# Design of Cislunar Navigation Constellation via Orbits with a Resonant Period
## 2 Fundamental Theory  
### Circular Restricted Three-Body Problem  
The orbital design for the constellation in this paper was established using the three-body problem, with the most typical simplified problem being the CRTBP. This problem is based on two important assumptions, as follows: (1) it is assumed that the primary bodies \(P_{1}\) and \(P_{2}\) have masses \(M_{1}\) and \(M_{2}\), respectively, and the mass of the target spacecraft \(P_{3}\) is \(m\), where \(m\) is much smaller than \(M_{1}\) and \(M_{2}\); (2) the motion of \(P_{1}\) and \(P_{2}\) around the system barycenter is a Keplerian circular orbit, meaning the distance between \(P_{1}\) and \(P_{2}\) is constant. The expressions for the CRTBP model are typically represented in a rotating coordinate system, which is defined as a dynamic system according to the motion of the two primary bodies. The origin \(O\) is located at the barycenter of the Earth-Moon system; the \(X\)-axis always points from the barycenter to \(P_{2}\), the \(Z\)-axis points in the direction of the system's angular momentum, perpendicular to the Earth-Moon motion plane and pointing towards the North Celestial Pole, and the \(Y\)-axis is determined using the right-hand rule.  
For computational convenience, it is necessary to perform normalization of the CRTBP problem. The normalized length unit is the distance between the two primary bodies, denoted as  
\[L^{*}=D_{1}+D_{2} \tag{1}\]  
In the Earth-Moon system, the characteristic length is defined as \(L^{*}=384,400\) km. The normalized mass unit is set as the sum of the two primary masses (Earth and Moon), denoted as  
\[m^{*}=M_{1}+M_{2} \tag{2}\]  
The normalized time is defined as  
\[t^{*}=\sqrt{\frac{L^{*}3}{Gm^{*}}} \tag{3}\]  
The unit of \(t^{*}\) is second. In cislunar dynamical modeling, the standard practice avoids the direct utilization of celestial mass parameters. Instead, gravitational parameters (GM) are adopted to eliminate uncertainties associated with the gravitational constant (G). As calibrated by the JPL DE440 ephemeris, the normalized GM values are defined as \(GM_{1}=398,600.435\) km\({}^{3}\)/s\({}^{2}\) for Earth and \(GM_{2}=4902.8001\) km\({}^{3}\)/s\({}^{2}\) for the Moon, demonstrating enhanced precision in orbital numerical integration. In subsequent analyses, all variables and dimensionless quantities undergo normalization through division by their characteristic reference values, except when explicitly accompanied by dimensional unit annotations.  
Additionally, a critical parameter in the CRTBP is the mass ratio \(\mu\), defined as the ratio of the secondary body's mass to the total system mass, expressed as  
\[\mu=\frac{M_{2}}{m^{*}}=\frac{GM_{2}}{GM_{1}+GM_{2}} \tag{4}\]The dimensionless equations of motion for the CRTBP in the rotating frame are provided as follows:  
\[\left\{\begin{array}{l}\ddot{x}-2\dot{y}-x=-\frac{(1-\mu)(x+\mu)}{\left\|\vec{r}_ {23}\right\|^{3}}-\frac{\mu(x-(1-\mu))}{\left\|\vec{r}_{23}\right\|^{3}}\\ \ddot{y}+2\dot{x}-y=-\frac{(1-\mu)^{3}}{\left\|\vec{r}_{23}\right\|^{3}}-\frac{ \mu y}{\left\|\vec{r}_{23}\right\|^{3}}\\ \ddot{z}=-\frac{(1-\mu)^{2}}{\left\|\vec{r}_{13}\right\|^{3}}-\frac{\mu^{2}}{ \left\|\vec{r}_{23}\right\|^{3}}\end{array}\right. \tag{5}\]  
where \(\vec{r}_{13}\) denotes the normalized vector from the primary body \(P_{1}\) to \(P_{3}\) in the rotating frame, \(\vec{r}_{13}=\left[\begin{array}{cc}x+\mu&y&z\end{array}\right]\). \(\vec{r}_{23}\) denotes the normalized vector from the primary body \(P_{2}\) to \(P_{3}\) in the rotating frame, \(\vec{r}_{23}=\left[\begin{array}{cc}x-(1-\mu)&y&z\end{array}\right]\).