Assessment of dynamical models for transitioning from the Circular Restricted Three-Body Problem to an ephemeris model with applications

Abstract

While the Circular Restricted Three-Body Problem (CR3BP) provides useful structures for various applications, transitioning the solutions from the CR3BP to a higher-fidelity ephemeris model while maintaining specific characteristics remains non-trivial. An analytical approach is leveraged to provide additional insight on the perturbations that are present in an ephemeris model. For the Earth–Moon CR3BP, pulsation of the Earth–Moon distance and solar gravity are identified as key components contributing to the additional accelerations, where patterns are illustrated through simplified mathematical relationships and graphics. Utilizing these findings, capabilities and limitations of two intermediate models, the Elliptic Restricted Three-Body Problem and the Bi-Circular Restricted Four-Body Problem, are assessed within the context of transitioning from the CR3BP to a realistic ephemeris model. A sample transition process for Earth–Moon L2 halo orbits is provided, leveraging the insight from the proposed analytical approach.

Appendices

Appendix A: Derivations of Equation (22)

The acceleration terms in Eq. (21) are rearranged to yield cancellations and a more useful form. First, note that the familiar relative acceleration of the Moon with respect to Earth is evaluated as follows,

$$\begin{aligned} \vec {A}_{EM}= & {} -\frac{{\tilde{\mu }}_E + {\tilde{\mu }}_M}{R_{EM}^3}\vec {R}_{EM}+ \sum _{j \in \mathcal {A}}{\tilde{\mu }}_j\left( \frac{1}{R_{jE}^3}\vec {R}_{jE} - \frac{1}{R_{jM}^3}\vec {R}_{jM}\right) \nonumber \\= & {} \frac{{\tilde{\mu }}_E + {\tilde{\mu }}_M}{l^2} \left( -\hat{x} + \sum _{j \in \mathcal {A}} \mu _j\left( \frac{1}{\rho _{jE}^3}\vec {\rho }_{jE} - \frac{1}{\rho _{jM}^3}\vec {\rho }_{jM}\right) \right) . \end{aligned}$$

(1)

Then, for the ephemeris model, $\ddot{\vec {\rho }}_{\mathcal {A}}$ is denoted as $\ddot{\vec {\rho }}_{\mathcal {A}, H}$, where each coefficient in Eq. (21) is evaluated with instantaneous ephemerides data. The $\hat{x}$-component of $\ddot{\vec {\rho }}_{\mathcal {A}, H}$ is evaluated as, $\ddot{\vec {\rho }}_{\mathcal {A}, H} \cdot \hat{x} = b_1 + (b_7-1)x+b_9y +b_8z - \sum _{j\in \mathcal {A}}\frac{\mu _j}{\rho _{jc}^3}(x - x_j)$. Here, note that $b_1$ is evaluated as,

$$\begin{aligned} b_1 =\sum _{j \in \mathcal {A}}\mu _j \left( \frac{1-\mu }{\rho _{jE}^3}(-\mu -x_j) + \frac{\mu }{\rho _{jM}^3}(1-\mu -x_j)\right) . \end{aligned}$$

(2)

Next, $b_7$ results in the following,

$$\begin{aligned} b_7-1= & {} -\frac{{l}''}{({t}')^2l}+ \frac{h^2}{({t}')^2l^4}-1 =-\frac{l(\vec {R}_{EM}\cdot \vec {A}_{EM}+V_{EM}^2)-{l}'(\vec {R}_{EM}\cdot \vec {V}_{EM})}{({t}')^2l^3}\nonumber \\{} & {} + \frac{l^2(V_{EM}^2-{l}'^2)}{({t}')^2l^4}-1 \nonumber \\= & {} -\frac{\vec {A}_{EM}\cdot \hat{x}}{({t}')^2l}-1 = \sum _{j \in \mathcal {A}} \mu _j\left( \frac{1}{\rho _{jE}^3}(\mu +x_j) + \frac{1}{\rho _{jM}^3}(1-\mu -x_j)\right) , \end{aligned}$$

(3)

where leveraging $\vec {R}_{EM} = l\hat{x}$, and $h =l \sqrt{V_{EM}^2-{l}'^2}$. Similarly, $b_8$ is evaluated as,

$$\begin{aligned} b_8 = - \frac{1}{{t}'^2l}\vec {A}_{EM}\cdot \hat{z} = \sum _{j \in \mathcal {A}} \mu _j\left( \frac{1}{\rho _{jE}^3}z_j - \frac{1}{\rho _{jM}^3}z_j\right) . \end{aligned}$$

(4)

Finally, $b_9$ results in,

$$\begin{aligned} b_9&= \frac{{h}'}{({t}')^2l^2} = \frac{\hat{z} \cdot (\vec {R}_{EM} \times \vec {A}_{EM})}{({t}')^2l^2} = \frac{1}{{t}'^2l} \vec {A}_{EM}\cdot \hat{y} = \sum _{j \in \mathcal {A}}\mu _j \left( -\frac{1}{\rho _{jE}^3}y_j + \frac{1}{\rho _{jM}^3}y_j \right) . \end{aligned}$$

(5)

Repeating a similar process for $\ddot{\vec {\rho }}_{\mathcal {A}, H} \cdot \hat{y}$ and $\ddot{\vec {\rho }}_{\mathcal {A}, H} \cdot \hat{z}$ results in the following expression,

$$\begin{aligned} \ddot{\vec {\rho }}_{\mathcal {A}, H}= & {} - \sum _{j\in \mathcal {A}}\mu _j\left( \frac{\vec {\rho }_{jc}}{\rho _{jc}^3}+\frac{\vec {\rho }_{Mc}\cdot \hat{x}}{\rho _{jE}^3}\vec {\rho }_{jE} - \frac{\vec {\rho }_{Ec}\cdot \hat{x}}{\rho _{jM}^3}\vec {\rho }_{jM} -\frac{\vec {\rho }_{Mc}\cdot \hat{y}}{\rho _{jE}^3}(\vec {\rho }_{jE}\times \hat{z}) \right. \nonumber \\{} & {} \left. + \frac{\vec {\rho }_{Ec}\cdot \hat{y}}{\rho _{jM}^3}(\vec {\rho }_{jM}\times \hat{z}) -(\frac{zz_j}{\rho _{jE}^3} - \frac{zz_j}{\rho _{jM}^3})\hat{x} \right) \cdots \nonumber \\{} & {} + \begin{bmatrix} 0 &{} 0 &{} 0 \\ 0&{} 0 &{} b_6 \\ 0 &{} -b_6 &{} 0 \end{bmatrix} \dot{\vec {\rho }} + \begin{bmatrix} 0 &{} 0 &{} 0 \\ 0 &{} b_{12}-b_{12a} &{} b_{11} \\ 0 &{} -b_{11} &{} b_{12}-b_{12a} \end{bmatrix} \vec {\rho }. \end{aligned}$$

(6)

The above equation is the same as Eq. (22), concluding the derivations.

Appendix B: Derivations of Equation (24)

Further simplification of Eq. (22) for $\ddot{\vec {\rho }}_{\mathcal {A}, H}$ is deduced from the additional assumptions. Assuming that $\rho _j>>1$, the inverse cubes of distance from j to the s/c, Earth, Moon, are linearized as,

$$\begin{aligned}&\frac{1}{\rho _{jc}^3} \approx \frac{1}{\rho _{j}^3} + \frac{3}{\rho _{j}^5}(x_jx +y_jy +z_jz) \end{aligned}$$

(7)

$$\begin{aligned}&\frac{1}{\rho _{jE}^3} \approx \frac{1}{\rho _{j}^3} + \frac{3}{\rho _{j}^5}x_j(-\mu ) \end{aligned}$$

(8)

$$\begin{aligned}&\frac{1}{\rho _{jM}^3} \approx \frac{1}{\rho _{j}^3} + \frac{3}{\rho _{j}^5}x_j(1-\mu ), \end{aligned}$$

(9)

respectively. Utilizing these linearized functions, the expressions for the terms in Eq. (22) are reduced to simpler formulations. For the $\hat{x}$-direction, the resulting expression is,

$$\begin{aligned} \ddot{\vec {\rho }}_{\mathcal {A}, H} \cdot \hat{x}&\approx \sum _{j \in \mathcal {A}}(\frac{3\mu _j }{\rho _{j}^5} \cdot 2 x_jy_jy) . \end{aligned}$$

(10)

Next, the approximate accelerations for the $\hat{y}$ and $\hat{z}$-directions are evaluated as,

$$\begin{aligned} \ddot{\vec {\rho }}_{\mathcal {A}, H} \cdot \hat{y}&\approx \sum _{j \in \mathcal {A}}\frac{3\mu _j }{\rho _{j}^5} \left( (y_j^2 - x_j^2)y + y_jz_jz\right) + (b_{12}-b_{12a})y + b_{11}z + b_{6}\dot{z} \end{aligned}$$

(11)

$$\begin{aligned} \ddot{\vec {\rho }}_{\mathcal {A}, H} \cdot \hat{z}&\approx \sum _{j \in \mathcal {A}}\frac{3\mu _j }{\rho _{j}^5} \left( y_jz_jy + z_jz_jz \right) - b_{11} y + (b_{12}-b_{12a}) z - b_6\dot{y}, \end{aligned}$$

(12)

where coefficients $b_6, b_{11}$, and $b_{12} - b_{12a}$ are also well approximated. From Table 2, the expressions for these coefficients consist solely of $\vec {A}_{EM}\cdot \hat{z}$ and $\vec {J}_{EM} \cdot \hat{z}$. The first term, $\vec {A}_{EM}\cdot \hat{z}$, is linearized as,

$$\begin{aligned} \vec {A}_{EM} \cdot \hat{z}&\approx {t}'^2 l \sum _{j \in \mathcal {A}} \frac{3\mu _j}{\rho _j^5} x_j z_j. \end{aligned}$$

(13)

Leveraging this expression, the second term, $\vec {J}_{EM}\cdot \hat{z}$, is linearized as,

$$\begin{aligned} \vec {J}_{EM} \cdot \hat{z}= & {} (\vec {A}_{EM} \cdot \hat{z})' - \vec {A}_{EM} \cdot \hat{z}' = (\vec {A}_{EM} \cdot \hat{z})' - \frac{h'}{h}(\vec {A}_{EM} \cdot \hat{z}) \nonumber \\{} & {} \approx K_1 \sum _{j \in \mathcal {A}} \frac{3\mu _j}{\rho _j^5} x_j z_j+ {t}'^2 l \sum _{j \in \mathcal {A}} \left( \frac{3\mu _j}{\rho _j^5} (x_j \dot{z}_j + \dot{x}_j z_j )\right. \nonumber \\{} & {} \left. -\frac{15\mu _j}{\rho _j^7}x_jz_j(\dot{x}_jx_j + \dot{y}_jy_j + \dot{z}_jz_j) \right) , \end{aligned}$$

(14)

where $K_1 = 2{t}'{t}''l + {t}'^2 {l}' - \frac{{h}'{t}'^2\,l }{h}$. These quantities are utilized to compute $b_6, b_{11}$, and $b_{12} - b_{12a}$, where it is noted that the linearized expressions contain $z_j$ and $\dot{z}_j$ within the summation, or, the out-of-plane position and velocity components of the additional bodies with respect to the pulsating–rotating frame. Additionally, if additional bodies are negligible except for the Sun, and assuming that $|z_S |\ll \rho _S$, or the $\hat{z}$-component of the solar position is negligible, the following linearized expression is produced,

$$\begin{aligned} \ddot{\vec {\rho }}_{\mathcal {A}, H} \approx \ddot{\vec {\rho }}_{\mathcal {A}, L} := \frac{3\mu _S }{\rho _{S}^5} \left( 2x_Sy_Sy\hat{x} +(y_S^2 - x_S^2)y \hat{y} \right) = \frac{3\mu _S }{\rho _{S}^3}y (\sin 2 \theta _S \hat{x} - \cos 2\theta _S \hat{y}), \end{aligned}$$

(15)

concluding the derivations of Eq. (24).