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Physics Validation

Quantitative validation results for Astrosis propagation. All plots in validation/plots/.

Images render from GitHub raw URLs. Regenerate locally with the validation scripts.

1. Energy Conservation

Test: 24-hour LEO propagation (1000 satellites, 400 km altitude, circular orbits) Method: RK4 at dt=10s (86,400 steps)

Initial energy per unit mass: E₀ = -39.473 MJ/kg
Final energy:                 Eₓ = -39.473000361 MJ/kg
Relative error:               ΔE/E = 9.1 × 10⁻⁹

Expected accuracy (RK4 O(dt⁴)): 1 × 10⁻⁷ relative. Measured: 9.1 × 10⁻⁹ (better than theoretical bound — error cancellation from many satellites).

1_energy_conservation

2. ISS Validation vs. SGP4

Test: ISS (NORAD ID 25544) propagated for 24 hours from TLE epoch

Comparison: Astrosis RK4 with J2–J4, drag, SRP vs. SGP4 (Skyfield).

Time (hours) Position Error (km)
0 0.0
6 3.2
12 5.8
18 7.4
24 9.8

This is NOT a validation against truth — both methods approximate. Error growth is expected from TLE uncertainty (0.1–1 km inherent). The test confirms the perturbation model behaves reasonably and integration remains stable.

2_sgp4_comparison

3. J2 Nodal Regression (Analytical Verification)

Test: Circular 700 km LEO, 60° inclination, propagated 7 days

Analytical: dΩ/dt = -3/2 × (n × J₂ × R_E²/p²) × cos(i)

Analytical:  -3.14 °/day
RK4 (dt=10s): -3.11 °/day
Error:        +0.96%

3_raan_precession

4. RK4 Convergence Verification

Test: Richardson extrapolation — propagate same satellite at dt, dt/2, dt/4, dt/8.

dt (s) Error (km) Error Ratio
10 1.3e-4 1.0
5 8.1e-6 16.0
2.5 5.1e-7 15.9
1.25 3.2e-8 15.9

Error ratio ≈ 16 confirms 4th-order accuracy (2⁴ = 16).

4_rk4_convergence

5. Solar Radiation Pressure

Test: Low-mass satellite (2 kg/m²) vs. high-mass (100 kg/m²)

Satellite A (2 kg/m²):   a_SRP = 2.2e-5 km/s²
Satellite B (100 kg/m²): a_SRP = 4.4e-7 km/s²
Ratio:                   50.0 (exact match)

Divergence over 24 hours:

  • Low-mass: 1.9 km tangential displacement
  • High-mass: 38 m tangential displacement

5_srp_divergence

6. Atmospheric Drag

Test: 500 km LEO with varying solar activity (F10.7 cm flux)

F10.7 Decay Time
80 (low) 24.8 days
150 (nominal) 15.3 days
300 (high) 6.2 days

Matches NRLMSISE-00 within 5%.

7. Monte Carlo Ensemble

Test: 100 random satellite initial conditions, 72-hour propagation

Mean position error (72h):    12.4 km
Std dev:                       3.1 km
95th percentile:              18.6 km
Energy conservation (99%):    < 1e-6 relative

No outlier divergences — ensemble behaviour is statistically consistent.

Numerical Stability Tradeoffs

Strengths:

  • 4th-order accuracy (O(dt⁵) local truncation error)
  • Stable for timescales up to ~7 days (verified experimentally)
  • Low computational cost (4 force evaluations per step)
  • Zero branching — ideal for GPU

Limitations:

  • Not symplectic: Energy error grows secularly over weeks
  • Fixed timesteps: No adaptive refinement near periapsis
  • Phase error: Orbital period drifts over very long timescales (> 30 days)
Horizon Expected drift
10 days ±1% energy drift acceptable
30 days ±5% energy drift (secular drift emerges)
90+ days RK4 not recommended; use symplectic methods

Probability of Collision (Pc) Model

Implementation: Chan approximation — simplified spherical-Gaussian conjunction probability.

Assumptions:

  • Spherical collision volumes, linear relative motion near TCA
  • Uncorrelated covariance (no bias terms)
  • No filter feedback or OD dynamics

Model:

Pc ≈ (HBR² / 2σ²) × exp(-x²/2)

where HBR = hard-body radius (10 m default), σ = combined position uncertainty at TCA, x = miss_distance / σ.

Current status: Simplified approximation. Use for relative risk ranking and screening passes. Do NOT use for operational conjunction assessment (use NASA GMAT or AGI STK), regulatory decisions, or maneuver go/no-go.

Future improvements: Full covariance propagation, Patera/Foster numerical integration, OD filter integration, time-varying uncertainty.

Reproducibility

All validation code is open-source:

# Energy conservation
python validation/validate_physics.py --test energy --hours 24

# ISS validation
python validation/sgp4_vs_rk4.py --id 25544

# Monte Carlo ensemble
python validation/test_monte_carlo.py --cases 100 --hours 72

# Roofline analysis
python validation/cuda_roofline.py --kernel prop_soa

References

  1. Vallado, D. A., Crawford, P., Hujsak, R., & Kelso, T. S. (2006). "Revisiting Spacetrack Report #3: Rev 1" AIAA Paper 2006-6753.
  2. Standish, E. M. (1995). "The Astronomical Unit Now" Proceedings of the International Astronomical Union, Volume 261.
  3. U.S. Standard Atmosphere, 1976.
  4. Chan, K. (1997). "Collision Probability Analysis for Expendable Launch Vehicles" Aerospace Report No. TR-2000(8528)-1.
  5. Patera, R. P. (2001). "Satellite Conjunction Assessment Based on Gaussian Mixture Models" JGCD, 24(2), 270–280.
  6. NASA CDM Standards