A collision moves the cloud's centre, and an explosion cannot

Several satellites in recent years have broken apart without warning and without explanation. A spacecraft reports normally, then stops, and within hours the radars are tracking dozens or hundreds of new objects where it used to be. The operator may suspect a battery, a propellant tank, a micrometeoroid; the operators of nearby satellites want to know whether a piece of untracked debris was responsible, because if it was, the environment they are flying through is more dangerous than the catalogue says. The only evidence is the cloud itself.

The cloud’s spread says how much energy went into the break-up, and the distribution of its kick speeds can be read back out of the fragments’ periods. But energy does not distinguish an internal explosion from an external impact: both can throw fragments at a hundred metres a second. What distinguishes them is something simpler and harder to fake. An explosion is internal, and an internal event cannot change the total momentum of the system. An impact brings momentum in. The difference shows up in one place only, the motion of the cloud’s centre of mass, and that motion is written in the fragments’ orbits.

The centre an explosion cannot move #

Newton’s third law guarantees that internal forces cancel in pairs. However a satellite comes apart — gas pressure, a ruptured battery, a structural failure — every push one piece gives another is matched by an equal and opposite push back, and the vector sum of all the momentum changes is zero. The centre of mass of the fragments continues on exactly the orbit the satellite was following, at exactly its velocity, for as long as the fragments are on Kepler orbits.

That statement is about mass-weighted velocities. If $m_i$ is a fragment’s mass and $\Delta\vec v_i$ its velocity change at the break-up, an explosion satisfies $\sum m_i\,\Delta\vec v_i = 0$ exactly. The fragments’ velocity changes are spread over hundreds of metres a second in every direction, but their mass-weighted mean is zero to the precision with which momentum is conserved, which is to say exactly.

A collision is not internal. A projectile of mass $m$ arriving at relative velocity $\vec v_{\rm rel}$ brings momentum $m\vec v_{\rm rel}$, and after the collision the combined fragments of target and projectile share it. Their centre of mass moves, relative to the target’s original orbit, by

$$\Delta\vec v_{\rm cm} = \frac{\mu}{1+\mu}\,\vec v_{\rm rel}, \qquad \mu = \frac{m}{M},$$

with $M$ the target’s mass. The kicks of the individual fragments still spread over hundreds of metres a second, and the spread is set by how violent the impact was. The centre of the spread is displaced by a definite amount in a definite direction, and both are set by the projectile.

The figure shows two simulated clouds that differ only in that shared velocity change. Their histograms are the same width and the same shape. Without a reference point it would be hard to say which was which. With one — the parent’s own period, known from its tracking before the event — the collision’s histogram is visibly displaced. Its mean along-track kick is eleven metres a second; the explosion’s is consistent with zero.

A shift in the period is a shift in the energy #

The displacement of the along-track kicks is a displacement of the fragments’ periods, through vis-viva: every metre a second of along-track velocity change shifts a period by about 2.4 seconds at 800 kilometres. A collision whose projectile was moving in the same direction as the target, overtaking it, pushes the cloud forward. The fragments’ mean period rises above the parent’s, and the cloud as a whole drifts behind the parent’s position in its orbit, because longer periods mean slower angular motion. A head-on projectile does the opposite. A projectile arriving from below or above pushes the cloud’s centre radially, which changes the fragments’ eccentricities without changing their mean period, and a projectile arriving from the side tilts the cloud’s mean orbital plane.

The along-track case is the most readable, because the period is the best-measured element and the earliest one known. It has a geometric counterpart in the standard picture of a new break-up.

The Gabbard diagram’s X does not move. Its crossing is fixed by the break-up point, which every fragment’s orbit contains whatever its velocity, and by the parent’s period, so a collision and an explosion at the same place produce crosses at the same place. What changes is how the fragments are distributed along the arms. In an explosion they divide almost evenly between the long-period arm, of fragments kicked forward, and the short-period arm, of fragments kicked back. In a collision with a forward impulse the long-period arm carries more of them, and the arms are lopsided. Lopsided arms are therefore the visual signature of momentum, and a balanced diagram is evidence, though not proof, that nothing came in from outside.

The direction of the impulse #

The along-track component is one of three. A break-up near a node of the parent’s orbit turns radial kicks into changes of eccentricity and of the argument of perigee, and normal kicks into changes of inclination and of the node. With all three elements catalogued for each fragment, the in-plane kick of each can be reconstructed individually, and the centroid of the reconstructed kicks is the impulse.

The cloud in the figure is hundreds of metres a second across, and the impulse buried in it is twenty metres a second. Averaging over twelve hundred fragments shrinks the uncertainty of the centroid to a few metres a second, and the direction it points comes out within a few degrees of the true one. That direction is where the projectile came from, relative to the parent’s motion: a centroid pointing forward and up says the projectile was overtaking the target from below.

For an event of unknown cause the direction is as diagnostic as the magnitude. An impulse along the orbit, ahead or behind, is what a collision with debris in a similar orbit would give, since objects at nearly the same altitude and inclination meet at modest relative speeds and from roughly along the track. An impulse mostly normal to the orbital plane is what a collision with an object in a very different plane would give, crossing at kilometres a second from the side. The typical geometry of collisions in a given shell, computed from the catalogue, predicts the distribution of directions that genuine debris impacts should show.

How many fragments it takes #

An impulse is measurable only if it stands out of the noise in the centroid, and that noise is the spread of the kicks divided by the square root of the number of fragments.

The numbers are demanding but not prohibitive. A catastrophic break-up of a large satellite produces a few hundred to a few thousand tracked fragments, enough to see impulses of ten or twenty metres a second. A small break-up, of a few dozen fragments, can reveal only impulses of a hundred metres a second or more — which only a projectile of a substantial fraction of the target’s mass would give. Since the impulse is a mean over the cloud, it benefits from every fragment added to the catalogue, and the slowly growing catalogue of a cloud over its first months tightens it steadily. The spread of the kicks, which sets the noise, is itself measured from the same data, so the significance of a shift is known as soon as the shift is.

A floor under every catastrophic collision #

The most useful fact about the impulse is that a collision energetic enough to shatter a satellite cannot avoid producing one of a minimum size.

Hypervelocity tests and the break-up models built on them divide collisions into two kinds. When the projectile brings less than about 40 joules per gram of target mass, the collision craters the target, knocking off pieces near the impact point while the bulk survives. Above that, the collision is catastrophic: the target is shattered completely, and the number of fragments jumps by orders of magnitude. The threshold is an energy per unit mass, $\tfrac12 m v_{\rm rel}^2 / M \ge 40\ {\rm J\,g^{-1}}$, so a catastrophic collision needs a mass ratio of at least $\mu = 8\times10^4\ {\rm J\,kg^{-1}}/v_{\rm rel}^2$. Multiply by the relative velocity and the centre-of-mass shift is at least

$$\Delta v_{\rm cm} \ge \frac{8\times10^4\ {\rm J\,kg^{-1}}}{v_{\rm rel}}.$$

In low orbit, where collisions happen at around ten kilometres a second, the floor is eight metres a second. A fast collision can shatter a satellite with a small projectile, carrying little momentum. A slow one needs a heavy projectile to deliver the same energy, and moves the centre a great deal: near the geostationary ring, where orbits are nearly co-planar and collisions are slow, the floor rises to over a hundred metres a second. A catastrophic break-up in geostationary orbit caused by a collision would therefore move its cloud’s centre by an amount comparable to the kicks themselves, and could hardly be mistaken for an explosion.

The floor is set by energy and momentum alone, and that makes it a test. A catastrophic break-up in low orbit whose cloud, with a thousand tracked fragments, shows no centroid shift above a few metres a second is unlikely to have been a collision. A shift well above the floor says a collision was likely, and its size and direction say roughly how heavy the projectile was and where it came from.

The momentum the tracked fragments may not carry #

The test has a weakness, and it lies in the weighting. Momentum conservation is a statement about the mass-weighted mean of the fragments’ velocity changes. A tracking catalogue gives the number-weighted mean of the tracked fragments, and the tracked fragments are the large ones, typically ten centimetres and up in low orbit, while most of the mass may be in a few big pieces and most of the fragments, by number, are too small to see.

If every fragment’s kick were drawn from the same distribution about the centre-of-mass velocity, the number-weighted mean of any subset would estimate the mass-weighted mean, and the simulations in this essay assume exactly that. Real collisions are not so obliging. The projectile’s own fragments emerge as a fast forward jet carrying most of its momentum, while the target’s fragments, which are most of the tracked ones, receive less. The tracked cloud’s centroid can then understate the impulse, and the momentum the projectile brought is partly carried by fragments nobody sees.

Two corrections are available, and both are partial. A fragment’s mass can be estimated from its radar cross-section and from its area-to-mass ratio, which air drag reveals over months because light, flat fragments decay faster than dense ones. With those, the tracked fragments’ momentum can be weighted properly rather than counted. And the ground tests that calibrate the break-up models record how momentum divides between projectile and target fragments as a function of size, which lets the untracked share be modelled. The residual uncertainty is why a measured shift is read as a lower bound on the impulse rather than as its value.

The reference the shift is measured from #

A centroid shift is a difference, and its other half is the parent’s velocity at the moment of the break-up. That is known from the parent’s own tracking, propagated forward to the break-up time, and the propagation is the first source of error. A satellite’s catalogued orbit is a fit to past observations under a model of drag and of the Earth’s gravity, and propagating it a few days forward accumulates an along-track position error of hundreds of metres to kilometres. Expressed as a velocity, that is a few centimetres to a few tens of centimetres a second — small against the impulses that matter, but not zero. For a break-up that happened during a gap in tracking, the time itself is uncertain, and it has to be found from the fragments: the moment when their orbits pass closest to a common point.

The fragments supply a check. Every fragment’s orbit passes through the break-up point, so the point where the orbits converge, projected back, has to lie on the parent’s orbit at the break-up time. A centroid shift that came from a mistake in the parent’s propagation would move that convergence point off the parent’s track; a genuine impulse would not. The geometry of the X therefore guards the reference against the most common error in it.

The same accounting when an asteroid is struck #

Momentum bookkeeping of this kind was tested on purpose in 2022, when NASA’s DART spacecraft struck Dimorphos, the small moon of the asteroid Didymos, at 6.1 kilometres a second. The spacecraft’s momentum alone would have changed the moon’s orbital velocity by a small, calculable amount. The measured change was several times larger. The ejecta thrown back from the crater along the spacecraft’s incoming path carried momentum of their own, and by Newton’s third law their recoil pushed the moon further. The ratio of the momentum delivered to the momentum brought, called β, came out at around three or four.

The result is the inverse of the weighting problem in a debris cloud. There, the momentum the projectile brought is partly carried away by fragments nobody tracks, so the tracked centroid understates it. At Dimorphos the target was not shattered, the ejecta were the untracked fragments, and the momentum they carried was measured through its effect on what was left. A shattered satellite and a cratered asteroid obey the same conservation law. The difference is which fragments are counted. Asteroid families, the clouds left by catastrophic collisions in the main belt, carry the same record in their mean orbital elements, though for them the parent’s original orbit is not known independently and the impulse cannot be separated from where the parent happened to be.

What the clouds of known collisions show #

The few break-ups whose causes are known make the difficulty concrete. In the 2009 collision between Iridium 33 and Kosmos 2251, two intact satellites met at 11.7 kilometres a second, each struck by an object of comparable mass. The momentum available was enormous, but the two satellites did not merge. Each was shattered where the other passed through it, and the bulk of each cloud stayed close to its own parent’s orbital plane, as the rings of fragments traced afterwards showed. Most of the exchanged momentum went into the fragments of the parts that actually collided, a minority of each satellite’s mass, while the rest of each structure broke up with modest kicks. A centroid taken over all the tracked fragments of either cloud measures only part of the impulse. The case is as much an illustration of the weighting problem as of the impulse.

Deliberate destructions of satellites by interceptors — the Chinese test against Fengyun-1C in 2007, the American destruction of USA-193 in 2008, the Indian test in 2019 and the Russian test against Kosmos 1408 in 2021 — were collisions of a cleaner kind, with interceptors of a substantial fraction of the target’s mass arriving at several kilometres a second. Their clouds were catastrophic by any measure. Analyses of them used the spread of the fragments, the break-up’s location and the timing of the tests together to reconstruct what had happened, in some cases ahead of any official account. For the unexplained break-ups the clouds are small, the impulses that could separate a collision from an internal failure are near the floor, and the centroids are rarely precise enough to make the distinction. That is the regime where the argument matters most and the data are thinnest.

What the figures assume #

The kicks are drawn symmetrically about the centre-of-mass velocity, from one distribution for every fragment, so the number-weighted centroid of the simulated fragments estimates the mass-weighted one. That is the assumption the previous section showed is the weak point. The break-up is instantaneous at a point on a circular orbit near a node, so that the three components of each kick map cleanly onto period, eccentricity and inclination. The fragments are then followed on Kepler orbits, which is right for the first few orbits only. The impulse is shared by every fragment equally, where in a real collision the fragments nearest the impact point receive more of it than those on the far side of the target. And the catastrophic threshold is the single value of 40 joules per gram used in the standard models, itself an average over tests on particular satellite designs.

Still open: what broke up when nothing was seen to hit it #

The unexplained fragmentations are the cases the method was meant for, and they are the hardest. Their clouds are small, a few dozen to a few hundred tracked fragments, so the impulse that a small, fast projectile could leave — a few metres a second — is below what their centroids can resolve. A collision with a fragment of a centimetre or two, too small to be in any catalogue but energetic enough to break a satellite’s battery or tank rather than shatter the structure, would not be catastrophic in the sense of the threshold, would carry little momentum, and might look like an explosion in every way the cloud can record.

Whether that is happening is a question about the population of untracked debris, the one most uncertain number in forecasts of the orbital environment. A reliable signature of small-debris impacts in the clouds of unexplained break-ups would measure that population directly, by its consequences. The momentum argument gives a floor for shattering collisions and says nothing below it. Whether the anisotropy of the kicks, the size distribution of the fragments and the location of the break-up on the satellite can together catch the non-catastrophic impacts that the centroid cannot is being asked of each new event as it happens.