A.1. The JH models
We derive here the main properties of the JH models that have been used in the numerical simulations and in the computations of and . The reported quantities are dimensionless, and the normalization constants are expressed in terms of the total stellar mass , the core radius and the gravitational constant G ; again and . The cumulative mass inside and the total potential are given by:
In the hydrodynamical simulations we used the globally isotropic stellar velocity dispersion profile, that we calculate here. For any two-component model the velocity dispersion profile is given by , where is the 1-dimensional radial velocity dispersion. The first term is the isotropic velocity dispersion for the Jaffe model:
and the second one describes the contribution due to the Hernquist dark halo:
Physically acceptable limits can be easily found for (thus mimicking a black hole at the center of a Jaffe model) and for .
The cumulative potential and kinetic energies inside r can be found analytically for this two-component model, but for simplicity we report here only their total values. The total potential energy is given by , where
By integrating over the whole galaxy Eq. (3), we have . The kinetic energy is again the sum of two different contributions, , where
The integration over the whole galaxy of Eq. (4) gives .
A.2. A centrally flat model
Here we derive the dynamical quantities corresponding to the density distribution
The mass contained inside r is given by
and the potential is
The globally isotropic velocity dispersion is given by:
As for the JH model, the potential and the kinetic energy inside the radius r can be expressed analytically, but here we give only their total values, being the only ingredient required in the definition of :
Finally, the projected distribution associated to is:
from which one obtains .
© European Southern Observatory (ESO) 1998
Online publication: April 20, 1998