Astronomy & Celestial Mechanics
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Where the mass is and where the light is

A population that grinds itself up settles into a size distribution with a fixed slope, and that slope puts almost all the mass in the largest bodies and almost all the cross-section in the smallest. So a debris disc's brightness measures a population whose mass it says nothing about, and the two are connected only by the exponent.

Assumes Asteroid families, Orbital debris and Non-gravitational forces.

18 min read 10 figures Scales that defeat intuitionMeasuring the unreachable

A population of solid bodies that collide with each other has one thing every member is doing: breaking. Dating one such population from the way its fragments have spread is a rung this collection has already climbed; this one is about what happens after a great many such events, when the fragments of fragments have themselves been broken. Fragments break again, their fragments break again, and the population runs downhill in size until something removes the smallest pieces. What is remarkable is that this process — chaotic, violent, and different in detail for every collision — reaches a steady state whose shape is fixed by a single requirement, and that the shape has a strange consequence for anybody trying to weigh such a population by looking at it.

Mass at the top, area at the bottom, 10 decades apart. Two moments of a collisional cascade's size distribution, per logarithmic interval of diameter, over 10 decades from a ten-micron grain to a hundred-kilometre parent body. Both axes are logarithmic and the vertical scale is arbitrary; only the slopes carry the argument. A population in which every collision makes fragments that go on to collide reaches a steady state where the same mass flows through every size per unit time, and that fixes the differential number distribution at an index of 3.5. The two consequences pull opposite ways. Mass per decade goes as the diameter to the power 0.5, so it climbs and almost all the mass is in the largest few bodies. Cross-sectional area per decade goes as the diameter to the power -0.5, so it falls, and almost all the area — which is what scatters light, what is detected, and what anything passing through gets hit by — is in the smallest. Over the range drawn the small end carries 10⁵ times the area of the large end and 10⁻⁵ times its mass. A disc's brightness therefore measures a population whose mass it says nothing whatever about, and the two numbers are connected only through the index of this line.
Fig. 1 Two moments of a collisional cascade’s size distribution, per logarithmic interval of diameter, over ten decades from a ten-micron grain to a hundred-kilometre parent body. Mass per decade climbs as the square root of the diameter, so almost all the mass is in the largest bodies. Cross-sectional area per decade falls as the same power, so almost all the area — which is what scatters light, what is detected, and what anything passing through gets hit by — is in the smallest. The two moments point opposite ways, and they are connected only through the index of the distribution.

The index is 3.5, in the differential number distribution, and it is not fitted to anything.

Why the exponent is what it is

The argument is a conservation argument and it does not require knowing anything about how a collision works.

Suppose the population is in a steady state: the number of bodies in any size interval is not changing. Bodies leave an interval by being smashed and enter it as the fragments of larger ones. If the process is self-similar — if the outcome of a collision depends only on the ratio of the colliding masses and their relative speed, and not on their absolute size — then the mass flux through every size must be the same. Mass is ground from the top of the distribution to the bottom at a constant rate, and no size is a bottleneck.

Writing that requirement out and demanding a power-law solution fixes the exponent at 3.5 for the differential number, and nothing else about the collision physics survives into the answer. Dohnanyi’s derivation of 1969 does the accounting carefully; the answer had already been guessed from the observed asteroid population. It is worth noticing what the derivation resembles: it is the same argument that fixes the initial mass function’s slope from a scale-free fragmentation process, and the two exponents are close for the same reason rather than by coincidence.

The two moments then follow by inspection. Number per logarithmic interval goes as the diameter to the power one minus the index, so it falls steeply. Multiply by the square of the diameter for cross-section, or the cube for mass, and the exponents become minus a half and plus a half.

Mass at the top, area at the bottom, 10 decades apart. Two moments of a collisional cascade's size distribution, per logarithmic interval of diameter, over 10 decades from a ten-micron grain to a hundred-kilometre parent body. Both axes are logarithmic and the vertical scale is arbitrary; only the slopes carry the argument. A population in which every collision makes fragments that go on to collide reaches a steady state where the same mass flows through every size per unit time, and that fixes the differential number distribution at an index of 3.3. The two consequences pull opposite ways. Mass per decade goes as the diameter to the power 0.7, so it climbs and almost all the mass is in the largest few bodies. Cross-sectional area per decade goes as the diameter to the power -0.3, so it falls, and almost all the area — which is what scatters light, what is detected, and what anything passing through gets hit by — is in the smallest. Over the range drawn the small end carries 1000 times the area of the large end and 10·10⁻⁸ times its mass. A disc's brightness therefore measures a population whose mass it says nothing whatever about, and the two numbers are connected only through the index of this line.
Fig. 2 The same two moments for a shallower index of 3.3, which is what a population whose members get harder to break as they get larger produces. The area still falls but far less steeply, and the mass rises more steeply. The sensitivity is worth noting: two tenths in the index moves the ratio of small-body area to large-body area by two decades over this range. Every statement about what a disc’s brightness implies about its mass is therefore a statement about an exponent that is measured, in the solar system, to about a tenth.

Where the solar system’s cascade can be counted

The asteroid belt is the one collisional cascade where individual members can be counted rather than inferred, and it is the calibration for everything else. The counting has a hard limit. A ten-kilometre asteroid at three astronomical units is a twentieth-magnitude object; a ten-metre one is thirty-two magnitudes fainter and has never been seen at that distance. So the observed distribution covers about three decades of the ten the cascade spans, and everything below is inferred from the crater record on surfaces that have been sitting in the belt collecting hits — a population counted by the holes it has left in something else.

A 128.8-million-year-old collision, dated from the shape of a scatter plot. The Erigone family: 165 members drawn at their diameters and their proper semi-major axes, with inverse diameter up the page. The cloud is a V, and the V is a clock. Each member has been drifting in semi-major axis ever since the collision at a rate that goes as one over its diameter, with a sign set by which way it spins — prograde outward, retrograde inward — so after 130 million years the small members have moved far and the large ones have barely moved at all. Plotted against 1/D that envelope is a straight line through the family's centre, and its slope is the drift rate for a one-kilometre body multiplied by the elapsed time. Fitting the two edges of the points actually drawn here returns 128.85 million years against the 130 the members were generated from. The rounding at the bottom is not an artefact: it is the ejection velocity, some 15 metres per second, which every member got at the moment of the collision and which is the same for all sizes. The picture cannot show the interlopers — background asteroids that happen to lie inside the V and have nothing to do with the family — and it cannot show the members that have drifted into a resonance and left the belt entirely, which is the reason the oldest families have the softest edges.
Fig. 3 The other thing a family carries: a clock. Fragments drift in semi-major axis at a rate that goes as one over their diameter, so plotted against inverse diameter the family is a V whose slope is the elapsed time. What matters for this essay is the axis rather than the age — the family’s members span a decade and a half in size in a single scatter plot, and the density of points along the vertical axis is the size distribution being measured directly.

What removes the small end

A pure cascade has no bottom: every size grinds to the next one down forever. Something has to remove the smallest pieces, and around a star there are two things that do, both of them consequences of light having momentum.

The first is radiation pressure. Below a size at which the force from starlight exceeds half the gravitational attraction to the star, a grain created at rest on a circular orbit is immediately on an unbound one and leaves. That blowout size is a few tenths of a micron around a Sun-like star and several microns around a luminous one, and it is a hard floor: nothing smaller stays.

The second is Poynting–Robertson drag, which acts on grains above the blowout size and spirals them inward on a timescale that grows with size. In a dense disc it loses to collisions — a grain is smashed before it can drift — so it matters only in tenuous systems like the zodiacal cloud. Which of the two wins is decided by a single comparison of timescales, and it is the same comparison that decides whether a small body’s thermal drift can carry it into a resonance before something hits it. The top end is set differently: by whether there has been time. A cascade in steady state requires that the largest bodies in it have had time to be destroyed, and the collision time grows with size. Above some diameter the population is still primordial, and the distribution breaks.

Mass at the top, area at the bottom, 10 decades apart. Two moments of a collisional cascade's size distribution, per logarithmic interval of diameter, over 10 decades from a ten-micron grain to a hundred-kilometre parent body. Both axes are logarithmic and the vertical scale is arbitrary; only the slopes carry the argument. A population in which every collision makes fragments that go on to collide reaches a steady state where the same mass flows through every size per unit time, and that fixes the differential number distribution at an index of 3.7. The two consequences pull opposite ways. Mass per decade goes as the diameter to the power 0.3, so it climbs and almost all the mass is in the largest few bodies. Cross-sectional area per decade goes as the diameter to the power -0.7, so it falls, and almost all the area — which is what scatters light, what is detected, and what anything passing through gets hit by — is in the smallest. Over the range drawn the small end carries 10⁷ times the area of the large end and 0.001 times its mass. A disc's brightness therefore measures a population whose mass it says nothing whatever about, and the two numbers are connected only through the index of this line.
Fig. 4 The same two moments at an index of 3.7 rather than 3.5 — steeper than the collisional steady state, which is what a population whose small members are being removed faster than they are supplied looks like. The mass curve barely moves and the area curve tilts noticeably, because the area integral is dominated by the small end and the mass integral by the large one. A two-tenths change in the exponent is a small change in the mass and a large one in what is seen, which is the asymmetry the next section is about.

The consequence for anybody looking at one

Now the point of the essay. A debris disc around another star is seen in one of two ways: scattered starlight, or thermal emission from the grains. Both are proportional to cross-sectional area — and dust does not destroy light so much as move it, which is why the same grains are seen twice, once as a deficit in the optical and once as an excess in the infrared. The area is in the smallest grains. The mass is in the largest bodies. So the observable is a measurement of a population that carries a negligible fraction of the mass, and the inferred mass depends on extrapolating a power law across up to ten decades. An error of a tenth in the index becomes an error of a factor of several in the mass.

This is not a defect in the observations. It is a structural feature of the object: the thing that emits and the thing that weighs are different populations, connected by a number that has to be assumed.

The same argument in low Earth orbit

There is a cascade a few hundred kilometres up, and it is the one where the population is tracked object by object.

The population peaks at 825 km, and so does the risk. Tracked objects per 25-kilometre shell against altitude, with the collision rate on a single object computed from each bin as nσv — a gas-kinetic rate, not an orbital calculation. The cross-section is not a satellite's area but an effective one, 3698 square centimetres, fixed by requiring the whole tracked population to produce the one catastrophic collision every 8 years that is observed — most tracked objects are fragments, and two fragments meeting make nothing new. The distribution is not smooth and its shape is history: the peak of 3,010 objects near 825 km is four decades of launches into sun-synchronous orbit plus the debris of two deliberate destructions, and the second rise past 1,300 km is the Soviet-era navigation constellation. At the peak one such object waits 46,409 years between strikes — which sounds safe until it is multiplied by the 3,010 objects sharing that shell, giving one collision every 31 years among them, and by the 23,680 in the whole of low orbit, giving one every 8. The rate on one object is reassuring and the rate on the population is not, and they are the same number.
Fig. 5 The catalogued population of objects in low Earth orbit, per twenty-five-kilometre shell, against altitude. The peak near eight hundred kilometres is four decades of launches into sun-synchronous orbit plus two deliberate destructions; the second bump near fourteen hundred is a different set of missions. This is a cascade in a very early stage: it is not in steady state, its source is not collisions but launches, and its sink at low altitude is atmospheric drag rather than radiation pressure.

The physics is the same and the accounting is different in one important way. The removal mechanism at the bottom of this cascade is atmospheric drag, whose timescale depends steeply on altitude, so there is a height above which nothing is removed at all on any human timescale.

A quadratic and a linear, crossing at 149 objects. The two rates that decide whether a shell at 900 km is stable, against how many objects are in it. Production goes as N² — every collision needs two objects, so the number of collisions is proportional to the square of the population, and each one is taken here to make 1600 trackable fragments. Removal goes as N, because drag acts on each object independently and takes 1,195 years to do it at this altitude. A quadratic and a linear cross exactly once, at 149 objects in this shell, and above that crossing the population grows with nothing launched. The shell presently holds about 2,280, which is 15 times the crossing. Every number on the production side is uncertain by a factor of a few — the fragment yield most of all, and the cross-section is calibrated against an observed collision rate rather than measured — so the position of the crossing carries that uncertainty with it. The shape does not, and the shape is the argument: a quadratic overtakes a linear once and never comes back, the crossing falls as the altitude rises because the lifetime is in the denominator, and what results is a threshold rather than a trend.
Fig. 6 The condition for the population to run away: the rate at which collisions create fragments against the rate at which the atmosphere removes them, as a function of the population’s own size. Below the crossing the population is self-limiting; above it, each collision produces enough new targets to cause more than one further collision, and the number grows without any further launches. The crossing is what the word cascade means in this context, and unlike the astrophysical case it has a policy attached to it.

The connection worth making is that the astrophysical cascade is what the orbital one looks like after it has run to completion. A collision rate can be computed with no collision ever having been observed — from a density, a cross-section and a relative speed — and that is exactly the calculation that both cases turn on.

What was actually measured

Three quantities, and they are measured very differently.

The index comes from counting. In the asteroid belt it is a direct count over the decades that can be seen, extended downward by crater counts on the surfaces of asteroids and moons — a crater population being itself a size distribution of impactors, read through an impact-scaling law. The observed index is close to 3.5 over the middle of the range and departs from it at both ends, in the directions the strength argument predicts.

The total cross-section in an extrasolar disc comes from photometry: the fraction of the star’s light that emerges in the infrared rather than the optical is the fraction intercepted by the disc, which is its cross-section divided by the area of a sphere at its radius. That fraction is a few parts in ten thousand for the brightest known discs and a few parts in a million for the solar system’s own.

The mass is not measured. It is inferred, by extrapolation, and the honest error bar on it is a factor of a few in the best cases.

Mass at the top, area at the bottom, 9 decades apart. Two moments of a collisional cascade's size distribution, per logarithmic interval of diameter, over 9 decades from a ten-micron grain to a hundred-kilometre parent body. Both axes are logarithmic and the vertical scale is arbitrary; only the slopes carry the argument. A population in which every collision makes fragments that go on to collide reaches a steady state where the same mass flows through every size per unit time, and that fixes the differential number distribution at an index of 3.5. The two consequences pull opposite ways. Mass per decade goes as the diameter to the power 0.5, so it climbs and almost all the mass is in the largest few bodies. Cross-sectional area per decade goes as the diameter to the power -0.5, so it falls, and almost all the area — which is what scatters light, what is detected, and what anything passing through gets hit by — is in the smallest. Over the range drawn the small end carries 3·10⁴ times the area of the large end and 3·10⁻⁵ times its mass. A disc's brightness therefore measures a population whose mass it says nothing whatever about, and the two numbers are connected only through the index of this line.
Fig. 7 The same distribution drawn over the range that a bright extrasolar disc’s observations actually constrain — from a micron to a kilometre — rather than over the full ten decades. Even here the mass and the area sit at opposite ends, and the extrapolation is over nine decades of number. What the observations pin is the height of the curve at one point near the bottom; everything else is the slope, assumed.

A worked case, because the abstraction hides the sizes

Take the numbers for a well-studied disc and follow them through, because the arithmetic is where the difficulty becomes concrete.

A disc reradiating one part in a thousand of its star’s light at a radius of fifty astronomical units has a total cross-section of about one part in a thousand of the area of a sphere at that radius — some ten to the twenty-two square metres. If that area is carried by grains a micron across, each contributing about ten to the minus twelve square metres, there are ten to the thirty-four of them, and at a density of two thousand kilograms per cubic metre they weigh together about ten to the eighteen kilograms: roughly a large asteroid.

Now extrapolate up. Ten decades of the hero figure’s mass curve multiply that by ten to the five, giving something of order a hundred Earth masses in bodies too large to see. Nine decades give ten times less. The disc’s brightness fixed the first number to within a factor of two and the second to within a factor of a hundred, and the second is the one anybody quotes.

The lesson is not that the extrapolation is illegitimate but that it should be reported as what it is. A disc’s measured quantity is an area; its mass is a model.

Three planets against one night's error bar. The same total error, with three transit depths laid across it as horizontal lines: a Jupiter at 10000 parts per million, a Neptune at 1100 parts per million, an Earth at 84 parts per million. The lines are flat because a transit depth is a ratio — the same fraction of the light whether the star is bright or faint — while the error bar is not. The marks are where a dip is seven times the error on one 3600-second measurement, which is a Jupiter down to V = 20.0, and a Neptune and an Earth nowhere at all. That last is the figure's real content: those depths are below the bright-end floor, so no single exposure of any star on any night contains them, however bright the star and however large the telescope. What the figure cannot show is the recovery, and it is the whole of how the planets were found: a transit is not one measurement but a few hundred through the dip and dozens of repeats, and folding them is exactly a √N.
Fig. 8 Why the observed range is three decades and not ten. The faintest object a given telescope can detect in a given exposure, against magnitude, with the noise sources separated. A ten-kilometre body in the asteroid belt sits near the bright end of this and a hundred-metre one is thirty magnitudes fainter — off the diagram entirely, and off every diagram that will be drawn this century. The size distribution below that limit is inferred from what it has done to surfaces, never from counting.
Mass at the top, area at the bottom, 8 decades apart. Two moments of a collisional cascade's size distribution, per logarithmic interval of diameter, over 8 decades from a ten-micron grain to a hundred-kilometre parent body. Both axes are logarithmic and the vertical scale is arbitrary; only the slopes carry the argument. A population in which every collision makes fragments that go on to collide reaches a steady state where the same mass flows through every size per unit time, and that fixes the differential number distribution at an index of 3.5. The two consequences pull opposite ways. Mass per decade goes as the diameter to the power 0.5, so it climbs and almost all the mass is in the largest few bodies. Cross-sectional area per decade goes as the diameter to the power -0.5, so it falls, and almost all the area — which is what scatters light, what is detected, and what anything passing through gets hit by — is in the smallest. Over the range drawn the small end carries 10⁴ times the area of the large end and 10⁻⁴ times its mass. A disc's brightness therefore measures a population whose mass it says nothing whatever about, and the two numbers are connected only through the index of this line.
Fig. 9 The same steady-state index over eight decades rather than ten — a millimetre grain to a ten-kilometre body, which is the range a well-observed extrasolar disc and a well-surveyed asteroid family have in common. Both moments keep their slopes, because a power law has no scale in it, and the two curves still cross nowhere. Truncating the range changes the totals and not the shape, and every mass quoted for a debris disc is an integral over decades that were never observed.

The distribution written on a surface

The count runs out at a size the telescopes cannot reach, and the record continues on any surface old enough to have been hit.

An airless body accumulates craters, and each crater’s size is a function of the impactor’s size and speed. So the size–frequency distribution of craters on a surface is the impactor distribution convolved with a scaling law — and since the scaling law is calibrated by explosion experiments and by hypervelocity impacts in the laboratory, the convolution can be undone.

That extends the measurement downward by many decades. Craters a few metres across on the Moon record impactors of centimetres, which no survey will ever detect directly in space, and the resulting distribution agrees with the telescopic counts where the two overlap.

The same data are read the other way round to date the surfaces. A surface exposed for longer accumulates more craters, so the crater density is a clock — calibrated absolutely at the handful of places where samples were returned and dated radiometrically, and applied everywhere else in the solar system by assumption.

The two uses pull against each other, and it is worth being explicit about the circularity. Reading a size distribution from craters assumes the surface’s age is known; reading an age from craters assumes the impactor flux is known. The literature keeps them separate by using different surfaces for each: the Moon’s dated terrains supply the flux, and the flux supplies ages everywhere else.

There is one observational handle on the index that does not require counting anything, and it is the reason submillimetre photometry of debris discs is worth the telescope time. A grain radiates efficiently at wavelengths shorter than its own size and poorly at longer ones, so the far-infrared and submillimetre spectrum of a disc falls more steeply than a blackbody by an amount that depends on how much of the emitting area sits in grains smaller than the observing wavelength. Measured across a decade of wavelength, that excess steepness is a measurement of the size index at the small end — the one end the counting can never reach. The values it returns cluster near the steady-state figure, which is the only independent check the exponent has in an extrasolar system. The check is weaker than it sounds, because the same steepening is produced by grains of a different composition — porous aggregates emit differently from compact spheres — so what is measured is a combination of the size index and a material property, and separating them needs the resolved images that only the nearest discs allow.

Why the largest body is not part of the cascade

A steady-state cascade requires that objects at every size be resupplied by the breakup of larger ones, and that cannot be true at the top.

The collisional lifetime of a body rises steeply with its size, because a larger target needs a larger projectile to disrupt it and larger projectiles are rare. Somewhere in the distribution the collisional lifetime crosses the age of the system, and above that size a body has not been broken since it formed.

So a size distribution has two regimes with a join between them. Below the crossing, the population is a steady-state cascade whose slope is set by the fragmentation physics and which has forgotten how it started. Above it, the population is primordial — a record of what accretion produced, modified only by whatever has been ground away underneath it.

For the asteroid belt the join is around a hundred kilometres, and the observed distribution does change slope near there. That is why the largest asteroids are treated as survivors of the original planetesimal population rather than as the top of the cascade, and why their number is used to constrain how many bodies accretion made rather than how efficiently they have since been destroyed.

A cascade is a steady state and a steady state has a source, and here the source is the slow erosion of a population that predates the process entirely.

What the small end is doing to a spacecraft

The distribution’s steep rise towards small sizes has a consequence for anything crossing it, and the engineering response is a good check on the numbers.

A particle of a hundred microns carries, at the tens of kilometres a second typical of an encounter, about the kinetic energy of a rifle bullet’s thousandth — enough to pit a surface and not to penetrate it. A particle of a centimetre carries the energy of a hand grenade and penetrates anything a spacecraft is made of.

The flux of each follows from the size distribution, and the two ends of that range are handled completely differently. Small particles are so numerous that they are treated as a continuous erosion: solar panels lose efficiency over years, optical surfaces are pitted, and thermal blankets are designed to be sacrificial. Large ones are so rare that the design accepts the risk and computes a probability of failure over the mission’s life.

In between is where the shielding lives. A double wall spaced a few centimetres apart is far more effective than a single wall of the same total mass, because the first sheet shatters the impactor into a spray that the second sheet absorbs over a wide area. That arrangement is standard on every crewed vehicle, and its design point is set directly by the part of the size distribution where the flux and the damage are both appreciable.

The returned surfaces of long-exposure spacecraft have been examined crater by crater, and the size distribution of the pits they carry matches the distribution measured astronomically. A steep power law counted through a telescope and a pattern of microscopic craters on a retrieved panel are the same measurement, which is about as direct a check on the small end as anybody is going to get.

The agreement also settles a question the astronomical counts cannot reach on their own, which is whether the distribution continues smoothly below the detection limit or turns over — and the returned surfaces say it continues.

It also gives the flux at one place and one time, which is the quantity a mission actually needs, rather than a distribution averaged over the whole population — and those two differ wherever a recent breakup has left a stream of debris on a particular orbit.

A stream of that kind is a transient departure from the distribution rather than a violation of it, and separating the two requires knowing where the debris is as well as how much of it there is.

The distinction has a practical form: a distribution measured over decades is a fair description of the background, and a spacecraft flying through a stream that formed last century meets a flux the background does not predict.

The orbital half of this argument has a threshold in it, and the threshold depends on a rate that is measured rather than modelled.

A quadratic and a linear, crossing at 75 objects. The two rates that decide whether a shell at 900 km is stable, against how many objects are in it. Production goes as N² — every collision needs two objects, so the number of collisions is proportional to the square of the population, and each one is taken here to make 1600 trackable fragments. Removal goes as N, because drag acts on each object independently and takes 1,195 years to do it at this altitude. A quadratic and a linear cross exactly once, at 75 objects in this shell, and above that crossing the population grows with nothing launched. The shell presently holds about 2,280, which is 31 times the crossing. Every number on the production side is uncertain by a factor of a few — the fragment yield most of all, and the cross-section is calibrated against an observed collision rate rather than measured — so the position of the crossing carries that uncertainty with it. The shape does not, and the shape is the argument: a quadratic overtakes a linear once and never comes back, the crossing falls as the altitude rises because the lifetime is in the denominator, and what results is a threshold rather than a trend.
Fig. 10 The same two rates at a launch rate twice as high. The quadratic production curve is unchanged — it is set by the physics of collisions — and the linear removal curve rises, so the crossing moves outward and the shell tolerates more objects before it runs away. What the shell can hold is therefore not a property of the shell.

That is a genuinely uncomfortable result and it is worth stating without softening. The critical number of objects in a shell depends on how fast objects are being removed from it, and objects are removed by atmospheric drag, which depends on solar activity, and by deliberate deorbit, which depends on policy. So the same shell is subcritical during solar maximum and supercritical during solar minimum, and it is subcritical under one disposal rule and supercritical under another.

The consequence is that “is this altitude stable?” has no answer independent of what is done about it. That is unlike almost every other threshold in this collection: the Chandrasekhar mass, the Roche limit and the Jeans criterion are all statements about a configuration, and this one is a statement about a configuration plus an operating practice.

It is also why the argument is usually made about a trend rather than a state. The observed number of tracked objects in the 800–1,000 km shells has grown faster than the launch rate for two decades, which is the signature of a production term that is quadratic, and that signature does not depend on knowing where the crossing is. A rate that outruns its own input is diagnostic on its own. What it cannot say is how far past the crossing the shell already is, because the tracked catalogue is complete only down to about ten centimetres and the population that does the damage extends well below that. The trend is measured on the objects that can be seen and the runaway is driven by the ones that cannot.

Where the ladder goes

Two directions, and both are about the assumption of self-similarity that produced the exponent.

The first is strength. Real bodies are not equally easy to break: small ones are held together by material strength and large ones by their own gravity, with a minimum in the middle around a few hundred metres. A cascade whose collision outcomes depend on absolute size is not self-similar, and its steady state is a broken power law with a knee at that minimum — which is observed, in the asteroid belt, at about the size the strength argument puts it.

The second is stirring. A cascade needs relative speeds high enough to shatter, and something has to supply them. In the asteroid belt it is the resonances with Jupiter; in a debris disc it may be a planet nobody has detected. A disc that is bright is therefore evidence of dynamical excitation as much as of solid material, and reading it as the second alone is the commonest mistake made with these systems.

Where this goes next

A 128.8-million-year-old collision, dated from the shape of a scatter plot. The Erigone family: 165 members drawn at their diameters and their proper semi-major axes, with inverse diameter up the page. The cloud is a V, and the V is a clock. Each member has been drifting in semi-major axis ever since the collision at a rate that goes as one over its diameter, with a sign set by which way it spins — prograde outward, retrograde inward — so after 130 million years the small members have moved far and the large ones have barely moved at all. Plotted against 1/D that envelope is a straight line through the family's centre, and its slope is the drift rate for a one-kilometre body multiplied by the elapsed time. Fitting the two edges of the points actually drawn here returns 128.85 million years against the 130 the members were generated from. The rounding at the bottom is not an artefact: it is the ejection velocity, some 15 metres per second, which every member got at the moment of the collision and which is the same for all sizes. The picture cannot show the interlopers — background asteroids that happen to lie inside the V and have nothing to do with the family — and it cannot show the members that have drifted into a resonance and left the belt entirely, which is the reason the oldest families have the softest edges. A collision dated by a scatter plot Nothing in the solar system carries a date. A collisional family does — because a force that depends on a body's size has been pushing its fragments apart ever since, so the cloud is a V whose slope is an elapsed time, and one of those dates is confirmed by fossil meteorites in Swedish limestone. A quadratic and a linear, crossing at 149 objects. The two rates that decide whether a shell at 900 km is stable, against how many objects are in it. Production goes as N² — every collision needs two objects, so the number of collisions is proportional to the square of the population, and each one is taken here to make 1600 trackable fragments. Removal goes as N, because drag acts on each object independently and takes 1,195 years to do it at this altitude. A quadratic and a linear cross exactly once, at 149 objects in this shell, and above that crossing the population grows with nothing launched. The shell presently holds about 2,280, which is 15 times the crossing. Every number on the production side is uncertain by a factor of a few — the fragment yield most of all, and the cross-section is calibrated against an observed collision rate rather than measured — so the position of the crossing carries that uncertainty with it. The shape does not, and the shape is the argument: a quadratic overtakes a linear once and never comes back, the crossing falls as the altitude rises because the lifetime is in the denominator, and what results is a threshold rather than a trend. A collision rate that needs no collision The flux through an orbital shell is a gas-kinetic calculation with no orbits in it. Production goes as the square of the population and removal goes as the first power, so a quadratic overtakes a linear once and never comes back — and which side of that a shell is on is decided by its altitude. The light dust removes at 0.56 µm comes back at 124 µm. One energy budget drawn twice, on a wavelength axis spanning four and a half decades. The upper curve is starlight from a 6500 K photosphere; the shaded region under it is the part removed by a magnitude of visual extinction, computed from the same CCM law the other modes here draw, and weighted towards the ultraviolet exactly as that law says. The curve on the right is what the grains do with it: a modified blackbody at 20 K with an emissivity index of 1.8, normalised so that the energy under it equals the energy under the shaded region. Integrating the two curves actually drawn returns a ratio of 1.000. The starlight peaks at 0.56 microns and the re-emission at 124, a factor of 220, so nothing about the two is recognisable as the same photons and everything about them is the same joules. The practical consequence is a rule about arithmetic: a galaxy's ultraviolet luminosity and its far-infrared luminosity are not two independent measurements of how many young stars it has. One is the light that escaped and the other is the light that did not, and adding them without noticing counts part of the population twice. The dust is not lost light, it is moved light An extinction curve says how much starlight dust removes. It does not say the light is gone. Every photon a grain absorbs heats the grain, which radiates it back out at twenty kelvin — so an extinction measurement and a far-infrared spectrum are two halves of one energy budget, and the area under them is the same area.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Asteroid familyCollisional cascadeCross-sectionDebris discDustOrbital debrisPower lawPoynting robertson dragRadiation pressureSize distributionSteady state