We construct dynamical models for the H_alpha+[NII] and H_beta kinematics that include a supermassive black hole, and in which the stellar mass distribution is constrained by the observed surface brightness distribution and ground-based stellar kinematics. In one set of models we assume that the gas rotates on circular orbits in an infinitesimally thin disk. Such models adequately reproduce the observed gas fluxes and kinematics. The steepness of the observed rotation velocity gradient implies that a black hole must be present. There are some differences between the fluxes and kinematics for the various line species that we observe in the wavelength range 4569 to 6819 Angstrom. Species with higher critical densities generally have a flux distribution that is more concentrated towards the nucleus, and have observed velocities that are higher. This can be attributed qualitatively to the presence of the black hole. There is some evidence that the gas in the central few arcsec has a certain amount of asymmetric drift, and we therefore construct alternative models in which the gas resides in collisionless cloudlets that move isotropically.
All models are consistent with a black hole mass in the range 1-4 x
10^8 solar masses, and models without a black hole are always ruled
out at high confidence. The implied ratio of black holes mass to
galaxy mass is in the range 0.4-1.5 x 10^{-3}, which is not
inconsistent with results obtained for other galaxies. These results
for the peculiar galaxy IC 1459 and its black hole add an interesting
data point for studies on the nature of galactic nuclei.
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