Circumnuclear Rings and Lindblad Resonances in Spiral Galaxies

Once the spectra were reduced, we proceeded to make extractions using one pixel wide apertures along the slit, covering the entire spatial range with Hα emission. Figure 1 shows one of these extractions, as an example, for the galaxy NGC 1512. This spectrum depicts the more conspicuous emission lines at this wavelength range, i.e., Hαλ6563, [N ii]λλ6548,6584, and [S ii]λλ6713,6731.

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Using the IRAF task splot, we fitted gaussians to each emission line to determine its parameters: intensity, FWHM, flux, and centroid. In the present work we will focus on the kinematical study derived from Hα. Once radial velocities of each extracted spectrum have been determined, we constructed the radial velocity curves for each galaxy. On Table 2 we summarized the main properties of the curves. Images of the galaxies, the location of the slit, and the radial velocity curves are shown in Figure 2. The adopted systemic radial velocity (V_syst) for each galaxy are 2621 km s⁻¹ for IC 1438, 2330 km s⁻¹ for IC 4214, 924 km s⁻¹ for NGC 1512, 2271 km s⁻¹ for NGC 2935, and 3160 km s⁻¹ for NGC 6753. These velocities are the ones that better symmetrize the radial velocity curves and they are in agreement with values found in the literature. For IC 4214 and NGC 2935, V_syst does not coincide with V_obs at the continuum peak, so an offset has to be applied to symmetrize the radial velocity curves. This offset, nevertheless, is always a fraction of the resolution element (∼1'') and we do not have enough spatial information to make conjectures about the possible off-centered nucleus, although we do not discard this possibility. The uncertainties in the radial velocities were calculated using the formula presented in Keel (1996).

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All radial velocity curves are relatively smooth, with the exception of NGC 2935, which deviates from circular rotation around the galaxy center from ∼1 to 3 kpc on the red side of the curve.

In order to determine the rotation curves of the galaxies, we calculated the inclination angle of each object. Through isophotal fitting to digitized sky survey (DSS)⁸ R-band images, we measured the major and minor axis of each galaxy and we calculated the inclination angle considering that cos $i={ \mathcal B }/{ \mathcal A }$, where ${ \mathcal B }$ and ${ \mathcal A }$ are the minor and major axis, respectively, of the outer isophote. We assumed the photometric position angle (P.A._phot) as the angle measured from north to east of the outer isophote major axis. Once the inclination angle is obtained, and assuming that PA_phot coincides with the kinematic P.A. (P.A._kin), the circular rotation velocity can be calculated. After a small correction due to the difference between PA_obs (position angle of the observation) and PA_kin, we are able to determine the circular velocity by

$$\begin{eqnarray}&&{V}_{c}=\displaystyle \frac{{V}_{\mathrm{obs}}-{V}_{\mathrm{syst}}}{\sin (i)},\end{eqnarray} \tag{ 1 }$$

where V_c is the circular velocity, V_obs is the observed radial velocity, and V_syst is the systemic radial velocity. While, in general, PA_kin is defined as the mean value of the radial evolution of the kinematic PA, our adopted PA_phot ∼ PA_kin was consistent with the values found in the literature for our objects.

We fitted the rotation curves of the galaxies considering two or three mass components, depending on the case, corresponding to axisymmetric Miyamoto–Nagai gravitational potentials (Miyamoto & Nagai 1975)

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

where Φ (R, z) is the Miyamoto–Nagai potential at (R, z) position, M is the total mass, and a and b are shape parameters. The ratio b/a ∼ 0.4 corresponds to a flattened disk distribution, b/a = 1 to an ellipsoidal distribution, and b/a ∼ 5 to a spherical distribution (e.g., Binney & Tremaine 1987).

The galaxies IC 4214, NGC 1512, NGC 2935, and NGC 6753 were fitted considering the presence of only two galactic subsystems (a central ellipsoidal component and a flattened disk) without the need for the external spherical component (halo). On the other hand, the IC 1438 rotation curve was fitted considering the presence of three galactic subsystems: a central ellipsoidal component, a flattened disk, and a massive halo.

Table 3 lists the obtained parameters that optimize the χ² minimization fit, such as the scale parameters and the total mass.

Considering the potential, the circular velocity can be inferred as ${V}_{c}=\sqrt{-R(\partial {\rm{\Phi }}/\partial R)}$ (see for example Binney & Tremaine 1987). Once the circular velocity has been obtained, the angular velocity can be easily calculated as Ω = V_c/R and the epicyclic frequency can be inferred (e.g., Elmegreen 1998) as

$$\begin{eqnarray}&&{\kappa }^{2}=4{{\rm{\Omega }}}^{2}[1+\displaystyle \frac{1}{2}(\displaystyle \frac{R}{{\rm{\Omega }}}\displaystyle \frac{d{\rm{\Omega }}}{{dR}})].\end{eqnarray} \tag{ 3 }$$

The location of the ILR and outer Lindblad resonance (OLR) can be calculated taking into account that Ω − Ω_p = −κ/2 at OLR and Ω − Ω_p = κ/2 at ILR (e.g., Binney & Tremaine 1987; Elmegreen 1998), where Ω_p is the pattern speed. In addition, the location of the corotation resonance (CR) can be inferred when Ω = Ω_p.

In order to construct the Linblad Ω − κ/2 and Ω + κ/2 curves for the galaxies, the first thing we did was determine the position of the CR. There is more than one way to calculate the radius of this resonance. For example, in order to do this Puerari & Dottori (1997) presented a method based on the Fourier analysis of azimuthal profiles (Aguerri et al. 1998; Díaz et al. 2003). On the other hand, Prendergast (1983) suggested that the dust lanes indicate the location of the shock regions in the spiral arms. This interpretation was first quantified by Roberts et al. (1979). Inside the corotation radius, dust lanes are located in the trailing side of the bar or the spiral arms perturbation pattern. In these locations the gas in circular orbits in the disk encounters the perturbation pattern, which is moving at lower angular velocity. Outside the corotation radius it is expected that the dust lanes are in the leading side of the perturbation pattern. The radii in which the dust lanes change from the trailing to the leading side of the perturbation pattern, indicate the approximate location of the CR. Taking into account all these considerations, we determined the CR radius by measuring directly from DSS images.

Considering the measured length of the bar (R_bar) and the corotation radius (R_CR), the dimensionless parameter ${ \mathcal R }$, defined as ${ \mathcal R }$ = R_CR/R_bar, is usually useful to characterize the bars as fast or slow. For the galaxies in our sample we obtained values for ${ \mathcal R }$ in the range 1.1–1.6, in agreement with previous results (e.g., Debattista & Sellwood 2000; Rautiainen et al. 2008; Aguerri et al. 2015; Guo et al. 2019).

The rotation curves corresponding to the fitted axial symmetric model are illustrated in the left panels of Figure 3, as well as the angular velocity curves and the Lindblad Ω − κ/2 and Ω + κ/2 curves (right panel) of the studied galaxies. Note that in all of the left panels of this figure, V_circ is 0 near the center. This is mostly likely due to limited resolution, taking into account that if there is a central mass concentration (e.g., a black hole), the rotation velocity should rise toward the center. The net effect of the finite spatial resolution is to average plus and minus values in both sides of nucleus along the major axis (Sofue 2013). The spectra have average spatial resolutions of 25, with a 102 spatial sampling. The innermost resolved rotation velocities are of the order of tens of km s⁻¹ at radii of 100–300 pc, without a change of slope toward the galactic core. This means that the gravitational sphere of influence of the supermassive black hole is not resolved in the rotation curve for the average distance of the sample.

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Table 4 lists where the different resonances of the studied galaxies are located (in units of kiloparsec), the pattern speed values, Ω_p (in units of km s⁻¹ kpc⁻¹), the radial length of the bar (in units of arcsec and kiloparsec), and the dimensionless parameter ${ \mathcal R }$.

Table 4.  Angular Velocity Pattern and Resonances

Galaxy	Ωp	ILR	CR	OLR	rbar	Rbar	${ \mathcal R }$
	(km s−1 kpc−1)	(kpc)	(kpc)	(kpc)	(arcsec)	(kpc)
1	2	3	4	5	6	7	8
IC 1438	27 ± 2	iILR = 0.13–oILR = 1.8 ± 0.2	5.5 ± 0.3	⋯	20.6 ± 0.9	3.5 ± 0.1	1.6 ± 0.1
IC 4214	38 ± 2	iILR = 0.21–oILR = 2.1 ± 0.1	6.6 ± 0.3	⋯	28.6 ± 0.9	4.1 ± 0.1	1.6 ± 0.1
NGC 1512	47 ± 1	iILR = 0.26–oILR = 0.87 ± 0.05	3.2 ± 0.1	⋯	60 ± 4	3.0 ± 0.2	1.1 ± 0.1
NGC 2935	26 ± 2	1.2 ± 0.2	5.2 ± 0.4	8.0 ± 0.4	24.3 ± 0.7	3.3 ± 0.1	1.5 ± 0.2
NGC 6753	35 ± 3	iILR = 0.72–oILR = 2.9 ± 0.2	9.4 ± 0.8	14.0 ± 0.8	⋯	⋯	⋯

Note. Column 1: galaxy name; column 2: pattern angular velocity; column 3: position of the ILR; column 4: location of the CR; column 5: position of the OLR; column 6: radius of the bar in arcsec; column 7: radius of the bar in kiloparsec; column 8: the dimensionless parameter defined as ${ \mathcal R }$ = R_CR/R_bar.

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To determine the CNR radius, we used the 2D spectra and generated the Hα spatial emission radial profiles (Figure 4). As can be seen in Figure 4, we perform an extraction by joining the emission peaks. This direction is oblique to both the spatial and spectral directions, so it is then necessary to project the profile over the spatial direction to have the distance in arcsec. In this way we are using the dispersive capacity of the diffraction grating to separate the emission and to be able to resolve and measure the rings radii. Unfortunately, we only have the information in the direction of the slit, so the P.A. of the ring is unknown. Although it is possible to measure the P.A. of the CNR from images (e.g., Pogge 1989; Buta & Crocker 1993; Comerón et al. 2010), the discrepancy in both P.A. and radius between different authors shows that it is not a simple task. In addition, in the case of spectra it is easier to correctly isolate the Hα emission to define the ring. Another question that requires a proper definition would be which point of the CNR range is considered its radius. For the present work we decided to measure the distance between the emission peaks on both sides. The nucleus of the galaxy is defined as the continuum emission peak in the Hα spectral region. In this way we avoid uncertainties due to diffuse emission in the internal or external region of the ring.

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The CNR of the five galaxies studied in this paper have been previously measured in other works. As we will see below (Section 4), these measurements vary from one work to another. Most of these previous studies use images in different bands to determine the CNR radius, obtaining distinct values. For completeness and looking for consistency in the measurement criteria, we have performed our own determinations of the CNR radio through the Hα profiles (Figure 4). It should be noted that we have information only at the direction of the observed P.A. and this may differ from the actual P.A. of the CNR. In the comparison with the values from the literature we carried out in Section 4, these differences are considered.

IC 1438. According to Buta et al. (2009), it is a weakly barred galaxy with a faint outer ring and presents enhanced star formation in nuclear and inner rings. Through isophotal fitting to DSS R-band images, we measured an inclination angle of 32°, in agreement with previous results (e.g., Böker et al. 2008). The parameters of the fitted Miyamoto–Nagai components imply the presence of three galactic subsystems such as an intermediate bulge−disk, a disk, and an external halo, with a total mass of 2.4 × 10¹¹ M_⊙. We measured a bar radial length of 206 ± 09, equivalent to (3.5 ± 0.1) kpc, in agreement with the value obtained by Comerón et al. (2010). We assumed that the CR is located at (5.5 ± 0.3) kpc from the center. Furthermore, this galaxy shows an angular velocity pattern of Ω_p = (27 ± 2) km s⁻¹ kpc⁻¹. Related to this, the dimensionless parameter ${ \mathcal R }$ results 1.6 ± 0.1, consistent with a slow bar (e.g., Debattista & Sellwood 2000; Guo et al. 2019). According to the Hα profile (Figure 4), the CNR radius is 245, equivalent to 0.41 kpc. However, other values are found in the literature. For example, Böker et al. (2008) obtained a CNR radius of 0.537 kpc with a P.A._CNR of 81°. In addition, according to Buta & Crocker (1993) and Comerón et al. (2010), the CNR semimajor axis is 0.650 kpc and 0.680 kpc, respectively. All these values show a dispersion of about 25% and it may be due to the different P.A.s considered. Assuming even the most extreme possible values, it can be said that the CNR is located between the iILR (0.13 kpc) and the oILR (1.9 kpc). Our data does not allow us to have information about the location of the OLR, but it could be located at around 10 kpc from the center, considering the end of the spiral arms. On the other hand, our slit P.A. is 127°, in agreement with the NED reported P.A._phot. However, Böker et al. (2008) consider a P.A. of 145° for this galaxy. This difference of 18° in P.A. would produce a negligible difference in the Ω_p value, and, as a consequence, in the location of the iILR, and a difference of 0.05 kpc in the location of the oILR.

IC 4214. It is a weakly barred galaxy (Moss et al. 1999) and according to the position of the galaxy in diagnostic diagrams based on line flux ratios, this object presents low-ionization nuclear emission line region (LINER) activity (Saraiva et al. 2001). These authors also report minimal star formation in the galaxy nucleus. Through isophotal fitting from DSS R-band images, we obtained an inclination angle of 54°, which is consistent with the value published by NED. The parameters of the fitted Miyamoto–Nagai components imply the presence of two galactic subsystems and a total mass of 1.6 × 10¹¹ M_⊙. We measured a radial length bar of R_bar = 286 ± 09, equivalent to (4.1 ± 0.1) kpc, which is in agreement with the value obtained by Comerón et al. (2010). We assumed that the CR is located at the end of the bar, at (6.6 ± 0.3) kpc from the center of the galaxy, with a dimensionless parameter of ${ \mathcal R }$ = 1.6 ± 0.1. Furthermore, this object presents an angular velocity pattern of Ω_p = (38 ± 2) km s⁻¹ kpc⁻¹. It can be seen in Figure 4 that the CNR is located about 0.6 kpc (38) from the nucleus. According to Buta & Crocker (1993), the major axis of the ring is 0146, equivalent to a semimajor axis of 0.632 kpc (438). Comerón et al. (2010) report a ring semimajor axis of 76, equivalent to 1.15 kpc. Despite the dispersion of reported values, and considering even the the extreme measurements, the CNR is located between the iILR (0.21 kpc) and the oILR (2.04 kpc). As in the previous case, we do not have any information about the location of the OLR. According to our measurements, the spiral arms seem to end at about 537 (∼7.75 kpc) from the center of the galaxy.

NGC 1512. Through isophotal fitting from DSS R-band images, our measurements of the major and minor axis of the galaxy give an inclination angle of 54°, in agreement with other works (see NED). The rotation curve fitting considering two components of the Miyamoto–Nagai potential gives a total mass of 2.7 × 10¹¹ M_⊙. According to our measurements, the bar has a total length of 2', equivalent to a radial length of 60'' ± 4'' and (3.0 ± 0.2) kpc. The CR was assumed to be at the end of the gap between the bar and the spiral arms, as mentioned before, at (3.2 ± 0.1) kpc from the galactic center. The system angular velocity pattern is Ω_p = (47 ± 1) km s⁻¹ kpc⁻¹. Related to this, the parameter ${ \mathcal R }$ is 1.1 ± 0.1, consistent with a fast bar (Debattista & Sellwood 2000; Rautiainen et al. 2008; Guo et al. 2019). According to Ma et al. (2017), this galaxy has a nuclear ring with a total stellar mass of ∼10⁷ M_⊙ and a young average age of ∼40 Myr. Figure 4 shows that the CNR has a radius of 85, equivalent to 0.42 kpc. According to Buta & Crocker (1993), the CNR has a major axis of 0280, equivalent to a semimajor axis of 840 or 0.420 kpc. In addition, Comerón et al. (2010) report a CNR with a radius of 0.480 kpc. These values present a much lower dispersion than the previous ones and indicate that the CNR is also located between the iILR (at 0.26 kpc) and the oILR (at 0.90 kpc). The extension of the rotation curve does not allow us to estimate the location of the OLR. It is also very difficult to determine the end of the spiral arms, since from a radius of 3', this galaxy presents very open arms, probably due to interactions, being possible to determine from DSS images an arm 85 from the center. This galaxy was also observed by the Hubble Space Telescope several times in the past. We used UV archive images in the F220W filter⁹ to measure the CNR radius and to compare with our measurements. The ring measurement in the F220W filter images gives R_CNR = 77 in acceptable agreement with our measurements, taking into account the spatial resolution and possible physical spatial differences between different bands.

NGC 2935. This galaxy exhibits some lobesideness in its morphology, and the radial velocity curve is asymmetric (see Figure 2). Florido et al. (2012) observe nonresolved emission and they found knots of gas at large radii, corresponding to H ii regions. According to our isophotal fitting from the DSS R-band image, this object has an inclination angle of 45°, consistent with previous results (see NED). The rotation curve was fitted considering the southeast part of the radial velocity curve, mainly because of the irregularity that the northwest part presents. The radial velocities of descending values in the range of 1–5 kpc (see Figure 2) are consistent with the detection of a strong incoming flux along the northwestern semimajor axis of the bar. Although the inflow dominates the inner regions on the northwestern side, there is also gas at higher radii with circular velocities with very similar values to those on the southeastern side. Considering that, the fit of only the southeastern side is perfectly representative of the circular velocities of the galaxy.

The two Miyamoto–Nagai fitted components under this assumption give a total mass of 4.3 × 10¹⁰ M_⊙. This object presents a bar with a radial length of R_bar = 243 ± 07, equivalent to (3.3 ± 0.1) kpc and we determined that the CR lies at (5.2 ± 0.4) kpc from the galactic nucleus. This yields an ${ \mathcal R }=1.5$ ± 0.2, indicating that we are in the presence of a slow bar (e.g., Debattista & Sellwood 2000; Rautiainen et al. 2008; Guo et al. 2019). The calculated Ω_p is (26 ± 2) km s⁻¹ kpc⁻¹ and according to the Hα profile (Figure 4), the CNR has a radius of 0.52 kpc (377), in agreement with Agüero et al. (2016), who report a radius of 0.45 kpc (326). Also, according to (Comerón et al. 2010), the CNR has a semimajor axis of 36, equivalent to ∼0.50 kpc, and Buta & Crocker (1993) measured a CNR diameter of 0106, equivalent to a radius of 318 and 0.44 kpc. The CNR is confined to a region between the center of the galaxy and the ILR, which is located at 1.2 kpc (see Figure 3). According to our estimation, the OLR is positioned at ∼8.3 kpc. The arms, from DSS images, end at 165 (∼13.56 kpc), away from the OLR. This could be probably due to the high asymmetry commented above that disturbed the galaxy disk. This object, however, presents a particular situation, taking into account what could generate a small variation in Ω_p. On the one hand, there would be no ILR if the pattern speed were increased (see Figure 3). On the other hand, if Ω_p were decreased, there would be two ILRs. In this case, the CNR would be located inside the iILR.

NGC 6753. It is a grand-design spiral galaxy that shows a very pronounced spiral structure (Windhorst et al. 2002). From DSS R-band images, our estimation of the major and minor axis gives an inclination angle of 40°, which is in agreement with previous results (see NED). The two Miyamoto–Nagai fitted components give a total mass of 4.2 × 10¹¹ M_⊙. We visually inspect 2MASS images of this galaxy and we do not see any bar, in agreement with Comerón et al. (2010). We estimated the OLR radius by assuming that the spiral arms end before the OLR. With this assumption the OLR is at (14.0 ± 0.8) kpc from the galactic nucleus. This way, the calculated Ω_p in this object is (35 ± 3) km s⁻¹ kpc⁻¹. According to our results, the corotation resonance is at (9.4 ± 0.8) kpc. This value is consistent with an obscured region in the vicinity where the spiral arms arise. According to the Hα profile (Figure 4), the CNR radius is 850, equivalent to ∼1.89 kpc, in agreement with previous results (Comerón et al. 2010; Agüero et al. 2016). The higher value found in the literature corresponds to a CNR with a diameter of 035, equivalent to a radius of 105 or 2.16 kpc (Buta & Crocker 1993). Even considering the extreme values, the CNR is located between the iILR (at 0.7 kpc) and the oILR, which is at (2.9 ± 0.2) kpc.