An Asymptotic Analysis of Spherically Symmetric Perfect Fluid Similarity Solutions

The asymptotic properties of self-similar spherically symmetric perfect fluid solutions with equation of state p = αμ (−1 < α < 1) are described. We prove that for large and small values of the similarity variable, z = r/t, all such solutions must have an asymptotic power-law form. They are associated either with an exact power-law solution, in which case the α > 0 ones are asymptotically Friedmann, asymptotically Kantowski-Sachs or asymptotically static, or with an approximate power-law solution, in which case they are asymptotically quasi-static for α > 0 or asymptotically Minkowski for α > 1/5. We also show that there are solutions whose asymptotic behaviour is associated with finite values of z and which depend upon powers of lnz. These correspond either to a second family of asymptotic Minkowski solutions for α > 1/5 or to solutions that are asymptotic to a central singularity for α > 0 . The asymptotic form of the solutions is given in all cases, together with the number of associated parameters. This forms the basis for a complete classification of all α > 0 self-similar solutions. There are some other asymptotic power-law solutions associated with negative α, but the physical significance of these is unclear.