Kinematic Clues to Bar Evolution for Galaxies in the Local Universe: Why the Fastest Rotating Bars are Rotating Most Slowly

To perform the measurements of the bar size and bar strength of each galaxy, we used images from different surveys according to their availability and quality (Table 1 lists the objects of the present study, and also includes in Column 3 the survey from which the images are obtained). To minimize dust effects on the morphology, we gave priority to images in the infrared, so that most of the images in our sample are 3.6 μm infrared images taken from the Spitzer archive⁸ , which offers the best infrared images publicly available online. We also found infrared images in the J, H, and Ks bands in the 2MASS survey⁹ for only five galaxies of our sample, but with a quality not sufficient to make the calculations, except for the galaxy UGC 2855, for which we used its image in the J band. For those galaxies for which no infrared image is available, we turned to images from the Sloan Digital Sky Survey in the r band (data release 12 of the SDSS III¹⁰ ). Finally, the images of only three galaxies of our sample (UGC 2080, UGC 11124, and UGC 12276) are taken from the ESO Digitized Sky Survey¹¹ (DSS2-red), as this is the only survey that provides data with high enough resolution to make reliable calculations for these objects.

The kinematical information is obtained from the high-resolution velocity fields of the ionized gas, which are extracted from a Fabry–Pérot datacube. The Fabry–Pérot interferometer maps the Hα emission line across the whole field covering the observed galaxy and produces a [x, y, λ'] or [x, y, v_l.o.s.] 3D datacube (where x, y are the two spatial coordinates, λ' and v_l.o.s. are the Hα redshifted wavelength and the velocity in the line of sight of the ionized gas, respectively) with high spectral and angular resolutions, after performing the phase calibration and the wavelength calibration.

The majority of velocity maps of the galaxies of this sample, 64 of a total set of 68, are taken from the GHASP survey (Gassendi HAlpha survey of SPirals¹² , Epinat et al. 2008). This survey consist of a large sample of 203 spiral and irregular galaxies observed with a Fabry–Pérot instrument at the 1.93 m telescope of the Observatoire de Haute de Provence in France, during the period 1998–2004. The data cubes with a pixel scale of 0.68 arcsec/pix obtained with this interferometer have an angular resolution limited by the seeing value with an averaged value of ∼3 arcsec, and the spectral resolution is ∼16 km s⁻¹ in Hα. The remaining four galaxies (UGC 3013, UGC 5303, UGC 6118, and UGC 7420) were observed with GHαFaS (Galaxy Hα Fabry–Pérot System, Hernandez et al. 2008). The observations were carried out in several runs at the William Herschel Telescope, Roque de los Muchachos Observatory, La Palma, Spain, in the period between 2010 and 2014. This instrument has a field of view of 3.4 arcmin² and gives data cubes with a spectral resolution of ∼8 km s⁻¹ in Hα, and with an average seeing-limited angular resolution of ∼1.2 arcsec.

In the present study we have measured morphological and kinematical properties of the bars and the disk galaxies that host the bars. The morphological parameters of the disk galaxies are the inclination angle, the geometrical center of the galaxy, and the position angle of the line of nodes; these parameters are needed to deproject the images. We also measured the maximum rotation velocity of the ionized hydrogen as the only kinematical parameter of the disk as a whole. Concerning the morphological properties of the bars, we determined the bar length and the position angle of the bar. The set of the measured bar parameters includes the bar strength, the bar corotation radius, and the pattern speed of the bar.

2.2.1. The Disk Properties

Several parameters that characterize the galactic disk are determined, such as the galactic center, the inclination angle, the position angle of the major axis of the disk galaxy, and the asymptotic circular velocity, which is defined as the value of the circular velocity that the rotation curve tends to when this curve is almost flat, which occurs for large galactocentric radii. The rotation curve is calculated using the ROTCUR task of the astronomical package GIPSY. This software fits a tilted ring model (Begeman 1987) to the velocity field, so that it is also possible to determine the geometrical parameters of the galaxy by allowing one single parameter to vary freely while the others are kept fixed; this is then repeated with another parameter, and so on. The values of these kinematical properties such as inclination angle, position angle, and asymptotic velocity can be found in Table 1, Columns 7, 8, and 9, respectively. The uncertainty of the position angle is taken to be 2° for all galaxies, and the uncertainty for the inclination angle is taken as 7°. For the asymptotic velocity, we assume a relative uncertainty of 10%. We also include in Table 1, Column 10, the values of r₂₅, which are taken from the NASA Extragalactic Database. The stellar masses of the galaxies, which are given in Column (11) of Table 1, are taken from the NASA Sloan-Atlas (NSA¹³ ), the masses of those galaxies that are not found in the NSA database are estimated following the linear fit of the expression between an arbitrary color and the V-band mass-to-light ratio given by Wilkins et al. (2013):

$$\begin{eqnarray}&&\mathrm{log}{{\rm{\Gamma }}}_{V}=\mathrm{log}[(M/{M}_{\odot })/({L}_{V}/{L}_{\odot })]={p}_{1}\cdot ({m}_{V}-{m}_{B})+{p}_{2},\end{eqnarray} \tag{ 1 }$$

where the relative magnitude in V is taken from NED database, and the color $B-V$ is obtained from the Hyperleda database.¹⁴ The uncertainties of the stellar mass are not included in Table 1, but can be estimated to be ∼60% of the mass of the galaxy (Mendel et al. 2014).

2.2.2. The Bar Length

We described in Section 1 some of the techniques that have been employed to measure the bar length in the past. We consider the method we have chosen here to be well adapted to the type of data used, mainly 3.6 μm images from Spitzer, supplemented with R-band SDSS images where Spitzer data were not available for the galaxies we had analyzed kinematically, and with a few DSS images where neither of the former types of observations was available. In this study, we adopt the technique developed by Erwin & Sparke (2003), which was applied in Erwin (2004) and discussed in detail in Erwin (2005). The technique consists of performing an ellipse fitting on the image of the galaxy, using a script based on the ELLIPSE task of the IRAF astronomical software package, in order to generate the radial dependence of the ellipticity and the position angle of ellipses. Initial values of the center, inclination, position angle, and ellipticity are estimated by eye inspection of the image. We then determine three different measurements that characterize the bar length, ${\rho }_{\epsilon }$, r_min, and r₁₀, which are defined as follows: ${\rho }_{\epsilon }$ is the radius where the peak of ellipticity has its maximum, while the position angle of the fitted ellipses is almost constant (Wozniak & Pierce 1991; Wozniak et al. 1995); r_min is the radial position of the first minimum in ellipticity just outside of the ellipticity peak; and r₁₀ indicates the radius where the position angle of the ellipses differs by at least 10° with respect to the angle measured at ${\rho }_{\epsilon }$. By definition, the two latter magnitudes, r_min and r₁₀, are larger than the first magnitude, which is taken as a lower limit of the bar size, therefore we define the upper limit of the bar length, ρ_bar, as the minimum of these two radii, i.e., ρ_bar = min(r_min, r₁₀), thus giving a bracketed value of the bar length, between the minimum and maximum defined here.

The position angle of the bar is determined as the position angle at the radius equal to ${r}_{\epsilon }$, this angle is needed to deproject the two measurements of the bar length, which are calculated according to the following expression:

$$\begin{eqnarray}&&{\rho }^{\mathrm{deproj}}={\rho }_{\mathrm{proj}}\cdot \cos {\theta }_{\mathrm{bar}}\ \cdot {({\tan }^{2}{\theta }_{\mathrm{bar}}\cdot {\sec }^{2}i+1)}^{1/2},\end{eqnarray} \tag{ 2 }$$

where ρ_proj is the measured projected bar length (i.e., ${\rho }_{\epsilon }$ and ρ_bar), θ_bar is the position angle of the bar with respect to the position angle of the disk galaxy, and i is the inclination angle of the galaxy. This expression is also used to calculate the corresponding uncertainties. A discussion of the effects of working on deprojected images to determine the bar length can be found in Gadotti et al. (2007).

Following this procedure, we give an upper and a lower limit for the bar length, in other words, the end of the bar should be placed between $[{\rho }_{\epsilon }^{\mathrm{deproj}},{\rho }_{\mathrm{bar}}^{\mathrm{deproj}}]$. However, in order to assign a single value to the bar length, we define r_bar as the mean value of the deprojected values of ${\rho }_{\epsilon }$ and ρ_bar, and the associated error is taken to be the half of the difference between these values. With an image of the galaxy, we have confirmed that the two limits to the bar length that we calculate bracket the bar size estimated by visual inspection, and we confirm that r_bar is a reliable measurement of the radius of the bar. In Table 2 (Columns 2 and 3) the deprojected values of ${\rho }_{\epsilon }$ and ρ_bar in kpc are given.

Table 2.  Bar Parameters of the Galaxies in the Sample

Name	${\rho }_{\epsilon }$	${\rho }_{\mathrm{bar}}$	rCR	${{\rm{\Omega }}}_{P}^{\mathrm{bar}}$	Sbar	${ \mathcal R }$	Γ
UGC	(arcsec)	(arcsec)	(arcsec)	(km s−1 kpc−1)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
508	46.8 ± 2.0	49.7 ± 2.0	51.8 ± 2.3	${32.6}_{-1.2}^{+1.3}$	0.37	1.07 ± 0.07	1.61 ± 0.20
763	40.3 ± 2.2	46.7 ± 2.1	51.9 ± 1.9	26.2 ± 0.6	0.30	1.20 ± 0.04	1.90 ± 0.11
1256	25.2 ± 2.2	57.4 ± 2.3	58.7 ± 3.0	26.5 ± 1.0	0.17	1.68 ± 0.09	2.37 ± 0.15
1317	15.5 ± 2.1	17.2 ± 2.1	26.3 ± 5.1	${34.7}_{-4.6}^{+6.2}$	0.45	1.61 ± 0.31	4.64 ± 0.73
1437	11.0 ± 2.0	11.2 ± 2.0	16.4 ± 5.6	${40.8}_{-10.5}^{+18.4}$	0.10	1.47 ± 0.50	4.57 ± 1.63
1736	30.2 ± 3.5	56.0 ± 5.1	56.1 ± 2.1	33.5 ± 0.9	0.15	1.43 ± 0.05	2.08 ± 0.37
1913	95.4 ± 2.0	117.3 ± 2.0	134.5 ± 4.6	16.2 ± 0.8	0.22	1.28 ± 0.04	2.18 ± 0.20
2080	23.4 ± 2.0	29.8 ± 2.1	32.1 ± 1.3	${46.1}_{-1.2}^{+1.3}$	0.35	1.22 ± 0.05	3.21 ± 0.52
2855	42.8 ± 2.5	44.0 ± 2.5	70.4 ± 6.3	${33.5}_{-2.7}^{+2.9}$	0.30	1.62 ± 0.15	1.63 ± 0.14
3013	72.3 ± 6.3	105.1 ± 8.2	100.1 ± 2.7	10.4 ± 0.3	1.21	1.17 ± 0.03	1.21 ± 0.07
3463	28.5 ± 2.7	32.0 ± 2.8	45.5 ± 7.5	${19.1}_{-2.4}^{+3.1}$	0.25	1.51 ± 0.25	1.93 ± 0.28
3685	23.2 ± 2.0	29.5 ± 2.0	27.7 ± 1.7	22.3 ± 0.3	0.40	1.07 ± 0.07	2.77 ± 0.23
3709	16.6 ± 2.7	21.8 ± 2.9	24.6 ± 1.7	${27.0}_{-1.5}^{+1.6}$	0.54	1.31 ± 0.09	1.59 ± 0.10
3740	17.2 ± 3.0	19.4 ± 3.1	33.3 ± 1.6	19.9 ± 0.5	0.30	1.82 ± 0.09	1.60 ± 0.19
3809	41.7 ± 4.3	53.2 ± 5.2	62.0 ± 4.9	${24.6}_{-1.6}^{+1.8}$	0.11	1.33 ± 0.10	3.23 ± 0.23
4165	32.5 ± 4.1	41.2 ± 4.8	48.9 ± 2.0	${27.3}_{-0.9}^{+1.0}$	0.34	1.35 ± 0.06	1.57 ± 0.19
4273	32.2 ± 3.2	44.6 ± 4.0	47.4 ± 3.2	${22.4}_{-1.1}^{+1.3}$	0.25	1.27 ± 0.09	1.35 ± 0.08
4325	58.6 ± 2.0	62.1 ± 2.0	67.2 ± 3.9	18.5 ± 0.6	0.22	1.11 ± 0.06	1.20 ± 0.10
4422	38.6 ± 2.8	43.6 ± 2.9	44.3 ± 1.5	${29.2}_{-0.9}^{+1.0}$	0.55	1.08 ± 0.04	2.21 ± 0.30
4555	9.3 ± 2.0	10.4 ± 2.0	13.0 ± 2.6	${41.1}_{-4.9}^{+6.3}$	0.15	1.32 ± 0.26	2.97 ± 0.47
4936	38.2 ± 2.0	43.0 ± 2.0	70.8 ± 2.1	${24.1}_{-0.5}^{+0.6}$	0.22	1.75 ± 0.05	2.46 ± 0.14
5228	17.7 ± 2.0	31.1 ± 2.0	30.4 ± 1.3	${32.3}_{-1.0}^{+1.1}$	0.45	1.35 ± 0.06	2.28 ± 0.11
5303	24.7 ± 2.0	27.8 ± 2.0	35.2 ± 6.2	${56.9}_{-7.7}^{+10.4}$	0.11	1.35 ± 0.24	2.69 ± 0.43
5319	10.4 ± 2.0	13.1 ± 2.0	16.2 ± 1.2	${51.3}_{-2.3}^{+2.5}$	0.30	1.40 ± 0.10	2.46 ± 0.34
5510	14.7 ± 2.0	17.6 ± 2.0	30.8 ± 1.6	${50.9}_{-1.9}^{+2.0}$	0.49	1.92 ± 0.10	2.49 ± 0.34
5532	6.9 ± 2.3	11.1 ± 2.3	16.3 ± 3.3	${115.9}_{-18.8}^{+26.6}$	0.13	1.92 ± 0.39	6.77 ± 1.34
5786	19.3 ± 4.9	23.1 ± 4.9	20.6 ± 2.1	${37.1}_{-6.2}^{+9.3}$	0.49	0.98 ± 0.10	2.96 ± 0.68
5840	29.8 ± 2.1	35.2 ± 2.1	60.9 ± 5.4	${84.6}_{-5.8}^{+6.7}$	0.12	1.89 ± 0.18	2.39 ± 0.19
5842	15.5 ± 2.1	22.0 ± 2.2	23.4 ± 1.5	39.0 ± 0.8	0.17	1.29 ± 0.08	2.26 ± 0.19
5982	14.0 ± 2.0	16.3 ± 2.0	19.5 ± 2.0	${73.3}_{-16.1}^{+31.4}$	0.17	1.29 ± 0.13	4.44 ± 1.44
6118	33.3 ± 2.0	43.9 ± 2.0	43.7 ± 1.7	58.9 ± 1.9	0.57	1.15 ± 0.05	1.90 ± 0.11
6537	44.4 ± 2.1	53.9 ± 2.1	59.8 ± 2.5	35.4 ± 0.8	0.24	1.23 ± 0.05	2.43 ± 0.12
6778	17.8 ± 2.0	20.5 ± 2.0	34.9 ± 4.2	${63.4}_{-5.1}^{+6.2}$	0.12	1.83 ± 0.22	2.86 ± 0.27
7021	25.3 ± 5.1	29.3 ± 5.6	30.0 ± 3.2	${48.2}_{-5.6}^{+6.7}$	0.86	1.11 ± 0.12	2.89 ± 0.40
7154	38.5 ± 3.8	44.7 ± 4.1	63.4 ± 2.7	19.8 ± 0.5	0.71	1.53 ± 0.07	1.89 ± 0.08
7323	56.4 ± 3.4	61.6 ± 3.6	92.4 ± 7.4	18.6 ± 0.9	0.47	1.57 ± 0.13	1.31 ± 0.13
7420	31.6 ± 2.3	36.1 ± 2.3	36.1 ± 3.1	${49.6}_{-2.9}^{+3.2}$	0.44	1.07 ± 0.09	5.26 ± 0.34
7766	17.9 ± 3.7	27.9 ± 5.1	37.3 ± 1.9	${39.4}_{-1.4}^{+1.5}$	0.17	1.71 ± 0.09	6.65 ± 0.35
7853	23.5 ± 3.2	43.7 ± 3.3	41.7 ± 1.1	19.4 ± 0.5	0.27	1.36 ± 0.04	1.69 ± 0.48
7876	16.4 ± 3.8	21.1 ± 4.2	28.7 ± 1.9	${36.4}_{-1.2}^{+1.3}$	0.20	1.55 ± 0.10	1.60 ± 0.13
7985	19.5 ± 3.2	21.4 ± 3.3	38.3 ± 5.3	${38.3}_{-3.7}^{+4.4}$	0.29	1.88 ± 0.26	1.83 ± 0.22
8403	10.6 ± 2.0	25.4 ± 2.1	30.8 ± 2.0	20.2 ± 0.4	0.32	2.05 ± 0.13	1.86 ± 0.09
8709	33.0 ± 2.0	44.4 ± 2.0	50.0 ± 1.4	24.1 ± 0.5	0.54	1.32 ± 0.04	3.33 ± 0.10
8852	17.4 ± 2.0	22.0 ± 2.0	26.8 ± 1.3	${44.5}_{-1.7}^{+1.8}$	0.12	1.38 ± 0.07	2.23 ± 0.11
8937	20.2 ± 2.2	36.6 ± 2.3	33.3 ± 3.0	${35.9}_{-3.4}^{+4.0}$	1.06	1.28 ± 0.12	2.04 ± 0.44
9179	61.6 ± 2.1	78.1 ± 2.1	74.3 ± 1.5	44.7 ± 1.1	0.30	1.08 ± 0.02	1.92 ± 0.32
9358	8.4 ± 2.0	17.3 ± 2.0	14.7 ± 1.9	${92.4}_{-8.4}^{+10.3}$	0.37	1.30 ± 0.17	5.84 ± 0.62
9366	25.4 ± 2.5	28.1 ± 2.6	36.9 ± 5.6	${34.1}_{-4.3}^{+5.6}$	0.25	1.38 ± 0.21	3.09 ± 0.45
9465	10.4 ± 1.6	13.1 ± 1.6	21.5 ± 2.0	26.7 ± 1.2	0.20	1.85 ± 0.17	2.37 ± 0.15
9736	8.3 ± 1.4	9.0 ± 1.4	12.1 ± 2.2	${46.6}_{-5.2}^{+7.1}$	0.34	1.40 ± 0.26	3.67 ± 0.51
9753	15.9 ± 2.0	38.7 ± 2.0	30.6 ± 1.9	${74.6}_{-4.4}^{+5.0}$	0.22	1.36 ± 0.08	4.06 ± 0.29
9943	16.2 ± 2.0	22.5 ± 2.0	34.4 ± 4.2	${39.5}_{-4.0}^{+4.9}$	0.27	1.83 ± 0.22	2.51 ± 0.29
9969	15.9 ± 1.4	19.0 ± 1.4	27.7 ± 2.8	${50.2}_{-2.4}^{+2.5}$	0.42	1.60 ± 0.16	4.64 ± 0.24
10075	10.3 ± 2.0	14.3 ± 2.0	16.1 ± 2.3	${71.6}_{-3.2}^{+3.5}$	0.17	1.34 ± 0.19	4.89 ± 0.26
10359	73.3 ± 2.0	90.1 ± 2.0	112.8 ± 1.4	14.8 ± 0.2	0.50	1.40 ± 0.02	1.52 ± 0.16
10470	35.2 ± 2.5	54.8 ± 2.9	57.8 ± 3.4	${24.9}_{-1.3}^{+1.4}$	0.92	1.35 ± 0.08	1.41 ± 0.18
10546	13.7 ± 2.0	15.4 ± 2.0	28.1 ± 1.5	36.3 ± 1.5	0.10	1.93 ± 0.11	2.93 ± 0.33
10564	32.8 ± 2.6	42.2 ± 2.8	45.6 ± 1.5	12.4 ± 0.3	0.67	1.23 ± 0.04	1.40 ± 0.08
11012	16.9 ± 4.0	17.8 ± 6.3	21.8 ± 1.8	107.7 ± 2.0	0.35	1.26 ± 0.10	5.02 ± 0.21
11124	22.7 ± 3.1	31.3 ± 3.7	48.3 ± 2.7	14.4 ± 0.3	0.30	1.84 ± 0.10	1.30 ± 0.10
11283	21.9 ± 2.0	26.1 ± 2.0	38.4 ± 2.6	${21.9}_{-1.9}^{+2.3}$	0.72	1.62 ± 0.11	1.05 ± 0.24
11407	49.6 ± 2.0	56.3 ± 2.0	64.9 ± 2.3	12.9 ± 0.3	0.40	1.23 ± 0.04	1.07 ± 0.10
11557	21.7 ± 1.6	26.0 ± 1.6	32.4 ± 2.6	18.4 ± 0.4	0.22	1.37 ± 0.11	1.10 ± 0.38
11861	28.5 ± 3.7	33.1 ± 4.1	37.8 ± 3.7	${24.8}_{-1.0}^{+1.1}$	0.34	1.23 ± 0.12	1.73 ± 0.20
11872	17.2 ± 3.3	19.3 ± 3.5	19.9 ± 2.8	${94.5}_{-9.6}^{+11.9}$	0.77	1.09 ± 0.15	4.20 ± 0.50
12276	14.2 ± 2.3	18.0 ± 2.4	21.8 ± 1.3	11.9 ± 0.6	0.17	1.37 ± 0.08	2.02 ± 0.41
12343	63.5 ± 5.0	108.4 ± 7.8	91.1 ± 4.3	18.4 ± 1.0	0.94	1.14 ± 0.05	1.33 ± 0.08
12754	47.3 ± 7.0	77.0 ± 10.0	69.6 ± 7.3	${36.6}_{-2.1}^{+2.4}$	0.50	1.19 ± 0.12	1.68 ± 0.13

Note. Columns (2) and (3) give the upper and lower limit, respectively, for the deprojected bar length of the galaxies named in Column (1), as described in the text. The bar corotation radius in arcseconds and the pattern speed of the bar calculated with the Font–Beckman method are given in Columns (4) and (5), respectively. The calculated values of the bar strength appear in Column (6) with an uncertainty of 0.04 for all galaxies. The rotational parameter, defined as the bar corotation length scaled by the bar size, is given in Column (7). Column (8) shows the angular velocity of the bar in units of the angular velocity of the disk.

Download table as:  ASCIITypeset images: 1 2

In order to characterize the effect of using images of different surveys taken in such different passbands (3.6 μm for Spitzer, R broadband for SDSS, and 0.85 μm bands for DSS2), we selected a small subset of six galaxies for which images from these three surveys are available, and we measured their bar lengths. In Figure 1 we plot r_bar values as measured with Spitzer images compared with those with SDSS images, and in the right panel we compare r_bar from Spitzer and DSS2 images. In the two plots the line marks the 1:1 proportion. We can see that the Sloan images give values of the bar length that match well, within error bars, with those calculated with infrared images, but the bar lengths calculated from DSS2 images are only just compatible with the Sloan bar lengths, showing a higher dispersion with respect to the 1:1 line for the longer bars than for the shorter ones. As the only three galaxies analyzed with DSS images do not have large bars, we can take the DSS2 bar lengths as reliable.

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2.2.3. The Bar Strength

In the introduction we give an outline description of some of the methods that have been used to define and measure the bar strength. Here we measure the strength of the bar by performing a Fourier decomposition of the galaxy image. In the updated version of the kinemetry code (Krajnović et al. 2006), the surface brightness image is written as a combination of a finite number of harmonic terms:

$$\begin{eqnarray}{\rm{\Sigma }}(r,\varphi ) & = & {A}_{0}(r)+\displaystyle \sum _{m=1}^{N}{A}_{m}(r)\cdot \sin (m\varphi )\\ & & +{B}_{m}(r)\cdot \cos (m\varphi ),\end{eqnarray} \tag{ 3 }$$

where φ is the azimuthal angle, and r is the length of the semimajor axis of the elliptical ring in which the code performs the harmonic fitting. With this we calculate the harmonic coefficients A_m and B_m as a function of the radius. Although the main contribution to the bar strength comes from the amplitude of the harmonic term m = 2, Ohta et al. (1990) showed that the contribution of the even terms m = 4, 6 is not negligible, therefore we include these higher order terms in the calculation of the bar strength, which is defined as the integration over a fixed radial range of the Fourier amplitude for the harmonics m = 2, 4, 6 relative to the m = 0 mode:

$$\begin{eqnarray}&&{S}_{b}=\tfrac{\displaystyle \sum _{m=\mathrm{2,4,6}}{\int }_{{r}_{1}}^{{r}_{2}}\sqrt{{A}_{m}^{2}+{B}_{m}^{2}}{dr}}{{\int }_{{r}_{1}}^{{r}_{2}}{A}_{0}{dr}},\end{eqnarray} \tag{ 4 }$$

in which r₁ and r₂ are arbitrary limits that characterize the bar region. In our calculations we take the integration limit r₁ to be half of the deprojected lower limit of the bar length ${\rho }_{\epsilon }$, and r₂ equal to the deprojected upper limit of the bar length ${\rho }_{\mathrm{bar}}$, in order to restrict the region that is dominated by the bar. This means that we exclude any contribution to the non-axisymmetric part from the spiral arms. To test our procedure, we have measured the bar strength taking only the amplitude of the harmonic m = 2, and we find a strong linear correlation between these values and the bar strength calculated using expression (4). Kinemetry also gives the uncertainty associated with each harmonic term. By propagating these uncertainties using expression (4), we can therefore estimate the uncertainty of the bar strength. In this way, we obtain approximately the same uncertainty of ∼0.04 for all galaxies.

In order to quantify how reliable the measurements of the bar strength are when we use an image in the R-band optical waveband from SDSS survey and in near-infrared from DSS2 survey (λ_eff = 0.85 μm) with respect to the values calculated from an infrared image, we perform the calculations of S_b of a subset of six random galaxies for which we have images in the three surveys. Results show that the bar strengths calculated from Sloan images do reproduce the values obtained from the Spitzer images, see left panel of Figure 2; the data show a small dispersion with respect to the 1:1 solid line. On the other hand, the S_b values from DSS2 images are veryy far from the infrared values, as shown in the right panel of Figure 2. When we examine the numerator of expression (4), we find that that the values for DSS2 are lower (nearly half) than those measured with the corresponding Spitzer images. Furthermore, while in the central regions the radial dependence of the term A0 (the denominator in expression (4)) is similar for the Spitzer and the DSS2 images, but farther out, A0_spitzer decays as a function of the radius, while A0_DSS is almost uniform. To cover this case, a linear fit to the data was made (plotted as a dashed line), and this is used to correct for the strength of the bar measured with DSS images:

$$\begin{eqnarray}&&{S}_{b}^{{Spitzer}}=0.009+3.125\cdot {S}_{b}^{\mathrm{DSS}}.\end{eqnarray} \tag{ 5 }$$

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The calculated values of the bar strength and their associated uncertainties are listed in column 6 of Table 2.

2.2.4. The Corotation Radius of the Bar and the Pattern Speed

2.2.5. The Scaled Properties

With the data listed in Table 2, we can construct the histograms of each measured parameter. We plot in Figure 3 the distribution of the rotational parameter (left panel) and the relative bar pattern speed (right panel). The histogram of the relative bar corotation clearly shows that the rotational parameter is somewhat larger than unity, within uncertainties, as theoretically predicted by Contopoulos (1980), who studied the response of a bar to the barred perturbations of density, deriving the result that the bar corotation resonance must occur beyond, but close to the bar end. The mean value of this parameter and its standard deviation for our sample is $\overline{{ \mathcal R }}=1.41\pm 0.26$, which is compatible within uncertainties with $\overline{{ \mathcal R }}=1.2\pm 0.2$ given by Athanassoula (1992). We also clearly distinguish four separated peaks in the histogram. The central positions of the peaks are found at ${{ \mathcal R }}_{1}=1.09$, ${{ \mathcal R }}_{2}=1.32$, ${{ \mathcal R }}_{3}=1.59$ and ${{ \mathcal R }}_{4}=1.85$, which means that the first two peaks are due to bars, which are traditionally termed fast rotators, as the limiting value to classify bars as fast/slow rotators is 1.4 (Debattista & Sellwood 2000). Adopting this classification, we find that two thirds of our galaxies host a fast rotator bar.

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It is widely accepted, from both observations and modeling, that the rotational parameter increases from early- to late-type galaxies (Aguerri et al. 1998; Rautiainen et al. 2008; Font et al. 2011, 2014a). This could induce a misinterpretation of the histogram of this parameter (left panel of Figure 3) in which each peak could be erroneously associated with a particular morphological type of galaxy if only the mean value of this parameter were taken into account. With the values of ${ \mathcal R }$ appearing in Column 7 of Table 2, we can see that in each peak we have a contribution of galaxies of many different morphological types. We can see in the right panel of Figure 3 that the four peaks in the dimensionless parameter ${ \mathcal R }$ have no equivalent in the distribution of the relative bar pattern speed, which shows two different populations: the bars that rotate over four times faster than the disk (a minority), and the majority of the galaxies, whose bar has an angular rate lower than four times the angular speed of the disk, with a mode value of twice the angular speed. In any case, we find that all bars rotate faster than their disks.

In this section we study the dependence of the bar properties, which are listed in Table 2, on the morphological type of the host galaxy. In Figure 4 we plot this variation for the relative bar length (panel a), the bar strength (panel b), the rotational parameter (panel c), and the relative bar pattern speed (panel d). In all plots, the red squares mark the mean value of the parameter for the specific morphological type, and the standard errors of the mean are indicated as error bars.

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In panel (a) of Figure 4 we see that the earlier galaxies of type T = 1, 2 host the largest bars, then the relative bar size of the bar drops from galaxies of ab morphological type down to cd-type of galaxies (T = 6), for which the shorter bars are found, and this is followed by a clear increase through the later type of galaxies. When we compare our bar length distribution as a function of the galaxy morphological type with the same distribution found in other studies, we find that in general they are in agreement, although our sample of 68 spiral galaxies is statistically poor especially for the earlier types; all these distributions show a local minimum of the relative bar length for galaxies of morphological type T = 6. This was also found by Martin (1995), who studied 136 spiral galaxies, Laurikainen et al. (2007), who analyzed a sample of 216 galaxies, and Díaz-García et al. (2016) with a very large sample of more than 600 galaxies belonging to the survey S4G.

The bar strength does not show large variations through galaxies of morphological type T $\geqslant \,4$ (see panel (b) of Figure 4). Laurikainen et al. (2007) plotted the distribution of bar strength as a function of galaxy morphological type for a sample of 216 galaxies. When they calculated the bar strength from the torque maps, they found a steady growth along the Hubble sequence. The same method was employed by Díaz-García et al. (2016) with a large sample of ∼600 galaxies, finding a similar result. Similarly, Seidel et al. (2015) used torques to estimate the bar strength and also found this growth of the bar strength from early to intermediate type of galaxies, with a sample of 16 galaxies.

The variation of the rotational parameter is displayed in panel (c) of Figure 4, where the horizontal dashed line marks the border between fast and slow rotators. The figure shows a significant rise of this parameter from early-type galaxies to intermediate-type galaxies, and then the parameter remains almost constant until a large drop for the SBm galaxies occurs, although the numbers here are small. The mean value of the ratio between bar corotation radius and the bar length and its standard deviation for early-type galaxies (SBa, SBab) is found to be ${\overline{{ \mathcal R }}}_{\mathrm{early}}=1.18\pm 0.12$, for intermediate-type galaxies (SBb, SBbc) is ${\overline{{ \mathcal R }}}_{\mathrm{inter}}=1.37\pm 0.27$, and for late-type galaxies ${\overline{{ \mathcal R }}}_{\mathrm{late}}=1.45\pm 0.25$. These results indicate that the corotation to the bar radius ratio shows a slight tendency to grow from morphologically earlier to late-type galaxies, as also pointed out in Aguerri et al. (1998), but these values are also consistent with the conclusion that this parameter shows no dependence on the morphological type of the galaxy, when the standard deviations are taken into account. The mean values of the rotational parameter for the three morphological types of galaxies determined in the present study are in agreement with those found in Font et al. (2014a) with a sample of smaller size, see Table 3. However, in that study the sizes of the bars were taken from the literature, which means that they were calculated by different authors using different methods. Modeling infrared images, Rautiainen et al. (2008) derived values of the corotation over the bar radius shown in Table 3, which are in agreement, within the uncertainties, with the values obtained from our sample. In disagreement with the values shown in Table 3, Aguerri et al. (2015), applying the Tremaine–Weinberg method to determine the pattern speed of 15 galaxies from the CALIFA survey plus 17 from the previous literature, found that this parameter shows no significant variation along the Hubble sequence.

Table 3.  The Rotational Parameter for Different Morphological Types of Galaxies

References	${\overline{{ \mathcal R }}}_{\mathrm{early}}$	${\overline{{ \mathcal R }}}_{\mathrm{inter}}$	${\overline{{ \mathcal R }}}_{\mathrm{late}}$	N
Rautiainen et al. (2008)	1.15 ± 0.25	1.44 ± 0.29	1.82 ± 0.63	38
Font et al. (2014a)	1.15 ± 0.28	1.30 ± 0.30	1.35 ± 0.28	32
This study	1.18 ± 0.12	1.37 ± 0.27	1.45 ± 0.25	68

Note. The mean values, and their standard deviations, of the corotation to the bar length ratio for earlier, intermediate, and later type of galaxies are given in the three central columns. In the first column we list the reference from which these values are taken, and in the last column the size of the sample is listed.

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In the last plot of Figure 4 (panel d) we show the variation of the scaled bar pattern speed with the morphological type of galaxy. We find that the bar of an earlier type galaxy (T = 1, 2) rotates slowly, then the parameter jumps and remains uniform for galaxies of Hubble type between 3 and 6, and finally, the bar angular rate drops to low values for galaxies with T $\geqslant$ 7. Numerical simulations of bar evolution predict the slowdown of the bar (Weinberg 1985; Athanassoula 2003). This braking of the bar from intermediate (3 $\leqslant$ T $\leqslant$ 6) to later types of galaxies (T $\geqslant$ 7) that we show in the distribution can therefore only be interpreted as a sign of galaxy evolution if we assume that late-type galaxies are more evolved than intermediate-type galaxies, which is potentially interesting, but beyond the scope of the present article.

We now show how the properties of the bar are affected by the stellar mass of the host galaxy. To do so, we plot the bar length (in kpc), the bar strength, the bar corotation scaled to the bar size, and the relative bar angular rate as a function of the stellar mass of the galaxy (in units of solar masses) in Figure 5, from panels (a) to (d), respectively.

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Kormendy (1979) showed that the bar size is well correlated with the galaxy mass (absolute magnitude); this dependence is shown in panel (a) of Figure 5. Although the total stellar mass is determined with rather large uncertainties, we show in this figure that the largest bars can only be found in the most massive galaxies, while the shortest bars are hosted in intermediate- and low-mass galaxies. This tendency is also found in Díaz-García et al. (2016).

In the plot of the stellar mass of the galaxy versus the bar strength (panel (b) of Figure 5), we can see that all data points are uniformly distributed in the upper half, confined by a lower limiting diagonal in this parametric plane; consequently, the lower half is a forbidden region. Interpreting the extent of the permitted region, we conclude that the strongest bars are only present in massive galaxies (although the most massive galaxies can also have weaker bars), and that the bars within the less massive galaxies can only be the weaker ones. Somewhat similar behavior is found in panel (d), where the galaxy masses are plotted against the bar angular speed. The figure shows that the bars that rotate fastest are only found in galaxies with intermediate mass (∼10¹⁰ solar masses), in other words, we do not detect any galaxy in our sample with a very low or a very high stellar mass that hosts a bar that spins with an angular speed greater than ∼50 km s⁻¹ kpc⁻¹. These fastest bars are only present in galaxies with an intermediate mass.

The plot of the stellar mass of the galaxies against the rotational parameter is shown in panel (c) of Figure 5. We do not find any particular trend between these two parameters. If we assume the conventional classification of bars as fast or slow rotators depending on whether ${ \mathcal R }$ is below or above 1.4, respectively (this critical value is marked in the plot as a vertical dashed line), we therefore find that the two types of bars are present in galaxies regardless of their stellar mass.

All data plotted in the four panels of Figure 5 are colored according to the morphological type of the galaxy. Early-type galaxies (in blue) include galaxies of types a and ab, intermediate types (in red) include b- and bc-type galaxies, and late-type galaxies (in green) are all the remaining galaxies. Comparing the intermediate- and late-type galaxies because the number of early-type galaxies is of relatively lower significance, we can therefore infer from panels (a) and (d) of Figure 5 that late-type galaxies are less massive than intermediate-type galaxies, and the bars in late-type galaxies are short, rotating with low angular rates compared with the bars hosted by intermediate-type galaxies. We cannot compare these two types of galaxies in terms of bar strength and rotational parameter as the population of these galaxies appears to be blended in the diagrams of panels (b) and (c) of Figure 5.

In Figure 6 we combine panels (a) and (d) of Figure 5 by plotting the bar angular rate against the bar length. We define three different groups of galaxies depending on their total stellar mass, and use this classification to color the data in the plot. We can see that each group of galaxies occupies a defined region in this (r_bar–Ω_bar) parametric space. We therefore infer from this figure that (i) the longest bars (${r}_{\mathrm{bar}}\gtrsim 10\,\mathrm{kpc}$) can only rotate with low angular speed (${{\rm{\Omega }}}_{\mathrm{bar}}\lesssim 40$ km s⁻¹ kpc⁻¹) and can only be found in the most massive galaxies (blue points in the figure). (ii) Those short bars (${r}_{\mathrm{bar}}\lesssim 3\,\mathrm{kpc}$) that rotate slowly (${{\rm{\Omega }}}_{\mathrm{bar}}\lesssim 40$ km s⁻¹ kpc⁻¹) are only present in the less massive galaxies (data in red of Figure 6). (iii) The bars that rotate with the highest angular speed are necessarily short (${r}_{\mathrm{bar}}\lesssim 3\,\mathrm{kpc}$), and can only be hosted by galaxies of intermediate mass (green points in the top left region of Figure 6). In general, Figure 6 shows that bars shorter than ∼3 kpc can only be present in galaxies of low or medium mass, depending on whether the pattern speed is lower or higher than ∼40 km s⁻¹ kpc⁻¹, respectively. Moreover, we find a linear correlation between the bar pattern speed and the total stellar mass of the galaxy for the smallest bars, as shown in Figure 7. In this figure, red dots correspond to bars with a radial length shorter than 3 kpc, and blue dots correspond to the remaining bars with ${r}_{\mathrm{bar}}\gt 3\,\mathrm{kpc}$ (uncertainty bars are omitted for clarity). The solid line in this figure shows the linear fit performed when only the smallest bars are considered (red dots), which follows the expression

$$\begin{eqnarray*}&&\mathrm{log}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right)=7.21+1.50\cdot \mathrm{log}({{\rm{\Omega }}}_{\mathrm{bar}}[\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{kpc}}^{-1}]).\end{eqnarray*}$$

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Figure 7 also demonstrates that we would find no overall relationship between these parameters if all bars were taken together.

It is known from many N-body simulations of barred galaxy evolution that as the galaxy evolves, the bar tends to grow and also to suffer a deceleration in its angular rotation rate (Weinberg 1985; Athanassoula 2003, 2014; Martínez-Valpuesta et al. 2006, 2017; Berentzen et al. 2007; Villa-Vargas et al. 2009, 2010). The galaxy as a whole also experiences a growth in mass, which has been monitored by the star formation and the activity of the active galactic nuclei (see Section 15.3 of Mo et al. 2010). In particular, the stellar mass of the galaxy in a dark matter halo grows while the galaxy is evolving (van de Voort 2016, and references therein). According to this scenario, an evolved barred galaxy is characterized by a high stellar mass and by harboring a slowly rotating and long bar (galaxies located in the bottom right region of Figure 6) compared with the initial values of these parameters before the galaxy evolution (top left region in Figure 6). The simulations of bar evolution show how these two stages are connected, as can be seen in Figure 8 (circles for model 1 and squares for model 2), where we plot the trajectory that a bar follows in the (r_bar, Ω_bar) parametric plane after the buckling of the bar, obtained from two simulations of the bar evolution by Martínez-Valpuesta et al. (2017). The evolution time, for 3 Gyr, proceeds from the red points (early stages) to the orange points (most evolved stages), the points are connected by a solid line to bring out the trajectory. We see that the two trajectories displayed in Figure 8 describe the slowdown and the growth of the bar, but for model 2 (squares), the path to lower pattern speeds and larger bar lengths is a very wide "S" shape, which is not seen in the case of the fly-by interaction (model 1), which is better described by a "decay-type" trajectory. This shows that the same galaxy at different evolutionary stages can populate different parts of the diagram.

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It would not be valid to directly identify the trajectories in the (r_bar, Ω_bar) parametric plane obtained from numerical simulations of a single galaxy with our data plotted in Figure 6, but we can use these trajectories to infer some general trends of bar evolution from this figure. Although the decay trajectory in the (r_bar, Ω_bar) plane for an interacting galaxy (circles in Figure 8), seems to reproduce in shape the distribution of our data in Figure 6 reasonably better than the S-shaped trajectory for the non-interacting galaxy, we cannot conclude that the fly-by interaction is the preferred scenario for the bar formation, as numerically the bar of model 2 is shorter and shows higher values of the pattern speed in the early stages than the bar of model 1. However, both mechanisms of bar production are broadly compatible with our observational measurements. Reading the data in Figure 6 in terms of the trajectories for the two models plotted in Figure 8, we see that the data on the left side of Figure 6 (galaxies with a short bar) can be associated with the early stages of barred galaxies, while the data in the bottom right region of Figure 6 are consistent with the most evolved galaxies.

The bar strength is another magnitude studied in the numerical simulations by tracing its evolution. In general, all simulations predict that the bar strength grows while the bar grows in size and reduces its angular velocity. With our measurements we can produce the plots of the bar strength versus the bar pattern speed relative to the disk velocity, and versus the relative bar size.

In Figure 9 we plot the relative bar pattern speed against the bar strength, coloring the data according the relative bar length. First we can see from Figure 9 that the scaled bar size is an appropriate parameter to distribute the data, as it is possible to distinguish the regions in the (Γ–S_b) parametric plane occupied by the short bars, the medium bars, and the large bars. The largest bars are located in the bottom region of the plot, where the bars rotate more slowly, while the shortest bars are at the top, where the relative angular rate is higher. We can also note that all points are confined to half of this parametric plane, as they are only distributed over the bottom triangle defined by the inverted diagonal, or to be explicit, we do not find any galaxy in our sample that harbors a strong bar rotating with a higher angular speed than the disk. We see from this figure that the strongest bars are those that are spinning with the lowest pattern speed (bottom right region), while the bars that rotate faster can only have lower values of the bar strength (top left region).

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It is not straightforward to see trends of bar slowdown and bar strength growth in Figure 9. Again, numerical simulations can provide a useful aid to the interpretation of our results in terms of bar evolution. In Figure 10 we plot the time evolution of these two magnitudes as trajectories in the parametric plane for three different simulations. In addition to the two models already described (models 1 and 2), we also include here a third model (model 3, diamonds in Figure 10, left panel) in which there is no interaction, as in model 2, but with a different distribution of matter in the disk. While in model 2 43% of baryonic matter is in the disk within a radius of 7 kpc (thus the remaining 57% in this part of the disk is the contribution of the dark matter), in model 3 the fraction of matter within a radius of 7 kpc of the disk is 92%, so there is almost no dark matter in that region of the disk. We denote the bar strength obtained in the numerical simulations as A₂ as it is calculated as the integrated Fourier amplitude of only the m = 2 mode normalized by the m = 0 mode. The values of the bar strength, A₂ from the simulations can be qualitatively compared with those we calculate using the amplitude of even higher modes, as there is a strong linear correlation between the bar strength calculated using expression (4) and calculated only considering the contribution of the m = 2 mode. In Figure 10 the evolution time flows from the green points to the gray points, in the same way as in Figure 9. Figure 10 therefore displays how the bar slows down and its bar strength increases while the bar evolves. Additionally, the bar growth can be clearly seen in the right panel of Figure 10, where the same data are color coded according to the relative size of the bar, showing that, regardless of the model, the bar is shorter in the early stages of the evolution (red points), and becomes larger after 3 Gyr of evolution (blue points). In the right panel of Figure 10, the bar of model 2 (squares) is left to develop for an additional time of 0.9 Gyr with respect to the same bar plotted in the left panel, in order to extend the evolution of these parameters, showing that they keep the same correlation as found in the left panel. We can recognize the same pattern in Figure 9, reading the points of that figure in diagonal (trajectories in Figure 10), so given one diagonal in Figure 9, the shortest bars and least evolved bars are those rotating with the highest pattern speed and having the lowest values of the bar strength, while the more evolved bars have larger bar strengths and bar lengths and rotate more slowly. This is also reproduced in the two simulations of the right panel of Figure 10.

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It is clear from the left panel of Figure 10 that the variation of some parameters of the numerical simulations (such as the mass distribution in the disk, the contribution of dark matter, or the fact of considering an interaction with a second galaxy) can extend the coverage of the simulations in the diagram, therefore our numerical results plotted in Figure 9 could be used to constrain the initial values of these parameters for further simulations of bar evolution. The results of the numerical simulations seem to indicate that the galaxies found in the short diagonal (close to the origin of the parametric plane) should contain a very low fraction of dark matter in the disk as model 3, with only 8% of dark matter within the region out to 7 kpc radius in the disk, reproduces the range of values of bar pattern speed and bar strength that we have found in our measurements. On the other hand, galaxies on the large diagonal (farthest from the origin of the parametric space) seem to contain a significant contribution of dark matter in that region of the disk, which can be estimated to be larger than ∼60% of the total mass, as model 2 simulates the evolution of a bar with values of these parameters close to that large diagonal.

In Figure 11 we plot the relative bar length versus the bar strength for the barred galaxies of our sample. In the plot, the data are coded in colors according to the morphological type of the galaxy, i.e., the earlier, intermediate, and later type, galaxies of type {a, ab, b}, {bc, c, cd} and {d, dm, m}, respectively. First we can see that although there is no clear correlation between these magnitudes, the strongest bars do tend to be the longest ones, while the shortest bars tend to have lower values of the bar strength. The plot also shows that the morphological type does not play an important role in determining these parameters, as the colored points are well mixed.

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We have used numerical simulations in order to help give a possible interpretation to our results. In Figure 12 we plot the evolution of the bar size and the bar strength, corresponding to models 1 and 2, in the left and right panel, respectively. The simulations indicate that the evolution of the two parameters is more or less correlated depending on the model considered. For the bar of model 1 (left panel of Figure 12) we see a strong correlation between the bar strength and the bar length, while the data of model 2 in the parametric plane (r_bar, S_bar) show a higher dispersion (right panel of Figure 12). In any case, when the bar is evolving, the bar strength in both models increases while the bar is growing in size. This tendency between the bar strength and the bar length is also found in Figure 11, where the largest bars tend to be the strongest ones and should be the most evolved bars according to the simulations (trajectories in Figure 12). Numerically, model 2 develops a larger and a stronger bar than model 1, but in the early stages of the formed bar it is shorter and rotates more slowly in model 1 than in the non-interacting model 2. Thus, our measurements at this do not favor one scenario of the bar formation of the two scenarios taken into account here (i.e., with and without interaction corresponding to models 1 and 2, respectively). The numerical simulations also offer an interpretation of our results plotted in Figure 11 in terms of more or less evolved bars. The former are strong and large bars, while the latter are shorter and weaker bars.

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The bar pattern speed and the bar corotation radius are linked by means of the angular velocity curve, which is calculated from the rotation curve by dividing the circular velocity by the galactocentric radius. In Figure 14, left panel, we show a typical rotation curve of a disk galaxy, and in the right panel we plot the derived angular velocity curve in which the relationship between the pattern speed and the corotation radius is illustrated in red at two different radial positions of the bar resonance. This latter panel illustrates the conventional definition of a fast or slow rotator bar, which has to be understood as follows: given a bar with a bar length, r_bar, marked as solid vertical line in the right panel of Figure 14, we define the critical corotation position at 1.4 times r_bar, this is marked in the plot as a vertical dotted line, which defines two different radial regions; between r_bar and r_critical filled in green in the plot (fast rotator range), and beyond r_critical in blue (slow rotator range). If the corotation of the bar occurs within the green region, marked as ${r}_{\mathrm{CR}}^{F}$, then the corresponding pattern speed is ${{\rm{\Omega }}}_{\mathrm{bar}}^{F}$, but if the corotation radius falls in the blue region (${r}_{\mathrm{CR}}^{S}$ in the figure), then the corresponding pattern speed is ${{\rm{\Omega }}}_{\mathrm{bar}}^{S}$. It is obvious from the right panel of Figure 14 that ${{\rm{\Omega }}}_{\mathrm{bar}}^{S}$ < ${{\rm{\Omega }}}_{\mathrm{bar}}^{F}$, therefore the bar rotates faster when the corotation radius is in the green region compared to the same bar for which the resonance is found in the blue region. When corotation occurs between r_bar and 1.4 times r_bar, that bar is therefore conventionally called a fast rotator bar (as first defined in Debattista & Sellwood 2000), while it is classified as a slow rotator bar if the resonance is located beyond 1.4·r_bar. However, this terminology of fast or slow rotator bars turns out to be confusing when we compare different bars for which we also know their pattern speeds, for example, UGC 1913 and UGC 9969. The former galaxy has a bar with a pattern speed of 16.2 km s⁻¹ kpc⁻¹ and the ratio between corotation and bar length is 1.28, which means that it is a fast rotator bar. The bar of UGC 9969 is rotating with an angular speed of 50.2 km s⁻¹ kpc⁻¹ and ${ \mathcal R }=1.60$ and is therefore classified as a slow rotator. This means we have a slow rotator bar (UGC 9969) that is spinning faster than a fast rotator bar (UGC 1913).

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Angular velocity curves can also be used to depict the effect of the bar evolution on the corotation. To do so schematically, we evaluate the bar at two different moments, t₀ and t₁ with t₁ > t₀. The angular braking of the bar implies that it rotates more slowly while it is evolving, from ${{\rm{\Omega }}}_{\mathrm{bar}}^{0}$ at t₀ to ${{\rm{\Omega }}}_{\mathrm{bar}}^{1}$ at t₁, where ${{\rm{\Omega }}}_{\mathrm{bar}}^{1}\lt {{\rm{\Omega }}}_{\mathrm{bar}}^{0}$, consequently, the corotation is pushed farther out, from ${r}_{\mathrm{CR}}^{0}$ to ${r}_{\mathrm{CR}}^{1}\gt {r}_{\mathrm{CR}}^{0}$. This is illustrated in all the panels of Figure 15, where we plot the same angular velocity curve as in Figure 14, and in red we indicate the position of the bar pattern speed and the corotation radius at t₀ and t₁. Thus, the slowdown of the bar implies that the corotation occurs at larger radii, so that the rate at which the corotation grows depends on the rate at which the bar angular rate decreases, which in turn depends on the angular momentum exchange rate with other structures of the galaxy (Weinberg 1985; Athanassoula 2003), mainly with the dark matter halo. On the other hand, it is also known that the bar grows in size, and this does not, in general, occur at the same rate as the increase of the corotation radius. The evolution of the ratio of the corotation radius to the bar length therefore depends on how fast or slow the growth of the former is compared to the latter. This leads to two possible scenarios. In the first scenario, the resonance radius grows faster than the size of the bar. When the initial condition is that the bar is a fast rotator, under these conditions, the bar will evolve to become a slow rotator. However, when the bar is initially a slow rotator, it will keep this status during the evolution. We can call these transitions FS and SS, respectively. In the second scenario, the growth of the corotation radius is slower than that of the bar. In that case, we can have SF and FF transitions depending on whether the bar is initially a slow or fast rotator, respectively. These four cases are schematically described in the four panels of Figure 15, where the bar size at t₀, ${r}_{\mathrm{bar}}^{0}$ is marked as a solid vertical line, and the size of the bar at t₁ is indicated as a dashed vertical line. The green regions cover the fast rotator range for each corotation indicated in the panel; this means that if a bar ends in this green region, it is classified (conventionally) as fast rotator, otherwise it is a slow rotator. It is known from numerical simulations (since the early work of Sellwood, see for example Sellwood 1981, until recent studies such as Villa-Vargas et al. 2009, 2010; Martínez-Valpuesta et al. 2017) that the bar growth rate is often higher than the slowdown rate, and in that case, the bar increases its size faster than its corotation is moving outwards. This is clearly illustrated in Figure 16, where we plot the time evolution of the bar length and the corotation radius for models 1 and 2 in the left and right panels, respectively. The green area shows the fast rotator region according to the conventional definition. When the values of the bar length fall within the green region, the bar is a fast rotator, otherwise it is a slow rotator. In the two models considered here, the bar is a slow rotator in the early stages, but it always ends as a conventionally fast rotator in spite of the braking of its angular rate. This implies that in the final stages of the bar evolution, when it has undergone considerable braking and is rotating slowly, this bar will conventionally be classified as a fast rotator. This is clearly in contradiction with the fact that bar has been substantially slowed down. Under these conditions, only the transitions SF and FF are possible (depicted in panels (c) and (d) of Figure 15, respectively). In the present study we find evidence of this contradiction in the terminology of bars in Figure 13, where the largest bars, which are most likely to be the most evolved bars, are rotating with the lowest angular velocities compared with the speed of the disk. This is in agreement with the predictions of the simulations, but all these bars would be conventionally classified, misleadingly, as fast rotators.

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