Self - gravitating discs : what can we learn from the dynamics of maser spots ?

For a few nearby active nuclei, the disc orbiting the black hole is traced by water maser emission. By combining a simple model together with observed velocity profiles, we show that it is possible to put constraints on the black hole mass and on the distribution of matter (shape, density, size) in the outer disc. We then report possible parameters for the non-keplerian disc and for the black hole in NGC 1068, and mention an uncertainty of at least 25% on the central mass in NGC 4258. 1. The inverse problem Finding the spatial distribution of matter ρ(r) that reproduces a given rotation curve Ω obs. (r) is a common inverse problem in Astrophysics. It can be addressed at the scale of AGN gaseous discs if one interprets the rotation curve of water masers detected in the core of some objects (Greenhill, 2002) in terms of a pure gravitational attraction. Efforts must therefore be made in developing reliable theoretical and/or numerical techniques to perform the inversion (e.g. Sibgatullin, Garcia & Manko, 2002). In the present situation, this must account for long-range interacting components (black hole, stellar cluster, bars, galactic bludge, etc.) as well as for objects populating the parsec-scale (disc, clouds, compact objects, torus, etc..) where masers are seen. Uncertainties are large, limiting our predictions and interpretations. We discuss here an example of inversion by considering a very simplified, two component system, made of a black hole and a disc, with an application to NGC 4258 and NGC 1068. 2. Assumptions, equations and method At the parsec-scale, the standard theory predicts that all discs are (strongly) self-gravitating (Collin & Huré, 2000) and so, they might evolve towards a state with a unity Q-Toomre parameter. Accounting for disc mass effects is therefore necessary to investigate these outer regions. But a reliable modeling of such discs is not permitted yet, given difficulties of various origins (nature of the flow, turbulent transport, numerical resolution, instabilities, boundary conditions), and some assumptions are necessary. The simplest way to tackle the question is a " minimal approach " which assumes that the velocity field has a pure gravitational origin. For a black hole with mass M BH and a disc only, the 1