The Frequency and Sizes of Inner Bars and Nuclear Rings in Barred Galaxies and Their Dependence on Galaxy Properties

Using a volume- and mass-limited (D < 30 Mpc, log (M_star/M_sun) $\geq 9.75$) sample of 155 barred S0-Sd galaxies, I determine the fraction with secondary structures within their bars. Some 20 +/- 3% have a separate inner bar, making them double-barred; an identical fraction have nuclear rings, with 11^{+3}_{-2}% hosting both. The inner-bar frequency is a strong, monotonic function of stellar mass: only 4^{+3}_{-2}% of barred galaxies with log (M_star/M_sun) = 9.75-10.25 are double-barred, while 47 +/- 8% of those with log (M_star/M_sun) > 10.5 are. The nuclear-ring frequency is a strong function of absolute bar size: only 1^{+2}_{-1}% of bars with semi-major axes < 2 kpc have nuclear rings, while 39^{+6}_{-5}% of larger bars do. Both inner bars and nuclear rings are absent in very late-type (Scd--Sd) galaxies. Inner bar size correlates with galaxy stellar mass, but is clearly offset to smaller sizes from the main population of bars. This makes it possible to define "nuclear bars" in a consistent fashion, based on stellar mass. There are eight single-barred galaxies where the bars are nuclear-bar-sized; some of these may be systems where an outer bar failed to form, or previously double-barred galaxies where the outer bar has dissolved. Inner bar size is even more tightly correlated with host bar size, which is likely the primary driver. In contrast, nuclear ring size is only weakly correlated with galaxy mass or bar size, with more scatter in size than is true of inner bars.