3 Bar torque and form family

Combining the sample in Table 1 with that in Table 1 of Buta & Block (2001), we have 75 galaxies for which $Q_{\rm b}$ is now available. Six of the galaxies in Table 1 are in common with the list in Buta & Block (2001). Except for NGC 4548, the values are in good agreement, with differences attributable to the quality of the images. We give preference in our analysis to the Table 1 values, because the INGRID images are superior in signal-to-noise to the images used by Buta & Block (2001). In the case of NGC 4548, a rebinning error caused Buta & Block (2001) to overestimate the bar strength; the Table 1 value is the actual bar torque in this galaxy.

Figure 4 shows $Q_{\rm b}$ versus the Hubble S and SB classifications, extracted from the "Revised Shapley-Ames Catalog'' (RSA; Sandage & Tammann 1981). Figure 5 is a similar plot, but for the de Vaucouleurs (1963) classifications. For a number of galaxies in Table 1, Martin (1995) lists an estimate of the deprojected visual bar axis ratio, $(b/a)_{\rm bar}$. Figure 6 shows a plot of our gravitational bar torque $Q_{\rm b}$ vs. $(b/a)_{\rm bar}$. Several points are noteworthy:

 

 

Table 1: Optical and near-infrared classifications. Column 1 lists the NGC number; Col. 2 the Hubble type extracted from the Revised Shapley Ames Catalogue (Sandage & Tammann 1981); Col. 3 the de Vaucouleurs (1963) form family; Col. 4 lists our dust penetrated (DP) classification at $K_{\rm s}$. The format used in Col. 4 is Hm (where m is the dominant Fourier harmonic), followed by the pitch angle class ($\alpha$, $\beta$ or $\gamma$), followed by the bar torque class.

Galaxy	RSA type	Form Family	DP Type
NGC 0210	Sb(rs)I	SAB	H2$\beta$1
NGC 0488	Sab(rs)I	SA	...1
NGC 0628	Sc(s)I	SA	H2$\beta$0
NGC 0864	Sbc(r)II-III	SAB	H2$\gamma$3
NGC 1042	Sc(rs)I-II	SAB	H2$\beta$3
NGC 1073	SBc(rs)II	SB	H2$\alpha$4
NGC 1169	SBa(r)I	SAB	H2$\gamma$3
NGC 1179	SBc(rs)I-II	SAB	H2$\alpha$3
NGC 1300	SBb(s)I.2	SB	H2$\alpha$5
NGC 2775	Sa(r)	SA	...0
NGC 2805	---	SAB	H2$\gamma$2
NGC 3184	Sc(r)II.2	SAB	H2$\beta$1
NGC 3344	SBbc(rs)I	SAB	H2$\alpha$1
NGC 3351	SBb(r)II	SB	...2
NGC 3368	Sab(s)II	SAB	...2
NGC 3486	Sc(r)I-II	SAB	H2$\gamma$0
NGC 3631	Sc(s)I-II	SA	H2$\gamma$0
NGC 3726	Sc(r)I-II	SAB	H2$\gamma$2
NGC 3810	Sc(s)II	SA	H2$\beta$1
NGC 4030	Sbc(r)I	SA	H2$\beta$1
NGC 4051	Sbc(s)II	SAB	H2$\gamma$2
NGC 4123	SBbc(rs)	SB	...4
NGC 4145	SBc(r)II	SAB	H2$\beta$2
NGC 4254	Sc(s)I.3	SA	H2$\gamma$1
NGC 4303	Sc(s)I.2	SAB	H2$\gamma$3
NGC 4314	SBa(rs) pec	SB	H2$\gamma$3
NGC 4321	Sc(s)I	SAB	H2$\beta$2
NGC 4450	Sab pec	SA	...2
NGC 4501	Sbc(s)II	SA	H2$\gamma$1
NGC 4535	SBc(s)I.3	SAB	H2$\beta$3
NGC 4548	SBb(rs)I-II	SB	H2$\gamma$2
NGC 4579	Sab(s)II	SAB	H2$\gamma$2
NGC 4618	SBbc(rs)II.2 pec	SB	H1$\gamma$2
NGC 4689	Sc(s)II.3	SA	H2$\beta$0
NGC 4725	Sb/SBb(r)II	SAB	...3
NGC 5247	Sc(s)I-II	SA	H2$\gamma$1
NGC 5248	Sbc(s)I-II	SAB	H2$\beta$0
NGC 5371	Sb(rs)I/SBb(rs)I	SAB	H1$\gamma$1
NGC 5850	SBb(sr)I-II	SB	...2
NGC 5921	SBbc(s)I-II	SB	H2$\gamma$4
NGC 5964	---	SB	H2$\gamma$5
NGC 6140	---	--	H1$\gamma$-
NGC 6384	Sb(r)I	SAB	H2$\beta$1
NGC 6946	Sc(s)II	SA	H2$\gamma$0
NGC 7741	SBc(s)II.2	SB	...5

Figure 4: Bar torque versus Hubble Classification as prescribed in the RSA for 64 galaxies, based on the combined sample from Table 1 and the similar table in Block & Buta (2001). Spirals classified as Sa, Sab, Sb, Sbc etc are all grouped into the unbarred "S'' bin; those of type SBa, SBab etc. into the barred SB bin.

Figure 5: Bar torque versus the de Vaucouleurs (1963) form family for 69 galaxies, again based on the combined sample. The plot excludes 6 galaxies whose form family is from other sources, as well as NGC 4618, classified by de Vaucouleurs as a magellanic barred spiral.

Figure 6: A comparison between gravitational bar torques $Q_{\rm b}$and deprojected bar axis ratio determined by Martin (1995). Highly elongated bars have low values of $b/a_{\rm bar}$, where a and b denote the bar major and minor axis respectively. An important point to note is that highly elongated "strong'' bars in the definition of Martin (1995) may have weak gravitational bar torques.

(i) Category S in Fig. 4 includes galaxies ranging from bar torque class 0 (e.g., NGC 628) to bar class 3 (e.g., NGC 1042). NGC 4321, a Hubble Sc prototype, has a bar class of 2. Likewise, NGC 4450 (Sab) is of bar class 2. NGC 4450 is illustrated in Panel 110 of Sandage & Bedke (1994), and there is a distinct visual impression of a bar. This is clearly evident in the near-infrared (see Fig. 7).

Figure 7: NGC 4450 is of de Vaucouleurs type SA. In the near- infrared, a bar of class 2 is identified, and our bar torque method finds the locations (indicated by four filled black squares) where the ratio of the tangential to the mean axisymmetric radial force reaches a maximum (in modulus), per quadrant. The hint of a bar is also indicated optically, in Panel 110 of "The Carnegie Atlas of Galaxies'' by Sandage & Bedke (1994).

Similarly, Hubble category SB in the RSA has a wide range of bar strengths. This category commences at a bar torque of class 1 (NGC 3344) and reaches bar class 5 (e.g., NGC 7741) in Table 1 and bar class 6 in Table 1 of Buta & Block (2001). In other words, the bar strengths of some RSA SB galaxies may be weaker than those found in RSA unbarred spirals such as NGC 1042 (Sc; near-infrared bar class 3). This is not due to the uncertainties in the $Q_{\rm b}$ method, but instead reflects the difficulties of making reliable bar strength judgments in the visual Hubble system. The work of Knapen et al. (2000) reaches this identical conclusion, using their independent definition of bar strength.

 

 

Table 2: Mean bar torque versus visual bar classifications for 64 RSA galaxies and 69 galaxies included in Appendix I of de Vaucouleurs (1963). Galaxies in Table 1 of this paper and in Table 1 of Buta & Block (2001) which do not have bar classifications from these sources are excluded from these means. NGC 4618, a low-luminosity magellanic spiral, is also excluded.

Classification	$<Q_{\rm b}>$ $\pm$ $\sigma$	N	range
RSA S	$0.11\pm0.08$	32	0.01-0.33
RSA SB	$0.28\pm0.13$	32	0.07-0.63
deV SA	$0.06\pm0.04$	14	0.01-0.14
deV SAB	$0.16\pm0.08$	32	0.02-0.33
deV SB	$0.33\pm0.13$	23	0.16-0.63

The gravitational influence of bars is thus poorly recognized by the Hubble classification scheme. Table 2 shows that, on average, RSA SB galaxies have relative bar torques only 2.5 times as strong as in RSA S galaxies, with a very large range in $Q_{\rm b}$ in each class. In this table, two galaxies of RSA type Sb/SBb have been included in the SB category.

(ii) Figure 5 and Table 2 show that the situation is better for de Vaucouleurs classifications. The mean value of $Q_{\rm b}$ changes smoothly with de Vaucouleurs family, and in fact verifies the continuity in bar strength embodied in de Vaucouleurs classifications. De Vaucouleurs SB galaxies have relative bar torques 5.5 times that of SA galaxies and twice that of SAB galaxies. However, the scatter is still very large in the SAB and SB categories. In Table 1, SAB galaxies encompass bar torque classes over the wide range 0 (e.g., NGC 6946) to 3 (e.g., NGC 4303), while SB galaxies encompass the range 2 (e.g., NGC 3351) to 5 (e.g., NGC 1300). NGC 7479 (Buta & Block 2001) is a type SB galaxy of bar class 6.

(iii) Figure 6 shows that $Q_{\rm b}$ correlates fairly well with Martin's (1995) $(b/a)_{\rm bar}$ parameter, confirming that bar ellipticity does provide a measure of bar strength. However, the scatter at a given bar axis ratio is still large. Buta & Block (2001) had noted that highly elongated bars (such as in M 83; an example of Martin's bar ellipticity class 7) may have weak torques. At $(b/a)_{\rm bar} = 0.5$, $Q_{\rm b}$ ranges from 0.1 to 0.5. Thus, apparently strong bars with significant ellipiticity (e.g., with elongations of $(b/a)_{\rm bar}\leq 0.5$) may be strong, weak, or intermediate as far as $Q_{\rm b}$ torque values are concerned. $Q_{\rm b}$ does not measure just the shape of an isolated bar; it also accounts for the disk in which the bar is embedded.

(iv) We suspect that some of the scatter seen in Figs. 4-6 could be due to dilution of the bar torque by a strong bulge. We might expect this because bars that are strong in terms of the m=2 Fourier component of the optical light distribution (Elmegreen & Elmegreen 1985) can have either small or large relative torques, depending on the relative mass of the bulge. If the bulge is weak, then even a weak bar can have a strong torque compared to the radial component of the force (e.g., NGC 1073). Bars that are long can have a strong torque because the end of the bar is far from the bulge (e.g., NGC 1300). This means that the simultaneous decreases in relative bulge strength and bar length with later Hubble type partially offset each other, giving a relative torque that can either go up or down, depending sensitively on the mass distribution.

However, when we replot Fig. 5 separating the points by Hubble type, we find that the galaxy-to-galaxy variation in the bar torque for each bar type is not entirely the result of a varying force dilution from the bulge. This is shown in Fig. 8, which includes the same galaxies as in Fig. 5 but with different symbols for early, intermediate, and late Hubble types. These subtypes reflect a variation in the relative strength of the bulge, with earlier types having stronger bulges in both barred and non-barred galaxies. Even within a subtype, some optically barred galaxies have smaller bar torques than some optically unbarred galaxies. The relative bar torque comes from a mixture of bar amplitude, radial profile and relative length, all combined with the bulge strength. These quantities vary in different ways along the Hubble subtype sequence, producing a wide range in relative bar torques.

Figure 8: Relative bar torque versus de Vaucouleurs bar type is shown with different symbols and a slight offset for the different Hubble subtypes. The wide range of relative bar torques for each bar type is present for all subtypes.

Variations in torque with bar type are not obvious from the morphology. If bars drive spirals, particularly in early Hubble types where the presence of a bar correlates well with grand design spiral structure (Elmegreen & Elmegreen 1989), they tend to do so only to the point of saturation, producing very strong arms after only a few revolutions. This is apparently true for both high and low bar torques, because even the low torques are enough to make strong spirals. Thus there is little sensitivity in spiral arm strength to the bar torque, aside from the known sensitivity of arm strength to the relative magnitude of the m=2 component of the infrared light (Elmegreen & Elmegreen 1985).

Several bars in Fig. 2 show a two-component morphology: a broad oval bulge or bar-like structure extending out to about half or two-thirds of the full bar length, and a thin spindle-like structure extending out further. NGC 4314 is an example; the thick component is outside the ILR in this case because there is a small ILR ring much closer to the center. These two bar components generally appear to be from two distinct populations of stars: a warm or hot population to make the thick bar, and a cool population to make the spindle. The two components could also have formed at different times, with the hot component being much older. In this case, it would be interesting to study these galaxies as possible examples where a relatively short bar formed first and dissolved by the instabilities discussed in Hasan et al. (1993), producing the oval we see today, and then another, larger bar formed afterwards out of a younger population of stars and gas, producing the spindle. We also note that some galaxies have the thick oval leading the thin spindle in the direction of rotation (e.g., NGC 1300), and other galaxies have it lagging (e.g., NGC 4123). This variation might indicate some dynamical interaction between the two bars, such as an oscillation about the equilibrium aligned configuration.