On the Connection between Spiral Arm Pitch Angle and Galaxy Properties

Figure 7 plots pitch angles versus the central velocity dispersion (σ_c). The correlation is strong, with Pearson correlation coefficient ρ = −0.64. For galaxies with σ_c ≳ 100 km s⁻¹, pitch angle decreases with σ_c with small scatter, reaching a mean value of $\varphi =9\buildrel{\circ}\over{.} 2\pm 2\buildrel{\circ}\over{.} 2$ at σ_c = 193 ± 11 km s⁻¹, the highest velocity dispersion covered by our sample. No galaxies with high σ_c show open spiral arms. By contrast, galaxies with σ_c ≲ 100 km s⁻¹ host arms with a wide spread in pitch angles, from values as high as φ ≈ 30° to as low as φ ≈ 15°. The pitch angle seems to remain at a roughly constant mean value of φ ≈ 23° for σ_c ≲ 100 km s⁻¹.

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Seigar et al. (2008) reported a strong inverse correlation between spiral arm pitch angle central stellar velocity dispersion for a sample of 27 galaxies. Here, we independently reexamine their results. Instead of using the same images used in Seigar et al. (2008), we use, whenever possible, images of higher quality: 7 images from CGS (Ho et al. 2011), 16 images from SDSS, and 3 images collected from the NASA/IPAC Extragalactic Database (NED).³ The data point for the Milky Way is not included, as its pitch angle may be unreliable. We uniformly analyze all the objects, using, for consistency, the 2D Fourier transformation method. Seigar et al. (2008) did not provide the projection parameters (e and PA) for the galaxies, and so measure them following the procedure described in Section 3.1. We successfully measure pitch angles for 21 of the 26 galaxies.

Direct comparison of our new pitch angle measurements with those published by Seigar et al. (2008) reveal that a significant fraction of them (6/21) were severely and systematically overestimated (by more than 8°). Figure 8 illustrates the four cases with the most serious discrepancy, whose pitch angles were overestimated by more than 10°. The panels in the left column present the unsharp-masked images overplotted with the synthetic arm with the pitch angle derived from this work (red solid line) and from the work of Seigar et al. (2008; orange dashed–dotted line). The panels in the right column plot the 2D Fourier spectra, with an arrow indicating the peak chosen to derive the pitch angle. It is obvious that the synthetic arms created using our pitch angles trace the spiral arms very well, whereas those that adopt the pitch angles from Seigar et al. (2008) do not. In the extreme case of NGC 3938 (top row), Seigar et al. (2008) quoted a pitch angle of $43\buildrel{\circ}\over{.} 4$, whereas we find $16\buildrel{\circ}\over{.} 7$. These four galaxies have clear spiral arms and a distinctly dominant Fourier mode. Their pitch angles can be measured rather straightforwardly and unambiguously. The large discrepancies with the published values cannot arise from errors in the measurement technique.

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With our updated pitch angle measurements in hand for the 21 galaxies reanalyzed by us, we redraw in Figure 9 the relation between φ and σ_c as originally published by Seigar et al. (2008), using σ_c given by those authors (their Table 1). The relationship between φ and σ_c is considerably less tight than claimed by Seigar et al. (2008), and instead closely resembles our results based on a much larger sample (Figure 7).

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Pitch angle correlates strongly with σ_c for σ ≳ 100 km s⁻¹. As σ_c is measured within the central $3^{\prime\prime}$ (∼1 kpc for our median sample distance of 68 Mpc), it must connect with certain global properties to influence galactic-scale spiral structure. It is well known that the luminosity of elliptical galaxies scales with stellar velocity dispersion following $L\propto {\sigma }^{4}$ (Faber & Jackson 1976), but the Faber–Jackson relation of bulges is not well-determined. It varies systematically with Hubble type (Whitmore & Kirshner 1981; Kormendy & Illingworth 1983) and between classical bulges and pseudobulges (Kormendy & Kennicutt 2004). The correlation between pitch angle and σ_c is not entirely consistent with that between pitch angle and bulge mass. The $\varphi \mbox{--}{M}_{* }^{\mathrm{bul}}$ relation clearly has larger scatter than the $\varphi \mbox{--}{\sigma }_{c}$ relation, especially at the high-mass end (Figure 6(c)). For σ_c ≲ 100 km s⁻¹, the mean value of pitch angle remains essentially constant, suggesting that in this regime another parameter determines the pitch angle. In order to study the effect of galaxy mass on pitch angle for a given central velocity dispersion, for each of the five bins of σ_c, we again split the data into two subsets according to the mean value of $\mathrm{log}({M}_{* }^{\mathrm{gal}})$, and then separately examine the behavior of each. As Figure 7 shows, when σ_c ≳ 100 km s⁻¹ stellar mass does not help to reduce the scatter in pitch angle, but in the low-σ_c regime more massive galaxies tend to have smaller pitch angles, consistent with the empirical relation found in Section 4.2. Thus, galaxy mass can account for part of the scatter in pitch angle.

The $\varphi \mbox{--}{M}_{* }^{\mathrm{gal}}$ relation and the $\varphi \mbox{--}{\sigma }_{c}$ relation are actually projections of a stronger three-parameter relation involving φ, ${M}_{* }^{\mathrm{gal}}$, and σ_c. Figure 10 plots $\mathrm{log}({M}_{* }^{\mathrm{gal}})$ against σ_c. To separate the data equally into five fan-shaped bins, numerically denoted 1 to 5, we scale the data by dividing them by (σ_c,max − ${\sigma }_{c,\min }$) and ($\mathrm{log}{M}_{* ,\max }^{\mathrm{gal}}$ − $\mathrm{log}{M}_{* ,\min }^{\mathrm{gal}}$) and then generate six radiant dotted lines in the scaled parameter space, with equal separation in orientation angle of 18°, originating from a reference point with maximum σ_c and minimum ${M}_{* }^{\mathrm{gal}}$. The counterparts of the dotted lines in the original (σ_c, $\mathrm{log}{M}_{* }^{\mathrm{gal}}$) space are shown in Figure 10. We calculate the mean and standard deviation of φ in each bin. The inset panel in Figure 10 shows the tendency of pitch angle progressively increasing from the fifth bin (φ = 93 ± 20) to the first bin (φ = 260 ± 60). Our results suggest that when σ_c or ${M}_{* }^{\mathrm{gal}}$ is high, φ is mainly determined by σ_c, whereas when σ_c or ${M}_{* }^{\mathrm{gal}}$ is low, φ is mainly determined by ${M}_{* }^{\mathrm{gal}}$. Figure 10 can explain the behavior of the $\varphi -{M}_{* }^{\mathrm{gal}}$ and $\varphi \mbox{--}{\sigma }_{c}$ relations. In the high-mass regime, the scatter in φ is large at a given ${M}_{* }^{\mathrm{gal}}$ (Figure 5(b)) because σ_c is high (∼130–210 km s⁻¹) and φ is mainly dictated by σ_c. In the low-σ_c regime, the mean value of φ remains nearly constant with σ_c (Figure 7) because φ is mainly determined by ${M}_{* }^{\mathrm{gal}}$ instead of σ_c. Therefore, our results suggest that two primary parameters—central velocity dispersion and galaxy mass—synergistically determine spiral arm pitch angle.

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The observed rotation curves can be grouped roughly qualitatively into several types according to their behavior in the central region (e.g., Keel 1993; Sofue et al. 1999). Kalinova et al. (2017) applied a principal component analysis to quantitatively classify the rotation curves of the CALIFA galaxies. The coefficient of the first eigenvector PC₁ describes the morphology of the rotation curve of the central region. Galaxies with high PC₁ ($\gt 0$) have a high-amplitude, centrally peaked rotation curve that attains a sharp maximum near the center, followed by a dip and then a broad maximum of the disk component; those with low PC₁ ($\lt 0$) have a low-amplitude, slow-rising rotation curve that rises gradually from the center in a rigid-body fashion. As the coefficient PC₁ is a measure of both the shape and amplitude of the rotation curve, PC₁ simultaneously reflects the mass of the central component and the mass of the disk, especially for bulgeless galaxies. As expected from the $\varphi \mbox{--}{\sigma }_{c}\mbox{--}{M}_{* }^{\mathrm{gal}}$ relation, φ shows a strong inverse correlation with PC₁ (Figure 11; ρ = −0.66).

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In addition to baryonic mass, the shape and the amplitude of the rotation curve also reflect the mass distribution of dark matter in galaxies. Hence, the strong correlation between pitch angle and morphology of the central rotation curve also implies that dark matter content might help to shape spiral arms.

The strong correlation between pitch angle and shear rate Γ originally suggested by Seigar et al. (2005, 2006) has not been substantiated by the recent study of Yu et al. (2018), who show that ∼one-third of the pitch angle measurements of Seigar et al. (2006) have been severely overestimated, as a consequence of which the correlation between φ and Γ is much weaker than previously claimed.⁴ We reassess the relationship between φ and Γ in Figure 12. Pitch angle does have a tendency to decline with increasing Γ (ρ = −0.49), although the scatter is large. Two physical mechanisms may explain this behavior.

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An association between φ and Γ was recently explored using numerical simulations by Grand et al. (2013) and Michikoshi & Kokubo (2014). N-body simulations of isolated stellar disks produce transient but recurrent local spiral arms (Carlberg & Freedman 1985; Bottema 2003; Fujii et al. 2011; Sellwood 2011; Grand et al. 2012a, 2012b; Baba et al. 2013; D'Onghia et al. 2013). D'Onghia et al. (2013) argue that spiral arms originate from the dynamical response of a self-gravitating shearing disk to local density perturbations. Differential motion tends to stretch and break up the spiral arms locally. In regions where self-gravity dominates, the disk is locally overdense and generates arm segments, which reconnect and make up the spiral arms. The simulations of Grand et al. (2013) and Michikoshi & Kokubo (2014), marked by green crosses and blue triangles in Figure 12, share the same general tendency for φ to decline with increasing Γ. However, a systematic offset can be clearly seen. For a given shear rate, the predicted pitch angles are larger than the observed values, typically by Δφ ≳ 8°, especially at the low-Γ end. This implies that, in terms of arm morphology, N-body simulations cannot yet generate realistic spiral arms even if the resolution of the simulations is very high.

Swing amplification (Julian & Toomre 1966; Goldreich & Tremaine 1978; Toomre 1981) is a mechanism of amplifying spiral arms when a leading spiral pattern rotates to a trailing one due to the shear in a differentially rotating disk. Swing amplification theory is reasonably consistent N-body simulations in terms of the predicted number of arms, which is approximately inversely proportional to the mass fraction of the disk component (Carlberg & Freedman 1985; Bottema 2003; Fujii et al. 2011; D'Onghia et al. 2013), and in terms of the relationship between the shear rate of the rotation curve and the pitch angle of the simulated spirals (Grand et al. 2013; Michikoshi & Kokubo 2014).

Michikoshi & Kokubo (2014) derived a theoretical relation between φ and Γ in the context of swing amplification theory. We give a brief summary here. We consider a material arm that swings from leading to trailing due to differential motion. The pitch angle evolves as (e.g., Binney & Tremaine 2008)

$$\begin{eqnarray}&&\tan \varphi =\displaystyle \frac{1}{2{At}},\end{eqnarray} \tag{ 3 }$$

where A is the first Oort constant and t = 0 means that the arm extends radially outward across the disk (φ = 90°). In the simulations of Michikoshi & Kokubo (2014), Toomre's Q parameter increases rapidly and exceeds 1.5 for Γ ≳ 0.5. In the linear approximation of swing amplification theory, if $Q\gt 1.5$, the maximum amplification is reached at ${t}_{\max }\backsimeq 3.5/\kappa$, where κ is the epicyclic frequency. They interpret the most amplified short wave as the spiral structures observed in their simulation. Combined with ${\rm{\Gamma }}=\tfrac{2A}{{\rm{\Omega }}}$, the relation between pitch angle and shear rate becomes (their Equation (15))

$$\begin{eqnarray}&&\tan \varphi =\displaystyle \frac{2}{7}\displaystyle \frac{\sqrt{4-2{\rm{\Gamma }}}}{{\rm{\Gamma }}}.\end{eqnarray} \tag{ 4 }$$

This is indicated by the solid line in Figure 12. The prediction from swing amplification theory is quantitatively consistent with the results from N-body simulations. Note that the predicted pitch angle for a given shear rate is still higher than our measurements (by ∼8°). This is because Equation (4) assumes $Q\gt 1.5$, which is indeed the case for the simulated spiral galaxies of Michikoshi & Kokubo (2014). However, as suggested by Bertin et al. (1989a), because of self-regulation of gas content, Toomre's Q parameter in the outer regions of galactic disks should be close to unity. On the other hand, Q may be substantially larger in the central regions of galaxies because of the presence of a bulge. Considering both the effects of the disk and the bulge, if Q is constrained to ∼1.2, we estimate from Figure 7 of Michikoshi & Kokubo (2014) that ${t}_{\max }\approx 5.5/\kappa$. For $Q\approx 1.2$, we obtain, from swing amplification theory,

$$\begin{eqnarray}&&\tan \varphi =\displaystyle \frac{2}{11}\displaystyle \frac{\sqrt{4-2{\rm{\Gamma }}}}{{\rm{\Gamma }}}.\end{eqnarray} \tag{ 5 }$$

This revised relation, presented as the dashed curve in Figure 12, is now consistent with our observational results. Our results have two important implications. First, if swing amplification theory is the correct framework to explain spiral structures in galaxies, then Toomre's Q should be roughly 1.2. And second, to generate more realistic spiral arms, simulations may need an effective cooling mechanism for the disk, perhaps by including the effects of gas, to lower Toomre's Q.

In the framework in which spiral structure constitutes transient material arms, the shape of the arms should reflect the effects of differential rotation alone. Although we present some evidence supporting this picture, the existence of other stronger empirical relationships between pitch angle and σ_c or PC₁, which have statistically stronger correlation coefficients, probably rules out the transient material arms scenario.

A more compelling, alternative explanation for the relationship between pitch angle and shear rate comes from density wave theory. As the shape of the rotation curve depends on the distribution of mass, the shear rate reflects the mass concentration. Consequently, the inverse correlation between φ and Γ is qualitatively consistent with the expectations of density wave models (Lin & Shu 1964; Roberts et al. 1975; Bertin et al. 1989a, 1989b). The predicted relation between φ and Γ is consistent with our observed $\varphi -{C}_{28}$ relation (Figure 5(b)).