FA-69811 / Orbital propagation / Open access

Co-orbital phasing manoeuvre: Phasing orbit speed uses swapped vis-viva radii · case 01

Burn magnitudes are wrong for every eccentric phasing orbit.

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

ROOT CAUSE

vis-viva is evaluated as mu(2/a - 1/r).

THE FAILURE

vis-viva is evaluated as mu(2/a - 1/r).

Unsuccessful approach: The circular speed at a_ph is not the speed at the apsis r.

Case contract

Input [r, theta, k, mu, rmin]: circular radius, target lead angle in degrees (positive = target ahead), k phasing revolutions. Phasing period Tph=T-theta_rad/(k n). a_ph from Tph; total dv = 2|v_ph(r)-v_circ| with vis-viva at r. Return None if the phasing perigee 2a_ph-r is below rmin, else [a_ph rounded 3, dv rounded 6, k*Tph rounded 3].

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,theta,k,mu,rmin=x
    n=math.sqrt(mu/r**3)
    T=2*math.pi/n
    Tph=T-math.radians(theta)/(k*n)
    aph=(mu*(Tph/(2*math.pi))**2)**(1/3)
    if 2*aph-r<rmin: return None
    vc=math.sqrt(mu/r)
    vph=math.sqrt(mu*(2/aph-1/r))
    dv=2*abs(vph-vc)
    return [round(aph,3),round(dv,6),round(k*Tph,3)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('co-orbital phasing manoeuvre [6778.0, 20.0, 3, 398600.4418, 6478.0]', [6778.0, 20.0, 3, 398600.4418, 6478.0], [6694.061, 0.096463, 16351.842]), ('co-orbital phasing manoeuvre [6778.0, 20.0, 1, 398600.4418, 6478.0]', [6778.0, 20.0, 1, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [6778.0, -30.0, 2, 398600.4418, 6478.0]', [6778.0, -30.0, 2, 398600.4418, 6478.0], [6964.994, 0.204521, 11569.7]), ('co-orbital phasing manoeuvre [42164.0, 10.0, 1, 398600.4418, 6478.0]', [42164.0, 10.0, 1, 398600.4418, 6478.0], [41379.525, 0.058569, 83770.138]), ('co-orbital phasing manoeuvre [42164.0, -45.0, 4, 398600.4418, 6478.0]', [42164.0, -45.0, 4, 398600.4418, 6478.0], [43037.904, 0.062119, 355424.729]), ('co-orbital phasing manoeuvre [6778.0, 90.0, 1, 398600.4418, 6478.0]', [6778.0, 90.0, 1, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [6778.0, 90.0, 5, 398600.4418, 6478.0]', [6778.0, 90.0, 5, 398600.4418, 6478.0], None)], [('co-orbital phasing manoeuvre [6778.0, -30.0, 2, 398600.4418, 6478.0]', [6778.0, -30.0, 2, 398600.4418, 6478.0], [6964.994, 0.204521, 11569.7]), ('co-orbital phasing manoeuvre [42164.0, 10.0, 1, 398600.4418, 6478.0]', [42164.0, 10.0, 1, 398600.4418, 6478.0], [41379.525, 0.058569, 83770.138]), ('co-orbital phasing manoeuvre [42164.0, -45.0, 4, 398600.4418, 6478.0]', [42164.0, -45.0, 4, 398600.4418, 6478.0], [43037.904, 0.062119, 355424.729]), ('co-orbital phasing manoeuvre [6778.0, 90.0, 1, 398600.4418, 6478.0]', [6778.0, 90.0, 1, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [6778.0, 90.0, 5, 398600.4418, 6478.0]', [6778.0, 90.0, 5, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [1.0, 60.0, 2, 1.0, 0.5]', [1.0, 60.0, 2, 1.0, 0.5], [0.944, 0.060642, 11.519]), ('co-orbital phasing manoeuvre [1.0, -60.0, 2, 1.0, 0.5]', [1.0, -60.0, 2, 1.0, 0.5], [1.055, 0.051305, 13.614])], [('co-orbital phasing manoeuvre [42164.0, 10.0, 1, 398600.4418, 6478.0]', [42164.0, 10.0, 1, 398600.4418, 6478.0], [41379.525, 0.058569, 83770.138]), ('co-orbital phasing manoeuvre [6778.0, 90.0, 5, 398600.4418, 6478.0]', [6778.0, 90.0, 5, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [1.0, 60.0, 2, 1.0, 0.5]', [1.0, 60.0, 2, 1.0, 0.5], [0.944, 0.060642, 11.519]), ('co-orbital phasing manoeuvre [1.0, -60.0, 2, 1.0, 0.5]', [1.0, -60.0, 2, 1.0, 0.5], [1.055, 0.051305, 13.614]), ('co-orbital phasing manoeuvre [1.0, 170.0, 1, 1.0, 0.1]', [1.0, 170.0, 1, 1.0, 0.1], [0.653, 0.630624, 3.316]), ('co-orbital phasing manoeuvre [6778.0, 20.0, 1, 398600.4418, 6478.0]', [6778.0, 20.0, 1, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [6778.0, 20.0, 3, 398600.4418, 6478.0]', [6778.0, 20.0, 3, 398600.4418, 6478.0], [6694.061, 0.096463, 16351.842])], [('co-orbital phasing manoeuvre [42164.0, -45.0, 4, 398600.4418, 6478.0]', [42164.0, -45.0, 4, 398600.4418, 6478.0], [43037.904, 0.062119, 355424.729]), ('co-orbital phasing manoeuvre [1.0, 170.0, 1, 1.0, 0.1]', [1.0, 170.0, 1, 1.0, 0.1], [0.653, 0.630624, 3.316]), ('co-orbital phasing manoeuvre [6778.0, 20.0, 1, 398600.4418, 6478.0]', [6778.0, 20.0, 1, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [6778.0, 20.0, 3, 398600.4418, 6478.0]', [6778.0, 20.0, 3, 398600.4418, 6478.0], [6694.061, 0.096463, 16351.842]), ('co-orbital phasing manoeuvre [6778.0, -30.0, 2, 398600.4418, 6478.0]', [6778.0, -30.0, 2, 398600.4418, 6478.0], [6964.994, 0.204521, 11569.7]), ('co-orbital phasing manoeuvre [42164.0, 10.0, 1, 398600.4418, 6478.0]', [42164.0, 10.0, 1, 398600.4418, 6478.0], [41379.525, 0.058569, 83770.138]), ('co-orbital phasing manoeuvre [6778.0, 90.0, 1, 398600.4418, 6478.0]', [6778.0, 90.0, 1, 398600.4418, 6478.0], None)], [('co-orbital phasing manoeuvre [1.0, 60.0, 2, 1.0, 0.5]', [1.0, 60.0, 2, 1.0, 0.5], [0.944, 0.060642, 11.519]), ('co-orbital phasing manoeuvre [6778.0, -30.0, 2, 398600.4418, 6478.0]', [6778.0, -30.0, 2, 398600.4418, 6478.0], [6964.994, 0.204521, 11569.7]), ('co-orbital phasing manoeuvre [42164.0, 10.0, 1, 398600.4418, 6478.0]', [42164.0, 10.0, 1, 398600.4418, 6478.0], [41379.525, 0.058569, 83770.138]), ('co-orbital phasing manoeuvre [42164.0, -45.0, 4, 398600.4418, 6478.0]', [42164.0, -45.0, 4, 398600.4418, 6478.0], [43037.904, 0.062119, 355424.729]), ('co-orbital phasing manoeuvre [6778.0, 90.0, 1, 398600.4418, 6478.0]', [6778.0, 90.0, 1, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [6778.0, 90.0, 5, 398600.4418, 6478.0]', [6778.0, 90.0, 5, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [1.0, -60.0, 2, 1.0, 0.5]', [1.0, -60.0, 2, 1.0, 0.5], [1.055, 0.051305, 13.614])]]
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 fixture	Actual	Expected	Outcome
co-orbital phasing manoeuvre [6778.0, 20.0, 3, 398600.4418, 6478.0]	`[6694.061, 0.191129, 16351.842]`	`[6694.061, 0.096463, 16351.842]`	Failed
co-orbital phasing manoeuvre [6778.0, 20.0, 1, 398600.4418, 6478.0]	`None`	`None`	Passed
co-orbital phasing manoeuvre [6778.0, -30.0, 2, 398600.4418, 6478.0]	`[6964.994, 0.417451, 11569.7]`	`[6964.994, 0.204521, 11569.7]`	Failed
co-orbital phasing manoeuvre [42164.0, 10.0, 1, 398600.4418, 6478.0]	`[41379.525, 0.115495, 83770.138]`	`[41379.525, 0.058569, 83770.138]`	Failed
co-orbital phasing manoeuvre [42164.0, -45.0, 4, 398600.4418, 6478.0]	`[43037.904, 0.126159, 355424.729]`	`[43037.904, 0.062119, 355424.729]`	Failed
co-orbital phasing manoeuvre [6778.0, 90.0, 1, 398600.4418, 6478.0]	`None`	`None`	Passed
co-orbital phasing manoeuvre [6778.0, 90.0, 5, 398600.4418, 6478.0]	`None`	`None`	Passed

SHA-256 / 0a2267708ef40ca367b9b66932b2b9f462140840d91a75d8e96eebc13305a206

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,theta,k,mu,rmin=x
    n=math.sqrt(mu/r**3)
    T=2*math.pi/n
    Tph=T-math.radians(theta)/(k*n)
    aph=(mu*(Tph/(2*math.pi))**2)**(1/3)
    if 2*aph-r<rmin: return None
    vc=math.sqrt(mu/r)
    vph=math.sqrt(mu/aph)
    dv=2*abs(vph-vc)
    return [round(aph,3),round(dv,6),round(k*Tph,3)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[('co-orbital phasing manoeuvre [6778.0, 20.0, 3, 398600.4418, 6478.0]', [6778.0, 20.0, 3, 398600.4418, 6478.0], [6694.061, 0.096463, 16351.842]), ('co-orbital phasing manoeuvre [6778.0, 20.0, 1, 398600.4418, 6478.0]', [6778.0, 20.0, 1, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [6778.0, -30.0, 2, 398600.4418, 6478.0]', [6778.0, -30.0, 2, 398600.4418, 6478.0], [6964.994, 0.204521, 11569.7]), ('co-orbital phasing manoeuvre [42164.0, 10.0, 1, 398600.4418, 6478.0]', [42164.0, 10.0, 1, 398600.4418, 6478.0], [41379.525, 0.058569, 83770.138]), ('co-orbital phasing manoeuvre [42164.0, -45.0, 4, 398600.4418, 6478.0]', [42164.0, -45.0, 4, 398600.4418, 6478.0], [43037.904, 0.062119, 355424.729]), ('co-orbital phasing manoeuvre [6778.0, 90.0, 1, 398600.4418, 6478.0]', [6778.0, 90.0, 1, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [6778.0, 90.0, 5, 398600.4418, 6478.0]', [6778.0, 90.0, 5, 398600.4418, 6478.0], None)], [('co-orbital phasing manoeuvre [6778.0, -30.0, 2, 398600.4418, 6478.0]', [6778.0, -30.0, 2, 398600.4418, 6478.0], [6964.994, 0.204521, 11569.7]), ('co-orbital phasing manoeuvre [42164.0, 10.0, 1, 398600.4418, 6478.0]', [42164.0, 10.0, 1, 398600.4418, 6478.0], [41379.525, 0.058569, 83770.138]), ('co-orbital phasing manoeuvre [42164.0, -45.0, 4, 398600.4418, 6478.0]', [42164.0, -45.0, 4, 398600.4418, 6478.0], [43037.904, 0.062119, 355424.729]), ('co-orbital phasing manoeuvre [6778.0, 90.0, 1, 398600.4418, 6478.0]', [6778.0, 90.0, 1, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [6778.0, 90.0, 5, 398600.4418, 6478.0]', [6778.0, 90.0, 5, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [1.0, 60.0, 2, 1.0, 0.5]', [1.0, 60.0, 2, 1.0, 0.5], [0.944, 0.060642, 11.519]), ('co-orbital phasing manoeuvre [1.0, -60.0, 2, 1.0, 0.5]', [1.0, -60.0, 2, 1.0, 0.5], [1.055, 0.051305, 13.614])], [('co-orbital phasing manoeuvre [42164.0, 10.0, 1, 398600.4418, 6478.0]', [42164.0, 10.0, 1, 398600.4418, 6478.0], [41379.525, 0.058569, 83770.138]), ('co-orbital phasing manoeuvre [6778.0, 90.0, 5, 398600.4418, 6478.0]', [6778.0, 90.0, 5, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [1.0, 60.0, 2, 1.0, 0.5]', [1.0, 60.0, 2, 1.0, 0.5], [0.944, 0.060642, 11.519]), ('co-orbital phasing manoeuvre [1.0, -60.0, 2, 1.0, 0.5]', [1.0, -60.0, 2, 1.0, 0.5], [1.055, 0.051305, 13.614]), ('co-orbital phasing manoeuvre [1.0, 170.0, 1, 1.0, 0.1]', [1.0, 170.0, 1, 1.0, 0.1], [0.653, 0.630624, 3.316]), ('co-orbital phasing manoeuvre [6778.0, 20.0, 1, 398600.4418, 6478.0]', [6778.0, 20.0, 1, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [6778.0, 20.0, 3, 398600.4418, 6478.0]', [6778.0, 20.0, 3, 398600.4418, 6478.0], [6694.061, 0.096463, 16351.842])], [('co-orbital phasing manoeuvre [42164.0, -45.0, 4, 398600.4418, 6478.0]', [42164.0, -45.0, 4, 398600.4418, 6478.0], [43037.904, 0.062119, 355424.729]), ('co-orbital phasing manoeuvre [1.0, 170.0, 1, 1.0, 0.1]', [1.0, 170.0, 1, 1.0, 0.1], [0.653, 0.630624, 3.316]), ('co-orbital phasing manoeuvre [6778.0, 20.0, 1, 398600.4418, 6478.0]', [6778.0, 20.0, 1, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [6778.0, 20.0, 3, 398600.4418, 6478.0]', [6778.0, 20.0, 3, 398600.4418, 6478.0], [6694.061, 0.096463, 16351.842]), ('co-orbital phasing manoeuvre [6778.0, -30.0, 2, 398600.4418, 6478.0]', [6778.0, -30.0, 2, 398600.4418, 6478.0], [6964.994, 0.204521, 11569.7]), ('co-orbital phasing manoeuvre [42164.0, 10.0, 1, 398600.4418, 6478.0]', [42164.0, 10.0, 1, 398600.4418, 6478.0], [41379.525, 0.058569, 83770.138]), ('co-orbital phasing manoeuvre [6778.0, 90.0, 1, 398600.4418, 6478.0]', [6778.0, 90.0, 1, 398600.4418, 6478.0], None)], [('co-orbital phasing manoeuvre [1.0, 60.0, 2, 1.0, 0.5]', [1.0, 60.0, 2, 1.0, 0.5], [0.944, 0.060642, 11.519]), ('co-orbital phasing manoeuvre [6778.0, -30.0, 2, 398600.4418, 6478.0]', [6778.0, -30.0, 2, 398600.4418, 6478.0], [6964.994, 0.204521, 11569.7]), ('co-orbital phasing manoeuvre [42164.0, 10.0, 1, 398600.4418, 6478.0]', [42164.0, 10.0, 1, 398600.4418, 6478.0], [41379.525, 0.058569, 83770.138]), ('co-orbital phasing manoeuvre [42164.0, -45.0, 4, 398600.4418, 6478.0]', [42164.0, -45.0, 4, 398600.4418, 6478.0], [43037.904, 0.062119, 355424.729]), ('co-orbital phasing manoeuvre [6778.0, 90.0, 1, 398600.4418, 6478.0]', [6778.0, 90.0, 1, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [6778.0, 90.0, 5, 398600.4418, 6478.0]', [6778.0, 90.0, 5, 398600.4418, 6478.0], None), ('co-orbital phasing manoeuvre [1.0, -60.0, 2, 1.0, 0.5]', [1.0, -60.0, 2, 1.0, 0.5], [1.055, 0.051305, 13.614])]]
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 fixture	Actual	Expected	Outcome
co-orbital phasing manoeuvre [6778.0, 20.0, 3, 398600.4418, 6478.0]	`[6694.061, 0.09586, 16351.842]`	`[6694.061, 0.096463, 16351.842]`	Failed
co-orbital phasing manoeuvre [6778.0, 20.0, 1, 398600.4418, 6478.0]	`None`	`None`	Passed
co-orbital phasing manoeuvre [6778.0, -30.0, 2, 398600.4418, 6478.0]	`[6964.994, 0.207286, 11569.7]`	`[6964.994, 0.204521, 11569.7]`	Failed
co-orbital phasing manoeuvre [42164.0, 10.0, 1, 398600.4418, 6478.0]	`[41379.525, 0.058016, 83770.138]`	`[41379.525, 0.058569, 83770.138]`	Failed
co-orbital phasing manoeuvre [42164.0, -45.0, 4, 398600.4418, 6478.0]	`[43037.904, 0.062753, 355424.729]`	`[43037.904, 0.062119, 355424.729]`	Failed
co-orbital phasing manoeuvre [6778.0, 90.0, 1, 398600.4418, 6478.0]	`None`	`None`	Passed
co-orbital phasing manoeuvre [6778.0, 90.0, 5, 398600.4418, 6478.0]	`None`	`None`	Passed

SHA-256 / 2c9e42fc70fbdc57d8cbbd070a466bcf73a3d3c760df9fafba66c1318bee30e2

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.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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 2026-09-29T14:48:14.764453+00:00.

Case digest / 04456464ddb7f25765d29706bf096e84e82c0f8ebe6959f761c5aa78419b08a5