Ephemeris Model

Author: Tianjiang Talk

Website: https://cislunarspace.cn

Definition

The Ephemeris Model is the dynamics model closest to the real gravitational field environment. It uses the N-body equations of motion to describe a spacecraft's trajectory under the gravitational influence of multiple celestial bodies. Unlike simplified models such as CRTBP and QBCP, the ephemeris model obtains the position and velocity information of celestial bodies from the JPL (Jet Propulsion Laboratory) planetary ephemerides (such as DE440), rather than simplified circular or elliptical orbit assumptions.

N-Body Dynamics Equations

In the J2000 Earth-centered inertial frame, assuming the central body is PcP_c, each perturbing body PiP_i is treated as a point mass, and the N-body dynamics equation for the spacecraft PsP_s is:

r¨cs=−Gmcrcs3rcs+G∑i=1Nmi(rsirsi3−rcirci3)\ddot{\mathbf{r}}_{cs} = -\frac{Gm_c}{r_{cs}^3}\mathbf{r}_{cs} + G\sum_{i=1}^{N}m_i\left(\frac{\mathbf{r}_{si}}{r_{si}^3} - \frac{\mathbf{r}_{ci}}{r_{ci}^3}\right)

where GG is the gravitational constant, mcm_c, msm_s, and mim_i are the masses of the central body, spacecraft, and each perturbing body, respectively. rcs\mathbf{r}_{cs}, rsi\mathbf{r}_{si}, and rci\mathbf{r}_{ci} are the relative position vectors between the corresponding bodies. rci\mathbf{r}_{ci} is provided by the ephemeris data.

Compact Form

Let the spacecraft state vector be X=[rT,vT]T\mathbf{X} = [\mathbf{r}^{\mathrm{T}}, \mathbf{v}^{\mathrm{T}}]^{\mathrm{T}}. The equations of motion can be written as:

X˙=[va]\dot{\mathbf{X}} = \begin{bmatrix} \mathbf{v} \\ \mathbf{a} \end{bmatrix}

The acceleration is contributed by all perturbing bodies:

a(r,t)=∑b∈Bab(r,t)\mathbf{a}(\mathbf{r}, t) = \sum_{b \in \mathcal{B}} \mathbf{a}_b(\mathbf{r}, t)

where B={Earth,Moon,Sun}\mathcal{B} = \{\text{Earth}, \text{Moon}, \text{Sun}\} is the set of perturbing bodies.

For the central body (Earth):

a⊕=−μ⊕r3r\mathbf{a}_{\oplus} = -\frac{\mu_{\oplus}}{r^3}\mathbf{r}

For non-central bodies bb (such as the Moon and Sun), the acceleration contribution includes both indirect and direct terms:

ab=−μb(r−rb∥r−rb∥3+rb∥rb∥3)\mathbf{a}_b = -\mu_b\left(\frac{\mathbf{r} - \mathbf{r}_b}{\|\mathbf{r} - \mathbf{r}_b\|^3} + \frac{\mathbf{r}_b}{\|\mathbf{r}_b\|^3}\right)

State Transition Matrix

To perform orbit correction and optimization, the State Transition Matrix (STM) needs to be computed. Linearizing the equations of motion yields the variational equations:

Φ˙(t,t0)=A(t)Φ(t,t0),Φ(t0,t0)=I6×6\dot{\boldsymbol{\Phi}}(t, t_0) = \mathbf{A}(t)\boldsymbol{\Phi}(t, t_0), \quad \boldsymbol{\Phi}(t_0, t_0) = \mathbf{I}_{6\times6}

where A(t)\mathbf{A}(t) is the Jacobian matrix of the dynamics equations with respect to the state. In numerical implementation, the 6-dimensional state vector and the 36 elements of the 6×66\times6 state transition matrix are concatenated into a 42-dimensional augmented state vector, integrated simultaneously with the equations of motion.

Coordinate Transformation

Computations in the ephemeris model are typically performed in the J2000 Earth-centered inertial frame. The transformation between this frame and the synodic (rotating) frame used in CRTBP is a critical step in orbit design. Let the Moon's position and velocity in the J2000 frame be RM\mathbf{R}_M and VM\mathbf{V}_M, respectively. Then:

  • Moon angular momentum: hM=RM×VM\mathbf{h}_M = \mathbf{R}_M \times \mathbf{V}_M
  • Moon angular velocity: ω=hM/∥RM∥2\boldsymbol{\omega} = \mathbf{h}_M / \|\mathbf{R}_M\|^2
  • Rotating matrix basis vectors: x^=RM/∥RM∥\hat{x} = \mathbf{R}_M/\|\mathbf{R}_M\|, z^=hM/∥hM∥\hat{z} = \mathbf{h}_M/\|\mathbf{h}_M\|, y^=z^×x^\hat{y} = \hat{z} \times \hat{x}

Relationship with Simplified Models

In practical orbit design, a "simplify first, then refine" strategy is typically employed:

  1. Obtain initial orbit solutions in simplified models such as CRTBP
  2. Convert the simplified model solutions to the ephemeris model using methods such as homotopy
  3. Perform high-precision correction and optimization in the ephemeris model

In the ephemeris model, strictly periodic orbits from CRTBP evolve into quasi-periodic orbits, requiring position and velocity correction via multiple-shooting methods.

References

  • Liu G. Study on Quasi-Periodic Orbit Design for Libration Points in the Earth-Moon System Using Ephemeris Model[D]. 2017.
  • Park R S, Folkner W M, Williams J G, et al. The JPL planetary and lunar ephemerides DE440 and DE441[J]. The Astronomical Journal, 2021, 161(3): 105.