This paper deducted the specific form of Cramer-Rao lower bound (CRLB) for the initial state estimation for the boost phase object based on the 8-state gravity turn model using space-based observations and illustrated the computation process with examples. In addition, the geometrical comprehension of the Fisher information matrix (FIM) of the single observation is presented, and how a target's launch point and shoot direction effects the CRLB based on the LOS measurements from sensor located on a geosynchronous earth orbit (GEO) satellite is investigated. Since shoot direction is of particular importance in early-phase warning, it is defined and computationally examined. Simulations are employed to testify the CRLB derivations and facilitate the understanding of the theoretical construct. The paper concludes with a summary of results and future work.