An asymptotic analysis of spherically symmetric perfect fluid self-similarsolutions

The asymptotic properties of self-similar spherically symmetric perfect flu id solutions with the equation of state p = alpha mu (-1 < <alpha> < 1) are described. We prove that for large and small values of the similarity vari able, z = r/t, all such solutions must have an asymptotic power-law form. S ome of them are associated with an exact power-law solution, in which case they are asymptotically Friedmann or asymptotically Kantowski-Sachs for 1 > alpha > -1 or asymptotically static for 1 > alpha > 0. Others are associat ed with an approximate power-law solution, in which case they are asymptoti cally quasi-static for 1 > alpha > 0 or asymptotically Minkowski for 1 > al pha > 1/5. We also show that there are solutions whose asymptotic behaviour is associated with finite values oft and which depend upon powers of ln z. These correspond either to a second family of asymptotically Minkowski sol utions for 1 > alpha > 1/5 or to solutions that are asymptotically Kasner f or 1 > alpha > -1/3. There are some other asymptotic power-law solutions as sociated with negative alpha, but the physical significance of these is unc lear. The asymptotic form of the solutions is given in all cases, together with the number of associated parameters.