7 numerical experiments validate the CR3BP physical model and integrator accuracy. All results are reproducible from source code.
Experiment
Key Result
01 Jacobi Drift
Iterated Störmer-Verlet preserves the Jacobi integral to 10−11 — 5 orders of magnitude better than Euler
02 Convergence (log-log)
Iterated Verlet: slope 1.97±0.03 (R²=0.999), error 0.13 m at dt=30 s
03 Lagrange Points
L1–L5 found numerically; analytical approximations give relative error 0.4–0.8%
04 Halo Orbit
Periodic 3D orbit near L1: period 12.2 days, closure error 3.6 km
05 Free Return
Lunar flyby at 3,399 km, Earth return at 8,081 km; sensitivity: +1 m/s → +89 km at Moon
06 Chaos near L1
Lyapunov exponent λ = 0.64 day−1, predictability horizon ~2 days
07 Low Thrust
Δv=0.72 m/s deflects from L1 by 101,000 km; fuel 0.12 kg (0.02% of m₀)
Jacobi Integral Drift
The Jacobi integral CJ is the only conserved quantity in CR3BP. Its drift is a direct measure of integrator accuracy. 3 integrators × 2 step modes are compared across three scenarios.
Configuration
Halo Orbit
Free Return
Chaos L1
Euler fixed
2.22×10−6
4.84×10−1
3.12×10−2
Euler adaptive
3.53×10−7
1.53×10−2
1.04×10−3
Verlet half-step fixed
1.10×10−6
1.60×10−2
2.70×10−4
Verlet half-step adaptive
2.39×10−7
2.21×10−4
4.15×10−6
Verlet iterated fixed
3.83×10−11
1.15×10−2
3.98×10−4
Verlet iterated adaptive
1.85×10−12
1.27×10−5
4.04×10−7
Integrator Convergence (log-log)
Position error relative to scipy RK45 when integrating a halo orbit for 100 h. The half-step Verlet degrades to O(h) due to the Coriolis force; iterated Verlet recovers O(h²).
Integrator
Slope
Error at dt=30 s
Relative error (d/dEL)
Euler
1.00
8,400 m
2.2×10−5
Verlet half-step
1.00
1,900 m
4.9×10−6
Verlet iterated
1.97
0.13 m
3.4×10−10
Lagrange Points L1–L5
Numerical computation of Lagrange points (bisection / Newton) and comparison with analytical approximations (Hill sphere).
Point
x, km
y, km
Error, km
Relative error
L1
323,696
0
2,649
0.82%
L2
446,531
0
1,977
0.44%
L3
−386,651
0
2,779
0.72%
L4
188,080
332,920
2,812
0.74%
L5
188,080
−332,920
2,812
0.74%
Residual acceleration at found points: < 10−18 m/s².
Apollo-13 type ballistic trajectory: launch from LEO, lunar flyby, Earth return with no engine use.
Parameter
Value
Launch angle
226.5°
TLI velocity (trans-lunar injection)
10,779 m/s
Lunar flyby
3,399 km (t = 63.6 h)
Earth return
8,081 km (t = 168.1 h)
Sensitivity: +1 m/s → Δr Moon
≈ 89 km
Sensitivity: +1 m/s → Δr Earth
≈ 242 km
Chaotic Sensitivity near L1
16 trajectories launch from L1 with the same speed of 10 m/s but in different directions. Same speed — qualitatively different fates.
Parameter
Value
Lyapunov exponent λ
0.64 ± 0.16 day−1
Lyapunov time τ = 1/λ
1.6 days
Predictability horizon
~2 days
Trajectory divergence is exponential (dashed line on the plot — eλt fit). Some trajectories escape toward the Moon (~380,000 km), others loop around the Earth (~80–120,000 km).
Low-Thrust Trajectory Effects
Even a small thrust (<1% of initial velocity) radically changes the trajectory in CR3BP.
Scenario
Δv
Fuel (chem.)
Result
Earth → Moon (5 N, 3 h)
108 m/s
18 kg (3.6%)
Lunar flyby: 2,494 km
Escape from L1 (0.05 N, 2 h)
0.72 m/s
0.12 kg (0.02%)
Departure: 101,000 km
For reference: ideal Hohmann transfer LEO → Moon requires Δv ≈ 3,134 m/s (fuel 328 kg out of 500 kg)
Fuel consumption via Tsiolkovsky equation (Isp=300 s, m0=500 kg). Low thrust is a correction maneuver, not a replacement for an impulsive transfer.
