We present the derivation of distribution functions for the first four members of a family of disks, previously obtained in (MNRAS, 371, 1873, 2006), which represent a family of axially symmetric galaxy models with finite radius and well behaved surf
ace mass density. In order to do this we employ several approaches that have been developed starting from the potential-density pair and, essentially using the method introduced by Kalnajs (Ap. J., 205, 751, 1976) we obtain some distribution functions that depend on the Jacobi integral. Now, as this method demands that the mass density can be properly expressed as a function of the gravitational potential, we can do this only for the first four discs of the family. We also find another kind of distribution functions by starting with the even part of the previous distribution functions and using the maximum entropy principle in order to find the odd part and so a new distribution function, as it was pointed out by Dejonghe (Phys. Rep., 133, 217, 1986). The result is a wide variety of equilibrium states corresponding to several self-consistent finite flat galaxy models.
An infinite family of new exact solutions of the Einstein vacuum equations for static and axially symmetric spacetimes is presented. All the metric functions of the solutions are explicitly computed and the obtained expressions are simply written in
terms of oblate spheroidal coordinates. Furthermore, the solutions are asymptotically flat and regular everywhere, as it is shown by computing all the curvature scalars. These solutions describe an infinite family of thin dust disks with a central inner edge, whose energy densities are everywhere positive and well behaved, in such a way that their energy-momentum tensor are in fully agreement with all the energy conditions. Now, although the disks are of infinite extension, all of them have finite mass. The superposition of the first member of this family with a Schwarzschild black hole was presented previously [G. A. Gonzalez and A. C. Gutierrez-Pi~neres, arXiv: 0811.3002v1 (2008)], whereas that in a subsequent paper a detailed analysis of the corresponding superposition for the full family will be presented.
The first fully integrated explicit exact solution of the Einstein field equations corresponding to the superposition of a counterrotating dust disk with a central black hole is presented. The obtained solution represents an infinite annular thin dis
k (a disk with an inner edge) around the Schwarszchild black hole. The mass of the disk is finite and the energy-momentum tensor agrees with all the energy conditions. Furthermore, the total mass of the disk when the black hole is present is less than the total mass of the disk alone. The solution can also be interpreted as describing a thin disk made of two counterrotanting dust fluids that are also in agreement with all the energy conditions. Additionally, as we will show shortly in a subsequent paper, the above solution is the first one of an infinite family of solutions.
