From two orbits to one number

A conjunction assessment ends with one number. Getting to it means taking two objects in three dimensions, each moving at nearly eight kilometres a second on its own orbit, each carrying its own uncertainty, and collapsing all of that into a flat two-dimensional integral over a small disc.

That collapse is the most interesting step in the whole pipeline, and it is the one that usually gets a sentence. I wrote a whole post about the number and referred to “the encounter plane” as though it were a given. It is not a given. It is a consequence, and the thing it is a consequence of is speed.

The encounter is over before anything can move

Take two objects on circular orbits at the same altitude. Both travel at the same speed v, and if their velocity vectors are separated by a crossing angle θ, the relative speed follows immediately:

|v_rel| = 2 · v · sin(θ/2)

At 400 km, v is 7.669 km/s. A perpendicular crossing gives 10.8 km/s. A head-on pass gives 15.34 km/s. Those are not unusual numbers; they are the ordinary condition in low Earth orbit, where objects share a shell and cross it in every direction.

Two approximations fall out of an encounter that lasts about a second, and neither is a fudge.

Relative motion is a straight line. Over a second, gravity bends each trajectory by a few metres, against a relative displacement of tens of kilometres. Curved paths become a straight-line flyby.

The covariance is frozen. Position uncertainty grows over hours and days. Across a second it does not meaningfully change, so a single covariance describes the entire encounter rather than a sequence of them.

Together these are the short-term encounter assumptions, and they are what make a closed two-dimensional integral legitimate instead of lazy. Where they fail — very low relative velocity, an encounter that lasts minutes rather than a second — the two-dimensional formula fails with them, and the literature has separate long-term methods for exactly that case.

The plane that makes it two-dimensional

If relative motion is a straight line, then the whole encounter is described by that line and the plane it punches through. Put the primary object at the origin. The encounter plane is the plane through it perpendicular to the relative velocity vector.

The secondary's relative position at closest approach lies exactly in that plane, and not by construction. Closest approach is the moment the separation stops shrinking, which is the moment the relative position is perpendicular to the relative velocity. Perpendicular to the relative velocity is the definition of the plane. The miss vector lands in it for free.

So: project the combined position covariance onto that plane, and you have a two-dimensional Gaussian. Draw a disc at the origin whose radius is the two objects' combined hard-body radius. Integrate the one over the other. That integral is the number.

Why along-track dominates the covariance

Uncertainty for an orbiting object is quoted in a frame that moves with it: radial, along-track, cross-track. In almost every case the along-track term is much the largest, often by an order of magnitude, and the reason is worth stating because it explains the rest of this post.

A small error in semi-major axis is a small error in orbital period. A small error in period is an error in where along the orbit the object is, and that error grows with every revolution. Radial and cross-track errors oscillate and stay bounded; the along-track error accumulates. Give it a day and you know an object's orbital path far better than you know its position on that path.

Two objects, two covariances. They are estimated independently, so for the encounter they add, and the sum is what gets projected.

Geometry decides how much of that uncertainty counts

Here is the part I find genuinely elegant, and it is a few lines of algebra.

The along-track direction is the velocity direction. For the two circular orbits above, the angle between the primary's velocity and the relative velocity works out to

v̂₁ · v̂_rel = −sin(θ/2)

Projection onto the encounter plane removes whatever lies along the relative velocity, so the surviving fraction of the along-track direction is cos(θ/2). For two objects with the same along-track sigma, the in-plane contribution is

σ_in-plane = σ_T · √2 · |cos(θ/2)|

Read what that says at the extremes. At θ = 180°, a head-on pass, the cosine is zero: the dominant uncertainty points straight along the relative velocity and projects entirely out of the plane. It contributes nothing. At small crossing angles the cosine is nearly one and the full along-track uncertainty, doubled across two objects, lands in the plane.

The operational consequence is not intuitive. A head-on conjunction at 15 km/s sounds like the frightening one, and in terms of what happens on contact it is. But it is also the geometry in which you know the most: the uncertainty that dominates your knowledge of both objects has been projected out of the plane where it would have mattered. The shallow crossing at 1 km/s, which sounds gentler, is the one where you know least.

Two conjunctions reported with the same miss distance and the same covariances are therefore not equally well determined. The geometry sits between the inputs and the number, and it is not recoverable from the number.

What this means for the number I wrote about before

Probability dilution is the observation that widening the covariance can lower the computed probability. Everything above says that the covariance being integrated is not the covariance that was estimated. It is that covariance after a projection whose severity is set by the crossing angle.

So the dilution curve is not a property of the two objects. It is a property of the two objects and the geometry of the encounter, and the same pair of objects with the same tracking quality sit at different points on it depending on how they happen to meet. Which is one more reason the single number, on its own, does not tell you what you need.

What I have left out

Enough that the figures should be read as a description of the geometry rather than an operational tool.

• Propagation. These are circular two-body orbits, which is exact for what they are and wrong for the real thing. Operational screening uses SGP4 or a special perturbations propagator with drag, the geopotential and third-body terms.
• The encounter is constructed. I place both objects at a point and set the geometry, rather than searching a catalogue for a real close approach. It makes the controls legible; it is not how screening works.
• Covariance is propagated too. Real covariance is mapped forward with the state and is neither static nor necessarily Gaussian in the tails, which is where the probability actually lives.
• Correlated errors. Adding the two covariances assumes the two estimates are independent. Two objects tracked by the same sensors with the same atmospheric model are not entirely independent.

The last one is the interesting one, and the honest summary of all four is that every number in this post is downstream of a covariance that somebody modelled. That is a different post.

References

1. D. A. Vallado. Fundamentals of Astrodynamics and Applications, 4th ed. Microcosm Press, 2013. The standard reference for everything in the first half.
2. M. R. Akella, K. T. Alfriend. “Probability of Collision Between Space Objects.” Journal of Guidance, Control, and Dynamics, 23(5), 2000, pp. 769–772. The short-term encounter formulation.
3. R. P. Patera. “General Method for Calculating Satellite Collision Probability.” Journal of Guidance, Control, and Dynamics, 24(4), 2001, pp. 716–722.
4. S. Alfano. “Review of Conjunction Probability Methods for Short-term Encounters.” Advances in the Astronautical Sciences, AAS 07-148, 2007.
5. K. Chan. Spacecraft Collision Probability. Aerospace Press, 2008.
6. V. T. Coppola. “Including Velocity Uncertainty in the Probability of Collision Between Space Objects.” Advances in the Astronautical Sciences, AAS 12-247, 2012. On where the frozen-covariance assumption starts to cost you.
7. H. Klinkrad. Space Debris: Models and Risk Analysis. Springer, 2006. Chapter 8 covers collision risk and avoidance.
8. F. R. Hoots, R. L. Roehrich. Spacetrack Report No. 3: Models for Propagation of NORAD Element Sets, 1980. What real propagation looks like.
9. J. L. Foster, H. S. Estes. A Parametric Analysis of Orbital Debris Collision Probability and Maneuver Rate for Space Vehicles. NASA JSC-25898, 1992.