Conjunction Assessment

Conjunction Assessment (space-systems/orbit-mechanics/conjunction-assessment)

Use when you must assess the collision risk between two space objects at close approach, for example an operational spacecraft screened against a debris object or a secondary spacecraft. From the relative position and velocity at the screening epoch this leaf computes the time of closest approach and the miss distance under a linear relative-motion model, projects the combined 1-sigma position uncertainty onto the encounter plane, estimates the probability of collision with a small hard-body approximation against the combined object radius, and returns the actionable verdict against a screen threshold. It pairs with space-systems/mission-design/radiation-debris for the long-term environment risk counterpart and gnc-autonomy/space/orbit-determination for the state source of the primary; the relative states themselves can come from a propagation of the two orbits.

Domain quick reference

• Time of closest approach: tca = -dot(r, v) / dot(v, v) when the relative velocity is constant, with r the relative position and v the relative velocity at the screening epoch. A closing geometry gives a positive TCA; a receding geometry gives a negative TCA, meaning closest approach already passed. Zero relative velocity is rejected.
• Miss distance: d_miss = |r + v * tca|, the norm of the relative position at the TCA epoch. The miss vector is orthogonal to the relative velocity at TCA, dot(r + v * tca, v) = 0.
• Encounter-plane sigma: sigma_enc = sigma_combined, the combined 1-sigma position uncertainty of both objects under the circular covariance approximation. A full 3x3 covariance projection onto the encounter plane is out of scope for this screening model.
• Probability of collision: Pc = exp(-d_miss^2 / (2 * sigma^2)) * (r_hb^2 / (2 * sigma^2)), the small hard-body approximation to the 2D Gaussian encounter integral with r_hb the combined radius of the two objects. The approximation is valid when r_hb / sigma is at most about 0.1; above that limit the value is only a rough screen indicator and analyze() reports valid_approximation False.
• Screen verdict: actionable when Pc >= threshold (default 1e-4); severity "high" when Pc >= 1e-3, "watch" when Pc >= 1e-4, else "green".
• Units are SI throughout: m, m/s, s; Pc is dimensionless.
• ECSS frames the space safety context of the screen; the relations above are standard engineering methodology, summary-only.

Workflow

1. Assemble the screening state: the relative position rel_pos_m and the relative velocity rel_vel_ms of the secondary in the primary-centered frame at the epoch, the combined 1-sigma position uncertainty sigma_combined_m, and the combined object radius hard_body_radius_m (default 5 m).
2. Compute the time of closest approach with tca_s; check the sign to confirm whether the encounter lies ahead of the epoch.
3. Evaluate the miss distance with miss_distance_m at that TCA and the encounter-plane sigma with encounter_sigma.
4. Estimate the probability of collision with probability_of_collision on the miss distance, sigma, and hard body radius.
5. Screen the result with screen_verdict against the actionable threshold (default 1e-4) to get the high, watch or green severity.
6. Run the full screen in one call with analyze, which returns the TCA, miss distance, sigma, Pc, actionable flag, severity, and the valid_approximation flag for the small hard-body model.
7. Confirm the deterministic checks with the contract test scripts/test_conjunction_assessment.py.

Worked example

A spacecraft is screened against a debris object. At the screening epoch the relative position is [5000, -3000, 2000] m and the relative velocity [-7.0, 1.0, -0.5] m/s (closing).

• dot(r, v) = -39000, dot(v, v) = 50.25, so tca = 39000 / 50.25 = 776.1 s (analyze returns 776.12 s).
• Miss: r + v * tca = [-432.8, -2223.9, 1611.9], so the miss distance is sqrt(7731268) = 2780.5 m (analyze returns 2780.53 m).
• Combined sigma 100 m, hard body radius 5 m: the exponent -2780.5^2 / 20000 is huge negative, so Pc is effectively zero (analyze returns 1.6e-171, below 1e-12), actionable False, severity "green", valid_approximation True.
• Near-miss case: relative position [10, 0, 0] m closing at 1 m/s gives tca 10 s, miss 0 m, and Pc = 1 * 25 / 20000 = 1.25e-3, actionable True, severity "high".
• Offset case: a 50 m miss with sigma 100 m and hard body 5 m gives Pc = exp(-2500 / 20000) * 0.00125 = 1.103e-3, actionable True, severity "high". The offset is realized with the relative position [0, 50, 0] m perpendicular to the closing velocity [-1, 0, 0] m/s (a closing velocity aligned with the relative position vector drives the miss to zero at TCA, so the perpendicular geometry is the one that produces a 50 m miss).
• Validity flag: hard body 15 m with sigma 100 m gives r_hb / sigma = 0.15 above the 0.1 limit, so valid_approximation is False while Pc still evaluates as a rough screen indicator.

Pitfalls

• Running the screen on a receding pair: a negative TCA means the closest approach already passed the screening epoch; treat it as a stale geometry rather than a live risk before acting on the verdict.
• Trusting Pc beyond the model's validity: the small hard-body approximation holds only for r_hb / sigma at most about 0.1; above that (a 15 m hard body against 100 m sigma) analyze reports valid_approximation False and the value is only a rough screen indicator.
• Feeding a zero relative velocity: the linear TCA model divides by dot(v, v), so a zero relative velocity raises ValueError - the closing-rate assumption is part of the model, not a detail.
• Building the miss with a velocity aligned to the position vector: a closing velocity along the relative position drives the miss to zero at TCA; only a perpendicular component produces the offset miss the Pc formula needs (the worked 50 m case).
• Mixing the sigma convention: the model uses the combined 1-sigma position uncertainty under the circular covariance approximation; a full 3x3 covariance projection onto the encounter plane is out of scope, so do not feed per-axis sigmas as if the projection were performed.
• Letting the exponent overflow silently: a distant pass (2780 m miss against 100 m sigma) returns Pc ~ 1.6e-171, so compare verdicts in log or threshold terms, not by the raw float's magnitude.

Verification

• Confirm tca_s([5000, -3000, 2000], [-7.0, 1.0, -0.5]) returns 776.1 s within 0.5 s and miss_distance_m at that TCA returns 2780.5 m within 1.0 m.
• Confirm probability_of_collision(0, 100, 5) returns exactly 1.25e-3 and probability_of_collision(50, 100, 5) returns 1.103e-3 within 1e-6, with severity "high" for both.
• Confirm the miss vector at TCA is orthogonal to the relative velocity: dot(r + v * tca, v) = 0.
• Confirm analyze reports valid_approximation False when the hard body radius exceeds 0.1 sigma (for example 15 m against 100 m).
• Confirm every non-positive sigma, negative hard body radius, and zero relative velocity raises ValueError.
• Run the contract test offline: python3 scripts/test_conjunction_assessment.py (35 tests, deterministic).

Related leaves

• space-systems/mission-design/radiation-debris: the long-term debris environment collision probability from flux, cross-section, and mission life, the environment-level counterpart of this screen.
• gnc-autonomy/space/orbit-determination: recovers the orbit state of the primary and secondary from observations as the source of the screening state vectors.
• space-systems/orbit-mechanics/clohessy-wiltshire: propagates the linearized relative motion of a deputy about a chief, a way to move the relative state to the screening epoch when the objects share a circular orbit.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_conjunction_assessment.py

The test covers the worked-example TCA and miss distance, the closing, receding, and perpendicular TCA geometries, the zero-relative-velocity rejection, the miss orthogonality identity at TCA, the circular encounter-plane sigma projection, the Pc formula at zero, 50 m, and 2780 m miss distances, the hard body squared and inverse sigma squared scalings, the high, watch, and green verdict bands with default and custom thresholds, the full analyze dict, the validity flag at and above the small hard-body limit, and ValueError rejection of non-positive sigma, negative hard body radius, negative probability, and zero relative velocity.

Compliance

• Standards referenced, not reproduced: ECSS space safety standards frame the conjunction screening context (ecss.nl/standards); the relations above are standard engineering methodology, summary-only per standards-map.yaml. No ECSS text is reproduced.
• compliance: STANDARDS-REF, gated: false.