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9.2. Linear and Relative Diameters

The diameters Dr of inner rings relative to isophotal and metric diameters of the parent galaxies are discussed in detail by de Vaucouleurs & Buta (1980b). The ratios X = log D0/Dr and Y = log Ae/Dr , where D0 is the ``face-on" corrected (RC2 formulation) isophotal blue diameter and Ae is the effective (half power) blue diameter, were found to depend on both family and stage as follows:

X = 0.54 - 0.13F + 0.036T 1

Y = 0.01 - 0.13F + 0.058T 2

where F is a family index (-1 for SA, 0 for SAB, and +1 for SB galaxies) and T is the stage index on the RC2 numerical scale. The formulae are applicable only to spirals in the type range 0 leq T leq 8 (S0/a to Sdm; see Figure 25), and they indicate that inner rings and pseudorings are relatively larger in barred galaxies than in nonbarred galaxies, and in earlier types than in later types. Although relative inner ring size could thus be taken as an indicator of apparent ``bar strength'' at a given type, or of type at a given family, the trends shown in Figure 25 reflect a complex mix of possible resonance types for nonbarred or weakly barred inner rings, differences in the mass distributions between early and late-type galaxies, and possibly even in the nature of rings in nonbarred galaxies (sections 15 and 17).

Figure 25. Mean logarithmic diameter ratios X' = log (D0 / Dr) + 0.13F and Y' = log (Ae / Dr) + 0.13F versus stage T, from de Vaucouleurs & Buta (1980b). Open circles are based on only one galaxy.
Figure 25

Absolute linear diameters Dr of inner rings and pseudorings have been studied by Kormendy (1979a), Pedreros & Madore (1981), and Buta and de Vaucouleurs (1982). Kormendy focussed only on a sample of 121 bright SB galaxies, and found that the basic correlation for linear inner ring and lens diameters is with the absolute magnitude MoB, and hence mass, of the parent galaxy. The correlation between log Dr and MoB had a slope of -5.0 within the uncertainties, which he concluded to mean that the mean surface brightness (as if all of the light is concentrated within the rings) of the galaxies is independent of luminosity. The lower mean surface brightness of later-type ringed galaxies was attributed to a variation of mass-to-light ratio with type, which he suggested implied that the mean mass density outside the core region is constant for all morphological types. He concluded that the total mass of a galaxy uniquely determines the size of a bar and all of its associated components. Kormendy also found a correlation between the sizes of outer rings, pseudorings, and lenses and absolute magnitude, but did not believe that these could provide a size scale that determines MoB , because the rings contain so little light. He suggested that the sizes of outer rings are fixed by the size of the bar.

De Vaucouleurs (1956) first suggested that inner ring and pseudoring diameters in barred spirals might be useful as extragalactic distance indicators. He saw definite advantages to the use of such structures at the time, because rings are defined by ridges in the light distribution, not isophotes, and hence can be measured with better internal and external precision than isophotal galaxy diameters (de Vaucouleurs 1959b). Pedreros & Madore (1981) and Buta & de Vaucouleurs (1982) calibrated the rings as distance indicators using distance-independent indices of absolute magnitude such as the type and luminosity class. Both studies showed that inner rings become smaller with advancing stage along the Hubble sequence and with fainter luminosity class.

Using distances from the luminosity index, Lambdac = (T + Lc) / 10, where T is the numerical stage (RC2 and RC3 scale) and Lc is the inclination-corrected luminosity class (also numerically coded; de Vaucouleurs 1979), and the B-band Tully-Fisher relation (Bottinelli et al. 1980), Buta & de Vaucouleurs (1982) derived the following formula for spirals of type Sab and later:

log Dr (pc) = 3.61 + 0.15F - 0.10(T - 4) - 0.05(Lc - 3) 3

This shows that the largest inner rings and pseudorings are found in early-type barred spirals of high luminosity. For an SB(r)ab I galaxy, the ring diameter averages about 11.5 kpc. However, for the average ringed galaxy of type SAB(r)bc II, the ring diameter is only about 4 kpc. Pedreros & Madore (1981) also tried a formulation using both type and luminosity class as separate parameters (rather than as the combined luminosity index) and found weakly significant differences between SB galaxies on one hand and SA+SAB galaxies on the other. Their general formulation for a moderately unbiased sample (SA+SAB+SB inclusive) gave the following formula for the apparent ring diameter (in arcseconds) reduced to a radial velocity of 5000 km s-1:

R(5000) = -2.05L - 1.31T + 29.46 4

where L is the raw numerical luminosity class (not inclination-corrected). This study gave a greater dependence on luminosity class than the formulation of Buta and de Vaucouleurs, but the difference may reflect differences in the sample characteristics and calibrating distances.

Buta & de Vaucouleurs (1982) also considered the sizes of rings in S0 galaxies and derived a formulation based only on the Hubble type and family of the galaxy, since luminosity classes are not defined for types earlier than Sab (T=2). The absolute linear diameters of inner rings over the range -2 leq T leq 7 can be derived from this alternative formulation:

log Dr (pc) = 3.81 + 0.15F + 0.05(T - 2.5), -2 leq T leq 2 5a

log Dr (pc) = 3.81 + 0.15F - 0.12(T - 2.5), 3 leq T leq 7 5b

Figure 26 shows that the reduced diameter, log Dro = log Dr - 0.15F, of inner rings and pseudorings from these formulae achieves a maximum of 6.1 kpc at stage Sab. A very similar trend is seen in Figure 7 of Pedreros & Madore (1981).

Figure 26. Mean logarithmic reduced inner ring diameter versus Hubble type, from Buta & de Vaucouleurs (1982).
Figure 26

Buta & Crocker (1993) have presented information on the diameters of inner, outer, and nuclear rings in a limited subset of galaxies which have at least a nuclear ring or related feature. Logarithmic means gave average diameters of 1.1, 9.5, and 22.4 kpc for nuclear, inner, and outer rings and pseudorings, respectively, in approx 20 SB galaxies. These are based on distances derived from radial velocities and an assumed Hubble constant of 100 km s-1 Mpc-1. The diameters of rings in weakly barred and nonbarred galaxies were also considered, and found to show greater scatter.

Many galaxies possess both inner and outer rings simultaneously. The co-existence of these features in the same galaxy provides a more dynamically interesting diameter ratio than those in equations 1 and 2. It was found by de Vaucouleurs (1956), Athanassoula et al. (1982), Kormendy (1982a), and Buta (1984, 1986a, 1995) that the ratio dR / dr (rings and pseudorings together) has a relatively small dispersion with a mean or median near 2.0. The largest samples, from the Catalog of Southern Ringed Galaxies (Buta 1995), gave the distributions in Figure 27. Since the rings are noncircular on average (see section 9.3), the plot shows the distributions of ratios rho of geometric mean diameters rather than major axis diameters. For more than 1000 galaxies of all families, the median ratio of geometric mean ring diameters is 2.19, very similar to the outer ring to bar diameter derived by Kormendy (1979a). When the sample is divided by family, little dependence is found: 2.29 for SA, 2.27 for SAB, and 2.13 for SB galaxies. As noted by Athanassoula et al. (1982), the scatter is larger for SA galaxies than for SB galaxies.

Figure 27. Distributions of ratio of geometric outer and inner ring/pseudoring diameters, from Buta (1995). The upper left panel is irrespective of family, while the other panels separate barred, weakly barred, and nonbarred galaxies.
Figure 27

Buta & Crocker (1993) have also considered ring ratios in systems with nuclear rings. For about 20 objects with strong bars, logarithmic averages gave ratios of 18.9, 8.7, and 2.2 for dR / dnr , dr / dnr , and dR / dr , respectively. No evidence for a significant type dependence of these ratios was found. Weakly-barred and nonbarred galaxies displayed considerably more scatter and possible weak type dependences in the same ratios.

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