The softened point mass model is commonly used in simulations of gaseous discs including self-gravity while the value of associated length lambda remains, to some degree, controversial. This ``parameter is however fully constrained when, in a discret
ized disc, all fluid cells are demanded to obey Newtons law. We examine the topology of solutions in this context, focusing on cylindrical cells more or less vertically elongated. We find that not only the nominal length depends critically on the cells shape (curvature, radial extension, height), but it is either a real or an imaginary number. Setting lambda as a fraction of the local disc thickness -- as usually done -- is indeed not the optimal choice. We then propose a novel prescription valid irrespective of the disc properties and grid spacings. The benefit, which amounts to 2-3 more digits typically, is illustrated in a few concrete cases. A detailed mathematical analysis is in progress.
We exploit our formula for the gravitational potential of finite size, power-law disks to derive a general expression linking the mass of the black hole in active galactic nuclei (AGN), the mass of the surrounding disk, its surface density profile (t
hrough the power index s), and the differential rotation law. We find that the global rotation curve v(R) of the disk in centrifugal balance does not obey a power law of the cylindrical radius R (except in the confusing case s = -2 that mimics a Keplerian motion), and discuss the local velocity index. This formula can help to understand how, from position-velocity diagrams, mass is shared between the disk and the black hole. To this purpose, we have checked the idea by generating a sample of synthetic data with different levels of Gaussian noise, added in radius. It turns out that, when observations are spread over a large radial domain and exhibit low dispersion (standard deviation less than 10% typically), the disk properties (mass and s-parameter) and black hole mass can be deduced from a non linear fit of kinematic data plotted on a (R, Rv 2)-diagram. For a deviation higher than 10%, masses are estimated fairly well from a linear regression (corresponding to the zeroth-order treatment of the formula), but the power index s is no longer accessible. We have applied the model to 7 AGN disks whose rotation has already been probed through water maser emission. For NGC3393 and UGC3789, the masses seem well constrained through the linear approach. For IC1481, the power-law exponent s can even be deduced. Because the model is scale-free, it applies to any kind of star/disk system. Extension to disks around young stars showing deviation from Keplerian motion is thus straightforward.
