5 Conclusions

The Fourier transform method provides us with a natural way of studying the different symmetries inside a galaxy. Spiral arms are quite well approximated by logarithmic spirals, so with a very few terms (usually two or three, and never more than four) it is possible to reproduce the structure and main features of galaxies, without losing three-dimensional information. It is also possible to quantify the relative importance of each component for a pre-fixed radial interval. With S_m(r) functions we can easily locate maxima and minima, and relate them with different resonances, where these exist, but the most important application we have found is that, with these functions, the change of dominant symmetry near or at corotation is (very) clear.

In all cases where corotation was clearly defined (Paper III) the Fourier transforms have confirmed this radius. In cases where corotation was not so firmly established, Fourier transforms have helped to locate different features, such as rings, breaks, changes in dominant mode, etc., and with these it was possible to find a reliable value for this resonance. Although corotation does not in all cases match a local minimum in S₂(r) (or S₄(r) in NGC 6814), it is usually close to one of them, close to maxima of secondary components (generated by the stochasticity of stellar orbits at this point), or close to a change in dominant mode, etc. With this information we have confirmed (or discarded) the location of corotation and the other main resonances.

Both methods (Considère & Athanassoula's and Puerari & Dottori's) have serious drawbacks in calculating the best pair of angles (PA and $\omega$) for a galaxy.

The Considère & Athanassoula method converges quite well to a pair of values, except for NGC 6814, where any PA is equally good, because the galaxy is almost face on, and NGC 6951, where the ratio |A(2,p)|/|A(0,p)| is not very high (it is strong barred, so p²=0 is important in any deprojection). But this pleasing convergence, which can be seen in Fig. 6, is somewhat spurious. If we represent the new |A(2,p)| coefficient vs. that calculated with our PA and $\omega$ values (Fig. 8), we can see that in the majority of cases the amplitude is lower (i.e. the arms are "less'' logarithmic) and the reason why the ratio is so high is because only the point with p=0 has a very low amplitude, close to zero (among 568 pairs the probability of finding such a case is finite). Furthermore, when galaxies are strongly barred and have faint arms, the correct deprojection could lead to $p^2_{_{\rm max}} = 0$, due to the influence of the bar. In this case this method also fails, because it chooses the first pair of values with $p^2_{_{\rm max}}\ne0$.

The Puerari & Dottori method has the opposite problem: while coefficients have a "normal'' shape (i.e. they do not have a gap in p=0) and in some cases are even higher than those calculated previously (Fig. 9), as is the case, for example, of NGC 7723, with more than twice the new amplitude, the convergence is worse (Fig. 7). It also presents the problem that galaxies with strong bars and faint arms have a very important contribution at p²=0, which is strongly favoured when the deprojection, even if wrong, is "aligned'' with the bar; i.e. when the new major axis is almost perpendicular to the bar. This is a cumulative effect. When the deprojection is wrong, with inclination angles larger than the correct value, the isophotes are elongated in the $\widehat{y}$-axis; hence, if the bar is also aligned with the $\widehat{y}$-axis, by varying the position angle the p² coefficient can reach values very close to zero. In this case also, the outer isophotes are not circular with the new deprojections.

Finally, both methods have the same problem in that they both optimize the best "view'' of a galaxy to obtain the most perfect logarithmic arms. This works only

  

Figure 10: Different projections. For each galaxy, with values found in Paper III (left), with those found with the Considère & Athanassoula method (centre) and with those found with the Puerari & Dottori method (right). From top to bottom: NGC 157, NGC 753 and NGC 895

 

Figure 10: (continued) From top to bottom: NGC 4321, NGC 6764 and NGC 6951

  

Figure 10: (continued) NGC 6814 (deprojected following m=2, upper, and m=4, bottom) Figure 10: (continued) NGC 7479 (top) and NGC 7723 (bottom)

for a very few galaxies, such as NGC 895, which has two long bi-symmetric arms (Fig. 5 and Fig. 2), whose fit to a logarithmic spiral is exceptionally good (Paper III). But this is not generally the case: logarithmic arms are only an approximation, and are not good enough when arms multiply or bifurcate, when one arm is much brighter than the others, when the pitch angle changes due to processes at corotation, etc. In such cases, both methods present a view of the galaxy which is not real or correct. Even the outer isophotes cease to be circular with these deprojections, and we think that it is much more physically plausible that the outer isophotes are almost circular than that the arms are perfectly logarithmic.

We have not found any relevant tri-symmetry component (except in NGC 7723) with the EEM method, much less four-symmetric components, even in those cases where Fourier transforms of such degrees dominate the S₂ image. This could be because different symmetries are mixed in the same image, while the Fourier transform method separates better the different components. Figure 5 shows a graphical comparison between the S₂ and S2-S4 images, demonstrating that symmetry separation is performed in a different way with each method. With the Fourier transform, points with a fixed flux that appear twice and only twice in the image are kept, whereas with EEM symmetrization, points that appear two, four, and presumably any multiple of two, times remain in the result, thereby smoothing the contrast, and do not allow the characteristic features of the bi-symmetric image to be found. We have not performed the symmetrization beyond S4, so we cannot assert that the same occurs with S3 and S4 images, but this is probably the reason for the lack of information on these images, which can be easily appreciated in S₃ and S₄. When there is a strong, or a wide bar, the FT component m=4 is "artificially'' enhanced, as pointed by Elmegreen et al. (1993). This is the case for NGC 6951 and NGC 7479. Other barred or pseudo-barred galaxies in the sample do not show this behaviour. But even in those cases, m=4 (or higher modes) is note the dominant component in the relevant part of the galaxies.

It is tempting, as is commonly done, to associate the flux minima to resonances (as Elmegreen & Elmegreen did in 1995), but this idea can lead to confusion. In some cases the brightest arms end near corotation (e.g. NGC 7479 and NGC 6951), but in others they are longer (NGC 895, NGC 4321...) or shorter (NGC 753) than the corotation distance. Finally, when there are anomalies, such as a four-armed pattern (NGC 6814), those minima are not related at all with main resonances.

Anyway, not all features occur in any single galaxy. In some cases there are local S₂(r) minima, or in the S2 image; in others there are changes in dominant mode; on other occasions pitch angle, skewness or the difference in mean position sign changes, etc. The most reliable case occurs when these features are present one at a time. One by one, they can then be produced for multiple physical processes, for example, lack of gas could provoke an S₂(r) or S2 minimum, but with no a change in pitch angle dominant mode.

Fourier decomposition does not show much difference between grand design and flocculent galaxies. In both flocculent cases the amplitudes are very low, but they are not the lowest in the sample. Also these galaxies present a rich structure, one with leading arms in a trailing main pattern, and the other with a very important S₃ image.

Acknowledgements

We are indebted to the referee, Dr. D.M. Elmegreen, for interesting remarks. This project was partially supported by the Spanish DGICYT grant No. PB94-0433 and by the Mexican CONACyT project No. F689E9411.