**9.2. Linear and Relative Diameters**

The diameters *D _{r}* 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* = 0.54 - 0.13*F* + 0.036*T*

*Y* = 0.01 - 0.13*F* + 0.058*T*

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 T 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).

Absolute linear diameters *D _{r}* 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

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, * _{c}* = (

*log D _{r} (pc)* = 3.61 + 0.15

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) = -2.05*L* - 1.31*T* + 29.46

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 T 7 can be derived from this alternative formulation:

*log D _{r} (pc)* = 3.81 + 0.15

*log D _{r} (pc)* = 3.81 + 0.15

Figure 26 shows that the reduced diameter, *log
D _{r}^{o}* =

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 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
*d _{R}* /

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 *d _{R}* /