3 Analysis of spiral structure

With the method of Fourier decomposition of spiral structure, an ideal logarithmic spiral presents a single-peaked spectrum of intensity normalized to unity, centred on a value of $p^m_{_{\rm max}}$ related with a pitch angle $p^m_{_{\rm max}}=-m/\tan i$. Real galaxian arms are approximations to logarithmic spirals, so that the corresponding peak is of lower amplitude, larger, and/or asymmetric. Also, if the deprojection is not correct, a spurious peak at p=0 can appear, with an intensity that increases with inclination angle, because the resulting distribution tends to a straight line as the inclination increases, and the Fourier transform of radial arms represent $i\sim 90^\circ$, i.e. $p^m_{_{\rm max}}\sim0$. When the bar is strong and arms are not symmetric and/or very weak, a component also appears at $p^2_{_{\rm max}} = 0$, even if deprojection is correct. Nevertheless, in cases where one arm is brighter than the other, the m=1 component gives a much more reliable pitch angle in better agreement with those calculated by other methods.

Through the S_m(r) functions of a given flux distribution, we can determine the angular periodicity which better represents the flux distribution at each radius, i.e. the predominant m for each galactocentric distance indicates that at this radius the intensity distribution shows a preferential angular separation of $2\pi/m$. Although it is tempting to associate the minima of the |S_m(r)| functions with resonances, it is necessary to add a word of caution, because there are several other ways in which such minima can occur. In fact, near corotation, we can expect not only a lower flux but also a gap, a change in pitch angle, skewness or a difference of sign in mean position in different bands (Paper III). We show in this paper that not only a local minimum occurs near corotation, but also that the dominant mode changes.

To apply the Fourier transform method, we have kept in the calculations only those pixels which are brighter than a given luminosity threshold and within a given deprojected distance from the galaxian center. Both these limits were determined with the help of isophotal contour maps, and the brightness threshold was taken so as to include not only the bright spiral arms but also the inter-arm and outer-disc regions. In fact, both conditions are equivalent because, once the image is deprojected, outer isophotes are roughly circular. So we must reach a compromise regarding most of the flux be included and do not take noise levels into account, or even the axisymmetric disc contribution. With all this in mind, we see that the chosen contours vary between $\sim \! 2.7\, \rm mag/{\hbox{$^{\prime\prime}$}}^2$ lower than limiting magnitude in NGC 157 (i.e. the equivalent contour to $23.11\, \rm mag/{\hbox{$^{\prime\prime}$}}^2$) and $\sim \!0.07\, \rm mag/{\hbox{$^{\prime\prime}$}}^2$ lower than limiting magnitude in the faintest galaxy, NGC 6764 (the contour equivalent to $24.94\, \rm mag/{\hbox{$^{\prime\prime}$}}^2$).

Also, to compare with the method developed by EEM, we have eliminated field stars because they introduce a lot of noise. Since they are usually randomly distributed over each frame and appear in all coefficients, and because of the finite width of these coefficients, when we reconstruct the S_m images, stars appear as small arcs, not as point-like structures. This phenomenon is not present in the EEM method, because all asymmetric features, such as stars, remain in the A2 image.

In order to reduce the noise level in the coefficients as much as possible, we remove the stars by using a simple technique. The flux around stars is calculated, and then the stars are substituted by this mean value (except when they are very close to the arms, in which case we prefer to leave them in the image). The results with and without field stars are shown in Fig. 3.

Although we are mainly interested in decomposing images in different angular modes, it is also possible, with the Fourier transform method, to estimate the inclination ($\omega$) and position angles (PA). This procedure assumes that the arms can be reasonably approximated by logarithmic spirals, so that their Fourier spectrum has a single peak centred around $p_{_{\rm max}}$. To calculate the best pair of values of PA and $\omega$, we can follow Considère & Athanassoula (1988) method of maximizing the value of the dominant component (usually m=2) while minimizing the component around p²=0, or we can follow Puerari & Dottori (1992), who maximize the signal-to-noise ratio of the dominant coefficient.

For each galaxy in our sample we first consider a wide range of angles, for for both PA and $\omega$ (an interval of $67^\circ$ centred on the values found in Paper II, assuming that the outermost deprojected isophotes are circular in Considère & Athanassoula 's method and $13^\circ$ for Puerari & Dottori's method, with a spacing grid of $\Delta$PA and $\Delta \omega =3^\circ$. Calculation of the spectra for these deprojection parameters allowed us to find a maximum, according to the criteria mentioned above. Centred on the new pair of values (PA, $\omega$) which maximizes the previous ratios, we re-calculated the Fourier coefficients using a finer grid, with steps of one degree, over a range of $7^\circ$ about these values. The results and comparisons with previous work are summarized in Table 2. Nevertheless, these methods present serious problems, which we will discuss later (Figs. 6, 7, 8 and 9).