  Many barred galaxies today harbor massive concentrations of gas in their
centers, and some are known to possess supermassive black holes (SBHs) and
their associated stellar cusps. Previous theoretical work has suggested that a
bar in a galaxy could be dissolved by the formation of a mass concentration in
the center, although the precise mass and degree of central concentration
required is not well-established. Here we report an extensive study of the
effects of central masses on bars in high-quality N-body simulations of
galaxies. We study models containing both strong and weak bars, and quantify
the change in bar amplitude as a mass concentration is grown in their centers.
We find that bars are more robust than previously thought. We have varied the
growth rate of the central mass, its final mass and the degree of concentration
of the mass. Our main conclusions are: (1) the central mass has to be as large
as several percent of the disk mass to completely destroy the bar; (2) for a
given mass, objects whose scale-length is a few pc or less cause the greatest
reduction in bar amplitude, while significantly more diffuse objects have a
lesser effect; and (3) the bar amplitude always decreases as the central mass
is grown, and continues to decay thereafter on a cosmological timescale. Thus
the masses of SBHs are probably too small, even when dressed with a stellar
cusp, to affect the bar amplitude significantly.
