Every fragment comes back through the same point

The vis-viva equation gives the speed of an orbiting body anywhere on its orbit, $v^2 = \mu(2/r - 1/a)$, and the eccentricity is not in it. Read the other way, it says something just as striking: the semi-major axis of an orbit — and with it the orbit’s energy and period — is fixed by the speed at one point and the distance from the centre, and not by the direction the body is moving in. Two bodies at the same place with the same speed have the same period, whether one is moving along the old orbit and the other straight up.

That fact organises one of the most consequential events in orbit: a break-up. When a satellite explodes, or two satellites collide, or a missile destroys one, the fragments leave the break-up point in every direction with a spread of speeds, and each follows its own Kepler ellipse from then on. Vis-viva decides what the cloud does next, and the answer has three parts that are not obvious from the picture of a spray of debris: the cloud re-forms at the break-up point once a period, it stretches along the orbit at a rate set by one component of each kick, and it closes into a ring round the Earth in a time that can be computed on the back of an envelope.

Every orbit contains the point it started from #

The pinch is geometry before it is dynamics. Every fragment was at the break-up point at the moment of break-up, so every fragment’s orbit passes through that point, and after one of its own periods each fragment is back there. The orbits are all different ellipses — some larger than the parent’s, some smaller, some tilted — but they share one point, as a bundle of ellipses through a common vertex.

Half a period later the fragments are at the far ends of their different ellipses, and those ends are widely spread: a fragment kicked forward has raised the far side of its orbit, one kicked backward has lowered it, and the cloud there is well over a thousand kilometres deep. A period after the break-up they are all back near the common point, and the cloud has pinched. The pinch is not the fragments being pulled back together; it is the orbits never having separated at that one place.

The picture is in the orbital plane, but the pinch has a second half out of it. A kick perpendicular to the plane tilts a fragment’s orbit about the line joining the break-up point to the Earth’s centre, so every orbital plane contains that line, and the fragments’ paths cross not only at the break-up point but also at the point diametrically opposite it, half an orbit later. The cloud pinches twice a period in the out-of-plane direction: at the break-up point and at the antipodal pinch line, where every fragment’s orbit crosses every other’s. That is where a break-up is most dangerous to anything else in orbit for the first days — the density of fragments there is highest — and it is why a satellite whose orbit happens to pass through a new cloud’s pinch point is at more risk than its altitude alone would suggest.

One component of the kick decides the period #

The cloud does not stay pinched in a single clump, because the fragments do not return to the break-up point at the same moment. Each returns after its own period, and the periods differ.

By vis-viva a fragment’s semi-major axis depends on its speed at the break-up point, and its speed there is the vector sum of the parent’s orbital speed and the kick. A kick $\Delta v$ at an angle $\phi$ to the direction of motion changes the speed to first order by $\Delta v\cos\phi$ — the along-track component — and changes it only at second order through the component across the track. The period depends on the semi-major axis as $a^{3/2}$, and the semi-major axis on the speed so that $\Delta a/a = 2\Delta v/v$ for a circular orbit, so

$$\frac{\Delta P}{P} = 3\,\frac{\Delta v\cos\phi}{v}.$$

The factor of three is worth pausing on, because it is large. An along-track kick of 100 metres a second is 1.3 per cent of the orbital speed, and it changes the period by four per cent — four minutes in a hundred. That is why a debris cloud stretches so quickly: a spread of kicks of a hundred metres a second becomes a spread of periods of several minutes, and each period the leading fragments gain several minutes on the trailing ones, which at 7.5 kilometres a second is a couple of thousand kilometres of orbit. It is the same leverage that makes a burn given at speed worth more than the same burn given slowly: the period depends on the energy, and the energy on the square of the speed.

It also explains the counter-intuitive behaviour of orbital rendezvous. A fragment kicked forward, with its speed increased, goes into a larger orbit with a longer period and falls behind the parent’s position; one kicked backward drops into a smaller, faster orbit and gets ahead. To catch up, a spacecraft slows down, and a debris cloud does the same thing in reverse: the fragments thrown forward become the rear of the cloud, and those thrown backward lead it.

The cloud seen from the parent #

The same dynamics look different, and in some ways clearer, from a frame that moves with the parent’s orbit — the frame in which rendezvous is planned, with one axis pointing away from the Earth and one along the orbit.

In that frame, and for kicks small compared with the orbital speed, the motion of a fragment is given by the Clohessy–Wiltshire equations, which are linear and can be solved exactly. A fragment kicked straight up — radially — does not drift at all. It moves on a closed ellipse round the parent, twice as long along the track as it is tall, and returns to the parent’s position every period. That relative orbit breathes in and out once a period and goes nowhere, which is the relative-motion statement of the fact that a radial kick leaves the period unchanged.

A fragment kicked along the track drifts. Its orbit has a different period, so in the parent’s frame it traces a series of loops that march steadily away along the track, backward if it was kicked forward and forward if it was kicked backward, by $6\pi\,\Delta v/n$ per orbit, where $n$ is the parent’s orbital frequency. At 800 kilometres, a kick of one metre a second along the track produces a drift of about eighteen kilometres per orbit. The whole along-track stretching of the cloud is that one term, summed over the fragments’ along-track kicks; everything else in the relative motion is periodic and returns to the pinch.

That decomposition is why the stretching is so predictable. A cloud’s shape at any later time is its drift, set by the along-track kicks, plus a breathing, set by the radial and cross-track kicks, and the breathing never accumulates. Every fragment’s relative orbit passes back through the parent’s position — the pinch — or would, if the drift did not carry it along the track first.

The X that every break-up draws #

The standard first look at a new break-up is a plot invented by John Gabbard at the North American Aerospace Defense Command in the 1970s: each fragment’s apogee altitude and perigee altitude, plotted against its orbital period.

The X is vis-viva again. A fragment with a longer period than the parent’s was kicked forward, gaining speed at the break-up point, so that point is the lowest point of its new orbit: its perigee is at the break-up altitude and its apogee rises with the period. A fragment with a shorter period lost speed there, so the break-up point is its highest point: apogee at the break-up altitude, perigee falling. The horizontal parts of the two arms are the fragments’ shared point; the sloping parts are how far each orbit reaches from it.

A Gabbard diagram therefore identifies the break-up altitude without anyone having seen the break-up: it is where the arms cross and where the flat parts lie. It measures the energy of the event from the spread of the arms. And it predicts the cloud’s future, because the fragments on the lower left, with the lowest perigees, are the ones the atmosphere will act on first. Over months their perigees drag down, their apogees follow, and the left arm of the X is eaten from the bottom.

That contrast is the physical basis of a policy argument. The destruction of the Chinese weather satellite Fengyun-1C by a missile in 2007, at about 865 kilometres, produced more than three thousand tracked fragments, most of which are still in orbit, because at that altitude drag takes decades to centuries. The Russian test that destroyed Kosmos 1408 in 2021 was at about 480 kilometres, and its fragments were largely gone within two or three years. The Gabbard diagram of each told, within days, which kind of event it had been.

How long until it is a ring #

The cloud stretches along the orbit at a steady rate set by the spread of periods. If the along-track kicks range over $\pm\Delta v$, the periods range over $\pm 3\Delta v/v$ of the orbit, and after $N$ orbits the leading and trailing fragments are separated by $6N\Delta v/v$ of a full circle. The cloud closes into a ring when that reaches one: after about $v/(6\Delta v)$ orbits.

For a collision or an explosion with typical kicks of a hundred metres a second, a ring forms in about half a day. That is fast compared with anything that can be done about it: by the time a break-up has been detected, tracked and catalogued, its fragments already surround the Earth at that altitude. A break-up is a local event for a few hours and a global hazard at its altitude for as long as the fragments stay up.

The same stretching makes a new cloud hard to catalogue. A tracking radar sees a fragment for a few minutes as it passes overhead, and to add it to a catalogue the fragment has to be recognised on a later pass, which means predicting where it will be. For the first orbits every fragment is near the parent’s path, and the radar sees a swarm it cannot separate; as the cloud stretches the fragments separate along the track, and each can be followed — but only once its period is known well enough to predict its return, which takes several passes. The catalogue of a new break-up therefore fills in over weeks, the fragments with the most distinctive periods first, and the ones kicked least, which stay close to the parent and to each other, last. The fragments that stretch least are the hardest to tell apart, and the early counts of any break-up are lower limits for that reason.

The collision that made the argument concrete #

On 10 February 2009 the active communications satellite Iridium 33 and the defunct Russian satellite Kosmos 2251 collided at about 790 kilometres above Siberia, at a relative speed of about 11.7 kilometres a second. It was the first accidental collision between two intact satellites, and the two clouds it produced have been tracked ever since.

Their evolution followed the sequence drawn here. Within hours each cloud was a tight clump at the collision point, pinching there each orbit; within a day each had stretched into a ring; the two rings lay in two different planes, the two satellites’ orbits having been inclined at about 86 and 74 degrees, and their pinch lines crossed at the collision point. Over the following years each ring fanned into a shell as the planes precessed. More than two thousand fragments were catalogued, and a large share are still in orbit, at altitudes where drag works slowly.

The first days after the collision were the most dangerous period for other satellites crossing the pinch points, and the decades after it are the most dangerous period in total, because a shell at 790 kilometres is crossed by everything in low orbit above it on the way up and down. Any object whose orbit passes through the altitude band of a shell has a collision probability per year proportional to the shell’s density, and the shells add. A collision rate needs no collision to be computed; it needs the density, and every break-up raises it. The collision itself was improbable; its products are a permanent part of the environment for everything that shares their altitude.

After the ring: planes and drag #

The ring is not the end of the spreading. Two slower processes take over, and each has a clock of its own.

The first is the precession of the orbital planes. The Earth’s equatorial bulge makes the node of an inclined orbit — where it crosses the equator — drift westward or eastward at a rate that depends on the orbit’s size, shape and inclination. Fragments with slightly different semi-major axes and inclinations precess at slightly different rates, so their planes fan out from the parent’s. Over months to a few years the ring, which lies in one plane, becomes a shell: fragments at every longitude of the node, filling the band of latitudes the parent’s inclination allows. The pinch line, where all the planes crossed, dissolves as the planes separate.

The second is drag, and it sorts the fragments. The deceleration from air drag depends on the ratio of a fragment’s area to its mass, which varies over orders of magnitude among the pieces of a broken satellite — a flake of insulating blanket and a chunk of a propellant tank at the same altitude feel very different drag. The high-ratio fragments come down within months; the dense ones stay for decades. The drag depends on an atmospheric density that varies with the solar cycle by a factor of ten at the relevant altitudes, so a break-up at solar maximum clears faster than one at minimum.

The debris environment at any altitude is therefore a superposition of shells of different ages, each the end state of a cloud that pinched, stretched into a ring, fanned into a shell and is slowly being sorted and drained. The rate of collisions it implies is computed from that superposition, and every new break-up adds a layer.

What the figures simplify #

The fragments move in the parent’s orbital plane. Out-of-plane kicks tilt the orbits without changing their periods to first order, so they do not change the along-track story; they add the second pinch and, over months, the fanning of planes by precession, which the figures leave to the prose.

The kicks are drawn from a single distribution. Real break-ups have kick distributions that depend on fragment size — small fragments are thrown faster — and on the geometry of the event: an explosion is roughly isotropic, a collision throws most fragments along the relative velocity. The NASA standard break-up model encodes those dependences, and a Gabbard diagram of a real event shows them as asymmetric arms.

The orbits are pure Kepler ellipses. The Earth’s oblateness, drag, the Moon and the Sun are all neglected, which is correct for the first few periods and progressively wrong afterwards; the ring’s closing time is computed before any of them matter much.

And the break-up is instantaneous at a point. A real collision takes milliseconds and throws fragments from an object metres across, so the “point” is a small volume and the pinch has a finite waist even in the ideal case — which the figures’ own waist of a few kilometres, from fragments that have not quite returned, somewhat exceeds.

Still open: what a cloud reveals about the event that made it #

A Gabbard diagram of a new break-up is a measurement of the event: the spread of the arms gives the distribution of kick speeds, their asymmetry the direction of the dominant impulse, and the number of fragments on each arm the kind of event. For explosions and collisions of known kinds the relation between the event and the diagram is calibrated by ground tests and by the tracked histories of past break-ups. For break-ups of unknown cause — a satellite that fragments without warning, as several have — the diagram is one of the few pieces of evidence, and reading the event from it is an inverse problem with limited information: the tracked fragments are only those larger than about ten centimetres, which are a small minority of the total and are biased towards the heavier, slower-thrown pieces. Whether the small fragments, which carry most of the collision risk and which radar can barely see, are thrown in the same pattern as the large ones, or faster and more isotropically, decides how the cloud nobody can see or shield against evolves, and it is the part of every break-up that has to be inferred rather than observed.