FAILURE MAP
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FA-69821 / Orbital propagation / Open access

Hyperbolic departure from a parking orbit: Periapsis speed uses the circular potential term · case 01

Departure burns are underestimated.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

vp is computed as sqrt(vinf^2 + mu/r).

THE FAILURE

vp is computed as sqrt(vinf^2 + mu/r).

Unsuccessful approach: Adding vinf to the escape speed is not energy conservation.

Case contract

Input [r, vinf, mu]: circular parking radius and required hyperbolic excess speed. Periapsis speed vp=sqrt(vinf^2+2mu/r); burn dv=vp-sqrt(mu/r); eccentricity e=1+r vinf^2/mu; turn angle delta=2 asin(1/e) in degrees. Return [vp, dv, e rounded 6, delta rounded 4].

Why this case matters

Orbit determination and mission planning chain many small conversions; one wrong branch or unit silently moves a spacecraft by kilometres.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    r,vinf,mu=x
    vp=math.sqrt(vinf*vinf+mu/r)
    dv=vp-math.sqrt(mu/r)
    e=1+r*vinf*vinf/mu
    delta=math.degrees(2*math.asin(1/e))
    return [round(vp,6),round(dv,6),round(e,6),round(delta,4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('hyperbolic departure from a parking orbit [6678.0, 3.0, 398600.4418]', [6678.0, 3.0, 398600.4418], [11.330366, 3.604526, 1.150783, 120.6792]), ('hyperbolic departure from a parking orbit [6678.0, 0.5, 398600.4418]', [6678.0, 0.5, 398600.4418], [10.937422, 3.211582, 1.004188, 169.5303]), ('hyperbolic departure from a parking orbit [7000.0, 6.0, 398600.4418]', [7000.0, 6.0, 398600.4418], [12.242787, 4.696734, 1.632212, 75.565]), ('hyperbolic departure from a parking orbit [42164.0, 1.5, 398600.4418]', [42164.0, 1.5, 398600.4418], [4.59969, 1.525023, 1.238005, 107.7536]), ('hyperbolic departure from a parking orbit [1.0, 1.0, 1.0]', [1.0, 1.0, 1.0], [1.732051, 0.732051, 2.0, 60.0]), ('hyperbolic departure from a parking orbit [2.0, 0.3, 1.0]', [2.0, 0.3, 1.0], [1.044031, 0.336924, 1.18, 115.8724]), ('hyperbolic departure from a parking orbit [6578.0, 11.0, 398600.4418]', [6578.0, 11.0, 398600.4418], [15.562519, 7.778176, 2.996832, 38.9853])], [('hyperbolic departure from a parking orbit [6678.0, 0.5, 398600.4418]', [6678.0, 0.5, 398600.4418], [10.937422, 3.211582, 1.004188, 169.5303]), ('hyperbolic departure from a parking orbit [42164.0, 1.5, 398600.4418]', [42164.0, 1.5, 398600.4418], [4.59969, 1.525023, 1.238005, 107.7536]), ('hyperbolic departure from a parking orbit [1.0, 1.0, 1.0]', [1.0, 1.0, 1.0], [1.732051, 0.732051, 2.0, 60.0]), ('hyperbolic departure from a parking orbit [2.0, 0.3, 1.0]', [2.0, 0.3, 1.0], [1.044031, 0.336924, 1.18, 115.8724]), ('hyperbolic departure from a parking orbit [6578.0, 11.0, 398600.4418]', [6578.0, 11.0, 398600.4418], [15.562519, 7.778176, 2.996832, 38.9853]), ('hyperbolic departure from a parking orbit [1.0, 0.0, 1.0]', [1.0, 0.0, 1.0], [1.414214, 0.414214, 1.0, 180.0]), ('hyperbolic departure from a parking orbit [6678.0, 3.0, 398600.4418]', [6678.0, 3.0, 398600.4418], [11.330366, 3.604526, 1.150783, 120.6792])], [('hyperbolic departure from a parking orbit [7000.0, 6.0, 398600.4418]', [7000.0, 6.0, 398600.4418], [12.242787, 4.696734, 1.632212, 75.565]), ('hyperbolic departure from a parking orbit [6578.0, 11.0, 398600.4418]', [6578.0, 11.0, 398600.4418], [15.562519, 7.778176, 2.996832, 38.9853]), ('hyperbolic departure from a parking orbit [1.0, 0.0, 1.0]', [1.0, 0.0, 1.0], [1.414214, 0.414214, 1.0, 180.0]), ('hyperbolic departure from a parking orbit [6678.0, 3.0, 398600.4418]', [6678.0, 3.0, 398600.4418], [11.330366, 3.604526, 1.150783, 120.6792]), ('hyperbolic departure from a parking orbit [6678.0, 0.5, 398600.4418]', [6678.0, 0.5, 398600.4418], [10.937422, 3.211582, 1.004188, 169.5303]), ('hyperbolic departure from a parking orbit [42164.0, 1.5, 398600.4418]', [42164.0, 1.5, 398600.4418], [4.59969, 1.525023, 1.238005, 107.7536]), ('hyperbolic departure from a parking orbit [1.0, 1.0, 1.0]', [1.0, 1.0, 1.0], [1.732051, 0.732051, 2.0, 60.0])], [('hyperbolic departure from a parking orbit [42164.0, 1.5, 398600.4418]', [42164.0, 1.5, 398600.4418], [4.59969, 1.525023, 1.238005, 107.7536]), ('hyperbolic departure from a parking orbit [6678.0, 0.5, 398600.4418]', [6678.0, 0.5, 398600.4418], [10.937422, 3.211582, 1.004188, 169.5303]), ('hyperbolic departure from a parking orbit [7000.0, 6.0, 398600.4418]', [7000.0, 6.0, 398600.4418], [12.242787, 4.696734, 1.632212, 75.565]), ('hyperbolic departure from a parking orbit [1.0, 1.0, 1.0]', [1.0, 1.0, 1.0], [1.732051, 0.732051, 2.0, 60.0]), ('hyperbolic departure from a parking orbit [2.0, 0.3, 1.0]', [2.0, 0.3, 1.0], [1.044031, 0.336924, 1.18, 115.8724]), ('hyperbolic departure from a parking orbit [6578.0, 11.0, 398600.4418]', [6578.0, 11.0, 398600.4418], [15.562519, 7.778176, 2.996832, 38.9853]), ('hyperbolic departure from a parking orbit [1.0, 0.0, 1.0]', [1.0, 0.0, 1.0], [1.414214, 0.414214, 1.0, 180.0])], [('hyperbolic departure from a parking orbit [1.0, 1.0, 1.0]', [1.0, 1.0, 1.0], [1.732051, 0.732051, 2.0, 60.0]), ('hyperbolic departure from a parking orbit [2.0, 0.3, 1.0]', [2.0, 0.3, 1.0], [1.044031, 0.336924, 1.18, 115.8724]), ('hyperbolic departure from a parking orbit [6578.0, 11.0, 398600.4418]', [6578.0, 11.0, 398600.4418], [15.562519, 7.778176, 2.996832, 38.9853]), ('hyperbolic departure from a parking orbit [1.0, 0.0, 1.0]', [1.0, 0.0, 1.0], [1.414214, 0.414214, 1.0, 180.0]), ('hyperbolic departure from a parking orbit [6678.0, 3.0, 398600.4418]', [6678.0, 3.0, 398600.4418], [11.330366, 3.604526, 1.150783, 120.6792]), ('hyperbolic departure from a parking orbit [6678.0, 0.5, 398600.4418]', [6678.0, 0.5, 398600.4418], [10.937422, 3.211582, 1.004188, 169.5303]), ('hyperbolic departure from a parking orbit [7000.0, 6.0, 398600.4418]', [7000.0, 6.0, 398600.4418], [12.242787, 4.696734, 1.632212, 75.565])]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
hyperbolic departure from a parking orbit [6678.0, 3.0, 398600.4418][8.287858, 0.562019, 1.150783, 120.6792][11.330366, 3.604526, 1.150783, 120.6792]Failed
hyperbolic departure from a parking orbit [6678.0, 0.5, 398600.4418][7.742002, 0.016163, 1.004188, 169.5303][10.937422, 3.211582, 1.004188, 169.5303]Failed
hyperbolic departure from a parking orbit [7000.0, 6.0, 398600.4418][9.640691, 2.094638, 1.632212, 75.565][12.242787, 4.696734, 1.632212, 75.565]Failed
hyperbolic departure from a parking orbit [42164.0, 1.5, 398600.4418][3.421048, 0.346382, 1.238005, 107.7536][4.59969, 1.525023, 1.238005, 107.7536]Failed
hyperbolic departure from a parking orbit [1.0, 1.0, 1.0][1.414214, 0.414214, 2.0, 60.0][1.732051, 0.732051, 2.0, 60.0]Failed
hyperbolic departure from a parking orbit [2.0, 0.3, 1.0][0.768115, 0.061008, 1.18, 115.8724][1.044031, 0.336924, 1.18, 115.8724]Failed
hyperbolic departure from a parking orbit [6578.0, 11.0, 398600.4418][13.475756, 5.691413, 2.996832, 38.9853][15.562519, 7.778176, 2.996832, 38.9853]Failed

SHA-256 / ba000403c71f28e1a25cdd9ac5a1203cbfbd1bd816a6fd0743f28ea0d54716d7

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    r,vinf,mu=x
    vp=vinf+math.sqrt(2*mu/r)
    dv=vp-math.sqrt(mu/r)
    e=1+r*vinf*vinf/mu
    delta=math.degrees(2*math.asin(1/e))
    return [round(vp,6),round(dv,6),round(e,6),round(delta,4)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('hyperbolic departure from a parking orbit [6678.0, 3.0, 398600.4418]', [6678.0, 3.0, 398600.4418], [11.330366, 3.604526, 1.150783, 120.6792]), ('hyperbolic departure from a parking orbit [6678.0, 0.5, 398600.4418]', [6678.0, 0.5, 398600.4418], [10.937422, 3.211582, 1.004188, 169.5303]), ('hyperbolic departure from a parking orbit [7000.0, 6.0, 398600.4418]', [7000.0, 6.0, 398600.4418], [12.242787, 4.696734, 1.632212, 75.565]), ('hyperbolic departure from a parking orbit [42164.0, 1.5, 398600.4418]', [42164.0, 1.5, 398600.4418], [4.59969, 1.525023, 1.238005, 107.7536]), ('hyperbolic departure from a parking orbit [1.0, 1.0, 1.0]', [1.0, 1.0, 1.0], [1.732051, 0.732051, 2.0, 60.0]), ('hyperbolic departure from a parking orbit [2.0, 0.3, 1.0]', [2.0, 0.3, 1.0], [1.044031, 0.336924, 1.18, 115.8724]), ('hyperbolic departure from a parking orbit [6578.0, 11.0, 398600.4418]', [6578.0, 11.0, 398600.4418], [15.562519, 7.778176, 2.996832, 38.9853])], [('hyperbolic departure from a parking orbit [6678.0, 0.5, 398600.4418]', [6678.0, 0.5, 398600.4418], [10.937422, 3.211582, 1.004188, 169.5303]), ('hyperbolic departure from a parking orbit [42164.0, 1.5, 398600.4418]', [42164.0, 1.5, 398600.4418], [4.59969, 1.525023, 1.238005, 107.7536]), ('hyperbolic departure from a parking orbit [1.0, 1.0, 1.0]', [1.0, 1.0, 1.0], [1.732051, 0.732051, 2.0, 60.0]), ('hyperbolic departure from a parking orbit [2.0, 0.3, 1.0]', [2.0, 0.3, 1.0], [1.044031, 0.336924, 1.18, 115.8724]), ('hyperbolic departure from a parking orbit [6578.0, 11.0, 398600.4418]', [6578.0, 11.0, 398600.4418], [15.562519, 7.778176, 2.996832, 38.9853]), ('hyperbolic departure from a parking orbit [1.0, 0.0, 1.0]', [1.0, 0.0, 1.0], [1.414214, 0.414214, 1.0, 180.0]), ('hyperbolic departure from a parking orbit [6678.0, 3.0, 398600.4418]', [6678.0, 3.0, 398600.4418], [11.330366, 3.604526, 1.150783, 120.6792])], [('hyperbolic departure from a parking orbit [7000.0, 6.0, 398600.4418]', [7000.0, 6.0, 398600.4418], [12.242787, 4.696734, 1.632212, 75.565]), ('hyperbolic departure from a parking orbit [6578.0, 11.0, 398600.4418]', [6578.0, 11.0, 398600.4418], [15.562519, 7.778176, 2.996832, 38.9853]), ('hyperbolic departure from a parking orbit [1.0, 0.0, 1.0]', [1.0, 0.0, 1.0], [1.414214, 0.414214, 1.0, 180.0]), ('hyperbolic departure from a parking orbit [6678.0, 3.0, 398600.4418]', [6678.0, 3.0, 398600.4418], [11.330366, 3.604526, 1.150783, 120.6792]), ('hyperbolic departure from a parking orbit [6678.0, 0.5, 398600.4418]', [6678.0, 0.5, 398600.4418], [10.937422, 3.211582, 1.004188, 169.5303]), ('hyperbolic departure from a parking orbit [42164.0, 1.5, 398600.4418]', [42164.0, 1.5, 398600.4418], [4.59969, 1.525023, 1.238005, 107.7536]), ('hyperbolic departure from a parking orbit [1.0, 1.0, 1.0]', [1.0, 1.0, 1.0], [1.732051, 0.732051, 2.0, 60.0])], [('hyperbolic departure from a parking orbit [42164.0, 1.5, 398600.4418]', [42164.0, 1.5, 398600.4418], [4.59969, 1.525023, 1.238005, 107.7536]), ('hyperbolic departure from a parking orbit [6678.0, 0.5, 398600.4418]', [6678.0, 0.5, 398600.4418], [10.937422, 3.211582, 1.004188, 169.5303]), ('hyperbolic departure from a parking orbit [7000.0, 6.0, 398600.4418]', [7000.0, 6.0, 398600.4418], [12.242787, 4.696734, 1.632212, 75.565]), ('hyperbolic departure from a parking orbit [1.0, 1.0, 1.0]', [1.0, 1.0, 1.0], [1.732051, 0.732051, 2.0, 60.0]), ('hyperbolic departure from a parking orbit [2.0, 0.3, 1.0]', [2.0, 0.3, 1.0], [1.044031, 0.336924, 1.18, 115.8724]), ('hyperbolic departure from a parking orbit [6578.0, 11.0, 398600.4418]', [6578.0, 11.0, 398600.4418], [15.562519, 7.778176, 2.996832, 38.9853]), ('hyperbolic departure from a parking orbit [1.0, 0.0, 1.0]', [1.0, 0.0, 1.0], [1.414214, 0.414214, 1.0, 180.0])], [('hyperbolic departure from a parking orbit [1.0, 1.0, 1.0]', [1.0, 1.0, 1.0], [1.732051, 0.732051, 2.0, 60.0]), ('hyperbolic departure from a parking orbit [2.0, 0.3, 1.0]', [2.0, 0.3, 1.0], [1.044031, 0.336924, 1.18, 115.8724]), ('hyperbolic departure from a parking orbit [6578.0, 11.0, 398600.4418]', [6578.0, 11.0, 398600.4418], [15.562519, 7.778176, 2.996832, 38.9853]), ('hyperbolic departure from a parking orbit [1.0, 0.0, 1.0]', [1.0, 0.0, 1.0], [1.414214, 0.414214, 1.0, 180.0]), ('hyperbolic departure from a parking orbit [6678.0, 3.0, 398600.4418]', [6678.0, 3.0, 398600.4418], [11.330366, 3.604526, 1.150783, 120.6792]), ('hyperbolic departure from a parking orbit [6678.0, 0.5, 398600.4418]', [6678.0, 0.5, 398600.4418], [10.937422, 3.211582, 1.004188, 169.5303]), ('hyperbolic departure from a parking orbit [7000.0, 6.0, 398600.4418]', [7000.0, 6.0, 398600.4418], [12.242787, 4.696734, 1.632212, 75.565])]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
Boundary fixtureActualExpectedOutcome
hyperbolic departure from a parking orbit [6678.0, 3.0, 398600.4418][13.925987, 6.200147, 1.150783, 120.6792][11.330366, 3.604526, 1.150783, 120.6792]Failed
hyperbolic departure from a parking orbit [6678.0, 0.5, 398600.4418][11.425987, 3.700147, 1.004188, 169.5303][10.937422, 3.211582, 1.004188, 169.5303]Failed
hyperbolic departure from a parking orbit [7000.0, 6.0, 398600.4418][16.671731, 9.125678, 1.632212, 75.565][12.242787, 4.696734, 1.632212, 75.565]Failed
hyperbolic departure from a parking orbit [42164.0, 1.5, 398600.4418][5.848235, 2.773568, 1.238005, 107.7536][4.59969, 1.525023, 1.238005, 107.7536]Failed
hyperbolic departure from a parking orbit [1.0, 1.0, 1.0][2.414214, 1.414214, 2.0, 60.0][1.732051, 0.732051, 2.0, 60.0]Failed
hyperbolic departure from a parking orbit [2.0, 0.3, 1.0][1.3, 0.592893, 1.18, 115.8724][1.044031, 0.336924, 1.18, 115.8724]Failed
hyperbolic departure from a parking orbit [6578.0, 11.0, 398600.4418][22.008723, 14.22438, 2.996832, 38.9853][15.562519, 7.778176, 2.996832, 38.9853]Failed

SHA-256 / bd1b4c72740618ec2a2b382d655d43a4fe3ca6537943106283094df9a10202db

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 7 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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Verification & scope

A deterministic toy two-body model with stipulated constants and conventions; not flight dynamics software or a validated SGP4 implementation. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:48:15.162667+00:00.

Case digest / 05e716349b47727063ff61683ac166045792ee0db4a8fc5f859098118c236234