Simple solutions of fireball hydrodynamics for self-similar elliptic flows

Simple, self-similar, elliptic solutions of non-relativistic fireball hydrodynamics are presented, generalizing earlier results for spherically symmetric fireballs with Hubble flows and homogeneous temperature profiles. The transition from one dimensional to three dimensional expansions is investigated in an efficient manner. Introduction. Recently, a lot of experimental and theoretical efforts have gone into the exploration of hydrodynamical behavior of strongly interacting hadronic matter in non-relativistic as well as in relativistic heavy ion collisions, see for example refs. [1] [2]. Due to the non-linear nature of the equations of hydrodynamics, exact solutions of these equations are rarely found. Those events, sometimes, even stimulate an essential progress in physics. One of the most impressive historical example is Landau’s one-dimensional analytical solution (1953) for relativistic hydrodynamics [3] that gave rise to a new (hydrodynamical) approach in high energy physics. The boost-invariant Bjorken solution [4], found more than 20 years later, is frequently utilized as the basis for estimations of initial energy densities reached in ultra-relativistic nucleus-nucleus collisions. The obvious success of hydrodynamic approach to high energy nuclear collisions raise interest in an analogous description of non-relativistic collisions, too. The first exact hydrodynamic solution of non-relativistically expanding fireballs was found in 1979 [5], that has been generalized for fireballs with Gaussian density and homogeneous temperature profiles [6] as well as for fireballs with arbitrary initial temperature profiles [7] and corresponding, non-Gaussian density profiles. All of these solutions have spherical symmetry and a Hubble-type linear radial flow. However, a non-central collision has none of the mentioned symmetries. Therefore, if we do not want to consider central collisions only, then we must attempt to generalize these solutions. The purpose of this Letter