Long-, Short-, and Dual-Period Motion near Triangular Libration Points

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

Definition

Linearising the CR3BP equations about either triangular libration point ( $L_4$ or $L_5$ ) yields two in-plane oscillatory modes plus an out-of-plane harmonic. The in-plane modes are the long-period and short-period modes; their combination is the dual-period case. Following Catlin & McLaughlin (2007), the in-plane frequencies are

s_{1,2} = \sqrt{\tfrac{1}{2}\Big(\tfrac{1}{2} \pm \sqrt{\tfrac{1}{4} - 27\,\mu(1-\mu)}\Big)},

where $\mu \approx 0.0121506$ for the Earth–Moon system. The $s_1 < s_2$ convention gives the long-period frequency $s_1$ and the short-period frequency $s_2$ ; $s_z = \sqrt{1 - \mu + \mu^2} \approx 0.999893$ is the (near-unit) out-of-plane frequency. With these, the periods and axis ratios (semimajor over semiminor of the relative-motion ellipse, $1/\bar\alpha$ ) are:

Mode	Period	Axis ratio $1/\bar\alpha$
Long-period ( $s_1$ only)	≈ 92 d	≈ 16/3 ≈ 5.33
Short-period ( $s_2$ only)	≈ 1 synodic month	≈ 2
Dual-period ( $s_1 + s_2$ )	≈ 458 d	≈ 16/5 = 3.2

The dual-period period of 458 d is not the period of either mode; it is the common recurrence time of the combined quasi-periodic motion at Earth–Moon $\mu$ .

Pure long-period and pure short-period motion

Either mode can be filtered out by judicial choice of the initial velocity (Catlin & McLaughlin 2007, Eqs. 6–7). When only the long-period mode is retained, the relative motion is an in-plane ellipse with axis ratio ≈ 16/3 and period ≈ 92 d; the required initial relative velocity is on the order of millimetres per second. When only the short-period mode is retained, the ellipse has axis ratio ≈ 2 and period ≈ one synodic month. These single-mode solutions are the practical building blocks for formation flight at $L_4/L_5$ : the motion is repeatable, planar, and amenable to analytical description.

A natural circular formation (axis ratio 1) is impossible in pure long-period motion — the required in-plane and out-of-plane frequencies do not coincide ( $s_z$ is far from $s_1$ ). It is only approximately realisable in short-period motion, where $s_2$ and $s_z$ differ by less than 0.05 non-dimensional frequency units. Even there, the planar approximation is not an accurate representation of CR3BP dynamics and must be supplemented by active control to hold circularity over long durations.

Dual-period motion and why it is rarely used

When both modes are present, the relative dynamics resolve into intricate three-dimensional curves (axis ratio ≈ 16/5, period ≈ 458 d). They are mathematically rich but practically inconvenient: the second satellite does not repeat a closed path relative to the first within any operationally useful interval, so most formation-flight concepts at $L_4/L_5$ deliberately filter one of the two modes.

Sensitivity to initial conditions

Single-mode motion at $L_4/L_5$ is not lost by instability (the linearised triangular points are stable for $\mu < \mu_\text{Routh} \approx 0.0385$ ) but by resurgence of the filtered mode: a small initial-condition error reintroduces the suppressed frequency, and over weeks the formation drifts away from its designed geometry. Catlin & McLaughlin's sensitivity analysis (2007, Table 3) shows that keeping the total relative-position error below 10% over 30 days requires initial-velocity knowledge at the $\sim 1$ µm/s level for long-period and short-period formations; the parallel (in-plane, phase-shifted) formation is dramatically more tolerant, accepting kilometre-scale initial-position errors. This is why uncontrolled formations at triangular libration points are unlikely to keep their geometry over mission-relevant timescales, and why active control is required for any practical concept.

Perturbations and model limits

The analysis above is CR3BP-only. In the real Earth–Moon system, solar gravity dominates the perturbation budget at $L_4/L_5$ : a short-period formation propagated for three years accumulates ≈ 110 km of relative-range error if solar gravity is neglected (≈ 1800% of the unperturbed amplitude). Solar-radiation pressure produces ≈ 14 m peak error (≈ 20%); Earth oblateness is negligible at the sub-millimetre level (Catlin & McLaughlin 2007, §IV). A faithful model therefore requires at least the bicircular or full-ephemeris formulation, not the pure CR3BP.

References

Catlin, K. & McLaughlin, C., 2007, "Earth–Moon Triangular Libration Point Spacecraft Formations," J. Guid. Control Dyn. — derivation of the long/short-period mode frequencies, axis ratios, formation designs, sensitivity analysis, and perturbation assessment (source of all numerical values quoted above).
Catlin & McLaughlin, 2004, "Relative motion of two spacecraft near the Earth–Moon triangular libration points" — earlier planar analysis.
Szebehely, 1967, Theory of Orbits, §§5.2–5.4 — linearisation about $L_4/L_5$ , characteristic roots, Routh stability criterion.
Hou & Liu, 2010, "On quasi-periodic motions around the triangular libration points of the real Earth–Moon system" — extensions into the ephemeris model.