We present a high-resolution (R~16,000) spectrum and a narrow-band image centered on the [NeII]12.8 micron line of the central kpc region of the starburst/Seyfert2 galaxy NGC 7582. The galaxy has a rotating circum-nuclear starburst disk, shown at great detail at a diffraction-limited resolution of 0.4 arcsec. The high spatial resolution allows us to probe the dynamics of the [NeII] gas in the nuclear regions, and to estimate the mass of the central black hole. We construct models of gas disks rotating in the combined gravitational potential from the stellar bulge and a central black hole, and derive a black hole mass of 5.5 x 10^7 solar masses with a 95% confidence interval of [3.6,8.1] x 10^7 solar masses. The black hole mass combined with stellar velocity dispersion measurements from the literature shows that the galaxy is consistent with the local M-sigma relation. This is the first time that a black hole mass in a galaxy except our own Milky Way system has been estimated from gas dynamics in the mid-infrared. We show that spatially resolved mid-infrared spectroscopy may be competitive with similar techniques in the optical and near-infrared, and may prove to be important for estimating black hole masses in galaxies with strong nuclear dust obscuration. The high spectral resolution allows us to determine the heliocentric systemic velocity of the galaxy to between 1614 and 1634 km/s. The mid-infrared image reveals several dense knots of dust-embedded star formation in the circum-nuclear disk, and we briefly discuss its morphology.
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