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9.1. Frequency

Perhaps the most logical next step after defining the different classes of rings is to determine how frequently these features occur in galaxies. The frequencies will help us to gauge how long-lived these ring patterns might be, and how important they are in normal galaxy structure. Unfortunately, definitive statistics, especially concerning nuclear rings, have yet to be derived owing to serious observational selection effects or inadequate samples.

The relative frequency of inner rings is the best determined of the three ring types. De Vaucouleurs & Buta (1980b) used revised Hubble classifications from the Second Reference Catalogue of Bright Galaxies (RC2, de Vaucouleurs et al. 1976) to deduce this frequency and attempted to account for selection effects. The main selection effects that would bias any frequency estimate are the inclination of a galaxy and the number of resolution elements in an image. The first effect is important mainly in highly inclined spiral galaxies where internal extinction prevents the ring from being detected. Any sample of randomly oriented objects would therefore underestimate the frequency of inner rings. An additional problem of high inclination is to cause an overestimate of the frequency of inner rings in nonbarred galaxies. This is because bars viewed end-on at high inclination are easily overlooked. An example is NGC 4274 in the Hubble Atlas (Sandage 1961), which is classified as Sa (nonbarred), but seems an obvious end-on bar case.

The resolution effect is important because classifications in RC2 are based on heterogeneous image material, some being large-scale plates taken with large reflectors, and others being small-scale Palomar Sky Survey prints or copy plates. The small-scale images are prone to overexposure and poor resolution, so that inner rings and bars are not recognized. Thus, it is best to restrict any frequency analysis to large galaxies of low inclination.

For a sample of 618 RC2 galaxies having corrected isophotal diameter D0 geq 2' and logarithmic axis ratio logR25 leq 0.2, de Vaucouleurs & Buta (1980b) deduced that 25% of galaxies of all types are classified as (r)-variety while 37% are classified as (rs)-variety. Table 3 shows statistics of varieties based on the same restrictions but using the larger database in RC3 (de Vaucouleurs et al. 1991). This table divides the sample according to family and stage, and also includes the percentages of galaxies satisfying the diameter and inclination restrictions but for which no variety is specified. Table 3 shows that among spirals, inner rings are found at the 19% level while inner pseudorings are found at the 39% level. The percentage of (r)-variety cases is higher among barred spirals than nonbarred or weakly barred spirals. Inner pseudorings and s-shapes are nearly equally frequent among SB galaxies. Among nonbarred (SA) galaxies, inner rings are much less frequent than s-shapes, while more than half of SAB galaxies are classified as having inner pseudorings. The indication from these results is that most normal giant galaxies have an inner ring or pseudoring, and that this is especially true if a galaxy has even a weak bar. Only a small percentage of spirals do not have a variety estimate for this kind of sample. The statistics are poorer for the lenticulars (S0 or L), and a higher percentage of them do not have a variety estimate. Note that the absence of a variety estimate for either an S0 or a spiral may also indicate that no variety is applicable (Buta 1995).


TABLE III. Relative frequencies of RC3 varieties for D0 geq 2', logR25 leq 0.2
Family f(r) f(rs) f(s) f(N.A.) N
SB 23.2% 32.4% 38.4% 6.0% 336
SAB 15.1 50.7 29.8 4.4 272
SA 15.8 32.1 46.7 5.5 165
All S 18.8 38.8 37.1 5.3 773
LB 36.1 19.4 25.0 19.4 36
LAB 13.5 16.2 35.1 35.1 37
LA 21.5 7.7 32.3 38.5 65
All L 23.2 13.0 31.2 32.6 138


In Table 4, we examine statistics of ring frequency using volume-limited samples of RC3 galaxies. This provides a somewhat different picture of the frequency of ring phenomena. We compute the relative frequency of inner rings and pseudorings for various limiting radial velocities v3K (reduced to the microwave background frame) from RC3, restricted to logR25 leq 0.2. For the lower redshift limits, the frequency of inner rings is about 12-13% and that of inner pseudorings is 28-34%. When the statistics are restricted by family, the frequency of rings is higher in SB galaxies than in SA galaxies, but the (s)-variety is still the dominant type. Nevertheless, volume-limited statistics still indicate that rings are observed in a large fraction of normal, massive galaxies along the Hubble sequence. Note that RC3 is not complete with regard to radial velocity measurements in most of the samples in Table 4, so the statistics become less complete in addition to less reliable with increasing limiting redshift.

The effect of inclination on spiral varieties is shown in Table 5, which gives the frequencies for logR25 > 0.2. As expected, the frequencies of rings and pseudorings are considerably less, and more galaxies lack a variety estimate, due mainly to detection difficulties. De Vaucouleurs & Buta (1980b) investigated the effects of inclination on ring frequencies and found a general decrease for large galaxies, but that frequencies of features seen in SA and SAB galaxies can increase due to misclassification. In RC2 lenticulars, ``ansae'' along the major axis of an apparently nonbarred example were attributed to edge-on views of inner rings, and as a consequence the frequency of the (r)-variety unrealistically increases with inclination for such objects. It is clear now, however, that some ``ansae'' are probably connected to bars and not rings.


TABLE IV. Percentages of RC3 spiral varieties versus redshift, for logR25 leq 0.2
v3K leq f(r) f(rs) f(s) f(N.A.) N
1400 12.0 33.7 45.4 8.8 249
2000 13.0 30.6 47.6 8.8 477
2500 13.3 28.4 49.3 9.0 647
3000 13.5 29.5 48.2 8.9 824
3500 14.5 29.7 46.5 9.4 1003
4000 14.6 31.6 44.5 9.4 1140
5000 16.7 30.8 40.4 12.1 1571
6000 17.6 30.1 39.0 13.2 1878
8000 19.2 28.5 37.6 14.7 2274
10000 19.8 28.2 36.7 15.3 2496
20000 20.8 27.8 35.8 15.6 2712


TABLE V. Percentages of RC3 spiral varieties versus redshift, for logR25 > 0.2
v3K leq f(r) f(rs) f(s) f(N.A.) N
1400 4.5 15.7 62.8 17.0 312
2000 5.5 17.5 60.5 16.5 595
2500 5.9 17.7 57.9 18.6 786
3000 6.7 18.8 56.4 18.1 954
3500 8.2 18.5 55.4 17.9 1130
4000 8.5 18.7 54.2 18.6 1277
5000 10.0 19.2 50.2 20.5 1622
6000 11.5 19.2 47.9 21.4 1858
8000 12.2 19.2 45.9 22.6 2098
10000 12.7 18.9 44.8 23.6 2245
20000 13.1 19.1 44.1 23.7 2375


Kormendy (1979a) provided another statistical study, but focussed only on SB galaxies and accounted for lenses which are not really recognized in RC2. He found that 76% of intermediate to late Hubble type SB galaxies have inner rings and pseudorings, while 54% of early type SB galaxies have inner lenses. In this analysis, inner rings and pseudorings were combined. Again, the indication is that most barred galaxies are ringed or lensed.

The distributions of inner rings and pseudorings with revised Hubble stage T (coded on the RC2 numerical scale) were described by de Vaucouleurs (1963) and de Vaucouleurs & Buta (1980b). These studies found that inner rings are most frequent among early-type galaxies while inner pseudorings are most frequent among later type galaxies. A double-peaked distribution for inner rings (with peaks at stages -1 and +2 or S0+ and Sab) was also found. It is possible that some of the excess at stage S0+ (T = -1) in those studies is due to inclusion of lenses (Kormendy 1981). In Figure 23, we show the distribution of inner rings and pseudorings with type for a sample of RC3 galaxies having v3K leq 2500 km s1 and logR25 leq 0.2. In such a volume-limited sample, the frequency of inner rings is nearly uniform in the range -2 leq T leq 5.

Figure 23. Relative frequency of inner rings (top) and pseudorings (bottom) as a function of RC3 Hubble type. The numbers of objects are indicated. The sample is restricted to v3K leq 2500 km s-1 and log R25 leq 0.2.
Figure 23

The frequencies of outer rings and nuclear rings are less well-established owing to more serious selection effects than for inner rings. De Vaucouleurs (1975a) first estimated the frequency of outer rings and pseudorings, and found that 4% of the galaxies in the Reference Catalogue of Bright Galaxies (RC1, de Vaucouleurs & de Vaucouleurs 1964) possess these features. The rings are most commonly observed around stage S0/a. However, though the rings seem rare, it is likely that the RC1 frequency estimate is too low. This is because of the wide range of surface brightnesses of outer rings. RC1 morphological types are based exclusively on large-scale prime focus plates, but outer rings are best detected on deep, small-scale Schmidt plates such as the SRC-J southern sky survey. Even on deep Schmidt plates, however, outer rings can be lost if there is significant galactic extinction. In Figure 24, we repeat de Vaucouleurs' calculations using the more complete RC3 data. This shows the relative frequency of closed outer rings (R) and outer pseudorings (R') versus revised Hubble stage. Rings classified as (R) are most common at stage S0/a, while pseudorings (R') are more common at later types, as would be expected. Both outer rings and pseudorings are nevertheless phenomena of early-type galaxies, being rare at stages later than Sc. Table 6 summarizes the percentage of outer rings and pseudorings versus radial velocity in RC3. For the lower redshift ranges, these features are found in about 10% of the entries over all types.

Figure 24. Relative frequency of outer rings (top) and pseudorings (bottom) as a function of RC3 Hubble type. The numbers of objects are indicated. The sample is restricted to v3K leq 20000 km s-1 and log R25 leq 0.2.
Figure 24


TABLE 6. Percentages of RC3 outer rings and pseudorings versus redshift, for logR25 leq 0.2 and all types
v3K leq f(R) f(R') No ring N
1400 4.1 5.2 90.7 387
2000 5.0 5.5 89.5 706
2500 4.6 5.2 90.1 953
3000 4.8 5.6 89.5 1177
3500 5.4 6.5 88.1 1404
4000 5.3 7.1 87.6 1600
5000 5.8 8.3 85.9 2176
6000 6.3 8.7 85.1 2574
8000 6.9 9.1 84.1 3119
10000 6.8 9.5 83.7 3430
20000 6.7 9.6 83.7 3749


For nuclear rings the problem of detection is most serious of all, not because these features are generally faint, but because they are located in the bright centers of galaxies that are easily overexposed on photographic images and where the luminosity gradient is very steep. Core overexposure and poor resolution are especially a problem with sky survey images, thus the true frequency of nuclear rings will have to be based on a CCD imaging survey of a well-defined galaxy sample. However, we believe that nuclear rings are not rare, especially in galaxies which have other rings. For example, Buta & Crocker (1991) found nuclear rings, some not previously known, in a small sample of galaxies chosen on the basis of the appearance of an outer ring or pseudoring. New examples of nuclear rings are turning up regularly in similar imaging surveys (e.g., Contini et al. 1995) or in radio continuum studies of active galaxies (Wilson et al. 1991; Forbes et al. 1994a, b; Storchi-Bergmann et al. 1996).

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