. The short arms visible in S2 hardly reach $R\sim 65\hbox{$^{\prime\prime}$}$, far away from corotation. In fact, there is no appreciable change at $R\sim 85\hbox{$^{\prime\prime}$}$. This is mainly due to the absence of a clearly defined second arm for $R\gt 65\hbox{$^{\prime\prime}$}$, which eliminates, in the symmetrizing process, any indication of the western arm.

In Paper III we established that corotation is at $R\sim 85\hbox{$^{\prime\prime}$}$, where we found changes in the skewness signs, both in the B and I bands, and in pitch angle, showing that the arm is "broken''. With the Fourier transform method we now find more evidence that supports our former result, such as a local (and global) minimum in the main component, S₂(r), local maxima of the other components and a change (in this case, definitive) of dominant mode.

4.9 NGC 7723

This galaxy, as occurs with NGC 6764, is Class 5 (Elmegreen & Elmegreen 1987), i.e. its arms are less defined than in the other galaxies. It can be seen that the arms are two faint small outlines that originate at the end of the bar and very soon bifurcate. One part (of each pair) is closed around the nucleus, and the other part is partially opened on the outer side. The "inner'' arms form an almost complete ring around the small bar.

Figure 4h shows a galaxy with a prominent bulge (8.4 kpc) and a bar $40\hbox{$^{\prime\prime}$}$ in length that dominate the S2 emission, which contains 90% of the total flux. The high percentage is not due to the arms, which are relatively short and faint, but to the bulge and bar. The asymmetric image, in addition to part of the bulge, contains "three'' moderately intense arms. Two of these correspond to the outer main arms and the third, the most open, one to the prolongation of one of the secondary arms. The high degree of symmetry could be due to a density wave generated by the bar, with a very low intensity, insufficient to trigger significant massive star formation processes along a very well defined spiral structure. The brightest part of S2 ends at $R\sim 35\hbox{$^{\prime\prime}$}$, where the sign of the difference in mean position changes (Paper III), while the arms in A2 are brighter from this point.

In contrast with the other galaxies, S3 has a relatively important weight in the structure of this galaxy. There is a ring in S3 at $R\sim 20\hbox{$^{\prime\prime}$}$.Corotation was not firmly defined in Paper III, but the difference in mean position, in one arm and the end of the brightest part in the other point to $R\sim 35\hbox{$^{\prime\prime}$}$ as a possible corotation radius. In this case, $R\sim 20\hbox{$^{\prime\prime}$}$corresponds to the 4:1 ILR. The second (Schwarz 1984) is in this resonance where rings are formed when bars are not very strong.

The Fourier coefficients are noisy and faint, even |A(2,p)| which, as for all barred galaxies, presents a sharp high peak, although of very low intensity (lower than the m=2 peak of the other class 5 galaxy, and comparable only to the m=4 maximum of NGC 6814). Also, as for almost all barred galaxies, $p^2_{_{\rm max}} = 0$, and with a Gaussian fit pitch angle change to $i^{\rm g} \sim 63^\circ$, still too much large for some small, tightly wound arms.

Unlike NGC 6764, this galaxy has a rich Fourier structure. Figure 2h is dominated by the image of S₂, which contains the bar and the brightest part of the arms. S₃ also has a clearly defined structure, which corresponds to the three brighter inner arms, and their continuation, much more open and faint, out the edge of the disc, with a pitch angle $i_3 \sim 20^\circ$, similar to that found by Paper III.

The ring already mentioned appears in S₁ and S₃ (incomplete in each frame, but nearly closed in the S₁+S₃ image). The S₄ component dominates S₂ in a small interval around corotation. At $R\sim 50\hbox{$^{\prime\prime}$}$ (the 4:1 outer resonance) S₃ becomes higher than S₂, out to $R\sim 58\hbox{$^{\prime\prime}$}$($\approx$ 3:1 outer resonance). The OLR, at $R\sim 70\hbox{$^{\prime\prime}$}$, marks the point at which the secondary components definitively overcome S₂.

In summary, no result, separated from the global context, is conclusive concerning CR, but all the results together lead us to the conclusion that CR is effectively located at $R\sim 35\hbox{$^{\prime\prime}$}$.

  

Table 4: Pitch angles calculated from Considère & Athanassoula method (marked with CA), from Puerari & Dottori method (marked with PD) and from our work (marked with OW). All values have been found with m=2 as main component, except those marked with $^\dagger$, that have been calculated with m=4 dominant mode