Amplitude Condition & Effective Phase (振幅条件与有效相位)

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

What Is the Amplitude Condition?

The linearized motion near CR3BP libration points decomposes into two approximately independent harmonic oscillations: one in the xy-plane and one out of plane. The in-plane frequency is $\omega$ and the out-of-plane frequency is $\nu$ — these are generally incommensurable, giving Lissajous-type trajectories as the linear approximation (Richardson 1980).

Halo-type periodic orbits require the in-plane and out-of-plane frequencies to become equal after nonlinear corrections. When Richardson (1980) constructed the third-order analytical approximation using the Lindstedt-Poincaré method, secular terms appeared in the third-order z-equation that could not be eliminated through frequency expansion alone. To complete a secular-term-free third-order expansion, a nonlinear algebraic constraint must be imposed between the in-plane amplitude $A_x$ and the out-of-plane amplitude $A_z$ — this is the amplitude condition (Richardson 1980):

l_1 A_x^2 + l_2 A_z^2 + \Delta = 0 \tag{18}

where $l_1$ , $l_2$ , and $\Delta$ are constant coefficients determined by the libration point position and the mass parameter $\mu$ (specific values are given in Richardson 1980, Appendix I; tabulated values for Sun-Earth libration points are in the paper's table). This condition implies that $A_x$ and $A_z$ are not independent parameters — choosing one constrains the other. When the amplitude condition is not satisfied, the orbit degenerates into a non-closed Lissajous-type trajectory with incommensurable in-plane and out-of-plane frequencies, rather than a periodic Halo orbit.

The amplitude condition also has two branches (n = 1 and n = 3), distinguished by the switch function $\delta_n = 2-n$ in the phase-angle constraint $\psi = \phi + n\pi/2$ — this corresponds to the bifurcation into the Northern and Southern Halo orbit families (Richardson 1980). Changing the sign of $A_z$ switches from the Northern to the Southern family.

The amplitude condition determines the minimum permissible $A_x$ for Halo orbits: when $A_z = 0$ , $A_{x,\min} = \sqrt{-\Delta/l_1}$ . For the Sun-Earth $L_1$ point, $A_{x,\min}$ is about 14% of the normalized distance, corresponding to approximately 200,000 km (Richardson 1980).

In-Plane and Out-of-Plane Amplitudes

Collectively, the two amplitude parameters of libration point orbits:

In-plane amplitude $A_x$ : the maximum excursion of a Lissajous/Halo orbit in the xy-plane (the orbital plane of the synodic frame), directly determining the spatial extent in the x-direction and the y-direction ( $y_{\max} = \bar{k} A_x$ , where $\bar{k}$ is a linear-system constant).
Out-of-plane amplitude $A_z$ : the maximum vertical oscillation of the orbit in the z-direction, determining how far the orbit rises above the libration point's orbital plane.

Canalias and Masdemont (2008) showed that in the Sun-Earth system, Lissajous orbits with large in-plane amplitude (above $\sim 10^5$ km) and small out-of-plane amplitude (below $\sim 1.5 \times 10^5$ km) are best suited as target orbits for cross-system (Sun-Earth to Earth-Moon) transfers, because the invariant manifolds of such orbits more readily allow low- $\Delta v$ inter-system patching.

Effective Phase and the Effective Phases Plane (EPP)

Belló et al. (2010) observed that the solution for a Lissajous orbit (and the linear approximation of Halos) takes the form:

\begin{aligned} x(t) &= A_x \cos(\omega t + \phi), \\ y(t) &= \bar{k} A_x \sin(\omega t + \phi), \\ z(t) &= A_z \cos(\nu t + \psi). \end{aligned}

The CR3BP is autonomous, so the origin of time is arbitrary. This leads to a key insight: when the amplitudes $A_x$ , $A_z$ are fixed, resetting time $t \to t - t_0$ is equivalent to phase shifts $\phi \to \phi + \omega t_0$ , $\psi \to \psi + \nu t_0$ . Hence one defines:

In-plane effective phase $\Phi = \omega t + \phi \pmod{2\pi}$ ,
Out-of-plane effective phase $\Psi = \nu t + \psi \pmod{2\pi}$ ,

which collapse time and the original phases into two angular variables. On a Lissajous orbit of given amplitudes, the state $(\vec{r}, \vec{v})$ is in one-to-one correspondence with the effective phase pair $(\Phi, \Psi)$ . From a dynamical-systems perspective, a Lissajous orbit is a 2D torus, and $\Phi$ , $\Psi$ are precisely its action-angle variables (Belló et al. 2010).

The Effective Phases Plane (EPP) is the 2D plane with coordinates $(\Phi, \Psi)$ . In the EPP:

A Lissajous trajectory is a straight line of slope $\nu/\omega$ (approximately 0.966–0.965 for Sun-Earth $L_1$ / $L_2$ ) that propagates with constant velocity components $\omega$ and $\nu$ in the $\Phi$ and $\Psi$ directions — when compactly represented in $[0, 2\pi] \times [0, 2\pi]$ , it becomes a periodically wrapping segmented line.
An exclusion zone (e.g., the 3° solar disk cone for Sun-Earth $L_1$ , or the Earth's penumbra disk for $L_2$ ) appears in the EPP as quasi-elliptic closed curves. The time to eclipse is simply proportional to the distance from the current point to the first intersection of the trajectory line with an exclusion-zone curve — converting eclipse prediction from a high-dimensional integration problem into a geometric calculation (Belló et al. 2010).

Maneuver Design via the EPP

In the EPP framework, an in-plane maneuver (xy-plane impulse) changes the in-plane effective phase $\Phi$ , while an out-of-plane maneuver (z-direction impulse) changes the out-of-plane effective phase $\Psi$ , without altering the amplitudes (i.e., $A_x$ , $A_z$ unchanged):

\phi_f - \phi_i = -2(\omega t_m - \beta + \phi_i) \pmod{2\pi}, \quad \psi_f - \psi_i = -2(\nu t_m + \psi_i) \pmod{2\pi}

where $\beta$ is a fixed directional angle determined by the linear-system constants $c$ and $\bar{k}$ . This is the mathematical foundation of Belló et al.'s LOEWE (Lissajous Orbit Ever Without Eclipse) strategy: performing a single-impulse maneuver near the corners of the Lissajous figure in the yz-projection (where velocities are smallest) can skip the exclusion zone without changing the orbit amplitudes. For a Lissajous orbit of the size used in the Sun-Earth Herschel/Planck mission, this strategy requires only about 15 m/s every 6 years (Belló et al. 2010).

Amplitude Correction Maneuver

When the Lissajous orbit reached via a zero-cost transfer (natural manifold entry) deviates significantly from the target amplitudes, an amplitude correction maneuver is required. In the EPP framework, the z-impulse to correct the out-of-plane amplitude from $A_z^{(i)}$ to $A_z^{(f)}$ is given by (Belló et al. 2010):

\frac{\Delta \dot{z}}{\nu} = A_z^{(i)} \sin(\nu t_m + \psi_i) \pm \sqrt{A_z^{(f)2} - A_z^{(i)2} \cos^2(\nu t_m + \psi_i)}

Increasing amplitude ( $A_z^{(f)} \ge A_z^{(i)}$ ) is possible at any time; decreasing amplitude is possible only when the current z-position does not exceed the target amplitude. The optimal maneuver epoch satisfies $\nu t_m + \psi_i = \pi/2 + k\pi$ , where the minimum fuel cost is $|A_z^{(f)} - A_z^{(i)}|$ . In-plane amplitude correction follows an analogous pattern, but additionally requires constraining the unstable mode ( $A_1$ component) to zero (Canalias and Masdemont 2008; Belló et al. 2010).

References

Richardson, 1980, Analytic construction of periodic orbits about the collinear points (derivation of the amplitude condition $l_1 A_x^2 + l_2 A_z^2 + \Delta = 0$ , third-order analytical solution expressions, tabulated coefficient values for Sun-Earth libration points)
Belló et al., 2010, Invariant manifolds, Lagrangian trajectories and space mission design, Ch. 5 (definition of effective phases and the EPP, LOEWE eclipse avoidance strategy, amplitude correction maneuver formulas)
Canalias and Masdemont, 2008, Computing natural transfers between Sun–Earth and Earth–Moon Lissajous libration point orbits, Acta Astronautica (impact of in-plane/out-of-plane amplitudes on cross-system transfer suitability)