On the Dynamical Relevance of Galaxy Spiral Arm Evolution. I. Arm Density Structure

In this work, we employ a set of simulations that includes models presented in Roca-Fàbrega et al. (2013); see Table 1 for further details on the model U1 and B1 parameters and a set of new simulations. The new models include isolated barred models that explore slightly different parameters from the ones in Roca-Fàbrega et al. (2013): an interacting disk galaxy and a self-consistent stellar disk inside a triaxial dark matter halo. All simulations presented here have been performed using either the GADGET (Springel 2005) or the ART (Kravtsov et al. 1997; Kravtsov 2003) codes. Initial conditions (ICs) in spherical halo models were generated following the Hernquist moments method (Hernquist 1993) and the Widrow & Dubinski (2005) distribution function method. The model with a triaxial dark matter halo has been obtained by using the Rodionov iterative method (Rodionov et al. 2009; Rodionov & Athanassoula 2011). The halo profile is a Navarro–Frenk–White (NFW; Navarro et al. 1997) with semiaxis ratios of b/a = 0.83 and c/a = 0.67. The spatial resolution ranges from 11 pc in the ART simulations to, in the triaxial GADGET case, 35 pc in the rest of the GADGET models. All galactic models have evolved at least 3 Gyr, and by this time, they have already produced spiral arms. Spiral arms have been produced through different mechanisms, depending on the properties of the simulated system, i.e., a central bar perturbation, other intrinsic disk instabilities, satellite interactions, and triaxial dark matter halo effects.

Aiming to understand how satellite interaction affects the formation and evolution of nonaxisymmetric structures, we have generated a new model based on U1 but with the addition of a companion satellite. We have decided to use model U1 in order to distinguish among the different trigger mechanisms of spiral arm formation. Model U1 does not develop a strong bar, and only a few short-lived spiral arms are generated in the disk (see Roca-Fàbrega et al. 2013).

The satellite ICs have been generated using the same procedure as for the main galaxy but changing the initial parameters: a smaller and ∼100 times less massive stellar disk and spherical dark matter halo (see Table 1). All of the satellite particles have the same mass. Once generated, the satellite particles have been added to the main galaxy simulation box (U1 model) at different locations and initial velocities in order to test the impact of the satellite orbital parameters in the outcome (see Table 2).

In this section, we explain in detail the techniques employed to obtain our main results. We also expose how the diagrams shown throughout the paper have been obtained.

2.2.1. Pitch Angle

To enhance the overdensity structure, we estimated the surface density over a polar grid. At every cell in the grid, we subtracted the azimuthal average density corresponding to its radial position, and we normalized the result by the same azimuthal average density value. In every simulation, we searched for a strong arm structure and kept track of it forward and backward in time until the structure vanished. We fitted a logarithmic function that matched the arm structure at three different snapshots: when the arm is born, when it reaches its peak strength, and when it disappears.

Figure 1 shows that by applying these techniques, we are able to (a) quantify the pitch angle from the very start of its formation until it has vanished and (b) estimate the pitch angle of different newborn strong arms within the same simulation.

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2.2.2. Spiral Arm Radial Density Law and Scale Lengths

To estimate the scale length of the arm, we selected the density maxima at each radius. At each annulus of the polar ring, we separate the arm contribution by subtracting the azimuthal average surface density at each radius to finally find that the arm contribution and the azimuthal average surface density follow an exponential function (e.g., Figure 2).

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2.2.3. Spiral Arm Vertical and Transversal Density Profiles

In order to study the 3D structure of spiral arms, we need to assess the vertical and transversal density laws. We calculate the particle density using a spherical constant-density kernel, including the 30 nearest neighbors.

To characterize the transversal/vertical structure of the spiral arms, for simplicity, we define the transversal/vertical direction of the arm as the corresponding direction of the ring that intersects the spiral. We proceed to subtract a symmetrized version of the disk density inside this ring. After this procedure, we consider the density residuals as the spiral arm's contribution. Finally, we build the density profile along the transversal/vertical directions (see Figure 3).

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In general, spiral arms in isolated simulations with spherical halos show arms that rip and vanish quickly after less than a galactic rotation. By the time the first arms vanish, new arms are already forming in such a way that the disk always presents strong spiral arms coexisting with weak fading arms. In this work, we fit a logarithmic spiral only to strong and well-defined spiral arms. Our fitting strategy is to locate the density peak of the spiral as a function of radius. In order to properly locate the spiral arm density peak, we analyze overdensities in the disk surface density after subtracting the radial average density (see Section 2.2 and Figure 4). In most cases, the fit has been performed only over the arm section with higher-density contrast. High-density contrast sections usually span a few kpc. In particular, these simulations that have no large-scale mechanism inducing spiral arm formation (i.e., bars, triaxial halos, or satellite interactions) show more fragmented arms with smaller-density contrast. These arms are highly transient and also show shorter lifetimes than the ones induced by large-scale mechanisms.

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Our results also show that in this scenario, the pitch angle of the spiral arms deviates just a few degrees, winding up at less than  8° before losing strength or ripping up. This can be seen in Figure 4. In this figure, we show how spiral arms in unbarred systems (U1, GU1, GU4) almost keep the pitch angle for all of their lifetime. This happens recurrently in the disk in all newly generated spiral arms, regardless of the simulation code (see U1, GU1, and GU4 in Table 3). This means that, even when there is a winding up (i.e., an effective reduction of the pitch angle), the reduction is very small prior to the dissolution of the spiral arm. Also, the amplitude density of the spiral arms reduces promptly as the pitch slightly decreases.

Table 3.  Measures of Pitch Angles of Different Arms along the Simulations of Isolated Disks in a Spherical Halo

Model	Time	Pitch			Hr	Hz	Ht	$\tfrac{{{\rm{\Sigma }}}_{\mathrm{arm}}}{{{\rm{\Sigma }}}_{\mathrm{prom}}}$
	(Gyr)	(1)	(2)	(3)	(kpc)	(kpc)	(kpc)
U1	0.096	38.0	38.0	⋯	⋯	⋯	⋯	⋯
	0.16	28.5	28.5	⋯	4.2	0.2	2.45	1.46
	0.24	28.0	28.0	⋯	⋯	⋯	⋯	⋯
	0.544	48.0	35.0	⋯	7.46	0.19	1.19	2.10
	0.592	35.0	33.0	34.0	⋯	⋯	⋯	⋯
	0.656	29.0	28.7	⋯	7.99	0.19	1.19	2.17
	2.48	33.5	39.0	⋯	5.22	0.27	2.48	1.66
	2.368	33.5	32.0	32.0	⋯	⋯	⋯	⋯
	2.528	34.5	32.5	32.5	3.91	0.24	2.17	1.66
B1	0.256	26.5	26.5	22.5	4.13	0.22	2.71	1.77
	0.352	32.0	32.0	28.0	3.72	0.24	8.89	1.86
	0.592	56.0	56.0	13.0	3.86	0.21	1.15	3.62
	0.688	57.0	46.0	⋯	13.1	0.19	1.8	3.27
	0.72	47.0	37.0	40.5	10.7	0.19	2.7	3.38
GU1	0.54	30.0	30.0	32.0	4.03	0.21	1.1	1.74
	0.6	33.0	25.0	25.0	⋯	⋯	⋯	⋯
	0.64	25.0	30.0	38.0	6.83	0.2	0.83	1.8
	1.11	32.0	34.0	30.0	⋯	⋯	⋯	⋯
	1.15	24.5	27.5	28.5	7.85	0.21	0.78	1.98
	1.2	27.0	33.0	33.0	⋯	⋯	⋯	⋯
GU4	0.95	29.0	29.5	24.0	9.18	0.24	0.88	1.50
	1.0	37.0	27.0	27.0	⋯	⋯	⋯	⋯
	1.05	29.0	27.0	⋯	5.95	0.21	1.22	1.82
	1.8	32.0	37.5	⋯	⋯	⋯	⋯	⋯
	1.86	32.0	34.0	⋯	7.72	0.2	1.05	1.7
	1.9	27.0	27.0	⋯	5.20	0.65	1.36	⋯
	2.34	30.0	30.0	⋯	6.43	0.21	1.71	1.54
GB4	0.68	25.0	26.0	26.0	⋯	⋯	⋯	⋯
	0.78	20.0	23.0	23.0	6.97	0.65	4.29	1.62
	0.95	21.0	23.0	22.5	⋯	⋯	⋯	⋯
	1.1–1.7		Buckling
	1.8	37.5	37.5	⋯	3.78	0.28	0.94	1.93
	1.9	33.5	31.0	⋯	8.57	0.24	1.74	1.62
	2.05	33.5	31.0	⋯	4.2	⋯	⋯	1.68
	2.75	30.0	39.0	39.0	⋯	⋯	⋯	⋯
	2.82	31.0	40.0	40.0	5.27	0.24	2.68	2.13
	2.9	33.0	36.0	36.0	⋯	⋯	⋯	⋯

Note. Radial, vertical, and transversal scale length was measured in just one of the arms at the respective time.

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In Figure 5, we show a zoom of the density structure in polar coordinates for four spiral arms taken at different times of the U1 and GU1 simulations. The upper panels show the surface density in polar coordinates. The lower panels show the disk density radial profile (blue stars) and spiral arm density profile (red stars); solid cyan and purple lines show the best exponential for disk and spirals, respectively. Spiral arms, as well as the disk, have a clear density exponential fall, but the scale radius is larger for the spiral arms than for the disk in all cases (see the scale radius in kpc in the legend in the bottom panels). This result indicates that the spiral arms are gravitationally and dynamically relevant far beyond the scale radius of the disk.

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Figure 6 corresponds to the vertical and transversal density profiles for the four spiral arms shown in Figure 5. The first panel of each mosaic is the same radial density scale presented in Figure 5, included here for reference purposes. The second and third panels of each mosaic show the vertical (i.e., perpendicular to the plane of the disk) and transversal density profiles of the respective spiral arm. In all panels of this figure, we present the best fit in solid curves and the respective standard deviation (σ). Scale length is indicated in the legend in each plot (all values in kpc). In these diagrams, we show that all arms have a vertical density profile that agrees with a sech² in the vertical and transversal directions, while for the radial component, i.e., along the spiral arm, an exponential fall provides the best fit. The scale length of the vertical component of the spiral arm is roughly similar to the corresponding one for the disk.

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Finally, we have calculated a density contrast for the spiral arms versus the disk in these nonbarred, spherical halo, isolated simulations. We find that the maximum surface density contrast of the spiral arms is higher by a factor of ∼2.1 with respect to the mean disk density. For a quantitative summary of the parameters obtained in this study, see Table 3.

As mentioned in the previous section, in general, spiral arms in the isolated simulations with spherical halos show arms that rip and vanish—in the nonbarred case, swiftly after less than a rotation to the galaxy. On the other hand, in the barred cases, some of the arms last a few rotations. In the barred simulations, the arms are connected and disconnected continually from the bar; some of them seem to be reinforced when the bar rotates half its period and again finds those arms and reconnects with them, again producing a spiral arm with a similar structure as the original (in pitch and density contrast). These types of arms are stronger and slightly more long-lasting than those in the nonbarred, spherical halo cases of Section 3.1. Some other arms are disconnected and slightly shifted outward; those arms will wind up and vanish quickly. Unlike the self-induced spiral arm case, in barred simulations, arms tend to conserve their pitch angle for longer times, as long as they are connected to the bar (this specific timescale depends on the physical characteristics of the bar, such as its mass and size). As soon as they disconnect from the bar, the ones that are not reinforced by the bar again wind up quickly and fade away. These results agree with the findings presented in Roca-Fàbrega et al. (2013).

An interesting feature we found in these simulations was the differences in the spiral arm characteristics before and after the buckling that show up in the first 3 Gyr. Before the buckling, we measure the pitch angles of four spiral arms at different times in the simulation previous to the buckling and find that the four different arms have pitch angles of approximately 26° that wind up a few degrees (with a maximum of 5°). However, after the buckling, we perform the same calculations over four different arms and obtain a systematic increase of the pitches for each one of around 10°. Preliminary analysis shows that this might be due to a redistribution of the mass of the bar (maybe an increase in the bar radial scale length) and the angular momentum. Such behavior is consistent with the discussion in Roca-Fàbrega et al. (2013), where spiral arms rotate rigidly with strong bars and wind up with weaker bars like the buckled ones. A careful study of these results will be developed in a future work.

In Figure 7, we show the surface density of the disk in polar coordinates. As in the previous section, the fitting is performed over the region of the arm where the density contrast is stronger, in this case, also several kpc (in general more than in the nonbarred case). In the case of arms induced by a central bar, they present at its maximum, higher-density contrasts with respect to the disk than in the barless case, with larger lifetimes (see Figure 7) compared to the ones without a bar.

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As in the unbarred case (with spherical halos), the spiral arms wind up by at most  6° and usually about  3° or less (see Table 3) while losing strength or ripping up. We find, for instance, in the simulation of Figure 7, an arm born at about 68 Myr that presents a pitch angle of 26°, while one born later, at approximately 95 Myr, presents 23°. In other simulations performed specifically for this work, we find even smaller (than in the barless case) changes in the pitch angle of newborn arms; see Table 3. This means that, even when there is a winding up (i.e., an effective reduction of the pitch angle), the reduction is very small prior to the fading of the spiral arm. Additionally, we noticed that the amplitude density of the spiral arms reduces swiftly as the pitch decreases (see Figure 4) only for the arms that definitely disconnect from the bar, while for those that reconnect continuously every time the bar comes back, the arms always keep approximately the same pitch and density contrast. This is important because a large reduction or increase of the pitch angle would imply a severe difference in the dynamical behavior of gas and stars (Pérez-Villegas et al. 2015a).

Figure 8 is a zoom of the density structure in polar coordinates for three spiral arms taken at different times of the simulation (GB4 and B1). The mosaics show (upper panels) the surface density in polar coordinates and (lower panels) the density fall of the disk (blue stars) and the spiral arms (red stars), as well as a fit for each one (cyan and purple lines, respectively). Bar-induced spiral arms show a clear density exponential profile with a scale radius (indicated in the lower panels at the top right corner in kpc) larger than the disk scale length in all cases. This means that the spiral arms are gravitationally and dynamically very important far beyond the scale radius of the disk, even more than the ones for the barless simulations.

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Figure 9 corresponds to the vertical and transversal density profiles for the spiral arms shown in Figure 8. The upper panel of the first mosaic is the same radial density scale presented in Figure 8, included here for reference purposes; the second and third panels of each mosaic show the vertical (i.e., perpendicular to the plane of the disk) and transversal density profiles of the respective spiral arm. In all panels of this figure, we present the best fit in solid curves, its standard deviation (σ, indicated at left), and the corresponding scale height/length (in kpc, indicated at upper right).

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Like in unbarred cases, our results show that the 3D arm density structures are well fitted by an exponential profile in the radial direction and by a sech² in the transversal and vertical directions. In general, the vertical height of the arms is lower than the one of the disk.

Finally, we have also calculated a density contrast for the spiral arms versus the disk in these barred simulations. We find that the density contrast, calculated at the maximum contrast of the spiral arms, is ∼3.3. For a quantitative summary of the parameters obtained in this study, see Table 3.

Unlike the isolated cases presented in the previous sections, in the simulations with a satellite galaxy (with a mass of about ∼3% of the main galaxy), great-design spiral arms with a larger density contrast are induced by the satellite interaction; this type of arm is only strong as long as the perturber is close to the pericenter of its orbit. Satellite-induced spiral arms fade away depending on the satellite trajectory (in agreement with Semczuk et al. 2017; Pettitt & Wadsley 2018). As the satellite slips away from the maximum approach to the main galaxy, the spiral arms suffer the largest and swifter winding up. For example, in Table 4, we can see how the pitch angles of the spiral arms in these simulations reach nearly 20° of change. These severe and fast changes in the pitch angle and their density contrast of the spiral arms will have a great impact on the stellar and gas dynamics of the disk, as long as the satellite is able to produce these kinds of grand-design spiral arms.

Table 4.  Measures of Pitch Angles of Different Arms Along the Simulations with Satellite Interaction and Isolated Disks in a Triaxial Halo

Model	Time		Pitch		Hr	Hz	Ht	$\tfrac{{{\rm{\Sigma }}}_{\mathrm{arm}}}{{{\rm{\Sigma }}}_{\mathrm{prom}}}$
	(Gyr)	(1)	(2)	(3)	(kpc)	(kpc)	(kpc)
U1S1	0.816	25.0	24.0	25.0	6.55	0.36	1.43	3.49
	0.976	12.2	23.5	22.0	⋯	⋯	⋯	⋯
	1.136	12.0	11.0	12.5	11.9	0.71	1.42	5.51
	1.296	12.5	11.0	12.5	⋯	⋯	⋯	⋯
	1.616	8.5	8.5	14.0	5.8	1.24	9.7	7.88
	1.712	8.5	6.5	6.5	⋯	⋯	⋯	⋯
U1S2	0.16	37.0	36.0	36.0	⋯	⋯	⋯	⋯
	0.32	24.9	25.5	23.5	⋯	⋯	⋯	⋯
	0.48	39.5	39.5	⋯	⋯	⋯	⋯	⋯
	0.64	40.5	40.5	⋯	8.34	0.39	2.21	2.23
	0.8	43.5	37.5	40.5	⋯	⋯	⋯	⋯
	0.96	43.5	42.5	39.5	6.9	0.21	1.24	2.04
	1.008	40.5	32.5	25.5	⋯	⋯	⋯	⋯
	1.12	33.5	25.5	37.5	8.29	0.65	1.22	2.20
	1.216	29.0	27.0	⋯	⋯	⋯	⋯	⋯
GT1	0.16	34.0	34.0	⋯	⋯	⋯	⋯	⋯
	0.32	19.0	20.0	34.0	3.8	0.65	24.32	1.98
	0.48	19.0	16.0	25.0	⋯	⋯	⋯	⋯
	0.64	12.5	12.0	12.5	4.16	0.63	⋯	1.86
	0.8	12.5	10.5	11.5	⋯	⋯	⋯	⋯
	1.12	9.5	9.5	8.6	4.45	0.65	5.15	2.25
	1.44	10.0	9.0	6.2	⋯	⋯	⋯	⋯
	1.92	10.0	6.0	5.0	4.45	0.65	5.15	2.98
	2.4	5.5	5.6	5.6	⋯	⋯	⋯	⋯
	3.2	4.7	3.6	3.5/4.5	⋯	⋯	⋯	⋯
	4.0	4.7	3.3	3.5	⋯	⋯	⋯	⋯
	4.8	3.2	3.0	3.5	5.99	0.73	2.33	3.56
	5.6	2.8	2.3	3.0	⋯	⋯	⋯	⋯
	7.2	2.9	2.2	2.4	7.85	0.81	8.3	4.35

Note. Radial, vertical, and transversal scale length was measured in just one of the arms at the respective time.

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In Figure 10, we show the surface density of the disk in polar coordinates. The fitting is performed over the region of the arm where the density contrast is stronger, in this case, the outskirts of the disk. The arms induced by a satellite have a higher-density contrast with respect to the disk than in the barless and barred cases, and they also show slightly larger lifetimes (see Figure 10).

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Figure 11 is a zoom of the density structure in polar coordinates for two spiral arms taken at different times of the simulation. The mosaics show (upper panels) the density contrast and (lower panels) the surface density profile of the disk (blue stars) and spiral arms (red stars), as well as a fit for each one (cyan and purple lines, respectively). Satellite-induced spiral arms also have a clear density exponential profile with a scale radius larger than the one for the disk (indicated in the lower panels at the top right corner in kpc), meaning that they are gravitationally and dynamically very important far beyond the scale radius of the disk.

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Figure 12 corresponds to the vertical and transversal density profiles for the two spiral arms shown in Figure 11. The first panel of each mosaic is the same radial density scale presented in Figure 11, included here for reference purposes; the second and third panels of each mosaic show the vertical (i.e., perpendicular to the plane of the disk) and transversal density profiles of the respective spiral arms. In all panels of this figure, we present the fitting in solid curves, with their respective standard deviation (σ) to the left and the scale length (in kpc) at the top right indicated for all cases.

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In these diagrams, we show that, like in the previous models, all arms are well fitted by a sech² in the vertical and transversal directions, while for the radial component, i.e., along the spiral arm, an exponential function provides the best fit. The scale length of the vertical component of the spiral arm is similar to the corresponding one for the disk.

Finally, we have calculated a density contrast for the spiral arms versus disk. We find that the density contrast, calculated at the maximum contrast of the spiral arms, is 7.88. For a quantitative summary of the parameters obtained in this study, see Table 4.

From all the simulations performed for this work, the most physically interesting case is by far the one with a triaxial halo. The outcome spiral arms from this simulation were very interesting because, among other things, the spiral arms were the most long-lasting and had the highest-density contrast compared to the disk. At the beginning, this type of spiral arm acts similarly to the ones induced by a strong galactic bar. However, in this specific simulation, a small bar is also raised at ∼2.6 Gyr.

The spiral arms wound up ∼16° before the formation of the bar; after that, the pitch angle of the arm was maintained over the rest of the simulation (see Table 4). The formation of a bar changed nothing in the spiral arm properties induced by the triaxial halo. This result clearly indicates that since the triaxial halo dominates the global potential, it gives place to the formation and long-lasting nature of these strong spiral arms.

Similar to the spiral arms presented in the previous sections, the radial density profile of the spiral arms is exponential and systematically has a scale length larger than that for the disk itself. In Figure 13, we show the density contrast of the disk in polar coordinates. The fitting is performed over the region of the arm where the density contrast is stronger and extends by several kpc. The arms induced by a triaxial halo tend to be coherent for a long time and maintain their general structure (density profiles: exponential in the radial direction and as a hyperbolic squared secant).

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Figure 14 is a zoom of the density structure in polar coordinates for two spiral arms taken at different times of the simulation. The mosaics show (upper panels) the surface density in polar coordinates and (lower panels) the density fall of the disk (blue stars) and spiral arms (red stars), as well as a fit for each one (cyan and purple lines, respectively). As before, the spiral arms follow an exponential profile with a scale length larger than the one of the disk, indicating that they are still dynamically important in the external regions of the disk.

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The 3D structure of the spiral arms is shown in Figure 15, where a sech² fits well in the vertical and transversal directions, while an exponential provides a radial fitting along the spiral arm. The scale height of the vertical component of the spiral arm is similar to the corresponding one of the disk.

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For illustrative purposes, we have selected the galaxy NGC 2543 with observations both in the optical CALIFA IFU survey (Sánchez et al. 2012) and in the near-infrared with the Spitzer Survey of Stellar Structure in Galaxies (S4G; Sheth et al. 2010). The observations at 3.6 μm are less affected by internal extinction and trace old stellar populations. This is particularly important for deriving the properties of disks and, in particular, for studying spiral arms. We retrieved directly from the IRSA website (https://irsa.ipac.caltech.edu/data/SPITZER/S4G) the results of the 2D structural decompositions using GALFIT (Peng et al. 2010). In order to isolate the spiral arms within the image, we subtracted the best bulge (Sérsic fit), disk (exponential fit), and bar (Ferrers bar) models from the original images.

Similar to what has been done with the results of the simulations, we use the Spitzer 3.6 μm images to trace the spiral arm loci and estimate their density mass profile by using the available CALIFA stellar density maps for NGC 2543. The upper panel in Figure 16 shows the contours of the Spitzer 3.6 μm image overplotted on the CALIFA mass density map after a proper spatial scale matching of these images. The lower panel shows the light profile along the spirals inferred from the Spitzer 3.6 μm image (left) and the corresponding stellar mass density profile from the CALIFA IFU data products (right). Notice that the spiral profile scale length is greater than the disk scale length in NGC 2543 after an exponential disk fit.

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