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 ( through 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 \sigma \lesssim 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 \sigma \gtrsim 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 .