Summary: Let \(G\) be a subgraph of \(K_n\). The graph obtained from \(G\) by replacing each edge with a 3-cycle whose third vertex is distinct from other vertices in the configuration is called a \(T(G)\)-triple. An edge-disjoint decomposition of \(3K_n\) into copies of \(T(G)\)-triple is called a \(T(G)\)-triple system of order \(n\). If, in each copy of \(T(G)\)-triple in a \(T(G)\)-triple system, one edge is taken from each 3-cycle (chosen so that these edges form a copy of \(G\)) in such a way that the resulting copies of \(G\) form an edge-disjoint decomposition of \(K_n\), then the \(T(G)\)-triple system is said to be perfect. The set of positive integers \(n\) for which a perfect \(T(G)\)-triple system exists is called its spectrum. Earlier papers by authors including E. J. Billington, C. C. Lindner, S. Küçükçifçi and A. Rosa determined the spectra for cases where \(G\) is any subgraph of \(K_4\). In this paper, we will focus in star graph and discuss the existence for perfect \(T(K_{1,k})\)-triple system. Especially, for prime powers \(k\), its spectra are completely determined.