Fomalhaut is a young, nearby star that is suspected to harbor an infant planetary system, interspersed with one or more belts of dusty debris. We present far-infrared images obtained with the Herschel Space Observatory with an angular resolution between 5.7 and 36.7 arcsec at wavelengths between 70 and 500 micrometer. The images show the main debris belt in great detail. Even at high spatial resolution, the belt appears smooth. The region in between the belt and the central star is not devoid of material; thermal emission is observed here as well. Also at the location of the star, excess emission is detected. We use a dynamical model together with radiative-transfer tools to derive the parameters of the debris disk. We include detailed models of the interaction of the dust grains with radiation, for both the radiation pressure and the temperature determination. Comparing these models to the spatially resolved temperature information contained in the images allows us to place strong constraints on the presence of grains that will be blown out of the system by radiation pressure. We use this to derive the dynamical parameters of the system. The appearance of the belt points towards a remarkably active system in which dust grains are produced at a very high rate by a collisional cascade in a narrow region filled with dynamically excited planetesimals. Dust particles with sizes below the blow-out size are abundantly present. The equivalent of 2000 one-km-sized comets are destroyed every day, out of a cometary reservoir amounting to 110 Earth masses. From comparison of their scattering and thermal properties, we find evidence that the dust grains are fluffy aggregates, which indicates a cometary origin. The excess emission at the location of the star may be produced by hot dust with a range of temperatures, but may also be due to gaseous free-free emission from a stellar wind.
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