FAILURE MAP
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FA-69376 / Orbital propagation / Open access

True anomaly from eccentric anomaly: Full angles are used in the half-angle formula · case 01

True anomaly runs twice as fast as the orbit.

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

ROOT CAUSE

The formula uses sin(E) and cos(E) while still doubling the result.

THE FAILURE

The formula uses sin(E) and cos(E) while still doubling the result.

Unsuccessful approach: Using half angles but dropping the doubling returns nu/2.

Case contract

Input [E, e] (radians, 0<=e<1, E any real). Return [nu, r/a] with nu = 2*atan2(sqrt(1+e) sin(E/2), sqrt(1-e) cos(E/2)) reduced to [0,2pi) and r/a = 1-e cos E, both rounded to 9 decimals.

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):
    E,e=x
    nu=2*math.atan2(math.sqrt(1+e)*math.sin(E),math.sqrt(1-e)*math.cos(E))
    nu%=2*math.pi
    return [round(nu,9),round(1-e*math.cos(E),9)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('true anomaly from eccentric anomaly [0.4, 0.1]', [0.4, 0.1], [0.440923623, 0.907893901]), ('true anomaly from eccentric anomaly [2.5, 0.3]', [2.5, 0.3], [2.663281246, 1.240343085]), ('true anomaly from eccentric anomaly [4.0, 0.3]', [4.0, 0.3], [3.789582293, 1.196093086]), ('true anomaly from eccentric anomaly [5.9, 0.7]', [5.9, 0.7], [5.418028431, 0.350765098]), ('true anomaly from eccentric anomaly [3.141592653589793, 0.5]', [3.141592653589793, 0.5], [3.141592654, 1.5]), ('true anomaly from eccentric anomaly [0.0, 0.9]', [0.0, 0.9], [0.0, 0.1]), ('true anomaly from eccentric anomaly [7.0, 0.2]', [7.0, 0.2], [0.860249203, 0.849219549])], [('true anomaly from eccentric anomaly [2.5, 0.3]', [2.5, 0.3], [2.663281246, 1.240343085]), ('true anomaly from eccentric anomaly [5.9, 0.7]', [5.9, 0.7], [5.418028431, 0.350765098]), ('true anomaly from eccentric anomaly [3.141592653589793, 0.5]', [3.141592653589793, 0.5], [3.141592654, 1.5]), ('true anomaly from eccentric anomaly [0.0, 0.9]', [0.0, 0.9], [0.0, 0.1]), ('true anomaly from eccentric anomaly [7.0, 0.2]', [7.0, 0.2], [0.860249203, 0.849219549]), ('true anomaly from eccentric anomaly [-1.0, 0.4]', [-1.0, 0.4], [4.892343298, 0.783879078]), ('true anomaly from eccentric anomaly [-4.0, 0.6]', [-4.0, 0.6], [2.691681202, 1.392186173])], [('true anomaly from eccentric anomaly [4.0, 0.3]', [4.0, 0.3], [3.789582293, 1.196093086]), ('true anomaly from eccentric anomaly [7.0, 0.2]', [7.0, 0.2], [0.860249203, 0.849219549]), ('true anomaly from eccentric anomaly [-1.0, 0.4]', [-1.0, 0.4], [4.892343298, 0.783879078]), ('true anomaly from eccentric anomaly [-4.0, 0.6]', [-4.0, 0.6], [2.691681202, 1.392186173]), ('true anomaly from eccentric anomaly [1.2, 0.0]', [1.2, 0.0], [1.2, 1.0]), ('true anomaly from eccentric anomaly [3.5, 0.95]', [3.5, 0.95], [3.199589845, 1.889633853]), ('true anomaly from eccentric anomaly [12.0, 0.15]', [12.0, 0.15], [5.630422415, 0.873421906])], [('true anomaly from eccentric anomaly [5.9, 0.7]', [5.9, 0.7], [5.418028431, 0.350765098]), ('true anomaly from eccentric anomaly [1.2, 0.0]', [1.2, 0.0], [1.2, 1.0]), ('true anomaly from eccentric anomaly [3.5, 0.95]', [3.5, 0.95], [3.199589845, 1.889633853]), ('true anomaly from eccentric anomaly [12.0, 0.15]', [12.0, 0.15], [5.630422415, 0.873421906]), ('true anomaly from eccentric anomaly [0.4, 0.1]', [0.4, 0.1], [0.440923623, 0.907893901]), ('true anomaly from eccentric anomaly [2.5, 0.3]', [2.5, 0.3], [2.663281246, 1.240343085]), ('true anomaly from eccentric anomaly [4.0, 0.3]', [4.0, 0.3], [3.789582293, 1.196093086])], [('true anomaly from eccentric anomaly [3.141592653589793, 0.5]', [3.141592653589793, 0.5], [3.141592654, 1.5]), ('true anomaly from eccentric anomaly [0.4, 0.1]', [0.4, 0.1], [0.440923623, 0.907893901]), ('true anomaly from eccentric anomaly [2.5, 0.3]', [2.5, 0.3], [2.663281246, 1.240343085]), ('true anomaly from eccentric anomaly [4.0, 0.3]', [4.0, 0.3], [3.789582293, 1.196093086]), ('true anomaly from eccentric anomaly [5.9, 0.7]', [5.9, 0.7], [5.418028431, 0.350765098]), ('true anomaly from eccentric anomaly [0.0, 0.9]', [0.0, 0.9], [0.0, 0.1]), ('true anomaly from eccentric anomaly [7.0, 0.2]', [7.0, 0.2], [0.860249203, 0.849219549])]]
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
true anomaly from eccentric anomaly [0.4, 0.1][0.874483786, 0.907893901][0.440923623, 0.907893901]Failed
true anomaly from eccentric anomaly [2.5, 0.3][4.694530571, 1.240343085][2.663281246, 1.240343085]Failed
true anomaly from eccentric anomaly [4.0, 0.3][2.011822058, 1.196093086][3.789582293, 1.196093086]Failed
true anomaly from eccentric anomaly [5.9, 0.7][4.753620334, 0.350765098][5.418028431, 0.350765098]Failed
true anomaly from eccentric anomaly [3.141592653589793, 0.5][0.0, 1.5][3.141592654, 1.5]Failed
true anomaly from eccentric anomaly [0.0, 0.9][0.0, 0.1][0.0, 0.1]Passed
true anomaly from eccentric anomaly [7.0, 0.2][1.635883773, 0.849219549][0.860249203, 0.849219549]Failed

SHA-256 / 5b54e38e057b31bce163134a8b553ed5ea23795dd8fb7719502821ed7172b347

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    E,e=x
    nu=math.atan2(math.sqrt(1+e)*math.sin(E/2),math.sqrt(1-e)*math.cos(E/2))
    nu%=2*math.pi
    return [round(nu,9),round(1-e*math.cos(E),9)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('true anomaly from eccentric anomaly [0.4, 0.1]', [0.4, 0.1], [0.440923623, 0.907893901]), ('true anomaly from eccentric anomaly [2.5, 0.3]', [2.5, 0.3], [2.663281246, 1.240343085]), ('true anomaly from eccentric anomaly [4.0, 0.3]', [4.0, 0.3], [3.789582293, 1.196093086]), ('true anomaly from eccentric anomaly [5.9, 0.7]', [5.9, 0.7], [5.418028431, 0.350765098]), ('true anomaly from eccentric anomaly [3.141592653589793, 0.5]', [3.141592653589793, 0.5], [3.141592654, 1.5]), ('true anomaly from eccentric anomaly [0.0, 0.9]', [0.0, 0.9], [0.0, 0.1]), ('true anomaly from eccentric anomaly [7.0, 0.2]', [7.0, 0.2], [0.860249203, 0.849219549])], [('true anomaly from eccentric anomaly [2.5, 0.3]', [2.5, 0.3], [2.663281246, 1.240343085]), ('true anomaly from eccentric anomaly [5.9, 0.7]', [5.9, 0.7], [5.418028431, 0.350765098]), ('true anomaly from eccentric anomaly [3.141592653589793, 0.5]', [3.141592653589793, 0.5], [3.141592654, 1.5]), ('true anomaly from eccentric anomaly [0.0, 0.9]', [0.0, 0.9], [0.0, 0.1]), ('true anomaly from eccentric anomaly [7.0, 0.2]', [7.0, 0.2], [0.860249203, 0.849219549]), ('true anomaly from eccentric anomaly [-1.0, 0.4]', [-1.0, 0.4], [4.892343298, 0.783879078]), ('true anomaly from eccentric anomaly [-4.0, 0.6]', [-4.0, 0.6], [2.691681202, 1.392186173])], [('true anomaly from eccentric anomaly [4.0, 0.3]', [4.0, 0.3], [3.789582293, 1.196093086]), ('true anomaly from eccentric anomaly [7.0, 0.2]', [7.0, 0.2], [0.860249203, 0.849219549]), ('true anomaly from eccentric anomaly [-1.0, 0.4]', [-1.0, 0.4], [4.892343298, 0.783879078]), ('true anomaly from eccentric anomaly [-4.0, 0.6]', [-4.0, 0.6], [2.691681202, 1.392186173]), ('true anomaly from eccentric anomaly [1.2, 0.0]', [1.2, 0.0], [1.2, 1.0]), ('true anomaly from eccentric anomaly [3.5, 0.95]', [3.5, 0.95], [3.199589845, 1.889633853]), ('true anomaly from eccentric anomaly [12.0, 0.15]', [12.0, 0.15], [5.630422415, 0.873421906])], [('true anomaly from eccentric anomaly [5.9, 0.7]', [5.9, 0.7], [5.418028431, 0.350765098]), ('true anomaly from eccentric anomaly [1.2, 0.0]', [1.2, 0.0], [1.2, 1.0]), ('true anomaly from eccentric anomaly [3.5, 0.95]', [3.5, 0.95], [3.199589845, 1.889633853]), ('true anomaly from eccentric anomaly [12.0, 0.15]', [12.0, 0.15], [5.630422415, 0.873421906]), ('true anomaly from eccentric anomaly [0.4, 0.1]', [0.4, 0.1], [0.440923623, 0.907893901]), ('true anomaly from eccentric anomaly [2.5, 0.3]', [2.5, 0.3], [2.663281246, 1.240343085]), ('true anomaly from eccentric anomaly [4.0, 0.3]', [4.0, 0.3], [3.789582293, 1.196093086])], [('true anomaly from eccentric anomaly [3.141592653589793, 0.5]', [3.141592653589793, 0.5], [3.141592654, 1.5]), ('true anomaly from eccentric anomaly [0.4, 0.1]', [0.4, 0.1], [0.440923623, 0.907893901]), ('true anomaly from eccentric anomaly [2.5, 0.3]', [2.5, 0.3], [2.663281246, 1.240343085]), ('true anomaly from eccentric anomaly [4.0, 0.3]', [4.0, 0.3], [3.789582293, 1.196093086]), ('true anomaly from eccentric anomaly [5.9, 0.7]', [5.9, 0.7], [5.418028431, 0.350765098]), ('true anomaly from eccentric anomaly [0.0, 0.9]', [0.0, 0.9], [0.0, 0.1]), ('true anomaly from eccentric anomaly [7.0, 0.2]', [7.0, 0.2], [0.860249203, 0.849219549])]]
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
true anomaly from eccentric anomaly [0.4, 0.1][0.220461812, 0.907893901][0.440923623, 0.907893901]Failed
true anomaly from eccentric anomaly [2.5, 0.3][1.331640623, 1.240343085][2.663281246, 1.240343085]Failed
true anomaly from eccentric anomaly [4.0, 0.3][1.894791146, 1.196093086][3.789582293, 1.196093086]Failed
true anomaly from eccentric anomaly [5.9, 0.7][2.709014216, 0.350765098][5.418028431, 0.350765098]Failed
true anomaly from eccentric anomaly [3.141592653589793, 0.5][1.570796327, 1.5][3.141592654, 1.5]Failed
true anomaly from eccentric anomaly [0.0, 0.9][0.0, 0.1][0.0, 0.1]Passed
true anomaly from eccentric anomaly [7.0, 0.2][3.571717255, 0.849219549][0.860249203, 0.849219549]Failed

SHA-256 / 272df4e5247963dfdaba04c3512cd94b377f2973e7f5079f6c410bca5258f421

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 2026-09-29T14:48:10.829885+00:00.

Case digest / 9d08ce37abfd5da09218e9511cfd1310ff3398cd91b26c20733f20b85e8d9755