3. Axisymmetric distributions
We assume that all three components have axisymmetric flat rotation curves in the outer regions of the disk. The quasi-stationary equilibrium rotation is jointly supported by the gravity of the halo and bulge, the self-gravity of the disk and the pressure gradient:
Here is the angular velocity in the disk, is the axisymmetric self-gravitating potential, and are the partial pressures of the components corresponding to their unperturbed density distributions.
In our simulations, we consider two types of the equilibrium rotation curves and surface density distributions. In the first experiment, we choose the exponentially decreasing surface density of the disk with density distributions of all three components given by the expression
with the normalization constants which are the masses of the gaseous, stellar and remnant component. The halo-bulge potential determining the rotational curve (16) is similar to that used by Vauterin and Dejonghe (1996). Namely, we assume that the external potential has the form
In a thin disk with sharp boundaries the potential diverges at its edges. To study to what extent the sharp cut-off in an unperturbed density distribution influences the results, we performed, additionally, experiments using a Gaussian-type density distribution vanishing at both boundaries:
Both halo and bulge potentials, described by the Eqs. (17) and (19) dictate the "flat" rotation in the outer region of the disk. The parameters and in Eqs. (17) and (19) determine the spatial scales of the halo and bulge density distributions, gives the asymptotic value of the rotational velocity, and and determine the masses of the halo and the central bulge.
© European Southern Observatory (ESO) 1999
Online publication: June 30, 1999