## 3 Optimal Bandwidth Table  
The proposed table-based adaptive DPLL algorithm (which will be discussed in Section 4) requires a pre-computed optimal bandwidth table that contains an optimal noise bandwidth value for each C/N\({}_{0}\) and dynamic stress. The optimal noise bandwidth is calculated using the theoretical measurement error modeling of the DPLL that incorporates the thermal noise, Allan deviation, vibration-induced oscillator phase noise, and dynamic stress error. This paper assumes that the order of the DPLL is third-order to effectively track the dynamics components leading up to acceleration. In addition, the receiver is assumed to track the pure pilot channel (i.e., phase transition due to the navigation data bit or secondary code does not exist) to track very weak signals such that the C/N\({}_{0}\) value is as low as a few dB-Hz. However, the other orders, including the data channel case, can be applied by an analogous method. The overall 1-sigma measurement error of the pilot DPLL is modeled as follows [1]:  
\[\sigma_{DPLL}=\sqrt{\sigma_{t}^{2}+\theta_{A}^{2}+\sigma_{v}^{2}}+\frac{\theta_ {e}}{3}\leq 30\;\text{deg} \tag{1}\]  
Figure 4: LOS jerk information of the Moon exploration spacecraft during the mission trajectory. Each color in the figure indicates a different GPS satellite. Figure 3: Line-of-sight (LOS) carrier-to-noise-density ratio (C/N\({}_{0}\)) information of the Moon exploration spacecraft during the mission trajectory. Each color in the figure indicates a different global positioning system (GPS) satellite. where \(\sigma_{t}\) is the thermal noise [deg], \(\theta_{A}\) and \(\sigma_{v}\) are the Allan deviation and vibration-induced oscillator phase noise [deg], respectively, and \(\theta_{e}\) is the LOS dynamic stress error [deg]. The rule-of-thumb for the tracking threshold is that the overall 3-sigma error must not exceed 1/4 of the pull-in range [1]. Because the pull-in range of the pilot DPLL discriminator is 360 degrees, the threshold of the 3-sigma error is 90 degrees, which indicates that the 1-sigma error must not exceed 30 degrees. The thermal noise of the pilot DPLL is modeled as follows [1]:  
\[\sigma_{t}=\frac{360}{2\pi}\sqrt{\frac{B_{n}}{C/N_{0}}}\ [\deg] \tag{2}\]  
where \(B_{n}\) is the noise bandwidth [Hz] of the loop filter and \(C/N_{0}\) is the carrier-to-noise density ratio converted to a linear scale [Hz]. Notably, the squaring loss cannot be found in (2) because this equation is modeled for the pilot channel. Because (2) shows that the thermal noise is proportional to the noise bandwidth, the noise bandwidth should be narrowed to reduce the thermal noise. The Allan deviation oscillator phase noise of the third-order DPLL is modeled as follows [26]:  
\[\theta_{A}=\frac{360}{2\pi}\sqrt{2\pi^{2}f_{c}^{2}}\left(\frac{\pi^{2}h_{-2}}{ 3\omega_{0}^{3}}+\frac{\pi h_{-1}}{3\sqrt{3}\omega_{0}^{2}}+\frac{h_{0}}{6 \omega_{0}}\right)\ [\deg] \tag{3}\]  
where \(f_{c}\) is the carrier frequency [Hz] of the oscillator; \(\omega_{0}\) is the natural frequency [Hz] of the loop filter, which can be calculated using a typical value for the third-order loop (\(\omega_{0}=1.27B_{n}\)); and \(h_{0}\) [s], \(h_{-1}\) [s/s], and \(h_{-2}\) [s/s\({}^{2}\)] are the clock parameters related to the Allan deviation. In this work, the oven-controlled crystal oscillator (OCXO) is assumed to be used for the spacecraft. The corresponding clock parameters used for the paper are presented in Table 1.  
The vibration-induced oscillator phase noise of the band-limited vibration with constant power spectral density (PSD) for the third-order DPLL is modeled as follows [26]:  
\[\sigma_{v}=\frac{360}{2\pi}\sqrt{\frac{2\pi f_{c}^{2}k_{g}^{2}G_{g}}{\omega_{0 }}