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Reproducible evidence

Checking the calculations.

These cases are generated by the same propagation and event-location functions used by the planner. The table is a compact view of behavior that is also covered by the project’s automated checks.

Reference case

Earth constants and thresholds.

Every case below uses the Earth constants in the body catalog and fixed initial states committed with the report generator. Distances are kilometres, velocities are kilometres per second, and time is seconds.

Earth radius
6378.137 km
Earth μ
398600.4418 km³/s²
Report generator
buildValidationReport()
Cases
5 representative checks

The analytical circular period and vis-viva maneuver calculations are independent references for the numerical propagation. They are not a second numerical implementation.

Cases generated from code

Representative checks.

  1. Circular orbit returns after one period

    pass

    Propagate a 7,000 km circular state for the analytical Kepler period.

    Analytical period
    5828.517 s
    Position error (limit 1e-5 km)
    1.373e-11 km
    Velocity error (limit 1e-8 km/s)
    0.000e+0 km/s
    Relative energy drift
    0.00e+0
    Relative angular-momentum drift
    0.00e+0

  2. Forward propagation reverses cleanly

    pass

    Propagate a 3D state forward 4,321.5 s, then propagate the result backward by the same duration.

    Position error (limit 1e-5 km)
    9.118e-12 km
    Velocity error (limit 1e-8 km/s)
    1.071e-14 km/s
    Forward solver
    converged
    Reverse solver
    converged

  3. Analytical prograde burn raises apoapsis

    pass

    Add 0.1 km/s at 400 km altitude and compare vis-viva prediction with propagation at the opposite apsis.

    Burn
    0.100 km/s prograde
    Predicted apoapsis
    765.5 km altitude
    Simulated apoapsis
    765.5 km altitude
    Absolute difference
    2.728e-12 km (limit 0.1 km)

  4. Near-parabolic, parabolic, and hyperbolic coasts

    pass

    Propagate three 20,000 s states around and above escape speed, requiring finite output and energy drift below 1e−9.

    Near-parabolic
    finite / converged
    Parabolic
    finite / converged
    Hyperbolic
    finite / converged

  5. Impact, escape, and solver failure are explicit

    pass

    Locate radius events and deliberately exhaust the subdivision budget; do not return a guessed state.

    Impact event
    643.836 s at Earth radius
    Escape display crossing
    12954.385 s at 80,000 km
    Failure reason
    non-convergence
    Returned state after failure
    none

Independent checks

A second numerical method.

SciPy 1.17.1 DOP853 integrates Cartesian acceleration independently of the universal-variable solver. Across inclined, eccentric, parabolic and hyperbolic cases, maximum position error must remain below 0.0001 km and velocity error below 0.0000001 km/s.

  • two-body-inclined: 1.185e-6 km; 1.411e-9 km/s. Pass.
  • two-body-eccentric: 1.580e-7 km; 1.683e-10 km/s. Pass.
  • two-body-parabolic: 9.950e-8 km; 1.326e-11 km/s. Pass.
  • two-body-hyperbolic: 1.107e-7 km; 1.311e-11 km/s. Pass.
Energy and angular momentum relative drift across twenty two-body periods; all samples below the one-billionth threshold.

Generating revision: 18ef9f4a958f. Exact source hashes, initial conditions, tolerances, and all samples accompany the report.

Download numerical report · Download reproducibility package

How to read this

Numerical checks answer a narrower question.

Small state and invariant errors show that the implementation behaves consistently for these scenarios. They do not show that a two-body model is physically complete. Numerical error and model error are different questions.

The Earth J2 investigation separates model divergence from numerical error using declared thresholds, independent reference cases, and step convergence.

Read the methods and assumptions →