  More than two thirds of disk galaxies are barred to some degree. Many today
harbor massive concentrations of gas in their centers, and some are known to
possess supermassive black holes (SMBHs) 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. We report an extensive study of the effects of central masses
on bars in high-quality N-body simulations of galaxies. We have varied the
growth rate of the central mass, its final mass and degree of concentration to
examine how these factors affect the evolution of the bar. Our main conclusions
are: (1) Bars are more robust than previously thought. The central mass has to
be as large as several percent of the disk mass to completely destroy the bar
on a short timescale. (2) For a given mass, dense objects cause the greatest
reduction in bar amplitude, while significantly more diffuse objects have a
lesser effect. (3) The bar amplitude always decreases as the central mass is
grown, and continues to decay thereafter on a cosmological time-scale. (4) The
first phase of bar-weakening is due to the destruction by the CMC of
lower-energy, bar-supporting orbits, while the second phase is a consequence of
secular changes to the global potential which further diminish the number of
bar-supporting orbits. We provide detailed phase-space and orbit analysis to
support this suggestion. Thus current masses of SMBHs are probably too small,
even when dressed with a stellar cusp, to affect the bar in their host
galaxies. The molecular gas concentrations found in some barred galaxies are
also too diffuse to affect the amplitude of the bar significantly.
