Deep space communication systems operate in the presence of high atmospheric attenuation and the capacity of the on-board energy generator is limited which produces a very low signal to noise ratio (SNR) at the reception. In order to satisfy the required transmission quality of deep space mission in terms of power and spectral efficiency, a coherent demodulation is mandatory. Coherent demodulation performance depends on the carrier phase synchronizer ability and its robustness against transmission constraints such as time-varying Doppler effect (Doppler rate). An efficient and well designed carrier phase recovery scheme is required to assure the synchronization. This paper analyses the performance of a blind carrier phase synchronizer derived from the maximum a posteriori criterion (MAP) and Laurent expansion for precoded Gaussian Minimum Shift Keying (GMSK) modulation. Due to the scarcity of spectral resources allocated for deep space transmission, the data-aided approach is not allowed. In the previous works, the carrier phase recovery system performance was evaluated in tracking phase without assessing its capacity in locking phase. In this paper, we present a more complete analysis of a GMSK carrier phase synchronizer covering the nonlinear acquisition phase and the theoretical operating limits in terms of SNR and input phase offset (Doppler, Doppler rate) in a closed-loop structure.