Let \(G\) be a subgraph of a complete graph \(k_n\) on \(n\) vertices. The configuration \(T(G)\)-triple is the graph obtained from \(G\) by replacing each edge \(ab\) of \(G\) with a 3-cycle \((a,b,c)\) where \(c\not\in V(G)\) and it does not appear in any other triple \(T(G)\). \(T(G)\) system is said to be of order \(n\) if there is an edge-disjoint decomposition of \(3k_n\) into copies of \(T(G)\). If one edge is taken from each 3-cycle in each copy of \(T(G)\) in such a way that the resulting copies of \(G\) form an edge-disjoint decomposition of \(k_n\), then \(T(G)\)-triple system is called perfect. The spectrum for perfect \(T(G)\) systems is the set of all positive integers \(n\) for which perfect systems of order \(n\) exists. The spectra for perfect \(T(G)\) systems, \(G\) being any subgraph of \(k_4\) or of \(k_5\) (with six edges or less) have already been obtained. In this paper, the authors address similar problem for star graphs \(k_{1,p}\) and for various of \(p\). It is shown that the spectrum is completely determined when \(p\) is an odd prime power.