A multiwavelength study of spiral structure in galaxies. II. Spiral arms in deep optical observations

The fitting of spiral structure in the sample galaxies was performed using averaged $gr$ DESI images. As noted above, this is done to i) increase the signal-to-noise ratio, which is of high importance in the low surface brightness outer region of a spiral galaxy, and ii) equally recover the shape of the blue and reddish arms in the periphery of the galaxy. In fact, S20 did not find any significant difference between the characteristics of the spiral structure in the $gri$ SDSS bands for galaxies in our sample, except for the arm fraction in total galaxy luminosity and the colour of the spiral structure, so this approach should not introduce any bias in our results.

A procedure for methodically characterising spiral arms is detailed in S20. We employ the same method but use deeper images from DESI rather than SDSS observations. With the deeper images, spiral arm features are more easily traced beyond the optical radius, allowing for more complete analysis. Below we summarise details of the algorithm used, but refer the reader to S20 and Mosenkov et al. (2020) for a more complete procedural explanation.

The core idea of the method is to analyse the light distribution in photometric cuts made across spiral arms at different points of the spiral structure. To do so, we need a set of regions along each traceable spiral arm, so in the first step, we utilised the SAO DS9 package (Joye & Mandel, 2003), to manually place circular regions covering each spiral arm to continue the mask of the spiral arms created in S20 for SDSS images. The radii of the regions were chosen to cover the whole width of the arm perpendicular to its direction at a given point, from the nearest inward to nearest outward interarm space. The arm was traced visually using the histogram equalization scale function in DS9 out to a radius where the spiral arm became indistinguishable from the background (see below how we selected the radius of the truncation for the traced spiral arms). To reduce the scattering of manually selected points, we then applied a moving window smoothing algorithm.

In the next step, the points on each arm were used to compute a set of perpendiculars to the arm at regular intervals and to make photometric cuts along these perpendiculars. Several subsequent cuts were averaged, and the resulted photometric profile was then fitted with an asymmetric Gaussian function given in eq. 1, where $w_{1}$ and $w_{2}$ are fit parameters describing the inner and outer arm half-widths, respectively, $I_{0}$ is the amplitude of the Gaussian profile and $r_{\mathrm{peak}}$ is the location of the brightness peak within each photometric cut:

We note that after this step, we refined the positions of the brightest points (calculated from the fitted values of $r_{\mathrm{peak}}$) along the arm forming an arm ridge line. Further analysis of the spiral arms utilises these new arm locations, not the manually placed regions. Slices were fitted (and stored in our results) until the peak intensity had become systematically lower than 3 $\sigma$ of the standard deviation of the residual (i.e. the data minus the model).

To calculate the pitch angle of a spiral arm approximated with a logarithmic spiral, the distribution of the points on the arm ridge line was then fit with a linear regression model. The slope of the regression line was recorded as the tangent of the pitch angle for that arm. For each galaxy, we also computed an average pitch angle (as the simple mean for all spiral arms in the galaxy) and its variation with radius, by fitting a linear regression model to individual pieces of the spiral structure as described in S20. The same was done for the entire width of the spiral arms $w=w_{1}+w_{2}$. In addition, we estimated how the width of a spiral arm changes with radius, approximating this behaviour with a linear function with a slope $s$. We estimated all these parameters within and outside the optical radius $R_{25}$, and at all radii both in SDSS and DESI where we traced the spiral arms.

The deep optical photometry from DESI indeed allows the spiral structure of the sample galaxies to be traced well beyond the optical radius, as illustrated in Fig. 1 for two sample galaxies, namely PGC 24929 and PGC 48023.

We also used the supplementary NUV observations to probe the extent of spiral structure based on young populations of stars that typically reside in spiral arms. Since the angular and image resolutions of the NUV images are quite poor for studying spiral structure in most of our sample galaxies, we decided to only determine the outer (maximum) radius of the well-defined spiral arms at a level of $3\,\sigma$ of the background.

The results of our fit for the GALEX and DESI data can be found in the supplementary material in the electronic version of the MNRAS; the columns of the data table are described in Table 2 in Appendix A.

To investigate the one-dimensional surface brightness distribution in our galaxies, we obtained azimuthaly averaged profiles for each deprojected galaxy image in the NUV and $r$ bands (the mask of foreground and background objects was taken into account). These profiles demonstrate the presence of galaxy structural components dominating in the corresponding filter: a relatively young stellar population in the NUV band and a mixed stellar population in the $r$ band. In Sect. 5, we compare how these profiles in the two different spectral regions compare with each other and are related to the characteristics of spiral structure.

To determine the type of disc profile (see below), we use a piecewise linear function with zero to three breaks on the surface brightness (expressed in mag arcsec^-2) distribution in the disc. We mask off the inner-galaxy region where a bulge, a bar, and/or an active galactic nucleus may reside. The presence of red/reddish central components is usually not apparent in the NUV because they are mainly dominated by Population II stars, but it is not negligible in the optical. Here, potential caveats of this approach include i) an underestimation of the scattered light, the outer profile of which may be misinterpreted as an up-bending disc profile in the periphery of the galaxy, and ii) the light from the central component(s) may influence the model of the disc. Although adding a central component in our fitting would result in a more reliable overall galaxy model, we decided to minimise the number of free parameters and focus on the disc parameters only. Also, the percentage of galaxies with a relatively high bulge-to-disc luminosity ratio should be low, since galaxies with Hubble stage earlier than Sbc constitute only $\approx 33$% of the sample (see figure 1 in S20). Full-fledged decomposition of the galaxy images will be carried out in our future paper.

The fitting of the galaxy profiles was conducted manually by fixing the number of breaks on the disc profile and specifying their first-guess locations. Although DESI galaxy profiles typically extend down to $\sim 30$ mag arcsec^-2 in the $r$ band, we do not consider surface brightness distributions beyond a level of $\sim 28$ mag arcsec^-2 because i) the possible scattered light from the central source may dominate the profile at lower surface brightness levels (the wings of the PSF may appear right beyond this level, Sandin 2015; Trujillo & Fliri 2016; Rich et al. 2019) and ii) this part of the profile may be dominated by a stellar halo (see e.g. Merritt et al., 2016; Merritt et al., 2020; Gilhuly et al., 2022), iii) galaxies with a large angular diameter may suffer from sky under/over subtraction. As we will see below, almost all galaxies in our sample possess spiral arms that truncate within $\sim 28$ mag arcsec^-2, so our classification and fitting of the disc profile should not be affected by this truncation of the galaxy profile.

During our fitting, the model is optimised using a Nelder-Mead simplex algorithm to find the minimum of the $\chi^{2}$ function. We minimised the number of breaks based on the BIC values after adding a new break, but the maximum number of breaks was three — this appeared to be sufficient to accurately fit galaxy profiles of a complex shape. The results of our fitting are provided in the supplementary material in the electronic version of the MNRAS; the columns of the data table are described in Table 3 in Appendix B.

Our classification of the disc surface brightness profiles is performed following the literature. Since the seminal works by Erwin et al. (2005) and Pohlen & Trujillo (2006), disc galaxies have been separated into three major classes (following and expanding upon the system devised by Freeman 1970). Type I discs demonstrate surface brightness profiles described well by a single exponential decline. Type II discs are best described by a broken exponential with a steeper decline in the disc outskirts. Finally, Type III discs are characterised by a broken exponential with a shallower decline in the outskirts. Although Type II disc breaks are the most common (Pohlen & Trujillo, 2006; Erwin et al., 2008; Laine et al., 2014), different modifications of Type II and Type III profiles are not rare (Gutiérrez et al., 2011; Comerón et al., 2018). Using the result of our fitting, all galaxy profiles were independently annotated in the NUV and $r$ bands.

In addition to the classification of the disc profiles, we quantify the bending strength of the disc profile using the following approach. We use the first (inner) model disc profile and compare the alignment of all model data points with respect to this first disc (it is extrapolated from the beginning of the disc profile, that is, the inner radius $R_{\mathrm{min}}$ where we start our disc fitting, down to the level $28$ mag arcsec^-2 at $R_{\mathrm{max}}$). Then the bending strength ($BS$) of the galaxy profile, where we consider the disc, is computed as

where $S_{1}$ is the area under the first model disc profile computed within the range [$R_{\mathrm{min}}$,$R_{\mathrm{max}}$] and $S$ is the area under the galaxy model profile within the same range of radii (see Fig. 2) – both areas are calculated for the profiles normalised by the central surface brightness of the first disc. From eq. 2, it is obvious that galaxies with Type I profiles should have $BS=0$, whereas Type II profiles have strongly negative $BS$ values and Type III profiles demonstrate positive values of $BS$ (note that the above areas are found for the mag arcsec^-2 profiles). The meaning of the $BS$ parameter is straightforward: it is the percentage of the total disc area covered by the extrapolated inner disc within the range [$R_{\mathrm{min}}$,$R_{\mathrm{max}}$] in the $r$–$\mu$ space. Note that the $BS$ indicator is only sensitive to the overall shape of the profile, so for profiles with multiple breaks, $BS$ accounts for the overall difference between the extrapolated inner disc profile and the outer disc profiles. This implies that the sign and value of the bending strength for multiple-break profiles depend on the overall up/down-bending degree for the entire disc profile. For example, if a Type II+III profile is dominated by the down-bending of the middle disc segment, the $BS$ value will be negative. However, if an outer up-bending segment makes a larger contribution to the bending strength than the middle down-bending profile, the $BS$ indicator will be positive.

To be able to compare the $BS$ indicator in the NUV band with that in the $r$ band, we use the same range of radii when computing the BS indicator, as estimated in the $r$ band. We ensured that the DESI and NUV profiles indeed traced the beginning and end of the disc dominance in approximately the same range of radii on the galaxy profiles.

In Fig. 3, we compare different characteristics of disc profiles in the NUV and $r$ bands: the bending strengths $BS$ (left upper plot), the break radii $R_{\mathrm{br}}$ (right upper plot – only galaxies, having the same type of disc profile in the optical and NUV, are displayed), and the first (inner) and second disc scale lengths. As we can see, the bending strengths show a moderate correlation, with $BS_{\mathrm{DESI}}$ being systematically slightly higher than $BS_{\mathrm{NUV}}$, especially for multiarmed spirals with $BS_{NUV}<-50$ (the uncertainty of the intercept is significantly lower than the intercept value). This indicates that the DESI disc profiles are, on average, somewhat more up-bending than the corresponding NUV profiles. We can notice that in the NUV, there are a number of multiarmed galaxies with a Type I disc profile ($BS=0$), whereas the same galaxies in the optical may possess up- or down-bending disc profiles. This difference in the NUV and optical profiles cannot be explained by the difference in the resolutions of the NUV images with $BS_{\mathrm{NUV}}\approx 0$ and $BS_{\mathrm{NUV}}\neq 0$: $R_{25}=43.2^{+30.8}_{-11.6}$ arcsec for $BS_{\mathrm{NUV}}\approx 0$ versus $R_{25}=48.9^{+36.7}_{-14.6}$ arcsec for galaxies with $BS_{\mathrm{NUV}}\neq 0$. Interestingly, the grand-design galaxies in our sample are located, on average, slightly higher above the regression line than the other galaxies and do not demonstrate extremely low bending strengths. The good trend between the two BS quantities measured in the $r$ and NUV bands serves as indirect evidence that the proposed method for estimating the bending of the disc profile is robust, which makes it useful for quantifying the type of disc profile for a further uniform comparison at different wavelengths in future studies (including comparisons with cosmological hydrodynamical simulations).

The break radii for discs that have identical types of disc profile in the NUV and $r$ bands (52 galaxies in total) are in very good agreement (see the upper right plot in Fig. 3). This suggests that approximately one-third of galaxies in our sample have very similar profiles in the wavebands under study. For these galaxies, the break radii of the disc appear to be very similar, with a 1-$\sigma$ scatter of 0.015 dex from the regression line that within the uncertainty follows the one-to-one relationship. However, two-thirds of the sample galaxies have quite different disc profiles in the NUV and $r$ bands, although the break radii in these two bands still follow the same trend but with a much larger scatter (0.1 dex), especially for the M-class spirals.

Finally, the comparison of the disc scale lengths in the NUV and $r$ bands for the innermost disc region exhibits an apparent trend with a 1-$\sigma$ scatter of 0.13 dex (bottom left plot in Fig. 3), with the optical scale lengths being, on average, systematically lower than the scale lengths in the NUV. This correlation results in the $h_{\mathrm{1,DESI}}$/$h_{\mathrm{1,NUV}}=0.74$ ratio, in perfect agreement with $h_{0.7\mu m}/h_{NUV}=0.74$ for a sample of 18 nearby spiral galaxies from Casasola et al. (2017). The ratio $h_{\mathrm{1,DESI}}$/$h_{\mathrm{1,NUV}}=0.74$ suggests that the galaxy discs build inside-out (Muñoz-Mateos et al., 2007; Pezzulli et al., 2015; Frankel et al., 2019). This is usually explained by the fact that the gas inflowing at later epochs has a higher angular momentum and, therefore, forms a more extended star-forming disc (Fall & Efstathiou, 1980; Mo et al., 1998). Interestingly, for the outer disc, the scale lengths in the NUV and the optical result in the different ratio $h_{\mathrm{2,DESI}}$/$h_{\mathrm{2,NUV}}=1.44$ (bottom right plot in Fig. 3). Although we admit that the number of data points is insufficient for drawing a reliable conclusion, the uncertainties of the slope and the intercept of the regression line indicate that the outer discs have a similar or even larger outer disc scale length in the optical making the actual formation process more complex because star formation in galaxies is also regulated by stellar feedback, feedback from active galactic nuclei, and external factors (van den Bosch, 2001; Agertz et al., 2011; Marinacci et al., 2014; Tacchella et al., 2016). Note, however, that these results may be somehow affected by the fact that the presence of spiral arms can slightly (insignificantly) influence the derivation of the inner disc scale length, but has more effect on the outer disc profile (see Sect. 5.4). In addition, we have neglected the effect of dust attenuation in galaxies, which can be quite severe at NUV-blue wavelengths (Gadotti et al., 2010). However, Baes et al. (in prep.) recently used the TNG50 simulation (Pillepich et al., 2019; Nelson et al., 2019), post-processed with the radiative transfer code SKIRT (Camps & Baes, 2015, 2020), and demonstrated that stellar population gradients are the main factor for the dependence of the galaxy effective radius on wavelength and the differential dust attenuation plays a lesser (but still significant) role. In principle, the estimation of the scale length for the inner disc should be more affected by dust than that of the outer disc because the dust is more concentrated towards the inner galaxy region and primarily obscures the central regions of galaxies (see e.g. Muñoz-Mateos et al., 2009; Pohlen et al., 2010; González Delgado et al., 2015; Mosenkov et al., 2018).

Finally, it is of great interest to compare the extent of spiral structure in the optical and NUV. From Fig. 4, we can ascertain that the correlation between the two measurements is very good, close to the one-to-one relation for more extended spirals and systematically declining from it toward smaller DESI radii for less extended spirals: the effect is significant which is reflected in the small uncertainties of the slope and the intercept of the regression line. Our results suggest that the distributions of galaxies of different class of spiral structure, morphological type, spatial environment, angular size, and of different colour of spiral arms do not depend on the position on this correlation. Therefore, the deviation of the short spirals from the one-to-one relationship is not clear and requires further investigation. We speculate that it may be related to the formation history of the galaxies: larger spiral galaxies more likely experienced multiple interactions than smaller spirals, which could affect the appearance of their spiral arms in both the NUV and optical.

Since in the previous subsection we mentioned the types of disc profiles, it is timely to discuss the statistics of the disc types in our sample and their relation to the classes of spiral structure. We also use the $BS$ indicator to quantify this comparison.

In Fig. 5, we present distributions of the $BS$ indicator for the three classes of spiral structure in both NUV and optical data. In these histograms, we do not see any strong correlation between the type of disc profile and the class of spiral structure: grand-design, multiarmed, and flocculent galaxies are not significantly different in the $BS$ distribution. We can notice, however, that in the optical the distributions appear almost symmetric and only slightly shifted toward negative values, whereas in the NUV the distributions are more dominated by negative bending strengths with overall down-bending profiles. This fact is also demonstrated in Table 1 where one can see, inter alia, the robustness of the $BS$ indicator in quantifying the shape of disc profiles. The column heights in Fig. 6 confirm that down-bending profiles are the most common type in all three classes of spiral structure, although grand-design galaxies have a higher fraction of Type III and Type II+III discs than the other two classes. Flocculent galaxies in the optical are significantly dominated by Type II profiles and have a tiny fraction of single-exponential discs. In general, the Type I profile is the least common type in our sample for all types of spirals (there is also a small fraction of Type III+II among multiarmed spirals). Although quantitatively the distributions by the disc type in the NUV and optical do not quite match (the fraction of Type II discs is appreciably higher in DESI, whereas in GALEX the fractions of other types are higher relative to the down-bending discs), the overall impression remains the same as we have discussed above.

Our results agree well with the literature: Pohlen & Trujillo (2006) determined that only 10% of their sample of late-type spirals belong to Type I discs, 60% have Type II profiles, and 30% demonstrate up-bending profiles. The distribution by types changes for early-type barred disc galaxies (Erwin et al., 2008): 27% of Type I, 42% of Type II, 24% of Type III, and 6% for combinations of Type II and Type III. Gutiérrez et al. (2011) studied the surface brightness profiles of early-type, unbarred disc galaxies and, based on previous studies by Pohlen & Trujillo (2006) and Erwin et al. (2008), concluded that Type I profiles have a higher fraction in early-type disc galaxies and are least frequent in Sd-Sm spirals. In contrast, the frequency of Type II profiles increases from lenticular to the latest-type spirals. They also provide the fractions of disc profile types for S0-Sm disc galaxies: 21% (Type I), 50% (Type II), 38% (Type II) including 8% with composite Type II+III profiles. These frequencies agree well with our results for DESI observations listed in Table 1 taking into account that our sample comprises a rather small fraction of early-type spirals. Note that in the mentioned studies, the authors considered surface brightness profiles down to levels below 27-28 mag arcsec^-2 in the $R$ band that are similar to the surface brightness limits that we set for our galaxy profiles.

Let us now turn to the main subject of this paper, the properties of spiral structure beyond the optical radius, as measured using the NUV and deep optical data.

For different arm classes and spatial environments, we did not find dramatic differences within and beyond the optical radius for the following characteristics of spiral pattern: the average pitch angle, asymmetry of the spiral arms, and pitch angle variations along the radius. For example, the average pitch angle of spiral arms beyond the optical radius may be several times greater or lower than the average pitch angle within the optical radius (half of the galaxies demonstrate an increase of the pitch angle outside of the optical radius, while the other half have lower pitch angles). Interestingly, this result holds for both isolated galaxies and galaxies in groups and clusters. Therefore, we can conclude that the pitch angle of galaxies may become systematically greater or lower in the galaxy periphery without any dependence on class of spiral structure, type of disc profile, global galaxy morphology, or spatial environment.

In Fig. 7, one can see a correlation between the extent of spiral arms and the nearest break radius to the edge of the spiral arms in the NUV (left plot) and DESI (middle plot) observations. Both correlations are tight (albeit with some outliers) which suggests that the extent of the spiral arms is always comparatively close to one of the disc breaks: $R_{\mathrm{br,NUV}}\simeq 0.73\,R_{\mathrm{arms,NUV}}$, $R_{\mathrm{br,DESI}}\simeq 0.92\,R_{\mathrm{arms,DESI}}$. In the NUV, the nearest break almost always lies within the radius of spiral structure, so $0.53\,R_{\mathrm{arms,NUV}}\lesssim R_{\mathrm{br,NUV}}\lesssim 1.02\,R_{\mathrm{arms,NUV}}$. This general trend reflects the fact that in more extended discs (and hence, for greater break radii), the spiral arms continue farther out than in galaxies with a smaller linear size. Therefore, this correlation represents a typical galaxy scaling relation: in galaxies with a larger physical size, all main characteristics scale relative to smaller galaxies. Note the systematic upward offset of the correlation for the NUV data with respect to the DESI data, which serves as another evidence that, on average, NUV spirals are typically more extended than optical ones. In the right plot of Fig. 7, we display the same scatter plot as in the middle one, but highlight different disc types. We can notice that galaxies with several breaks (Type II+III and Type III+II) lie closer to the general regression line, whereas discs with only one break (Type II and Type III) demonstrate a much larger scatter along the trend. Furthermore, the spiral arms in Type II discs are, on average, slightly more extended than those in Type III discs for the same break radius, although the intercepts of the respective regression lines overlap within their uncertainties. Specifically, the spiral arms embedded in Type II discs end, on average, at $0.95\,R_{\mathrm{br,DESI}}\lesssim R_{\mathrm{arms,DESI}}\lesssim 1.29\,R_{\mathrm{br,DESI}}$, whereas Type III discs extend to $0.86\,R_{\mathrm{br,DESI}}\lesssim R_{\mathrm{arms,DESI}}\lesssim 1.17\,R_{\mathrm{br,DESI}}$. In galaxies with several disc breaks, the nearest break radii are located even closer to the radius of spiral structure: $1.01\,R_{\mathrm{br,DESI}}\lesssim R_{\mathrm{arms,DESI}}\lesssim 1.15\,R_{\mathrm{br,DESI}}$. However, note that when the number of disc breaks increases, the tightness of the correlation between $R_{\mathrm{arms}}$ and $R_{\mathrm{br}}$ also increases, even if these breaks are orientated randomly along the disc. In Fig. 8, one can see, however, that this “artificial” effect cannot reproduce the very tight correlation we observe for galaxies with two to three breaks (note that only six galaxies in our sample demonstrate three profile breaks and 28 galaxies have two breaks). This plot provides visual evidence that at least some breaks in disc profiles should be closely related to the truncation of spiral arms.

Using Spitzer 3.6 $\mu$m observations, Laine et al. (2014) explored how the environmental density of galaxies and the tidal interaction strength can affect the shape of the disc profile. They found that most Type II profiles are observed in galaxies with apparent rings, lenses, and spiral arms. Similarly to Pohlen & Trujillo (2006), they did not find a notable correlation between the galaxy environment and the type of disc profile, although they note a positive correlation between the disc scale length and the tidal interaction strength for Type III profiles.

Pranger et al. (2017) estimated the frequency rates of Type I, Type II and Type III profiles in field and cluster environments and reported that the abundance of Type I profiles in clusters is higher ($15_{-4}^{+7}$%) than in the field ($6\pm 2$%) which may be explained by environment-driven mechanisms: in clusters, Type II discs are converted to Type I discs over time.

Watkins et al. (2019) investigated Spitzer 3.6 $\mu$m surface brightness profiles of disc galaxies, in particular, Type III disc break-hosting galaxies. They showed that most Type III breaks are caused by morphological asymmetries of the host galaxies, which, in its turn, is the result of tidal disturbance because these galaxies reside in dense environments. Staudaher et al. (2019) also note that some of the surface brightness profile breaks may be due to the presence of stellar haloes. Watkins et al. (2019) point out that Type III disks are often associated with outer spheroidal components as a result of, for example, galaxy harassment. Finally, some Type III breaks, residing in low density environments, are related to extended spiral arms or star formation (secular evolution features), and in such galaxies down-bending breaks are often found at larger radii. These conclusions are confirmed in a recent study by Pfeffer et al. (2022) who used EAGLE (Schaye et al., 2015) cosmological hydrodynamical simulations to explore the effects of galaxy mass and spatial environment on types of galaxy profiles. Down-bending breaks highlight truncations of star-forming discs, which, in their turn, imply radial gradients in the galaxy stellar populations. Consistent with observations, the frequency of Type II discs in simulations increases with Hubble stage. Type III profiles may form through several scenarios, the main of which is galaxy mergers. Also, in some Type III galaxies a shorter inner disc may have formed at later cosmological epochs. It comes as no surprise why the frequency of Type III galaxies increases with galaxy mass, since for more massive galaxies, galaxy mergers have been more frequent. The high fraction of Type I galaxies in galaxy clusters is explained by star formation quenching, which takes place when a star forming Type II galaxy enters a galaxy cluster and its young stellar population fades away converting it to a typical Type I galaxy.

To investigate how the current shape of our sample galaxies may be affected by tidal interactions, we examined their deep DESI images (our classification is provided in Table 2). We found that only a tiny fraction of our spiral galaxies (12 out of 155 galaxies, or $\approx 8$% of the sample) demonstrate signs of low-surface brightness structures brighter than 27 mag arcsec^-2 in the $r$ band (loops, arcs, bridges, stellar streams, etc.). This subsample comprises 7 multiarmed galaxies and 5 grand-design spirals, and no flocculent galaxies. Also, in accordance with Martínez-Delgado et al. (2023), some galaxies in our sample (for example, PGC 22957, PGC 24996, PGC 36902) exhibit galactic feathers produced by interactions with low-mass companions that perturb the disc of the host and lead to the formation of narrow tidal tails resembling the outer structure of asymmetric spiral arms in some galaxies from our sample. A similar mechanism involves interactions with a more massive neighbour that may induce the formation of grand-design spiral arms in the interacting galaxy (see e.g. Holmberg, 1941; Toomre & Toomre, 1972; Eneev et al., 1973; Günthardt et al., 2016). In this scenario, tidal-bridge tails are induced by fly-bys of nearby companions, which later wind-up and transform into grand-design spirals. However, these structures dissipate within 1 Gyr due to the decrease of their pitch angle with time, as they continue to wind up (Oh et al., 2008, 2015). The formation of grand-design spiral arms is also possible in the potential of a galaxy cluster, without any perturbations from close companions (Semczuk et al., 2017).

In Fig. 9, we consider the extent of spiral arms in the sample galaxies. In the left plot, we can see that grand-design galaxies are, on average, more extended than flocculent and multiarmed galaxies. We can also conclude that the maximum extent of the spiral arms in our sample for all three classes reaches 55–60 kpc, with an exception of PGC 22957 with $R_{\mathrm{arms,DESI}}=88.3$ kpc. In the middle plot of Fig. 9, one can see a correlation between the optical radius and the extent of spiral arms. For most galaxies, the deviation from the one-to-one relation does not exceed 0.15 dex which translates into $0.71\,R_{25}\lesssim R_{\mathrm{arms,DESI}}\lesssim 1.41\,R_{25}$. Interestingly, grand-design galaxies typically lie above the general trend – their spiral arms extend beyond the optical radius. This is also illustrated in the right plot of Fig. 9, where the extent of spiral arms is normalised by the optical radius of each galaxy. While the other two classes of spiral structure peak at $R_{\mathrm{arms,DESI}}\simeq(1.0\pm 0.2)\,R_{25}$, the distribution of grand-design galaxies by this parameter is asymmetric and shifted towards $R_{\mathrm{arms,DESI}}\simeq 1.4\,R_{25}$. This suggests that grand-design galaxies are more likely to have been stretched by past tidal interactions with neighbouring galaxies than the other two classes of spirals. In the same right plot of Fig. 9, we show the maximum extent of the spirals as measured by S20 using SDSS observations (colour-filled histograms). As one can see, the increase of the arms’ radius in deep images $\langle R_{\mathrm{arms,DESI}}/R_{\mathrm{arms,SDSS}}\rangle$ is moderate for flocculent ($1.12\pm 0.16$) and multiarmed ($1.18\pm 0.24$) galaxies, whereas for G-arms it is more appreciable ($1.28\pm 0.36$). Therefore, deep photometry is especially important for studying the extent of grand-design galaxies, whereas most flocculent and multiarmed spirals can be almost entirely traced in ordinary SDSS observations.

Fig. 10 shows a distribution by the overall azimuthal angle of spiral arms in our sample, separated by class of spiral structure (grand design, multiarmed, and flocculent). The data demonstrate that spiral arms in grand-design galaxies exhibit slightly larger maximum azimuthal angles, on average, than the other two classes, whereas flocculent galaxies typically have less than one full turn of spiral arms. As in the case with the maximum radius of spiral arms, the advantage of using deep photometry for studying spiral structure is evident for grand-design ($\Delta\phi_{\mathrm{DESI}}/\Delta\phi_{\mathrm{SDSS}}=1.18\pm 0.24$) and multiarmed spirals ($1.18\pm 0.27$), while flocculent galaxies show a slightly lower increase ($1.13\pm 0.19$).

The measurements of arm width, taken at each point beyond the optical radius, were averaged to obtain a single arm width value (see the distribution of $\langle w\rangle/R_{25}$ in Fig. 11, left plot). Although the arm width distributions, obtained for SDSS images alone, do not display much scatter (see figure 16 in S20), the deeper DESI images show that grand-design galaxies have, on average, wider spiral arms than the other two classes of spirals. Tidal interactions could also explain this increase in arm width beyond the optical radius. Fig. 11, right plot, compares the measured width of spiral arms beyond the optical radius for isolated galaxies versus galaxies in groups and clusters. The outer arm widths beyond the optical radius are normalised by the arm width within the optical radius. The data show that isolated galaxies tend to have smaller spiral arm widths, probably because isolated galaxies do not experience tidal effects. Interestingly, we did not find a significant difference between the extent of spiral arms (normalised by the optical radius) and the spatial environment, perhaps because our sample comprises only a small fraction of truly interacting galaxies with tidally induced arms.

In view of the above discussion, we conclude that there is a link between the spiral structure of grand-design galaxies and interactions with neighbouring galaxies. This suggests that tidal interactions can be the main formation mechanism for grand-design spirals, while the spiral arms in multiarmed and flocculent galaxies may be predominantly formed by other mechanisms.

Since for 67 galaxies in our sample, all main spiral arms have been completely traced, we can assess the accuracy of our fitting for the discs as we mask off the spiral arms. Measuring this effect is of great importance for understanding how spiral structure can influence the disc parameters retrieved in galaxy modelling. Of course, a more accurate decomposition with an additional component for spiral arms is required to explore this question in greater detail (Chugunov et al., in prep.), but this simple approach can give us a general sense of potential biases introduced by spiral arms. Fig. 12 illustrates a comparison of the fit parameters for the inner and outer (if present) discs with and without a mask of the spiral arms. As we can see, there may be a systematic overestimation of the first disc central surface brightness for some galaxies when the spiral arms are not masked out (note, however, that most galaxies yet follow the one-to-one relation, which is reflected in the values of the regression line coefficients and their uncertainties). Interestingly, the effect becomes more pronounced for the outer disc in half of the galaxies: $\Delta\mu_{0,1}\simeq 0.45$ versus $\Delta\mu_{0,2}\simeq 0.85$. The influence of spiral arms on the fitting of the first disc scale length is much less dramatic, but may lead to a systematic underestimation of the second disc scale length for galaxies with bright spiral arms: $\lg\,h_{1}\simeq 0.93\,\lg\,h_{1}^{\mathrm{mask}}$ for the inner disc and $\lg\,h_{2}\simeq 0.84\,\lg\,h_{2}^{\mathrm{mask}}$ for the outer disc. In other words, the outer discs with unmasked spirals may appear to be less extended than when a mask of the spiral arms is taken into account. The reason for these systematic errors in decomposition without masking of the spiral arms stems from the fact that the spiral arms make some contribution to the total galaxy luminosity (their fraction is typically higher in grand-design galaxies, see figures 19 and 20 in S20). Therefore, if we neglect the presence spiral arms in our modeling, their luminosity density is “assigned” to the stellar disc, which makes the disc brighter and shorter. This effect becomes very significant for estimating the parameters of the outer disc in grand-design galaxies with bright spiral arms.

As shown in Gao & Ho (2017), to obtain accurate decomposition models for the bulge, nuclear and inner morphological features (lenses, rings, discs, bars) should be taken into account, while outer secondary morphological features (such as rings and spirals) do not influence the retrieved parameters of the bulge significantly. However, according to Sonnenfeld (2022), if our goal is to fit a galaxy with a single Sérsic function (a widely used approach to quantify the morphology of a galaxy, see, e.g. Mosenkov et al. 2019), a bright spiral structure can significantly affect the retrieved Sérsic parameters for galaxies with a prominent bulge. Therefore, accurate photometric decomposition is of great importance for studying the structural properties of galaxies and may be critical for studying the galaxy scaling relations (see e.g. D’Onofrio et al. 2021 and an exhaustive list of references therein), especially if objects with non-axisymmetric features are considered.