Author: CislunarSpace

Source: https://cislunarspace.cn

NRHO Stability and Station-Keeping

Initial Condition Sensitivity and Lyapunov Exponents

Although NRHOs appear as stable quasi-periodic orbits in the CR3BP model, they exhibit significant sensitivity to initial condition errors and external perturbations in a real ephemeris environment. This sensitivity can be quantified using the Lyapunov Exponent.

The Lyapunov exponent λ\lambda characterizes the exponential separation (or convergence) rate of neighboring trajectories in phase space:

  • λ>0\lambda > 0: Orbital errors grow exponentially (unstable)
  • λ<0\lambda < 0: Errors are suppressed (stable)
  • λ=0\lambda = 0: Neutral (marginal stability along the orbital direction)

For L1/L2 NRHOs, Floquet modal analysis of the linearized equations shows that unstable modes with λ>0\lambda > 0 exist along certain directions. This means even minute initial errors are amplified over several orbital periods.

A typical NRHO Lyapunov exponent is on the order of λ∼10−2\lambda \sim 10^{-2} day−1^{-1}, corresponding to an e-folding time of approximately 50-100 days.

ΔV Station-Keeping Budget

Station-keeping for an NRHO requires periodic low-thrust corrections. The typical maintenance ΔV\Delta V budget depends on:

  • Orbital location (L1 vs. L2)
  • Mission duration
  • Propulsion system type (electric vs. chemical)

Typical values:

  • L1 NRHO: ΔV≈30−50\Delta V \approx 30-50 m/s/year
  • L2 NRHO: ΔV≈40−80\Delta V \approx 40-80 m/s/year (slightly higher than L1 due to stronger solar gravitational perturbations)

For comparison, DRO station-keeping budgets are approximately 5-20 m/s/year, demonstrating their higher inherent stability.

Station-Keeping Strategies

Impulsive Station-Keeping

Using low-thrust engines (e.g., hydrazine thrusters) for periodic impulsive corrections. Each correction typically requires ΔV≈1−5\Delta V \approx 1-5 m/s. The optimal correction time is usually at the apoapsis or periapsis (points of minimum velocity) to maximize correction effectiveness.

Continuous Thrust Station-Keeping

For electric propulsion systems, continuous low-thrust corrections can be employed by adjusting the thrust direction to compensate for perturbations. This requires more complex attitude-orbit coupled control but achieves higher orbital maintenance precision.

Optimal Correction Timing

The optimal station-keeping strategy must balance correction frequency against correction accuracy. Too-low correction frequency leads to accumulated orbital deviations, increasing the ΔV\Delta V required per correction; too-high frequency increases propellant consumption and mission scheduling complexity.

Effects of External Perturbations

Solar Gravitational Perturbation

Solar gravity is the primary external disturbance source for NRHO station-keeping. Near L2 NRHOs, solar gravitational perturbation is especially significant because the gravitational gradient in the L2 direction is weaker.

Lunar Non-Spherical Perturbation

The Moon's non-spherical gravitational terms (J2 term, C22C_{22} term, etc.) have non-negligible effects on the long-term evolution of NRHOs. For low-inclination NRHOs in particular, the lunar J2 term causes drift in the Right Ascension of the Ascending Node (RAAN).

Solar Radiation Pressure

For spacecraft with large solar panels or high area-to-mass ratios, Solar Radiation Pressure (SRP) is also a perturbation source that requires dedicated modeling.

Simulation Experiments

In the Satellite Orbit Simulation Laboratory, you can set NRHO initial conditions and add perturbation models to observe long-term orbital evolution and drift trends.