Properties of Fast and Slow Bars Classified by Epicyclic Frequency Curves from Photometry of Barred Galaxies

Stellar orbits in weak non-axisymmetric potentials can be described by the epicycle theory of nearly circular orbits in an axisymmetric potential (Binney & Tremaine 1987). The circular orbital frequency, namely the angular velocity Ω(r), is derived from the gravitational potential Φ(x, y) as follows:

$$\begin{eqnarray}&&{{\rm{\Omega }}}^{2}(r)=\left\langle \displaystyle \frac{1}{r}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial r}\right\rangle .\end{eqnarray} \tag{ 1 }$$

The epicyclic frequency κ(r) with which stars move inward and outward in the circular motion is determined as

$$\begin{eqnarray}&&{\kappa }^{2}(r)=\left\langle \displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {x}^{2}}+\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {y}^{2}}+2{\left(\displaystyle \frac{1}{r}\displaystyle \frac{\partial {\rm{\Phi }}}{\partial r}\right)}^{2}\right\rangle \end{eqnarray} \tag{ 2 }$$

where $\left\langle \right\rangle$ stands for an azimuthal average (Binney & Tremaine 1987; Pfenniger 1990; Michel-Dansac & Wozniak 2006; Schmidt et al. 2019). The corotation radius is defined as the radius where the stellar angular velocity becomes the same as the bar pattern speed, Ω = Ω_bar. Resonances occur when the difference between the stellar angular velocity and the pattern speed multiplied by an integer becomes the epicyclic frequency: m(Ω − Ω_bar) = ±κ for integer values of m (Binney & Tremaine 1987; Elmegreen 1998). Although we cannot be sure whether the stellar orbits become resonant when the axisymmetry is broken by a strong bar, the epicyclic approximation has been widely used to estimate resonance locations (Schwarz 1981; Combes & Elmegreen 1993; Byrd et al. 1994; Buta & Combes 1996; Combes 1996; Buta et al.1999; Buta 2002; Michel-Dansac & Wozniak 2006; Schmidt et al. 2019; Williams et al. 2021).

The gravitational potential can be constructed with the assumption of a constant mass-to-light ratio (M/L) (Quillen et al. 1994). The photometric surface brightness distribution is translated into the mass distribution, which yields the two-dimensional potential through the Poisson equation. The bar strength is the ratio of the transverse force to the radial one (force ratio, hereafter) and can be calculated from the potential as well (Buta & Block 2001; Lee et al. 2020).

3.1.1. From Light To Mass

The determination of M/L allows us to translate the photometric luminosity to the stellar mass. Bell & de Jong (2001) first explored the relation between M/L and color by comparing the observed colors with the stellar population synthesis (SPS) model. van de Sande et al. (2015) developed the relation into the dynamical M/L by direct stellar kinematic mass measurements, estimated from the effective radius, Sérsic index, and velocity dispersion measurements (Cappellari et al. 2006). This does not depend on any assumptions, such as the metallicity, stellar initial mass function (IMF), or the SPS model.

We construct our mass maps from the color and the absolute magnitude using the following formula:

$$\begin{eqnarray}&&{{\rm{log}}}_{10}\displaystyle \frac{{M}_{{\rm{dyn}}}}{{M}_{\odot }}=-1.27+1.75(g-i)-0.4({M}_{i}-{M}_{i,\odot }),\end{eqnarray} \tag{ 3 }$$

which is derived from the relation between the dynamical M/L ratio and the color (van de Sande et al. 2015, see Table 3) by comparison with the absolute magnitude of the Sun. The dynamical mass within each pixel is determined by its i-band absolute magnitude from i_P1-band images and the g − i color of the galaxy. We adopt the mean g − i color within a scale length h_r in the radial color profile as the g − i color of a galaxy. We use the PS1 i-band solar absolute magnitude of M_i,⊙ = 4.52 (in AB, Willmer 2018). However, we note that the relation in van de Sande et al. (2015) was explored for massive quiescent galaxies with a mass limit of M_* > 10¹¹ M_⊙ and with a color selection of U − V > (V − J) × 0.88 + 0.59 (Williams et al.2009).

3.1.2. Calculation of Potential Map

3.1.3. Frequency Curves

Figure 1 shows examples of the frequency curves (right panels) for IC 1438 and NGC 2935 obtained from our photometric approach with our adopted dynamical M/L ratio. We display Ω − κ/2, Ω − κ/4, Ω, and Ω + κ/2 as green, gray, blue, and red curves, respectively. The bar pattern speed Ω_bar determines the corotation radius where Ω_bar intersects the circular orbital frequency Ω(r). In the same way, the locations of the ILR, UHR, and OLR are determined by Ω_bar = Ω − κ/2, Ω_bar = Ω − κ/4, and Ω_bar = Ω + κ/2, respectively (Binney & Tremaine 1987). The pattern speeds Ω_bar of these two galaxies were estimated by Schmidt et al. (2019) using photometric methods mentioned above. Ω_bar is displayed as a horizontal line of orange color. R_ILR (green), R_UHR (gray), R_CR (blue), and R_OLR (red) are presented with uncertainties on the horizontal line of Ω_bar.

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We estimate the uncertainties from (1) the scatter of the relation between the dynamical M/L and the color (van de Sande et al. 2015) and (2) the difference between the assumptions of a thick disk (h_r /h_z = 4) and a thin one (h_r /h_z = 9). The scatter of ${\rm{log}}({M}_{{\rm{dyn}}}/L)$ is larger than the orthogonal scatter of the best-fitting line by a factor of 1.5 (van de Sande et al. 2015, see Figure 3). Therefore, we measure the uncertainties of ${\rm{log}}({M}_{{\rm{dyn}}}/{L}_{i})$ by multiplying by 1.5 the mean orthogonal scatter of the best fitting (i.e., 1.3) in the i band. The green, sky blue, and orange columns indicate R_ILR, R_CR, and R_OLR with error ranges calculated by Schmidt et al. (2019).

Schmidt et al. (2019) calculated the frequency curves for five spiral galaxies including IC 1438 and NGC 2935. They constructed the radial velocity curves from Hα emission line observations and calculated the gravitational potential by fitting the rotational curves to the axisymmetric Miyamoto–Nagai gravitational potential,

$$\begin{eqnarray}&&{\rm{\Phi }}(R,z)=-\displaystyle \frac{{GM}}{\sqrt{{R}^{2}+{\left[a+\sqrt{{z}^{2}+{b}^{2}}\right]}^{2}}},\end{eqnarray} \tag{ 4 }$$

where Φ(R, z) is the Miyamoto–Nagai potential at (R, z) and M indicates the total mass (Miyamoto & Nagai 1975). The parameters a and b are shape parameters, which represent a flattened disk distribution with the ratio b/a ∼ 0.4, an ellipsoidal distribution with b/a = 1, and a spherical distribution with b/a ∼ 5 (Binney & Tremaine 1987; Schmidt et al. 2019). They estimated the potential considering two or three components designated by b/a and calculated the angular velocity with Equation (1). They used the equation for epicyclic frequency κ(r) as

$$\begin{eqnarray}&&{\kappa }^{2}=4{{\rm{\Omega }}}^{2}\left[1+\displaystyle \frac{1}{2}\left(\displaystyle \frac{R}{{\rm{\Omega }}}\displaystyle \frac{d{\rm{\Omega }}}{{dR}}\right)\right]\end{eqnarray} \tag{ 5 }$$

(Elmegreen 1998).

Figure 1 shows the galaxies in common between Schmidt et al. (2019) and this study. For IC 1438 (top row), our measurements of R_CR and R_OLR are consistent with the estimates of Schmidt et al. (2019, see Figure 3) within errors, while R_ILR is located inward compared to theirs. When comparing R_ILR, we note different ways to deal with a bulge: they adopted an ellipsoidal distribution of b/a = 1, while we considered the bulge region to be exponentially distributed in the z direction like the disk. In the case of NGC 2935 (bottom row), our estimate of R_ILR is similar to that of Schmidt et al. (2019), whereas R_CR and R_OLR are larger than theirs (but still similar considering the uncertainty).

The right panels of Figure 1 show that IC 1438 and NGC 2935 have all kinds of resonances, including ILR, UHR, CR, and OLR. In the left panels, their color index maps show blue features appearing in the circumnuclear, inner, and outer rings. It is interesting that three kinds of rings are located near the ILR, UHR, and OLR, in sequence (Schmidt et al. 2019). Outer rings are occasionally associated with the outer 4:1 resonance located between CR and OLR according to their subclass, R₁ or $R{{\prime} }_{1}$ (Buta 2017).

To provide the result for a larger sample, we present Figure 2, which shows the comparison of our estimated corotation radii (R_CR) and those in the literature obtained from spectroscopy (${R}_{{\rm{CR}}}^{\ast }$) (Aguerri et al. 2015; Schmidt et al. 2019; Guo et al. 2019; Cuomo et al. 2019; Garma-Oehmichen et al. 2020). As noted in Section 2, the bar pattern speeds Ω_bar as determined from the TW method (Tremaine & Weinberg 1984) are taken from the literature, except for IC 1438 and NGC 2935 (green filled stars). The different colors indicate different literature sources. Garma-Oehmichen et al. (2020) (red circle) modeled the angular velocity with VELFIT, while the rest (Aguerri et al. 2015; Cuomo et al. 2019; Guo et al. 2019) estimated the corotation radius from R_CR = V_flat/Ω_bar by assuming a flat rotation. We find that our measurement of R_CR agrees with ${R}_{\mathrm{CR}}^{* }$ regardless of the variety of ways employed in the literature.

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4.1.1. Corotation Radius versus Bar Length

As an indicator of the bar pattern speed, the distance-independent ratio ${ \mathcal R }={R}_{\mathrm{CR}}/{R}_{\mathrm{bar}}$ has been used in hydrodynamical simulations to model gas and shocks (Lindblad et al. 1996; Weiner et al. 2001). Debattista & Sellwood (2000) suggested the limit of ${ \mathcal R }\leqslant 1.4$ for a fast bar that ends its slowdown in a maximum disk (minimum halo). The upper limit to where a bar can extend was suggested as ${ \mathcal R }=1$ in orbital calculations (Contopoulos 1980). On this basis, observational studies have classified barred galaxies into slow (${ \mathcal R }\gt 1.4$), fast ($1\lt { \mathcal R }\leqslant 1.4$), and ultrafast (${ \mathcal R }\lt 1$) bars (Aguerri et al. 2003, 2015; Guo et al. 2019; Cuomo et al. 2019; Garma-Oehmichen et al. 2020).

The bar length has usually been estimated by ellipse fitting (Martin 1995; Wozniak et al. 1995; Jogee et al. 2004) or Fourier analysis (Ohta et al. 1990; Laurikainen & Salo 2002). Because bars do not end with sharp edges, it is not trivial to define their full length. Although there have been lots of efforts to find the best way to determine the length of a bar, each method has its own strengths and weaknesses (Athanassoula & Misiriotis 2002; Michel-Dansac & Wozniak 2006; Cuomo et al. 2021). The widely used way is to measure the radius where the profile of the ellipticity, , or the normalized Fourier amplitude, A₂, reaches a maximum, at R or ${R}_{{A}_{2}}$, even though it cannot estimate the full length of the bar (Wozniak et al. 1995; Athanassoula & Misiriotis 2002; Laurikainen & Salo 2002). Similarly, we can define the bar radius where the radial profile of the transverse-to-radial force ratio has a plateau or a maximum peak, ${R}_{{Q}_{b}}$ (Lee et al. 2020; Cuomo et al. 2021).

Figure 3 shows the relation between R_CR and R_bar measured from R , R_A2, and R_Qb, together with the regimes of slow, fast, and ultrafast bars classified by the ratio ${ \mathcal R }$. The dashed and solid lines represent R_CR = 1.4R_bar and R_CR = R_bar, respectively. We present the mean value and standard deviation of R_bar and R_CR as the black square and error bars. The mean value of ${ \mathcal R }$ is given with the standard deviation at the top left in each panel. In panel (d), R_CR and ${R}_{\mathrm{bar}}^{* }$ are adopted from the literature (Aguerri et al. 2015; Guo et al. 2019; Cuomo et al. 2019; Garma-Oehmichen et al. 2020).

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Basically, the classification by ${ \mathcal R }$ depends on the method used to measure the bar length. In our measurement, galaxies are categorized into 29 slow bars, 9 fast bars, and 40 ultrafast bars when R is adopted for R_bar (Figure 3(a)). Five galaxies are rejected because they have no R_CR intersected by Ω_bar in their angular velocity curves because of their rapid pattern speeds. Ultrafast bars with ${ \mathcal R }\leqslant 1$ cannot exist theoretically, but they have existed in observations (Cuomo et al. 2019, 2021; Guo et al. 2019). The possibility of real ultrafast bars was raised also by Zhang & Buta (2007). When we use R_A2 (Figure 3(b)) or R_Qb (Figure 3(c)) instead, the number of slow bars increases to 39 galaxies, and the number of fast bars increases to 15 or 13 galaxies. The number of ultrafast bars decreases to 24 or 26 galaxies. On the other hand, when we adopt the values directly from the literature (Figure 3(d)), sample galaxies are classified into 20 slow bars, 23 fast bars, and 40 ultrafast bars. Except for the sample of Guo et al. (2019) (orange circle), most galaxies lie within the regimes of the fast and ultrafast bars.

The mean values of ${ \mathcal R }$ are 1.37, 1.68, and 1.45, respectively for R , R_A2, and R_Qb. They are somewhat larger than ${ \mathcal R }=1.01\pm 0.79$ in the literature due to the smaller bar length in our measurements. The mean bar length is 7.7 kpc in the literature, while it is measured to be 5.9, 4.9, and 4.7 kpc by R , R_A2, and R_Qb, respectively. We obtain the smallest ${ \mathcal R }$ from R and the largest ${ \mathcal R }$ from R_A2. This can more or less reconcile the difference in ${ \mathcal R }$ between observations and simulations. Observations have preferentially used the ellipse fitting method to measure the bar length (Aguerri et al. 2015; Cuomo et al. 2019; Guo et al. 2019; Garma-Oehmichen et al. 2020), whereas simulations have utilized Fourier analysis (Lindblad et al. 1996; Debattista & Sellwood 2000). Although observations often examined the bar length from the ratio of bar to interbar intensity based on Fourier analysis (Aguerri et al. 2015; Cuomo et al. 2019; Guo et al. 2019), this is different from the way that the bar length is usually calculated in simulations (Lindblad et al. 1996; Debattista & Sellwood 2000). We will discuss the effects of bar length measurements and the bar evolution on ${ \mathcal R }$ in Sections 5.1 and 5.2, respectively.

4.1.2. Bar Pattern Speed versus Frequency Curves

Here, we take a different approach to utilize the bar pattern speed more effectively. That is to compare the bar pattern speed with the Lindblad precession frequency curves directly. If the bar pattern speed decreases as a result of the exchange of angular momentum between a bar and a dark halo (Athanassoula 2002, 2003; Debattista & Sellwood 2000), then the radii of all resonances will increase with time. It will cross more frequency curves from Ω to Ω − κ/4 or Ω − κ/2, in turn producing a CR, a UHR, and one or two ILRs. Using these resonances, Byrd et al. (1994) suggested a more physical classification to define fast, medium, and slow bars when the bar pattern speed sets up CR, UHR, and ILR in sequence. Although we cannot trace the process of the slowdown of the bar for a given galaxy, we can glean "snapshots" for galaxies with fast, medium, and slow bars from observational data.

Figure 4 displays example galaxies with images (top row) for each class. The second row shows the frequency curves with the pattern speed for a fast (f), a medium (g), and a slow bar (h). The horizontal lines represent the bar pattern speed with uncertainties. The red, blue, gray, and green curves describe Ω + κ/2, Ω, Ω − κ/4, and Ω − κ/2, respectively. We show the corotation radius as a dotted vertical line. Although we indicate the bar length, R (red), R_A2 (green), and R_Qb (blue) by vertical markers together, this classification is irrelevant to the bar length, which is different from the classification based on the ratio ${ \mathcal R }$. In the bottom row, we display the radial profiles of force ratio for each galaxy, as introduced in Lee et al. (2020). This helps us to understand the evolution of bars, which we will discuss in Section 5.3.

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Among 83 galaxies, we find five galaxies with higher bar pattern speed, not intersecting the stellar angular velocity curve Ω (Figure 4(e)). All of them are also classified as nonbarred galaxies from the analysis of the ratio map (Lee et al. 2020). According to the definition of Byrd et al. (1994), a fast bar is one hosting a CR and an OLR because of a high pattern speed crossing Ω and Ω + κ/2 (Figure 4(f)). A medium bar in a galaxy is defined with UHR, CR, and OLR when the pattern speed crosses the frequency curves of Ω − κ/4, Ω, and Ω + κ/2 (Figure 4(g)). A slow bar has resonances of all kinds (Figure 4(h)). Byrd et al. (1994) used the simulations and found a hint in evolutionary stages for fast, medium, and slow bars. For example, a fast bar shows an outer ring, while a medium bar has outer and inner rings; a slow bar shows a nuclear ring as well as outer and inner rings.

According to our classification scheme, there are eight slow bars, 59 medium bars, and 11 fast bars. Five galaxies without a CR are categorized as nonbarred galaxies. van Albada & Sanders (1982) showed models with one ILR or two ILRs when an x₂ family extends to the center or stops before the center. However, we do not find any galaxies with two ILRs in our sample, probably because the data we analyze are not good enough to resolve the central region within ∼1 kpc from the center where any inner ILR is likely to be located. We also note that the measured ILRs in this study appear slightly smaller than those in Schmidt et al. (2019), which might be caused by different ways of dealing with a bulge in the calculation of the potential (even though the difference is not larger considering the uncertainty). There could be some slow bars missed in this study for a similar reason.

Figure 5 shows the relation between the bar length and the corotation radius. This plot is similar to Figure 3, but galaxies here are classified into fast (blue triangles), medium (green squares), and slow bars (solid red circles) according to the relation between the bar pattern speed and the frequency curve. The newly defined fast, medium, and slow bars are placed in a similar sequence of ultrafast, fast, and slow bars classified by the ratio ${ \mathcal R }$, though they do not correspond perfectly to each other. The newly defined fast bars fall in the region occupied by the ultrafast bars defined by ${ \mathcal R }$.

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We present the mean values of ${ \mathcal R }$ for newly defined fast, medium, and slow bars in Table 1. Although there are some differences according to the measurements of the bar length, they increase from 0.62 to 1.48 and 2.82, on average, from fast to medium and slow bars. We intend to investigate the properties of barred galaxies in terms of the newly defined classes of fast, medium, and slow bars in Sections 4.2 and 4.3.

Table 1. The Mean ${ \mathcal R }$ for Newly Defined Fast, Medium, and Slow Bars with Different Measurements of Bar Length

Classification	R	RA2	RQb
Fast bar	0.54 (±0.16)	0.71 (±0.27)	0.61 (±0.15)
Medium bar	1.42 (±1.30)	1.60 (±1.10)	1.43 (±0.52)
Slow bar	2.16 (±0.83)	3.50 (±1.16)	2.81 (±0.59)

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Figure 6 shows the bar length and strength as a function of the disk circular velocity V_circ, which is a parameter tightly correlated with the galaxy luminosity through the Tully–Fisher (TF) relation (Tully & Fisher 1977). We obtain the disk circular velocity derived from spectroscopy in the literature (Guo et al. 2019; Cuomo et al. 2019; Garma-Oehmichen et al. 2020). We calculate the bar length R_bar (top row) and strength S_bar (bottom row) using the ellipse fitting (R and ${\epsilon }_{\max }$ on the left), Fourier analysis (R_A2 and A₂ in the middle), and force ratio (R_Qb and Q_b on the right). All the calculations are conducted following Lee et al. (2020) except for the definition of A₂, which is

$$\begin{eqnarray}&&{A}_{2}=max\left(\frac{\sqrt{{a}_{2}^{2}+{b}_{2}^{2}}}{{a}_{0}}\right)\end{eqnarray} \tag{ 6 }$$

where a₀, a₂, and b₂ are the Fourier coefficients; this is to compare the results with more studies (Athanassoula 2013; Seo et al. 2019). We note another indicator of bar strength, max(Δμ), the maximum difference between luminosity profiles along and perpendicular to the bar axis, which correlates well with A₂ (Buta 2017; Kim et al. 2021).

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The top row shows that the bar length depends on the circular velocity of its host galaxy. The dependence becomes strongest when we measure the bar length by the force ratio R_Qb (Figure 6(c)). Previous studies reported that the bar length depends on several galaxy properties including galaxy luminosity (Kormendy 1979; Cuomo et al. 2020), effective radius, disk scale length (Ann & Lee 1987; Erwin 2019), and stellar mass (Díaz-García et al. 2016; Erwin 2019). In particular, Erwin (2019) showed that the bar length is a strong function of galaxy size in terms of effective radius R_e or disk scale length h_r. He also showed an additional dependence of the bar length on galaxy mass for massive galaxies.

On the other hand, in the bottom row, we find hardly any dependence of the bar strength on the circular velocity of the host galaxy. In previous studies, the maximum ellipticity ${\epsilon }_{\max }$ appears constant across the Hubble type sequence (Marinova & Jogee 2007; Menéndez-Delmestre et al. 2007; Díaz-García et al. 2016; Lee et al. 2020). Díaz-García et al. (2016) and Lee et al. (2020) reported the opposite tendencies of A₂ and Q_b at both ends of the Hubble sequence. This is because the measurements of A₂ and Q_b are influenced in opposite directions by a large bulge (Lee et al. 2020). However, we find similar distributions of A₂ and Q_b on the circular velocity for our sample constrained by T ≤ 5 due to the limit for applying the TW method (Figures 6(e) and (f)). Cuomo et al. (2020) also reported no correlation between the bar strength and the galaxy luminosity estimated with A₂ for their sample galaxies analyzed by the TW method. However, it is interesting that strong bars measured with A₂ and Q_b are prominent in galaxies with lower velocity, V_circ ∼ 150 km s⁻¹ (Figures 6(e) and (f)). This seems different from the distribution of long bars, which are hosted by galaxies with higher velocity, V_circ >250 km s⁻¹ (top row).

Figure 7 displays other important properties of bars—bar pattern speed Ω_bar and corotation radius R_CR–as functions of Hubble type T and disk circular velocity V_circ. We present the mean values for the Hubble type (gray solid lines) and linear fits with disk circular velocity (blue dotted line). We find that the bar pattern speed has no significant dependence either on the Hubble type or on the disk circular velocity (Figures 7(a) and (b)), even though there is an S0 galaxy that has an exceptionally high pattern speed. On the other hand, the corotation radius shows a weak correlation with the disk circular velocity (Figure 7(d)). When it comes to the Hubble type, Figure 7(c) shows that the earlier-type spirals (0 ≤ T ≤ 3) have a larger corotation radius than the later-type spirals (4 ≤ T ≤ 5).

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When we investigate this in terms of the new classification of fast (blue triangles), medium (green squares), and slow (solid red circles) bars, Figure 6 does not show any significant difference in the correlation between bar properties (the length and the strength) and the disk circular velocity. Galaxies of different types have a wide range of disk velocities. However, in Figure 7, fast bars are distinguished by having a higher bar pattern speed and a smaller corotation radius than slow bars. In particular, slow bars have the largest corotation radius for a given Hubble type bin or a specific velocity of host galaxies (Figures 7(c) and (d)).

Panels (a) and (c) in Figure 7 show that fast bars are concentrated in the later-type spirals (T ≥ 3), whereas slow bars are distributed throughout the Hubble sequence. Rautiainen et al. (2008) reported the opposite results: earlier-type spirals have only fast bars, whereas later-type spirals host both fast and slow bars, though the definitions of fast and slow bars are not the same. In terms of ${ \mathcal R }$, they showed that later-type spirals have a larger value of ${ \mathcal R }$. In Figure 8, we also compare ${ \mathcal R }$ with the Hubble type using different measures of bar length—ellipse fitting (a), Fourier analysis (b), and force ratio (c). Figure 8(d) shows the galaxies based on the measurements of ${R}_{\mathrm{CR}}^{* }/{R}_{\mathrm{bar}}^{* }$ from the literature. In our measurements, the mean ${ \mathcal R }$ appears slightly larger in earlier-type spirals (T ≤ 1), even though later-type spirals have a wider range of ${ \mathcal R }$. On the other hand, other studies have reported no correlation between ${ \mathcal R }$ and the Hubble type (Aguerri et al. 2015; Cuomo et al. 2020; Garma-Oehmichen et al. 2020). We seem to require a much larger sample size to better understand the relation between the ratio ${ \mathcal R }$ and the Hubble type.

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Figure 9 displays relations between the bar pattern speed Ω_bar and other properties of bars including corotation radius R_CR, bar length R_bar, and strength S_bar. We present the bar lengths on an absolute scale (top row) and on a scale normalized by the disk scale length h_r (middle row). The relation between the bar pattern speed and the disk scale length is displayed in the leftmost panel in the middle row.

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First, we find that the bar pattern speed Ω_bar is anticorrelated with other properties of bars: as the bar pattern speed decreases, the values of other parameters increase. Figure 9(a) shows that the pattern speed is anticorrelated with the corotation radius (confirmed by ρ = −0.73). This is expected because the disk angular velocity decreases in proportion to r^−3/2 so that a large corotation radius corresponds to low pattern speed. On the other hand, the pattern speed has weak anticorrelations with the bar length and the strength with ρ ∼ −0.35. This appears to support the concept of bar growth in terms of length and strength through the slowdown of a bar by losing its angular momentum to a dark halo (Debattista & Sellwood 2000; Athanassoula 2003; Seo et al. 2019).

However, if the relation between the bar pattern speed and the frequency curves gives a hint for the evolutionary stage of barred galaxies, we can investigate the relations between the bar pattern speed and other bar properties with another view. In Figure 9, we present the mean values with error bars for fast (blue triangles), medium (green squares), and slow bars (solid red circles). Panel (a) shows that slow bars have lower pattern speed and larger corotation radius than fast and medium bars, as expected. However, we cannot find any increase in the bar length from fast bars to medium or slow bars (top row). In the case of R_Qb, it even decreases from fast bars to slow bars (Figure 9(d)). When we investigate the normalized bar length, it shows a tendency of larger bar lengths for slowly rotating bars (middle row). However, this is caused by the decrease in disk scale length from fast to slow bars, as shown in panel (e). We note that the disk scale length could also change between fast and slow bars. For the bar strength, we find a weak tendency of increasing bar strength from fast to slow bars, even though the increases are within error bars (bottom row).

To examine the difference among the new subclasses of fast, medium and slow bars, we perform the Anderson–Darling (A-D) test for each combination of subclasses. We list the relevant p-value in each panel, which indicates the probability that the two samples are drawn from the same parent distribution. First, p-values of the A-D test for the pattern speed distributions (Ω_bar) between fast and medium bars and between medium and slow bars are 0.009 and 0.002, respectively. This means that the newly defined subclasses of bars are relevant to the differentiation of the pattern speed. Second, the probability from the A-D test for each property index—R_CR, h_r, R_bar, R_bar/h_r, and S_bar—is noted as p in each panel and shows that the three subclasses show different distributions in corotation radius but do not show differences in other properties. We will discuss these results on the bar evolution in Section 5.3.

In this work, we have used three measures of bar length defined by the radius—R , R_A2, and R_Qb—where the ellipticity (), Fourier amplitude (A₂), and force ratio (Q_b) reach their maxima in the radial profiles. Figure 10 shows correlations between bar length measurements. All the three measures of bar length are strongly correlated with each other; in particular, ${R}_{{A}_{2}}$ and ${R}_{{Q}_{b}}$ are similar to each other, resulting in the slope of 1 for the correlation between the two (Figure 10(c)). However, they are measured to be shorter than R by 20% (Figures 3(a) and (b)). Díaz-García et al. (2016) reported that R is the best indicator of the visually estimated bar length.

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Figure 4 shows four example galaxies with three bar length measurements overlaid on their images. We present the positions of the tip of the bar measured from ellipse fitting (red), Fourier analysis (green), and force ratio (blue) on the images, in the same manner as Cuomo et al. (2021). When we investigate the bar length measurements on images one by one, all three measurements are compatible for galaxies with simple structures such as shown in Figure 4(a). However, when a galaxy hosts a pseudo-ring or ring around a bar, R tends to be located on the ring (Figures 4(b)–(d)): the ellipticity gradually increases up to the ring radius. This could make the bar length overestimated (Cuomo et al. 2021). In contrast, in Figure 4(d), ${R}_{{A}_{2}}$ and ${R}_{{Q}_{{\rm{b}}}}$ are located in a very inner region despite a long bar that is visible in the image. This may mean that the maximum radius of A₂ and Q_b cannot reflect the bar length in the case of a strong bar, in particular. We also note that ${R}_{{A}_{2}}$ could be easily affected by spiral arms (Laurikainen & Salo 2002; Laurikainen et al. 2004a). Buta et al. (2003) introduced an extrapolation method to separate a bar from spiral arms on the Fourier amplitude profile. In this work, we set a radius limit in finding the maximum A₂ to avoid the contamination from spiral arms through a visual inspection.

From N-body simulations, Michel-Dansac & Wozniak (2006) showed that the bar lengths measured by different methods are correlated with different resonances. They investigated various radii available to define the bar length where a maximum, a minimum, or a transition between a bar and a disk appears on the ellipticity or Fourier amplitude profile. They showed that the bar length defined as the radius of a minimum ellipticity corresponds to the CR, while the bar length defined by the transition between a bar and a disk is located close to the UHR. They reported that the length measured by force ratio, R_Qb, is located in the circumnuclear region, and the length measured by maximum ellipticity does not show any correlation with dynamical resonances. As a result, they considered that the bar length measured by the maximum ellipticity, R , is not a proper estimator of the bar length.

Therefore, we compare the correlations between the bar length estimates and the dynamical resonance locations for our whole sample in Figure 11. R_ILR, R_UHR, R_CR, and R_OLR are measured where the pattern speed of a bar Ω_bar intersects the frequency curves, Ω − κ/2, Ω − κ/4, Ω, and Ω + κ/2, in sequence. The plot shows R_ILR, R_UHR, R_CR, and R_OLR from top to bottom and the bar length measurements by R , R_A2, and R_Qb from left to right. We display Spearman's correlation coefficient (ρ) with the significance (P) at the top right and the best-fit relation between the resonance radii and the bar length with the solid line.

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The plot shows that all the dynamical resonances are strongly correlated with the bar lengths regardless of the method by which they are measured. The correlations between the bar length and the resonance locations seem to be tighter for the resonances such as OLR or CR. The bar lengths measured by R_A2 and R_Qb are mostly located near CR (Figures 11(h) and (i)), but evolved (slow) bars seem to be located far away from this trend. Compared to the results in Michel-Dansac & Wozniak (2006), our resonance radii tend to be located further in, because both the minimum ellipticity and the transition between a bar and a disk occur after the maximum ellipticity. In conclusion, we hardly find any different links with specific dynamical resonances for different bar length measurements. The simulations may not be sufficient to compare with observations because few are available. We need more simulation models with various properties for detailed comparison.

In Section 4.1.1, we mentioned that most barred galaxies with measured bar pattern speeds belong to fast bars in terms of ${ \mathcal R }$ (Aguerri et al. 2015; Cuomo et al. 2019; Garma-Oehmichen et al. 2020). The observed galaxies with lower ${ \mathcal R }$ or with a small number of slow bars have led to concerns about a less concentrated dark matter halo or inefficient angular momentum exchange between a bar and a dark halo (Pérez et al. 2012; Aguerri et al. 2015). First, we suggest that the different bar length measurements can more or less explain the discrepancy of ${ \mathcal R }$ between observations and simulations, because observations usually used the ellipse fitting method to measure the bar length (Aguerri et al. 2015; Cuomo et al. 2019; Guo et al. 2019; Garma-Oehmichen et al. 2020), while simulations obtained the bar length from Fourier analysis (Lindblad et al. 1996; Debattista & Sellwood 2000; Athanassoula 2013; Seo et al. 2019). In our measurements, R_A2 from Fourier analysis is shorter than R by 20%, which can explain a larger ${ \mathcal R }$ in simulations.

Second, we argue that a certain criterion of ${ \mathcal R }=1.4$ may not be appropriate to classify barred galaxies into fast and slow bars or to constrain the density of a dark halo. The simulation that suggested ${ \mathcal R }\leqslant 1.4$ only for a dark halo with low density did not consider the existence of gas in their simulations (Debattista & Sellwood 2000). However, Athanassoula (2014) showed that barred galaxies even in a dense halo evolve within ${ \mathcal R }=1.4$ when they have enough gas initially. Moreover, the shape or spin of a halo influences the evolution of barred galaxies: triaxial or fast spinning halos drive lower ${ \mathcal R }$ (Athanassoula 2014; Long et al. 2014; Collier et al. 2018).

Nevertheless, observations show large differences from the simulation results, including the EAGLE and the IllustrisTNG projects. Roshan et al. (2021) obtained ${ \mathcal R }\sim 2.5$, on average, for simulated galaxies of EAGLE and Illustris TNG by measuring the bar pattern speed and the bar length. They used the TW method for the pattern speed and the ellipse fitting plus Fourier analysis for the bar length. The simulated galaxies have a much larger corotation radius, over 10 kpc, and a smaller bar length (〈R_bar〉 = 3.1 kpc) than those in observations. Algorry et al. (2017) also reported that strong bars in EAGLE simulations have corotation radii larger than 10 times the bar length. However, we cannot find such highly evolved barred galaxies in observations. The IllustrisTNG simulations also yield a slower bar pattern speed 〈Ω_bar〉 = 25.2 km s⁻¹ kpc⁻¹ (Roshan et al. 2021) than those of our sample galaxies, which have 〈Ω_bar〉 = 44.1 ± 29.1 km s⁻¹ kpc⁻¹.

In Section 4.2, we examined the dependence of the bar length on the disk circular velocity of the host galaxy, and found that rapidly rotating disks host long bars (Figure 6). The Tully–Fisher relation (Tully & Fisher 1977) dictates that rapidly rotating disks are luminous and massive. The observation of longer bars in brighter, massive, and larger galaxies in the local universe (Kormendy 1979; Ann & Lee 1987; Erwin 2019; Cuomo et al. 2020) could be an outcome of various processes. Long bars could inherit their size from their host galaxies; larger and massive galaxies could make their bars evolve longer effectively. Host galaxies and bars could evolve together by mutual interactions. In any case, we need to consider the disk velocity when we investigate the evolution of barred galaxies.

In Figure 4, we showed examples of a nonbarred galaxy along with those of fast, medium, and slow bars. We selected them by fixing their velocity V_circ = 190 ± 10 km s⁻¹ except for nonbarred galaxies. In our sample, nonbarred galaxies without a CR are mainly slowly rotating with V_circ < 150 km s⁻¹. The circular velocity of UGC 3944 is 148 km s⁻¹ (Figure 4(a)). In the bottom row, we present the radial profile of the force ratio for each galaxy. Lee et al. (2020) introduced a way to analyze a force ratio map defined as the transverse-to-radial force ratio by investigating the radial and azimuthal profiles. From a comparison with simulations, they suggested an evolution process of barred galaxies on the radial profile of force ratio. Galaxies grow from a plateau (type P) to a maximum peak (type M) on the radial profile with increasing force ratio Q_b (Lee et al. 2020, see their Figure 19). The galaxies in Figure 4 seem to follow the evolution process suggested in Lee et al. (2020): the radial profiles show a plateau for a nonbarred galaxy (Figure 4(i)), whereas fast, medium, and slow bars have a maximum peak on the radial profile (Figures 4(j)–(l)). They show increasing force ratios Q_b from a fast bar to a slow bar. On the other hand, we hardly find any increase in bar length from a fast bar to a slow bar. In particular, the maximum radii of A₂ and Q_b seem to be located further in than the bar end in the slow bar.

In Figure 9, the anticorrelation between the bar pattern speed and the bar length seems to show the growth of the bar length as the bar pattern speed decreases through an exchange of angular momentum between a bar and a dark halo (Athanassoula 2002, 2003). However, we are concerned that all of the longer bars with R_bar > 10 kpc are found only in rapidly rotating systems, namely massive galaxies. When we investigate them by classifying into fast, medium, and slow bars, we cannot find any difference in the bar length between fast and slow bars. Although the normalized bar length shows a trend to increase from a fast bar to a slow bar, this is caused by the decrease in the disk scale length. Therefore, long bars in massive galaxies seem to inherit their size from their host galaxy where they form. The bar instability in a massive disk galaxy may yield a large massive bar from the beginning of bar formation. When we normalize the bar length by indicators of the galaxy size, we need to be careful that the disk scale length could be changed as well during the bar evolution. When it comes to the bar strength, we can find a hint of increase by evolution, but the increase is not large.

In conclusion, we do not find the increase in bar length and strength for the bar evolution predicted by numerical simulations (Debattista & Sellwood 2000; Athanassoula 2003; Seo et al. 2019). This is in line with previous observations that most galaxies stay in the phase of fast bars in terms of ${ \mathcal R }$ (Aguerri et al. 2015; Cuomo et al. 2019; Guo et al. 2019). The recent observational study of Kim et al. (2021) supports little secular evolution of barred galaxies as well. They investigated the evolution of bar length and strength at 0.2 < z ≤ 0.835 from HST/COSMOS data. They showed that the absolute and normalized bar lengths have barely changed over the last 7 Gyr. They found only a slight increase in the bar strength over cosmic time. Therefore, they discussed the cases of simulation models in which bars could experience very little secular evolution, including gaseous disk, triaxial halo (Athanassoula 2013), or increasing dark halo spin (Long et al. 2014). Okamoto et al. (2015) also showed that bars do not always grow by evolution in self-consistent hydrodynamical simulations for two galaxies of Milky Way mass in a cosmological context.