Search for Slow Bars in Two Barred Galaxies with Nuclear Structures: NGC 6951 and NGC 7716

The pattern speed of bars, Ω_bar, is a fundamental quantity for understanding the dynamics and evolution of barred disk galaxies (J. Binney & S. Tremaine 1987; V. P. Debattista & J. A. Sellwood 2000). As of now, various methods have been proposed for measuring Ω_bar, utilizing photometric, spectroscopic, or modeling techniques, each of which is based on distinct characteristics or assumptions. For instance, some approaches explore the relation between resonance locations and galaxy rings (E. Athanassoula et al. 1982; C. Muñoz-Tuñón et al. 2004; I. Pérez et al. 2012), while others examine the connection between periodic orbits and gas flow (T. S. van Albada & R. H. Sanders 1982; E. Athanassoula 1992). Additionally, variations in galaxy properties near the CR, R_CR, have also been investigated, including changes in stellar population (I. Puerari & H. Dottori 1997; R. J. Buta & X. Zhang 2009; S. Scarano & J. R. D. Lépine 2013), spiral morphology (B. Canzian 1993; A. A. Marchuk et al. 2024), and streaming motion (J. Font et al. 2011, 2014). However, despite these advancements, consistent results across different methods remains elusive, making the reliable measurement of Ω_bar difficult (V. S. Kostiuk et al. 2024; A. A. Marchuk et al. 2024).

Among these available methods, the TW method stands out as the only model-independent and direct approach for measuring Ω_bar. It has been applied more extensively than any other method across a large number of galaxy samples (J. A. L. Aguerri et al. 2015; V. Cuomo et al. 2019a, 2022, 2024; R. Guo et al. 2019; L. Garma-Oehmichen et al. 2020, 2022; T. Géron et al. 2023). This method assumes a well-defined pattern speed and requires that tracers, such as stars, satisfy the continuity equation (S. Tremaine & M. D. Weinberg 1984). Ω_bar is derived from the line-of-sight velocity distribution (LOSVD) through multiple long-slit spectroscopic observations, as shown in the bottom panels of Figure 1, where the slit positions are marked in red. The slits are positioned along the major axis of the disk, spanning across the bar. For each slit, when the luminosity-weighted velocity and position are integrated, the contributions from the disk are canceled out, leaving only the asymmetries introduced by the bar, which provide information about the rotation of the bars (Y. Zou et al. 2019). Further details will be discussed in Section 4.2.

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We selected our target galaxies from the AINUR catalog (S. Comerón et al. 2010), which includes 107 nearby galaxies identified as hosting nuclear rings. These galaxies, located within 80 Mpc, are well suited for detailed studies of nuclear structures using high-resolution HST images. From the 78 barred galaxies, we chose two targets, NGC 6951 and NGC 7716, as shown in Figure 1, which satisfy the following criteria and have observational visibility. The selection criteria required galaxies to have a moderate inclination of 20° < i < 60° and an offset between the position angles (PAs) of the bar and the disk, with 10° < ∣PA_disk − PA_bar∣ < 80° (J. A. L. Aguerri et al. 2015; V. Cuomo et al. 2019a; R. Guo et al. 2019; Y. Zou et al. 2019; L. Garma-Oehmichen et al. 2020). These are the necessary geometric conditions for applying the TW method, avoiding configurations where bar contributions appear symmetric with respect to the disk (Y. Zou et al. 2019).

Therefore, accurate measurements of the galaxy disk orientation parameters (PA_disk and ellipticity, _disk) as well as PA_bar are essential for selecting targets. In particular, errors in PA_disk are the dominant source of systematic uncertainty in the application of the TW method (V. P. Debattista 2003; Y. Zou et al. 2019; L. Garma-Oehmichen et al. 2020), and, more generally, inaccurate orientation parameters significantly affect the bar property measurements, including the bar length, strength, and pattern speed (S. Comerón et al. 2010; L. Garma-Oehmichen et al. 2020; Y. H. Lee et al. 2020; T. Géron et al. 2023). This issue becomes especially critical when dealing with large samples, where inaccurate orientation parameters can lead to the misclassification of bars or introduce major uncertainties in the bar property measurements (Y. H. Lee et al. 2020). However, by visually inspecting the variation of isophotes for each galaxy, potential misjudgments can be significantly reduced.

Orientation parameters are typically derived from isophotes in the sky plane using ellipse fitting (S. Jogee et al. 2002; S. Laine et al. 2002; J. A. L. Aguerri et al. 2015; V. Cuomo et al. 2019a; R. Guo et al. 2019). In previous studies, the reference isophote has been chosen in different ways, such as using the 25 mag arcsec⁻² surface brightness contour (S. Jogee et al. 2002; S. Laine et al. 2002; Y. H. Lee et al. 2019) or averaging over the disk region where the PA and appear roughly constant (V. Cuomo et al. 2019a, 2019b). In this study, we determined the orientation parameters of the galaxy disks and PA_bar by performing an IDL-based ellipse fitting on sky-subtracted PS1 i-band images with bright stars masked, using the Bar Image astro-Arithmometer (BarIstA;⁹ Y. H. Lee et al. 2019, 2020). The masking was performed automatically using a built-in BarIstA routine, which identifies and removes foreground stars or nearby galaxies by analyzing the spatial variation in brightness across the image. The disk orientation parameters were measured as the mean PA and within the radial range where both profiles remain approximately constant. The inclination was derived from the mean using , and PA_bar was measured at the radius of maximum . The isophotes used for the disk and the bar are shown as the large and small gray dotted ellipses, respectively, in the bottom panels of Figure 1. These measurements, along with the basic properties of the target galaxies, are listed in Table 1. The standard deviations of PA_disk within this selected region are 17 for NGC 6951 and 13 for NGC 7716 and are adopted as the uncertainties in the PA measurements.

NGC 6951 and NGC 7716 were observed using long-slit spectroscopy with the GMOS instrument on Gemini-North (Program ID: GN-2023B-Q-303; PI: Y. H. Lee). Since the field of view of GMOS in Integral Field Unit mode is limited to 5″ × 7″, we adopted the long-slit mode to achieve spatially resolved spectroscopy over a wider area. For each galaxy, five slits were positioned along the major axis of the disk for each galaxy, as shown in red in the bottom panels of Figure 1: one at the center and four symmetrically placed across the bar. For NGC 6951, the slits were aligned at a PA of 1525 along the disk major axis, offset by across the bar major axis. For NGC 7716, the slits were aligned with a PA of 2250, with offsets of .

The TW method requires a wavelength window free of strong emission lines in order to trace the stellar kinematic signal reliably (V. Cuomo et al. 2024). Absorption features such as the Mg I triplet (5167, 5173, and 5184 Å) have been commonly used for this purpose (V. Cuomo et al. 2019b, 2024; C. Buttitta et al. 2022). Recently, V. Cuomo et al. (2024) demonstrated that the Ca II triplet (8498, 8542, and 8662 Å) yields consistent results with those obtained using the Mg I triplet, showing that the TW method provides robust measurements regardless of the choice of absorption lines. Considering the grating efficiency and spectral resolution of GMOS, we selected the Ca II region and used the R831 grating centered at 8600 Å, covering a wavelength range of 7411–9892 Å. A slit width of was employed, yielding a spectral resolution of ∼2900, corresponding to an FWHM of ∼ 3 Å at 8600 Å. The long slit spans a length of , with a spatial sampling of 01614 pixel⁻¹. Given the observational conditions,¹⁰ we achieved a signal-to-noise ratio (S/N) of ∼3 in regions with , indicated by the lengths of the red lines in the bottom panels in Figure 1. This slit length is sufficiently long to obtain a reliable Ω_bar, which will be discussed in Section 4.2. The exposure times per slit were 5300 and 5425 s for NGC 6951 and NGC 7716, respectively, totaling approximately 7.5 hr for each galaxy.

We reduced the data using the Data Reduction for Astronomy from Gemini Observatory North and South (Dragons) software (K. Labrie et al. 2019), incorporating bias subtraction, flat-field correction, and wavelength calibration using arc exposures. Flux calibration was achieved using standard star observations. The exposures were dithered along the spectral directions, and the final 2D exposures were combined to remove artifacts and bad pixels.

Figure 2 shows the velocity profiles derived from the five long-slit spectra for NGC 6951 (left panel) and NGC 7716 (right panel). From top to bottom, the panels correspond to the slits arranged from left to right in the bottom panels of Figure 1. The spectra have been sky-subtracted and Voronoi-binned along the spatial direction to achieve S/N > 3 (M. Cappellari & Y. Copin 2003). Velocities were measured by fitting Gaussian profiles to the Ca II triplet lines, with uncertainties estimated using chi-square minimization. We will discuss these results in Section 4.2.2.

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We examine the nuclear features of the target galaxies using HST/WFPC F606W images, which offer high spatial resolution, with a pixel scale of for WF (left panel) and for PC (right panel), as shown in Figure 3. To highlight the nuclear structures, we apply unsharp mask processing (bottom panels), a technique that enhances detail and contrast by dividing the original image with its smoothed version (D. F. Malin 1977; D. F. Malin et al. 1983; P. Erwin & L. S. Sparke 2003; P. Erwin 2004). This approach effectively suppresses the large-scale smoothed light distribution while amplifying faint and low-contrast features. The smoothed images were generated using 10″ sized boxes for the WF images and 5″ sized boxes for the PC images.

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The nuclear ring of NGC 6951 is clearly resolved in both the PC image (Figure 3(b)) and the unsharp masked image (Figure 3(f)), consistent with previous studies (H. Wozniak et al. 1995; D. Friedli et al. 1996; J. H. Knapen et al. 2006). NGC 7716 displays a nuclear bar within the inner ring in the PC image (Figure 3(d)), with this feature further accentuated in the unsharp masked image (Figure 3(h)). This observation aligns with earlier studies that classified NGC 7716 as a double-barred galaxy (S. Laine et al. 2002; P. Erwin 2004).

We further analyze the nuclear structures using the HST/PC images by fitting ellipses with BarIstA (Y. H. Lee et al. 2019, 2020), as shown in Figure 4. The three bottom panels show the radial profiles of intensity, ellipticity, and PA, from top to bottom. First, the sizes of the nuclear ring and nuclear bar are identified as the locations of maximum ellipticity in the ellipticity profile (P. Martin 1995; H. Wozniak et al. 1995; S. Jogee et al. 2004), marked by the red solid lines. The corresponding ellipses are overlaid with red solid lines on the HST/PC unsharp masked image in the top panel. In our target galaxies, the sizes of the nuclear structures determined by the maximum ellipticity are likely underestimated compared to those estimated from visual inspection, as similarly found in previous studies (E. Laurikainen et al. 2002; P. Erwin & L. S. Sparke 2003; J. A. L. Aguerri et al. 2009). To mitigate this effect, we also indicate the sizes of the nuclear structures as the radius with minimum ellipticity for NGC 6951 and as the radius identified through visual inspection for NGC 7716. These are represented by the red dotted lines.

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The sizes of the nuclear structures are then deprojected and summarized in Table 2. The deprojected radii can be estimated using the following relation:

where r_obs is the observed radius, ϕ is the angle between the major axis of the nuclear structures and that of the galaxy disk, and i is the inclination of the galaxy (P. Martin 1995; D. A. Gadotti et al. 2007). However, the values of r_deproj and r_obs for our target galaxies are nearly identical, as the nuclear structures are closely aligned with the major axes of their respective main disks.

Table 2. Measurements of the Galactic Structures

Galaxy	Unit	Nuclear Substructure		Nuclear Disk		Bar Length			Bar Strength			Disk
		(RNR, RNB)		(RND)		(Rbar)			(Sbar)
		R	Rvisual	Rkin	v/σ	R	RA2	RQb		A2	Qb	R25
NGC 6951	arcsec	3.9	5.0	5.9	2.8	40.0	47.7	44.9	0.55	0.42	0.36	126.7
	kpc	0.39	0.49	0.59	⋯	3.92	4.77	4.49	⋯	⋯	⋯	12.66
NGC 7716	arcsec	3.2	4.5	6.5	2.1	24.5	17.5	17.8	0.28	0.13	0.07	61.4
	kpc	0.58	0.81	1.16	⋯	4.30	3.18	3.22	⋯	⋯	⋯	11.12

Note. The sizes of the structures are represented by R_NR, R_NB, R_ND, and R_bar, corresponding to a nuclear ring, nuclear bar, nuclear disk, and primary bar, respectively. The subscripts in the third row indicate the measurement methods used. The nuclear structures of NGC 6951 and NGC 7716 are a nuclear ring (NR) and a nuclear bar (NB), respectively.

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Last, we note that NGC 7716 was selected from the AINUR catalog, which compiled galaxies with nuclear rings (S. Comerón et al. 2010). Historically, G. de Vaucouleurs & R. Buta (1980) identified NGC 7716 as hosting an inner ring. However, subsequent studies reclassified this structure as a nuclear ring, due to the detection of a larger bar extending beyond the radius of the ring (J. H. Knapen et al. 2006; L. M. Mazzuca et al. 2008). This reclassification is why NGC 7716 is included in the AINUR catalog (S. Comerón et al. 2010).

However, considering the bar length of 33″ reported in the AINUR catalog, the winding spiral arms, filled with a reddish hue and indicated by the white arrow in Figure 1(b), could have been interpreted as the bar. The deprojection using the orientation parameters from the catalog further accentuates this feature. This serves as a representative example of how erroneous orientation parameters can significantly affect the measurement of the bar length and strength, as discussed in Section 2.2. In contrast, the parameters we derived suggest that NGC 7716 hosts a weak bar (18″ in length, measured using the same method as in the AINUR catalog), surrounded by an inner ring, with a nuclear bar at its central region, as shown in Figure 3(h). Further details will be discussed in Sections 4 and 5.2.

The LOSVDs can be modeled using Gauss–Hermite functions with four parameters: stellar velocity (v), velocity dispersion (σ), and the asymmetric (h₃) and symmetric (h₄) deviations from a Gaussian (R. P. van der Marel & M. Franx 1993). These parameters serve as powerful tools for understanding the stellar structures in galaxies (D. A. Gadotti et al. 2020). To derive these stellar kinematics, we applied penalized pixel fitting (pPXF; M. Cappellari & E. Emsellem 2004; M. Cappellari 2017) to the sky-cleaned long-slit spectra obtained along the major axes of the disks. These spectra are the same as those used in the third panels of Figure 2, but are Voronoi-binned to achieve S/N ≥ 20 (M. Cappellari & Y. Copin 2003). We used the wavelength range of 8500–8750 Å, which covers the Ca II triplet lines at z < 0.01 for our target galaxies, while masking the 8630–8670 Å region to remove instrumental noise. The best-fit templates were constructed using the E-MILES library of stellar spectra (A. Vazdekis et al. 2016). The uncertainties were estimated using Monte Carlo simulations with 100 iterations.

Figure 5 presents v_rot, σ, h₃, h₄, and v/σ, from top to bottom. The rotation curves (Figure 5(a)) are derived from the observed stellar velocity as follows:

where v_rot, v_obs, and v_sys represent the rotation velocity, observed velocity, and systemic velocity, respectively, while i denotes the inclination of the galaxy. v_sys was determined at the kinematic center, identified as the point where the velocity curve exhibits approximate symmetry between the approaching and receding sides, yielding 1414 ±2 km s⁻¹ for NGC 6951 and 2576 ±2 km s⁻¹ for NGC 7716. The heliocentric systemic velocities and distances of these galaxies are summarized in Table 1. The velocity dispersion (Figure 5(b)) is determined after correcting for instrumental broadening during the pPXF process. The deprojected sizes of the nuclear structures are represented as the red dotted vertical lines, as measured in Section 3.1.

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The complex nuclear structures of our target galaxies are clearly evident across all kinematic profiles presented in Figure 5. The top panels show that the v_rot profiles exhibit peaks around the nuclear structures, as indicated by the red vertical lines. These characteristics likely suggest the presence of dynamically decoupled components within the galaxies (E. Emsellem et al. 2001). Such components are typically identified by a sudden change in v, a drop in σ, an anticorrelation between h₃ and v, or elevated values of h₄ (E. Emsellem et al. 2001; H. Wozniak et al. 2003; A. Chung & M. Bureau 2004; J. Falcón-Barroso et al. 2006; D. A. Gadotti et al. 2020). These signatures of dynamically decoupled components are evident in both of our target galaxies.

The left panel of Figure 5(b) shows a broad decrease in the central region of the σ profile in NGC 6951, indicating the presence of a rotationally supported structure, a nuclear cold disk (H. Wozniak et al. 2003; H. Wozniak & N. Champavert 2006; D. A. Gadotti et al. 2020). While the central region of the σ profile in NGC 7716 (right panel) initially appears as a peaked nucleus, the zoomed-in profile in Figure 6 reveals a flat central region, characteristic of a σ-drop (J. Méndez-Abreu et al. 2014). A σ-drop typically appears as a flat section or a dip in the central region of the σ profile (E. Emsellem et al. 2001; I. Pérez et al. 2009; J. Méndez-Abreu et al. 2014). Meanwhile, the upturn observed within the σ-drop in NGC 6951 may be attributed to kinematically hot spheroids or large-scale outflows driven by active galactic nucleus (AGN) activity (I. Márquez et al. 2003; I. Pérez et al. 2009; A. Bittner et al. 2020). We note that NGC 6951 has been classified as either a Seyfert 2 galaxy (L. C. Ho et al. 1997) or a low-ionization nuclear emission-line region galaxy (T. Storchi-Bergmann et al. 2007).

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Additionally, near-circular orbits are characterized by an anticorrelation between h₃ and v, while elongated orbits exhibit a positive correlation between the two (A. Chung & M. Bureau 2004; M. Bureau & E. Athanassoula 2005; F. Iannuzzi & E. Athanassoula 2015; D. A. Gadotti et al. 2020). In both galaxies, we observe an anticorrelation between h₃ (Figure 5(c)) and v_rot (Figure 5(a)), which further supports the presence of a nuclear stellar disk. A zoomed-in view of the h₃ profile of NGC 7716 is also shown in Figure 6. The small distortion in the central region of h₃ in NGC 7716 could be attributed to the presence of the nuclear bar. In the case of NGC 6951, the elevated h₄ (Figure 5(d)) further suggests the superposition of structures with distinct LOSVDs (R. Bender et al. 1994; D. A. Gadotti et al. 2020).

As a result, the kinematic profiles of both target galaxies consistently indicate the presence of dynamically decoupled nuclear stellar disks. The v/σ profile in Figure 5(e) provides the kinematic radius of the nuclear disk (R_kin), where the profile peaks due to the rapidly rotating disk (D. A. Gadotti et al. 2020). We present R_kin as the red vertical solid lines across all profiles in Figures 5 and 6, and we summarize both R_kin and the corresponding v/σ values at R_kin in Table 2. In our observations, both a nuclear ring and a nuclear bar, as indicated by the red dotted lines, are located within R_kin. Further discussion on the nuclear disk will be provided in Sections 5.1 and 5.2.

The bar length and strength were initially estimated through visual inspection (P. Nilson 1973; G. de Vaucouleurs et al. 1991; H. B. Ann et al. 2015; R. J. Buta et al. 2015), a practice still widely used thanks to contributions from citizen scientists participating in the Galaxy Zoo project (C. J. Lintott et al. 2008; K. L. Masters et al. 2011; B. D. Simmons et al. 2014, T. Géron et al. 2025). More recently, this visually driven approach has evolved into a scalable framework through machine learning trained on Galaxy Zoo classifications. Machine learning methods have shown remarkable performance in processing large-scale survey data, including those from the Dark Energy Spectroscopic Instrument (DESI; T. Géron et al. 2023; M. Walmsley et al. 2023).

On the other hand, automated techniques have also been developed to analyze the bar length and strength, enabling quantitative measurements and serving as powerful tools for detailed investigation of bar structures. To this end, we adopted three widely used methods: ellipse fitting (P. Martin 1995; H. Wozniak et al. 1995; S. Jogee et al. 2004), Fourier analysis (K. Ohta et al. 1990; E. Laurikainen et al. 2002, 2004), and the force ratio map (R. Buta & D. L. Block 2001; E. Laurikainen & H. Salo 2002; S. Díaz-García et al. 2016; Y. H. Lee et al. 2020). The bar lengths obtained from these methods are defined as R, , and , respectively, while the bar strengths are characterized by , A₂, and Q_b. Ellipse fitting and Fourier analysis are described in detail in Y. H. Lee et al. (2019), while the force ratio map method is explained in Y. H. Lee et al. (2020).

We applied all three methods using BarIstA (Y. H. Lee et al. 2019, 2020) on sky-subtracted PS1 i-band images. Figures 7 and 8 illustrate the procedure for measuring the bar length and strength. The observed image (panel (a)) was masked and deprojected, based on the orientation parameters listed in Table 1, as outlined by the gray dotted ellipses. The resulting image is shown in panel (b), where the disk outline has been transformed into a perfect circle. The red ellipse traces the bar identified through ellipse fitting, while the four crosses mark the four corners of the bar structure determined using the force ratio map, as detailed below.

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The results from ellipse fitting (panel (c)), Fourier analysis (panel (d)), and the force ratio map (panel (e)) are presented from left to right. In general, the bar strengths ( and A₂) are defined by the maximum values in the and A₂ profiles (P. Martin 1995; E. Laurikainen & H. Salo 2002; E. Laurikainen et al. 2002), as shown in the middle row. The bar lengths (R, R_A2, and R_Qb) are determined by the radii where the bar strengths are measured. The PA and phase (ϕ) are typically constant across the bar structures, remaining within ±10° (J. A. L. Aguerri et al. 2009; I. Marinova et al. 2009; Y. H. Lee et al. 2019), as shown in the bottom row. All the red solid lines in Figure 7 depict the measured bar lengths. For NGC 6951, however, appears along a spiral arm beyond ∼50″, aligned with the bar PA. Therefore, we adopted the local maximum near ∼40″ to determine the bar length and strength.

In Fourier analysis, A₂ is defined as follows:

where a₀, a₂, and b₂ are the Fourier coefficients (E. Athanassoula 2013; W.-Y. Seo et al. 2019; Y. H. Lee et al. 2022). We note that this value is half of that used in some previous studies, based on their definition (K. Ohta et al. 1990; E. Laurikainen et al. 2002; Y. H. Lee et al. 2019, 2020).

The rightmost panels (column (e)) illustrate the procedure of analyzing the force ratio map, which represents the transverse-to-radial force derived from the potential map (F. Combes & R. H. Sanders 1981; R. Buta & D. L. Block 2001; Y. H. Lee et al. 2020). When displayed in polar coordinates (r, ϕ), as shown in the top panel, the bar structures appear as four horizontal thick slabs, aligned with similar length (r) and width (ϕ) (Y. H. Lee et al. 2020). The force ratio is defined as follows:

where the radial force is described by

and the transverse force is given by

Here, Φ₀ represents the m = 0 Fourier component of the gravitational potential (F. Combes & R. H. Sanders 1981; R. Buta & D. L. Block 2001).

The force ratio map (top panel) can be analyzed in both the radial (middle panel) and azimuthal (bottom panel) directions. The radial profile (middle panel) of the force ratio map, Q_T(r), is obtained by averaging Q_T over all azimuthal angles at each radius. This profile initially decreases in the bulge-dominated region and then rises to a maximum, which defines the bar length (R_Qb), marked by the red solid vertical line. The azimuthal profile (bottom panel), on the other hand, is constructed by plotting the Q_T values across azimuthal angles at a fixed radius. Notably, when the azimuthal profile is examined at R_Qb, it reveals four distinct peaks, as shown in the bottom panel. These four peaks correspond to the four corners of the bar, which also align with the four slabs in the force ratio maps (top panel) and are marked as the red crosses on the deprojected images (panel (b)). The bar strength (Q_b) is calculated as the mean Q_T value at these four peaks (Y. H. Lee et al. 2020). All measured bar lengths (R, R_A2, and R_Qb) and strengths (, A₂, and Q_b) are provided in the relevant panels of Figures 7 and 8, as well as in Table 2. The errors of the bar lengths are estimated from the widths of the peaks in the , A₂, and Q_T(r) profiles, determined at the radii where , A₂, and Q_T reach 95% of their peak values, following the approach of V. Cuomo et al. (2021). For comparison, the deprojected sizes of the nuclear structures (the red dotted vertical lines) and the galaxy disk, measured at 25 (R₂₅, the gray solid vertical lines), are also included.

The three methods described above for measuring bar length and strength are useful for identifying barred galaxies (Y. H. Lee et al. 2019, 2020). While additional characteristics, such as the transition between a bar and a disk, and a constant PA or phase, should also be considered (H. Wozniak et al. 1995; E. Laurikainen et al. 2002; S. Jogee et al. 2004; I. Marinova & S. Jogee 2007; K. Menéndez-Delmestre et al. 2007; J. A. L. Aguerri et al. 2009; Y. H. Lee et al. 2019, 2020), each method provides specific criteria for the bar strength that aid in identifying barred galaxies. These criteria can be summarized as , A₂ ≥ 0.15, and Q_b ≥ 0.15 (E. Laurikainen et al. 2002; I. Marinova & S. Jogee 2007; F. D. Barazza et al. 2008; I. Marinova et al. 2009; Y. H. Lee et al. 2019, 2020). While E. Laurikainen et al. (2002) suggested a criterion of I₂/I₀ > 0.3, this can be interpreted as A₂ > 0.15, since I₂/I₀ is larger than A₂ by a factor of 2. We overlay these bar strength criteria as the red horizontal dotted lines in the relevant panels of Figures 7 and 8. Additionally, we plot criteria of , A₂ ≥ 0.4, and Q_b ≥ 0.25 (S. Jogee et al. 2004; V. Cuomo et al. 2019a; Y. H. Lee et al. 2019, 2020), which have been used to indicate strongly barred galaxies.

Therefore, given the criteria for the bar strength, NGC 6951 can be classified as strongly barred, with , A₂ = 0.42, and Q_b = 0.36. On the other hand, NGC 7716 only barely meets the requirement for barred galaxies with the criterion of and rarely satisfies the criterion of Fourier analysis. Moreover, the results from the force ratio map do not support the presence of a bar. Specifically, the four peaks in the azimuthal profile of the force ratio map (the bottom rightmost panel in Figure 8) are significantly low. Therefore, it is difficult to conclude that the oval feature surrounded by the inner ring of NGC 7716 represents a weak bar. We note that both of our target galaxies, NGC 6951 and NGC 7716, have been visually classified as SAB galaxies (i.e., weakly barred) in the Third Reference Catalog of Bright Galaxies (G. de Vaucouleurs et al. 1991), as summarized in Table 1.

The bar length tends to depend on the properties of the host galaxies, such as stellar mass or galaxy size (V. Cuomo et al. 2019a; P. Erwin 2019, 2024; T. Kim et al. 2021; J. A. L. Aguerri et al. 2023), which differs from the bar strength (S. Díaz-García et al. 2016; V. Cuomo et al. 2020; Y. H. Lee et al. 2022). In particular, the bar length is tightly correlated with the galaxy size, including the scale length (h_r), effective radius (R_eff), as well as the radii enclosing 50% (R₅₀) and 90% (R₉₀) of the total galaxy luminosity (H. B. Ann & S.-W. Lee 1987; T. Kim et al. 2021). As a result, the bar length is often examined as a relative size with respect to galaxy size, such as h_r or R₂₅ (B. G. Elmegreen & D. M. Elmegreen 1985; P. Martin 1995; K. Menéndez-Delmestre et al. 2007; J. A. L. Aguerri et al. 2009; Y. H. Lee et al. 2020). For our target galaxies, when investigating R_bar/R₂₅, the bar size of NGC 6951 ranges from 0.32 to 0.38, while NGC 7716 ranges from 0.29 to 0.40, depending on the methods used. This will be discussed further in Section 5.3.

The bar pattern speed, Ω_bar, can be directly obtained from multiple long-slit observations using the TW method (S. Tremaine & M. D. Weinberg 1984; M. R. Merrifield & K. Kuijken 1995), which is expressed as follows:

where i, v_sys, and x₀ denote the inclination, systemic velocity, and center of the galaxy, respectively. The functions l(x) and represent the mean luminosity and observed LOS velocity of stars at position x along the slit, respectively (S. Tremaine & M. D. Weinberg 1984; M. R. Merrifield & K. Kuijken 1995). The luminosity profile l(x) was derived from spectra collapsed along the wavelength axis. Consequently, 〈X〉 and 〈V〉 correspond to the luminosity-weighted position and velocity along each slit. In practice, each slit yields a pair of 〈X〉 and 〈V〉, and a linear fit to these pairs across multiple slits directly gives , as shown in the middle panels of Figure 9.

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Figure 9 illustrates the procedure for measuring Ω_bar in our target galaxies. The leftmost panels show the positions of the five long slits, marked with the red solid lines, aligned with the major axis of the disk. Luminosity-weighted positions, 〈X〉, are shown as the cyan triangles on each slit, while the black or white squares mark the slit midpoints. In the background, the g − i color maps highlight bluer features, such as nuclear rings, inner rings, or spiral arms. The bar shapes are outlined with dotted ellipses in panels (a) and (d).

For NGC 6951, in panel (a), the five 〈X〉 positions closely follow the major axis of the bar. Notably, they align with the dust lane, which appears as a lighter-colored region. In contrast, panel (d), for NGC 7716, shows that the 〈X〉 positions do not align with the major axis of the bar. This discrepancy may indicate that the contribution of the bar to the luminosity along each slit is not significant in NGC 7716. This may be partly due to the small angular offset between the major axis of the bar and that of the disk.

The luminosity-weighted velocity, 〈V〉, was directly derived by measuring the velocity from spectra collapsed along the spatial direction, which effectively weights the luminosity, as pixels with higher signal contribute more to the resulting 1D spectrum (M. R. Merrifield & K. Kuijken 1995; E. M. Corsini et al. 2007; J. A. L. Aguerri et al. 2015; V. Cuomo et al. 2019b). This method has the advantage of yielding higher S/N measurements compared to integrating the velocities measured at each pixel, weighted by the corresponding luminosity (Y. Zou et al. 2019). However, when collapsing over ∼400 pixels, we found that instrumental noise also accumulates. Although the noise per pixel is negligible, its combined effect can sometimes hinder pPXF. Therefore, we measured the velocity by fitting Gaussian profiles to the Ca II triplet lines, with the uncertainties estimated through chi-square minimization.

Finally, the five long-slit observations yield five data points for 〈X〉 and 〈V〉, as shown in the middle column of Figure 9. The slope of the best-fitting straight line, shown by the red solid line, corresponds to . Using the galaxy inclination listed in Table 1, Ω_bar is calculated as for NGC 6951 and for NGC 7716, which are summarized in Table 3. The uncertainties were estimated using Monte Carlo simulations with 1000 iterations. The errors in 〈X〉 are not considered, as they are negligible (J. A. L. Aguerri et al. 2015; R. Guo et al. 2019).

Table 3. Bar Pattern Speeds of Sample Galaxies

Galaxy	Unit	One Component	Two Components
		Ωbar	Ωbar	ΩNSa
NGC 6951		−3.2 ± 0.7	−2.5 ± 0.8	−10.9 ± 6.4
	km s−1 kpc−1	−31.9 ± 7.1	−25.5 ± 7.8	−108.7 ± 63.5
NGC 7716b		−32.1 ± 4.8	⋯	⋯
	km s−1 kpc−1	−177.3 ± 26.6	⋯	⋯

Notes.

^aΩ_NS is measured at r ∼ 10, where it begins to converge. ^bAlthough Ω can technically be measured in NGC 7716, it may be influenced by external factors and may not represent the actual Ω_bar.

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As described so far, this method is direct and straightforward. However, asymmetries caused by other structures, such as spiral arms or lopsidedness, could influence the measurements. It is therefore crucial to confirm that the contributions to the integrals primarily originate from the bar and not from other structures or artifacts (V. Cuomo et al. 2019b; Y. Zou et al. 2019). The positions of 〈X〉 in the leftmost column of Figure 9 demonstrate that 〈X〉 for NGC 6951 effectively reflects the contribution of the bar, whereas for NGC 7716, the bar contributes minimally to the luminosity integrals. In the middle column, the data points deviate from the straight-line fit, differing from the typical trend observed in Ω_bar measurements (M. R. Merrifield & K. Kuijken 1995; V. Cuomo et al. 2019b, 2024). Therefore, further investigation is required, which will be discussed in Sections 4.2.1 and 4.2.2.

In addition, various factors that could influence the measurement of Ω_bar have been examined, including the alignment of the slits (PA_slit), slit length, slit width, number of slits, wavelength range, and others (V. P. Debattista 2003; V. Cuomo et al. 2019b, 2024; R. Guo et al. 2019; Y. Zou et al. 2019; L. Garma-Oehmichen et al. 2020). Among these, the misalignment between the slit position (PA_slit) and PA_disk introduces the most significant uncertainties (V. P. Debattista 2003; Y. Zou et al. 2019; L. Garma-Oehmichen et al. 2020). In such cases, the axisymmetric light from the disk fails to cancel out, amplifying the error (L. Garma-Oehmichen et al. 2020). Thus, obtaining an accurate measurement of PA_disk is essential.

Regarding the slit length for spatial coverage, the integrals in Equation (7) theoretically extend from −∞ to ∞. However, in practice, a low S/N in the outer regions can introduce false signals (R. Guo et al. 2019), and beyond the bar, potential contributions from spiral arms must also be considered. Fortunately, Ω_bar generally converges to a reliable value when the slit length slightly exceeds the bar length, R_bar (R. Guo et al. 2019; Y. Zou et al. 2019; L. Garma-Oehmichen et al. 2020). Therefore, we assess the convergence of Ω_bar as a function of the slit length, as shown in the rightmost column of Figure 9. The red plus symbols show the variation of Ω_bar with increasing slit length, while the blue plus symbols represent the case where Ω_bar is analyzed with two components, as will be discussed in Section 4.2.1. The bar length, R_bar, measured from the observed images in the leftmost panels, is indicated by the red solid vertical lines, while the gray dotted solid lines indicate the slit length at which S/N > 3 per pixel is achieved. We find that Ω_bar begins to converge when the slit length approaches R_bar in both NGC 6951 and NGC 7716. The values of Ω_bar remain stable to some extent, even when the S/N at certain pixels does not reach 3. The leftmost and middle columns correspond to cases where the slit length ensures S/N > 3 per pixel. Last, factors such as the slit width, the number of slits, and the wavelength range generally do not introduce significant differences in Ω_bar measurements (V. Cuomo et al. 2019b, 2024; Y. Zou et al. 2019).

4.2.1. Two Pattern Speeds in NGC 6951

4.2.2. Uncertain Bar Pattern Speed of NGC 7716

Fast and slow bars are commonly distinguished using the rotation ratio, (J. A. L. Aguerri et al. 2003, 2015; V. Cuomo et al. 2019a, 2020; R. Guo et al. 2019; L. Garma-Oehmichen et al. 2020, 2022; T. Géron et al. 2023), with R_CR denoting the CR, the location at which stars rotate at the same angular velocity as Ω_bar. A value of serves as the threshold, defining fast bars as and slow bars as , based on the simulation by V. P. Debattista & J. A. Sellwood (2000) using halos with different densities. R_CR is typically calculated under the assumption of flat rotation (J. A. L. Aguerri et al. 2015; V. Cuomo et al. 2019a, 2024; R. Guo et al. 2019), while some studies have utilized rotation curves fitted with various functions, including arctan and tanh functions (S. Courteau 1997; L. Garma-Oehmichen et al. 2020; Y. Yoon et al. 2021; T. Géron et al. 2023; D. Jeong et al. 2025).

Figure 10 represents the rotation velocity of NGC 6951, obtained from pPXF applied to the same spectrum used in Figure 5. To extend the rotation curve to larger radii, the spectrum was binned to achieve an S/N of 10. The rotation curve on the blueshifted side provides data at greater distances. As discussed in Section 3.2, NGC 6951 hosts a dynamically decoupled nuclear disk superposed on the main disk. Stars in the nuclear disk rotate faster than stars in the main disk at the same radius (D. A. Gadotti et al. 2020). Therefore, the rotation curve of NGC 6951 cannot be effectively modeled using an arctan function. We instead modeled it with two components: a Gaussian fit for the nuclear region and a multiparameter function (S. Courteau 1997) for the residuals, as shown by the black solid line in Figure 10. The multiparameter function is described as follows:

where x = 1/R = r_t/r − r₀. Here, v_sys represents the systemic velocity and r₀ is the galaxy center. v_c denotes the asymptotic velocity, while r_t is the transition radius between the rising and flat parts of the rotation curve. The term γ controls the sharpness of the turnover, and β accounts for the dropoff or steady rise in the outer part of the rotation curve (S. Courteau 1997). The uncertainties are estimated using Monte Carlo simulations with 1000 iterations, as indicated by the gray solid curve in Figure 10.

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The anticipated flat rotation is depicted by the cyan solid horizontal line, with a value of 179 ± 18 km s⁻¹, calculated as the average velocity from the maximum near 40″ to the endpoint of the obtained curve, with the uncertainty representing the 1σ error. Previous studies based on Hα observations have reported a similar flat rotation trend in NGC 6951 beyond 50″ (I. Marquez & M. Moles 1993; M. Rozas et al. 2002; S. Haan et al. 2009).

The bar rotation is also shown in Figure 10, where the red and blue solid lines represent the one-component and two-component models for Ω_bar, respectively, as discussed in Section 4.2.1. R_CR is determined at the intersection of the bar rotation (the red or blue solid line) and the rotation curve (the black or cyan solid line). We derived the errors of R_CR by accounting for the uncertainties of both Ω_bar and v_rot.

The rotation rate is then calculated as the ratio of R_CR to R_bar. The uncertainties are estimated by propagating the uncertainties in both R_CR and R_bar, as summarized in Table 4. Figure 11 visualizes the data from Table 4, showing how varies depending on whether a one- or two-component model for Ω_bar is used, as well as on the methods applied to measure R_bar. The flat rotation and the rotation curve derived using a multiparameter function yield similar results. When we adopt the criterion of , NGC 6951 can be classified as a slow bar with the two-component bar model. In the case of the one-component bar model, only the ellipse-fitting method for R_bar barely falls within the slow-bar regime. Further discussions will be provided in Section 5.3.

Table 4. CR and Rotation Rate in NGC 6951

Ωbar	vrot	RCR
		(arcsec)	RCR/R	RCR/RA2	RCR/RQb
One component	Flat
	Multiparameter
Two components	Flat
	Multiparameter

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Figure 5 demonstrates the presence of a nuclear stellar disk in each of our target galaxies. In general, the σ profiles of galaxies follow an exponential law, peaking at the nucleus, where stars near the bottom of the potential well are expected to have the highest σ (E. Emsellem et al. 2001; S. Comerón et al. 2008; E. Portaluri et al. 2017). However, a central decrease in σ, known as a σ-drop, is observed in about 50% of disk galaxies when spectroscopic data with high spatial resolution are available (E. Emsellem et al. 2001; I. Márquez et al. 2003; J. Falcón-Barroso et al. 2006; S. Comerón et al. 2008; J. Méndez-Abreu et al. 2014). The σ-drop is understood as evidence for the presence of a cold nuclear disk with larger rotational support than the underlying main disk (H. Wozniak et al. 2003; H. Wozniak & N. Champavert 2006; E. Portaluri et al. 2017; D. A. Gadotti et al. 2020), often associated with nuclear rings, nuclear bars, pseudo-bulges, or AGN activities (E. Emsellem et al. 2001; I. Márquez et al. 2003; S. Comerón et al. 2008; E. Portaluri et al. 2017).

Nuclear disks are believed to form through star formation by dissipative gas funneled into the central regions, primarily driven by bars (R. H. Sanders & A. D. Tubbs 1980; S. M. Simkin et al. 1980; E. Athanassoula 1992, 2013; H. B. Ann & P. Thakur 2005; D. R. Cole et al. 2014; W.-Y. Seo et al. 2019; D. A. Gadotti et al. 2020). External gas accretion or mergers can also transport gas to the center, leading to nuclear disk formation (L. Mayer et al. 2008; M. C. Eliche-Moral et al. 2011; D. Chapon et al. 2013), but these mechanisms may leave distinct imprints on the resulting nuclear disk (M. C. Eliche-Moral et al. 2011; D. A. Gadotti et al. 2020).

The Time Inference with MUSE in Extragalactic Rings project (D. A. Gadotti et al. 2019) reported that bar-driven nuclear disks exhibit characteristic features, including a σ-drop, an anticorrelation between h₃ and v, or elevated h₄ (D. A. Gadotti et al. 2020). They also showed that bar-driven nuclear disks range in size (R_kin) from 200 to 1000 pc, which is strongly related to bar properties, including the bar length, strength, and the bar-to-total ratio (D. A. Gadotti et al. 2020).

In one of our target galaxies, NGC 6951, we find clear evidence for a bar-driven nuclear disk, including a σ-drop, an anticorrelation between h₃ and v, and a central increase in h₄, as shown in the left column of Figure 5. The measured values of R_kin and v/σ are 590 pc and 2.8, respectively, as listed in Table 2. These values fall within the typical range of observed bar-driven nuclear disks (D. A. Gadotti et al. 2020). Compared with the bar length and strength in Table 2, the nuclear disk of NGC 6951 follows the expected trends between R_kin and the bar properties as illustrated in Figure 8 of D. A. Gadotti et al. (2020). Interestingly, this nuclear disk appears as a superposed feature on the rotation curve in Figure 10, which may result from the superposition of different LOSVDs. This characteristic aligns with the elevated h₄ observed in Figure 5(d).

Nuclear disks often contain substructures such as nuclear rings, bars, or spiral arms (H. B. Ann & H. M. Lee 2000; D. A. Gadotti et al. 2020). Nuclear rings are closely associated with bar properties, showing that their size is correlated with bar length but there is an anticorrelation with bar strength (J. H. Knapen 2005; S. Comerón et al. 2010; W.-T. Kim et al. 2012; Z. Li et al. 2015; M. C. Sormani et al. 2018; D. A. Gadotti et al. 2020). These relations are similar between nuclear disks and bar properties; this suggests a fundamental process in common, which is bar-driven (D. A. Gadotti et al. 2020). Specifically, nuclear rings (R_NR) are typically located on the outer edges of nuclear disks (R_ND) and are considered to be part of them (D. R. Cole et al. 2014; A. Bittner et al. 2020; D. A. Gadotti et al. 2020). The left column of Figure 5 shows that R_NR (the red dotted line) and R_ND (the red solid line) are consistent in NGC 6951 as well.

NGC 7716 is also likely to host a nuclear disk with near-circular orbits. As shown in the right column of Figure 5 and in Figure 6, it exhibits a σ-drop and an anticorrelation between h₃ and v. However, the nuclear disk in NGC 7716 appears larger than the typical R_ND observed in bar-driven nuclear disks. Its size, 1.2 kpc, slightly exceeds the typical range of 200–1000 pc (D. A. Gadotti et al. 2020). Using the values of R_bar from Table 2, we find that the ratio R_ND/R_bar ranges from 26% to 37%, depending on the method used to measure R_bar. This is significantly higher than the typical value of less than 10% (D. A. Gadotti et al. 2020).

For the nuclear bar within the nuclear disk in NGC 7716, the length (R_NB) is measured to be 810 pc, based on visual inspection of the unsharp mask, as shown in Figure 4. While R_NB is generally positively correlated with the stellar mass of the host galaxy (P. Erwin 2024), the nuclear bar in NGC 7716 is significantly longer than expected from the nuclear bar–stellar mass relation. Instead, it falls within the lower regime of the primary bar–stellar mass relation, suggesting that its size is more comparable to a short primary bar than a typical nuclear bar (see Figure 9 in P. Erwin 2024), assuming a stellar mass of logM_* = 10.17 ± 0.08M_⊙ (E. Aquino-Ortíz et al. 2018).

On the other hand, A. de Lorenzo-Cáceres et al. (2020) reported a bimodal distribution of R_NB/R_bar, suggesting distinct formation or evolutionary pathways. For instance, based on TNG50 simulations, M. Semczuk et al. (2024) proposed that nuclear bars may form first, either in isolation via instabilities or through galaxy interactions. This mechanism could explain how nuclear bars grow longer relative to the primary bar, as observed in A. de Lorenzo-Cáceres et al. (2020). Notably, NGC 7716 hosts a large nuclear bar, with R_NB/R_bar ∼ 18%–26%, depending on the method used to measure R_bar. These values could increase when using the decomposition method employed by A. de Lorenzo-Cáceres et al. (2020).

When it comes to the primary bar of NGC 7716, it appears as an oval structure surrounded by an inner ring (Figure 1(b)) and has been visually classified as a weak bar (G. de Vaucouleurs et al. 1991). This classification is supported by the ellipse-fitting method (Figure 8(c)). However, Fourier analysis and the force ratio map (Figures 8(d) and (e)) make it challenging to classify NGC 7716 even as a weak bar. In particular, the bar strength is notably low, with Q_b = 0.07, which is well below the typical threshold of Q_b = 0.15. Additionally, the minimal contribution of nonaxisymmetric components to the luminosity within each long-slit spectrum further supports the weakness of the bar structure (Figure 9(d)).

Therefore, the growth of a large nuclear bar in NGC 7716 may be facilitated by the absence or weakness of the primary bar, similar to the longer nuclear bars observed in TNG50 simulations (M. Semczuk et al. 2024). Considering the tidal stream revealed in the deep image (G. Á. Bakos et al. 2018), NGC 7716 may have undergone a more complex evolutionary history.

The possibility of two pattern speeds in NGC 6951, as discussed in Section 4.2.1, is also an intriguing aspect of this study. Previous N-body simulations have shown that galaxies can host independent structures, such as bars, spirals, and nuclear bars, each rotating with a different pattern speed (J. A. Sellwood & L. S. Sparke 1988; P. Rautiainen & H. Salo 1999; I. Minchev & A. C. Quillen 2006; I. Minchev et al. 2012; J. Baba 2015; K. Zouloumi et al. 2024). This finding is supported by observations detecting multiple R_CR (I. Puerari & H. Dottori 1997; J. A. L. Aguerri et al. 1998) or multiple pattern speeds (S. E. Meidt et al. 2009). Slowly rotating spiral arms compared to a bar have been observed in the Milky Way (W. S. Dias & J. R. D. Lépine 2005; O. Gerhard 2011) and in other spirals (J. C. Speights & P. C. Rooke 2016).

In the TW method, a different pattern speed appears as a slope change in the straight-line fitting to the TW integrals (E. M. Corsini et al. 2003; V. Cuomo et al. 2021). A rapidly rotating nuclear bar was first observed in NGC 2950 using the TW method (E. M. Corsini et al. 2003). In Figure 9(b), we observe a slope change similar to that seen in NGC 2950, suggesting the presence of two pattern speeds in the central region of NGC 6951. Considering the offset between slits (), we expect the presence of a bar-like structure around r ∼ 10″. We also found several pieces of evidence supporting this structure. First, the and PA profiles show a bar-like structure at a comparable radius in both HST/PC and PS1 photometry (Figures 4(c)–(d) and 7(c)); Fourier analysis further supports this feature (Figure 7(d)). Second, the rapidly rotating pattern speed converges when the slit length approaches r ∼ 10″, similar to the convergence of Ω_bar near R_bar. Nevertheless, neither the PC image nor the corresponding unsharp masked image (Figures 3(b) and (f)) provide clear evidence of a bar-like structure at this radius.

Similarly, several previous studies anticipated the presence of a nuclear bar in NGC 6951 but were unable to detect it (D. Friedli et al. 1996; E. Pérez et al. 2000; M. Rozas et al. 2002; S. Haan et al. 2009). E. Pérez et al. (2000) suggested the possibility of a gas-driven structure, such as a nuclear bar, based on the relatively high molecular-to-total mass fraction within r ≲ 6″, which is consistent with the size of the kinematic nuclear disk, , measured in Figure 5(e). M. Rozas et al. (2002) and S. Haan et al. (2009) inferred it from the detection of strong noncircular motion near the nucleus. Therefore, the bar signature at r ∼ 10″ and the two pattern speeds observed in NGC 6951 in this study are consistent with the properties reported in previous studies (E. Pérez et al. 2000; M. Rozas et al. 2002). Instead of a bar-like structure at r ∼ 10″, previous studies have reported nuclear spirals and a nuclear oval within the nuclear ring (E. Pérez et al. 2000; S. García-Burillo et al. 2005; T. Storchi-Bergmann et al. 2007; T. P. R. van der Laan et al. 2011), which are likely smaller in size given the slit positions. Although we do not observe a distinct feature in the images, our results suggest that an oval structure of r ∼ 10″ could be responsible for inducing the rapidly rotating pattern speed.

When we adopt the two pattern speeds of NGC 6951, the galaxy is classified as a slow bar ( in Table 4). Moreover, all three methods used to measure the bar strength—ellipse fitting, Fourier analysis, and the force ratio map—consistently indicate that NGC 6951 hosts a strong bar (, A₂ = 0.42, Q_b = 0.36), despite its visual classification as a weak bar, as discussed in Section 4.1.

On the other hand, assessing whether a bar is long is somewhat complex. For instance, while visual inspection often relies on the relative bar length to distinguish between SBs and SABs (H. B. Ann et al. 2015; R. J. Buta et al. 2015; Y. H. Lee et al. 2019), the relative bar lengths measured using the three methods (R/R₂₅, R_A2/R_R25, and R_Qb/R₂₅) do not show significant differences in their distributions (Y. H. Lee et al. 2020). Moreover, the bar length depends on the morphological type or bulge-to-total fraction, with early-type spirals generally hosting longer bars in both absolute and relative terms (B. G. Elmegreen & D. M. Elmegreen 1985; P. Martin 1995; K. Menéndez-Delmestre et al. 2007; J. A. L. Aguerri et al. 2009; D. A. Gadotti 2011; T. Kim et al. 2014). More specifically, the relative bar length exhibits a bimodal distribution, increasing toward both ends of the Hubble sequence (E. Laurikainen et al. 2007; S. Díaz-García et al. 2016; J. Font et al. 2017, 2019; Y. H. Lee et al. 2020). Accordingly, when we compare the relative bar length of NGC 6951 to the distribution of relative bar lengths across the Hubble sequence (Figure 14 in Y. H. Lee et al. 2020), it is classified as a long bar in an Sbc galaxy (G. de Vaucouleurs et al. 1991). The relative bar lengths of NGC 6951, measured using three different methods, are higher than the mean values for SBbc galaxies. In particular, R_A2/R₂₅ and R_Qb/R₂₅ exceed the mean by nearly a factor of 2.

As a result, based on all analyses presented in this study, we classify NGC 6951 as a galaxy with a slow, long, and strong bar. Additionally, NGC 6951 is a deeply isolated galaxy, with no nearby galaxies within 1 Mpc in projected distance, implying no significant gravitational influence from other galaxies for at least the past 10⁹ yr (I. Marquez & M. Moles 1993). Therefore, we suggest that NGC 6951, with a nuclear ring, could be an ideal candidate for a slow, long, and strong bar evolved through interactions with a dark matter halo, as predicted in simulation studies (J. A. Sellwood 1980; V. P. Debattista & J. A. Sellwood 2000; E. Athanassoula 2002, 2003, 2014; W.-Y. Seo et al. 2019; D. Jang & W.-T. Kim 2023, 2024). This slow, long, and strong bar may significantly contribute to the formation of the nuclear structure, including its nuclear disk and nuclear ring.