A kick speed read off the slope of a histogram

When a satellite breaks up, the first thing anyone wants to know is how violent the event was. The answer decides how many fragments too small to track were produced along with the ones that are, how long the cloud will stay, and — for an event of unknown cause — whether it was an explosion of leftover propellant, a battery failure, or something hitting it. Nobody saw the event. What exists, within days, is a catalogue of orbits: a few hundred to a few thousand fragments, each with a period, an eccentricity and an inclination, each determined by tracking radars from a handful of passes.

The Gabbard diagram of those orbits is the standard first look. Every fragment’s orbit passes back through the break-up point, so the cloud draws an X, with the arms crossing at the parent’s period and altitude, and the spread of the arms says, roughly, how hard the fragments were thrown. This essay takes the word roughly out of that sentence. The spread of the periods is not just related to the kick speeds; for an isotropic event it is a precise, invertible transform of their distribution, and the transform comes from a theorem Archimedes proved about spheres.

One speed, every direction, a flat spread #

A fragment’s period is fixed by its energy, and by vis-viva its energy at the break-up point is fixed by its speed there. A kick of $\Delta\vec v$ added to the parent’s orbital velocity $v_0$ changes the speed squared by $2v_0\,\Delta v_\parallel + |\Delta\vec v|^2$, and since the kicks are tens to hundreds of metres a second against an orbital speed of seven and a half kilometres a second, the first term dominates. To first order the period depends on one number: the component of the kick along the direction of motion. A radial or sideways kick leaves the period alone.

Now suppose every fragment is thrown at the same speed $v$, in a direction chosen at random over the sphere. How is the along-track component $v\cos\theta$ distributed? The tempting answer is that most kicks are roughly sideways, so the component should cluster near zero. It does not. Archimedes showed that the area of a sphere between two parallel planes depends only on the distance between the planes, not on where they cut — the zone near the equator and the cap near the pole have the same area if they have the same height. For directions spread uniformly over the sphere, the component along any fixed axis is therefore uniformly distributed between $-1$ and $+1$.

The consequence is clean. A break-up that threw every fragment at 100 metres a second would spread their along-track kicks — and so, through vis-viva, their periods — evenly across a band 200 metres a second wide. The edges of the band would be sharp, and their position would be the kick speed. A Gabbard diagram of such an event would have arms of uniform density ending abruptly.

A period is a very fine ruler #

The sensitivity of the period to the along-track kick is what makes the whole method possible, and it is worth putting a number on it. Differentiating vis-viva and Kepler’s third law gives $\Delta P/P = 3\,\Delta v_\parallel/v_0$ for small kicks: a fractional change of speed changes the period by three times as much, because the speed enters the energy squared and the energy enters the period to the power three-halves. At 800 kilometres the orbital speed is 7.45 kilometres a second and the period 100.9 minutes, so each metre a second of along-track kick changes the period by about 2.4 seconds. A fragment thrown forward at 100 metres a second has a period four minutes longer than its parent’s.

Tracking determines periods far better than that. Within a few days of a break-up, a fragment’s period is known to a small fraction of a second, which corresponds to along-track kicks of hundredths of a metre a second. Compared with the kicks themselves, which run from a few metres to a few hundred metres a second, the period is effectively an exact measurement of one component of each fragment’s velocity change. The precision runs out elsewhere: in the second-order terms, in the assumption of a single break-up point, and in the slow perturbations that start to move the periods once the cloud is days old.

The second-order terms are easy to see and easy to correct. A kick’s sideways and radial components do not change the period to first order, but they do change the speed squared, by $|\Delta\vec v_\perp|^2$, which raises every fragment’s energy slightly whatever its direction. A fragment thrown purely sideways at 200 metres a second has its period raised as if it had been kicked forward by about 2.7 metres a second. That shifts the whole along-track distribution towards positive values by a few metres a second for a vigorous event, and the inversion has to remove it. The size of the shift is set by the same speed distribution the inversion is recovering, so the correction is applied iteratively, and it converges in one or two steps.

Many speeds, a stack of flat bands #

Real break-ups throw fragments at many speeds, and the speeds depend on the fragments’ sizes: small pieces are thrown faster than large ones, the spread covering a factor of ten or more. Each speed contributes its own flat band, of width set by its value, and the along-track distribution is the stack of all of them. If $p(v)$ is the distribution of kick speeds, the along-track kicks have the density

$$g(u) = \int_{|u|}^{\infty} \frac{p(v)}{2v}\,dv,$$

because a fragment thrown at speed $v$ lands with equal probability anywhere in $[-v, v]$, and contributes to $u$ only if $v > |u|$.

The stack is peaked at zero, because every speed contributes there and only the fast ones contribute far out, and it has wings set by the fastest fragments. That shape is why a debris cloud’s Gabbard diagram is densest near the crossing of the X and thins along the arms, and why the period distribution of real clouds looks roughly bell-shaped even though the physics behind it contains no bell. The histogram in the figure is computed from the fragments’ periods alone, inverted through vis-viva, and it follows the stacked prediction to within a fraction of a metre a second per fragment. The periods are a faithful record of the along-track kicks.

Reading the speeds back off the slope #

The relation between $g$ and $p$ can be turned around. Differentiating the integral with respect to its lower limit gives

$$p(v) = -2v\,g'(v),$$

so the distribution of kick speeds is read off the slope of the along-track distribution: wherever the histogram of along-track kicks falls steeply, many fragments were thrown at that speed. The relation is exact for an isotropic event, and it needs only the periods.

The figure shows the inversion at work on simulated clouds. With three thousand fragments — the size of the catalogue from the largest break-ups, such as the destruction of the Fengyun-1C weather satellite in 2007 — the recovered distribution follows the true one closely, and its median lands on the true median. With three hundred, typical of a moderate break-up, the recovery is noisy point by point but still places the bulk of the distribution correctly. The noise is the price of differentiating: the derivative of a measured distribution amplifies its fluctuations, and the second derivative in the method’s working form amplifies them twice. Smoothing helps, and more fragments help more, since the noise falls as the square root of their number.

The relation also contains a test. Since $p(v)$ cannot be negative, $g$ can never increase with $|u|$: for any isotropic event, the distribution of along-track kicks must be highest at zero and fall monotonically on both sides. The test runs only one way. A cloud whose along-track histogram has a second peak away from zero, or a shoulder that rises outward, cannot have been produced by a single isotropic event, however its speeds were distributed. It was either beamed, or it is two events superposed — a primary break-up followed by a secondary one in the cloud, or a collision whose projectile’s fragments and target’s fragments carry different momenta. The inversion is therefore not only a measurement but a diagnostic, and a negative recovered density at some speed is a signal rather than a numerical nuisance.

The same inverse problem appears in a very different place. When an asteroid is shattered, its fragments are thrown into orbits whose semi-major axes record the along-track component of their ejection velocity in exactly this way, and the spread of an asteroid family in semi-major axis has been used to estimate the speeds at which its members left the parent body — before the slow drift from the Yarkovsky effect smears the record out over hundreds of millions of years. A satellite break-up is the same experiment, done a few hundred kilometres up, and read within days rather than within a few million years.

The same transform, pointed at stars #

The inversion is older than space debris. In 1950 Chandrasekhar and Münch faced the same problem with stellar rotation. A spectrum measures a star’s rotation only as $v\sin i$, the equatorial speed projected onto the line of sight, and for a population of stars with randomly oriented axes the projected speeds are a stack of distributions, one for each true speed, just as a cloud’s along-track kicks are. They showed that the true distribution of rotation speeds can be recovered from the projected one by an integral inversion of the same family as $p(v) = -2v\,g'(v)$, under exactly the same assumption: the axes, like the kicks, point in random directions.

The kernels differ, because $\sin i$ is not distributed like $\cos\theta$. A random direction’s component along a fixed axis is uniform, while its component perpendicular to that axis — which is what $\sin i$ measures — piles up near one, since most random axes are closer to perpendicular to the line of sight than to parallel with it. That is why projected rotation speeds are usually close to the true ones, and why projected spin–orbit angles preserve the spread of the true ones when the angles are small. The geometry is the same in all three cases, a random direction seen through one component, and in each the deconvolution needs a large, unbiased sample and returns a distribution rather than any individual value.

The debris problem is the easiest of the three, because the component it measures is the uniformly distributed one, and the uniform kernel makes the inversion a single derivative. It is also the one where the assumption of random directions is most likely to fail, since a collision has a preferred direction and a rotating population of stars does not.

Three elements, one kick at a time #

The inversion uses one number per fragment and an assumption about the event. A tracking catalogue supplies more than one number. The eccentricity of each fragment’s orbit records the in-plane kick: an along-track kick of $\Delta v_\parallel$ raises the eccentricity by about $2\Delta v_\parallel/v_0$, and a radial kick by about $\Delta v_r/v_0$, the two adding in quadrature. The inclination records the out-of-plane kick, most directly for a break-up near a node, where a sideways push tilts the orbit by about $\Delta v_\perp/v_0$ radians. With the period fixing the along-track component, the eccentricity gives the radial component and the inclination the normal one.

With all three elements each fragment’s kick is measured individually, up to the signs of the radial and normal components, and no assumption about isotropy is needed. The figure shows that catalogue precision is enough: eccentricities to $10^{-4}$ and inclinations to a few thousandths of a degree correspond to kick components of about a metre a second, far below the kicks themselves. The scatter grows at the slow end, where a kick of ten metres a second is only ten times the catalogue’s resolution.

The individual reconstruction is more demanding in practice than the figure suggests. It needs the break-up time and location, which can be found from the Gabbard crossing and from the point where the fragments’ orbits intersect, and it needs elements determined soon after the event, before the Earth’s equatorial bulge precesses the orbital planes apart at rates that depend on each fragment’s own elements and before air drag begins to sort the cloud. For a break-up observed within a few days, both conditions hold.

When the event was not isotropic #

The isotropic inversion is a first look because explosions are roughly isotropic and collisions are not. A projectile striking a satellite drives most of the debris forward along its own direction of travel, in a cone. If that direction happens to lie along the parent’s orbit, the kicks put more of their speed into the period than isotropy would; if it lies across the orbit, less.

The periods alone cannot tell a fast event thrown sideways from a slow one thrown isotropically. An inversion that assumes isotropy reads the first as gentler and the second as more violent than they were. The eccentricities and inclinations can tell the difference, since a sideways cone puts its speed into the inclinations, and that is why the element-by-element reconstruction is the measurement and the isotropic inversion is a quick estimate of it. The comparison between the two is itself informative. A cloud whose periods imply slower kicks than its inclinations do was thrown preferentially out of the plane, which is the signature of a collision rather than an explosion, and its direction says where the impact came from.

What a tracking catalogue actually supplies #

The figures treat the catalogue as complete and clean. A real one is neither. Radars in low orbit reliably detect objects larger than about ten centimetres, so the catalogue of a break-up is a census of its large fragments, which are the slow ones. The first orbits are determined from a few passes and are refined over weeks; fragments that were close together at the break-up are hard to separate in the first days and are sometimes catalogued late or swapped. And the elements distributed publicly are mean elements fitted under a simplified model of the perturbations, good to perhaps a kilometre in position and correspondingly coarser in eccentricity and inclination than the best internal solutions.

None of this damages the period-based inversion much. The periods are the best-determined elements, the most robust to the fitting model, and the first ones known. It does damage the element-by-element reconstruction, which leans on eccentricities and inclinations precise enough to resolve kicks of a few metres a second. In practice the two are used in sequence. The isotropic inversion of the periods gives the event’s scale within days. The full reconstruction follows once the elements have settled and the break-up time and location have been pinned down from the converging orbits, and it is the one that can reveal a cone.

What the fragments’ sizes add #

The kick speeds of a real break-up are not one distribution but a family of them, one for each size of fragment. The standard model used for debris forecasting, derived from ground tests in which satellites were shattered in vacuum chambers and from the tracked histories of past events, makes the speed a log-normal distribution whose median rises with the fragment’s area-to-mass ratio: flakes of insulation and thin panels are thrown faster than dense chunks of structure. The tracked fragments, a few centimetres and larger, are the slow end of that family. The pieces too small to track, which outnumber them by orders of magnitude and are the ones nobody can shield against, are faster and further spread.

That is where the inversion earns its keep. The kick-speed distribution of the tracked fragments, recovered from their periods, calibrates the model’s speed law for that event, and the model extends it to sizes the radars cannot see. A forecast of how many untracked fragments cross a given altitude, and when, starts from the distribution the figure recovers. An error in the median kick of the tracked pieces propagates into the predicted spread of the untracked ones, which sets the collision rate every other satellite in the shell faces for years.

What the figures assume #

The kicks are added instantaneously at a point, on a circular parent orbit, and the fragments then follow pure Kepler ellipses. That is correct for the first few orbits and becomes progressively wrong as the planes precess and drag acts. The kick speeds are drawn from a single log-normal distribution, independent of fragment size and of direction, except where the beamed figure says otherwise. The catalogue errors are idealised and the same for every fragment, where real tracking quality varies with size, altitude and the number of passes observed. And the inversion’s smoothing — needed to differentiate a measured distribution twice — trades resolution in speed for noise, a trade that would be tuned to each real event.

None of these changes the underlying result: the along-track kicks of an isotropic event are uniformly spread at each speed, the periods record them through vis-viva, and the slope of their distribution is the distribution of speeds.

Still open: whether a cloud says what hit it #

The kick-speed distribution measures how violent an event was. Whether it measures what caused the event is a harder question. Explosions of residual propellant, battery ruptures, and hypervelocity collisions with small untracked debris produce different numbers of fragments and different speed distributions, and the ground tests that calibrate those differences were made on a few satellite designs decades ago. Several satellites have fragmented in recent years without any announced cause, and their clouds are the only evidence. Whether the speed distribution, the degree of anisotropy and the number of fragments together can distinguish a collision with a small object from an internal explosion — and whether a collision’s direction can be read from where its cone pointed — is being asked of real clouds without a sample of known cases large enough to answer it. One quantity separates the two cleanly in principle, because it depends on nothing but conservation of momentum: the motion of the cloud’s centre.