Astronomy & Celestial Mechanics
the sky, computed
Orbits

The drag that sorts a disc by size

Starlight does three different things to a dust grain depending on how big it is — blows it out of the system, drags it inward over millennia, or ignores it entirely. Which one happens is decided by a single length, and whether it happens at all is decided by how crowded the disc is.

Assumes Collisional cascade and Non-gravitational forces.

17 min read 8 figures Scales that defeat intuitionSmall perturbations, large consequences

A dust grain in orbit around a star is not simply a very small planet. Gravity acts on its mass and starlight acts on its cross-section, and those two scale differently with size — as the cube of the radius and as the square. Their ratio is therefore proportional to one over the radius, and below some size the starlight wins.

That size is a single number, fixed by the star’s luminosity and the grain’s density and nothing else. Above it a grain is bound and its fate is decided by a competition between two clocks. Below it a grain is not in orbit at all.

Below 0.46 microns a grain is not in orbit at all. The ratio of the radiation force to the gravitational force on a dust grain, against the grain's radius, for three densities. Every line has slope exactly −1 because gravity acts on the mass and radiation on the cross-section, and the ratio of a volume to an area is a length. Two horizontal lines matter and they are different statements. At β = 1 the star does not attract the grain at all. At β = 1/2 a grain released at rest from a circular orbit is already unbound, because it keeps the speed appropriate to the full stellar mass while feeling only half of it — and since dust is made by breaking up larger bodies that were on circular orbits, the lower line is the one that applies. For rock at 2500 kilograms a cubic metre that is 0.46 microns; for ice it is 1.15, and for iron 0.15. A collisional cascade that grinds material finer runs into this floor and stops, and the material that would have been finer leaves the system on a hyperbola.
Fig. 1 The ratio of the radiation force to gravity, against grain radius, for three densities. Every line has slope exactly minus one, because a volume divided by an area is a length. The line that matters is not the one where the two forces are equal but the one at half that: a grain made by breaking up a body on a circular orbit inherits that body’s speed while feeling only a fraction of the star’s pull, and is already unbound when the fraction drops below one half. For rock around the Sun that is 0.46 microns.

The size that is not a size of anything

It is worth pausing on how strange the blowout size is as a quantity. It is a length, and it is not the length of anything. No grain has that radius in particular; nothing about the material picks it out; it is not a wavelength or a mean free path or a distance between anything and anything else.

It is a ratio of a luminosity to a density, with a handful of constants attached, and it comes out in metres because the constants do. Written out,

sblow=3LQpr8πGMcρ,s_{\rm blow} = \frac{3LQ_{\rm pr}}{8\pi GMc\rho},

with QprQ_{\rm pr} the radiation-pressure efficiency, which is near one for grains larger than the wavelength of the light and falls for grains smaller. The distance from the star does not appear, because both forces fall as the inverse square and the ratio is the same everywhere. That is a genuinely useful accident: a grain that is unbound at one astronomical unit is unbound at a hundred, so the blowout condition is a property of the grain rather than of where it is.

The consequence for a debris disc is that it has a hard floor. A cascade of collisions grinds material to smaller and smaller fragments, and the grinding stops at the blowout size — not because the physics of fragmentation changes, but because everything smaller leaves. The material that would have been the fine end of the distribution is streaming out of the system on hyperbolic orbits, and around the Sun those grains have a name: beta meteoroids, detected directly by dust instruments on interplanetary spacecraft, moving outward at tens of kilometres a second.

Below 3.50 microns a grain is not in orbit at all. The ratio of the radiation force to the gravitational force on a dust grain, against the grain's radius, for three densities. Every line has slope exactly −1 because gravity acts on the mass and radiation on the cross-section, and the ratio of a volume to an area is a length. Two horizontal lines matter and they are different statements. At β = 1 the star does not attract the grain at all. At β = 1/2 a grain released at rest from a circular orbit is already unbound, because it keeps the speed appropriate to the full stellar mass while feeling only half of it — and since dust is made by breaking up larger bodies that were on circular orbits, the lower line is the one that applies. For rock at 2500 kilograms a cubic metre that is 3.50 microns; for ice it is 8.75, and for iron 1.12. A collisional cascade that grinds material finer runs into this floor and stops, and the material that would have been finer leaves the system on a hyperbola.
Fig. 2 The same construction around an A-type star of sixteen solar luminosities and a bit over two solar masses. The lines have shifted up by nearly an order of magnitude, because the luminosity has risen much faster than the mass, so the blowout size for rock is 3.5 microns rather than 0.46. That is why the discs around bright young stars are made of visibly coarser material than the solar system’s, and it is a prediction with no free parameters in it: the ratio of the two blowout sizes is the ratio of the two luminosity-to-mass ratios.

There is a second consequence, and it is about how the dust is made rather than about how it leaves. A grain does not appear at rest; it appears as a fragment of a parent body that was itself on a nearly circular orbit. At the instant of release it has that orbit’s velocity and suddenly feels a reduced central mass, so its orbit is instantly transformed into an ellipse with the release point at perihelion — or into a hyperbola, if the reduction is more than half. This is why the threshold is at β=1/2\beta = 1/2 rather than at β=1\beta = 1, and it is also why grains a little above the threshold have eccentric orbits they did not inherit. A population of small grains released from a circular ring is therefore not a ring: it is a set of nested ellipses all sharing a perihelion distance, which spreads outward and is visible in resolved images as a halo beyond the parent ring.

Something similar sets the practical floor on how small a grain can be observed, as opposed to how small it can exist. A grain much smaller than the wavelength of the light it is being looked for in radiates inefficiently at that wavelength, so the infrared emission of a disc is dominated by grains of about the wavelength in size. The blowout size and the observing wavelength are unrelated numbers, and whether a disc’s smallest grains are visible at all depends on which of them is larger. For the Sun and a mid-infrared survey the two happen to be within a factor of a few of each other, which is a coincidence that has caused a good deal of confusion.

The second force, which is the first one seen from the grain

A grain above the blowout size is bound, and it still does not stay where it is. The same radiation that pushes it outward also drags it backwards, and the drag is a relativistic effect of the most ordinary kind.

In the star’s frame the grain absorbs radially and re-emits isotropically in its own frame. In the grain’s frame the incoming starlight is aberrated forward by the angle v/cv/c, so it arrives slightly from ahead rather than exactly radially. The forward component is a drag. Its magnitude is β\beta times v/cv/c times the gravitational force, which is small twice over, and it acts continuously for as long as the grain exists.

The result is a slow inward spiral on a timescale

tPR≈400 (a/au)2β years,t_{\rm PR} \approx 400\,\frac{(a/\text{au})^2}{\beta}\ \text{years},

which for a ten-micron grain at three astronomical units is a few hundred thousand years. That is short compared with the age of the solar system by four orders of magnitude, which is the whole reason the zodiacal cloud requires a continuing supply: every grain now visible arrived recently, and the cloud is a steady state rather than a relic.

Two clocks, and only one of them knows how big the grain is. The time a grain at 3 astronomical units takes to spiral into the star under radiation drag, against its size, with the time between collisions drawn for three disc densities. The drag time rises exactly in proportion to the size, because the drag is proportional to β and β is inversely proportional to the size — a big grain has too much inertia for the starlight to move. The collision time is flat, because a collision does not care how big the target is, only how much material is in the way. Where the two cross is the size above which a grain is destroyed before it can migrate. In the solar system's zodiacal cloud, at an optical depth near 10⁻⁷, the crossing is at 200 microns and almost everything smaller than a sand grain reaches the Sun; in a debris disc a thousand times denser it falls to 0.03 microns, and nothing at all arrives. The same physics, the same star, and two completely different discs.
Fig. 3 The two clocks at three astronomical units. The drag time rises exactly in proportion to the grain size, because the drag is proportional to the radiation-to-gravity ratio and that ratio is inversely proportional to the size — a large grain has too much inertia for starlight to move. The collision time is flat, because a collision does not care how big the target is, only how much material is in the way. Where they cross is the size above which a grain is destroyed before it can migrate, and the crossing moves by orders of magnitude with the density of the disc.

Which clock is shorter is a property of the disc

Nothing so far has been about the disc. The blowout size is a property of the star and the grain; the drag time is a property of the star, the grain and the orbit. The competition that decides what actually happens introduces the one quantity that belongs to the population.

In a disc of vertical optical depth τ\tau, a grain on a randomly inclined orbit encounters other grains at a rate that makes the mean time between collisions about the orbital period divided by 4πτ4\pi\tau. The optical depth is the fraction of the sky, seen from inside the disc, that is covered by material — a purely geometric quantity, and the one an observation of a disc most directly measures.

The solar system’s zodiacal cloud has τ≈10−7\tau \approx 10^{-7}. A young star’s debris disc has τ≈10−3\tau \approx 10^{-3}, four orders of magnitude denser. Four orders of magnitude in the collision time moves the crossing point in the previous figure by four orders of magnitude in size, and that is enough to change the character of the system completely.

Two clocks, and only one of them knows how big the grain is. The time a grain at 40 astronomical units takes to spiral into the star under radiation drag, against its size, with the time between collisions drawn for three disc densities. The drag time rises exactly in proportion to the size, because the drag is proportional to β and β is inversely proportional to the size — a big grain has too much inertia for the starlight to move. The collision time is flat, because a collision does not care how big the target is, only how much material is in the way. Where the two cross is the size above which a grain is destroyed before it can migrate. In the solar system's zodiacal cloud, at an optical depth near 10⁻⁷, the crossing is at 72 microns and almost everything smaller than a sand grain reaches the Sun; in a debris disc a thousand times denser it falls to 0.02 microns, and nothing at all arrives. The same physics, the same star, and two completely different discs.
Fig. 4 The same competition out at forty astronomical units, where a Kuiper belt or a resolved debris ring lives. The drag time has risen by more than two orders of magnitude, because it goes as the square of the distance, while the collision time has risen only by the orbital period, which goes as the three-halves power. Collisions therefore win more easily far from the star than near it, and the crossing size falls accordingly. A disc can be a flow in its inner region and a mill in its outer one, with the transition at a radius that depends on nothing but its own surface density.

In the collision-dominated case, essentially no dust reaches the star. Every grain is shattered into smaller grains, which are shattered again, until the fragments fall below the blowout size and leave. The disc is a machine for converting large bodies into escaping micron-sized grains, and the star at the centre receives almost nothing.

In the drag-dominated case, grains spiral inward largely intact, and the inner solar system is fed a steady stream of material from the asteroid belt and from comets. That stream is what is seen as the zodiacal light, and its inward flux at one astronomical unit is measurable: some tens of tonnes a day arrive at the Earth’s orbit, of which a few tens of tonnes a day strike the Earth.

The distinction between the two regimes has a consequence that sounds backwards until it is stated carefully. A brighter disc delivers less material to its star. Brightness is emitting area, which is optical depth; higher optical depth means shorter collision times; shorter collision times mean grains are destroyed before they can migrate. So the discs that are easiest to detect are precisely the ones in which the transport this essay is about does not operate, and the discs in which it does operate are too faint to image. Everything known about drag-dominated transport comes from the one such disc that can be observed from inside it.

The same argument bounds what a disc can deliver in absolute terms. There is a maximum inward mass flux that radiation drag can sustain, because raising the surface density to increase the flux also shortens the collision time and destroys the carriers. The maximum is reached at an optical depth of a few times 10−610^{-6}, and it is remarkably low — a few hundred tonnes a day in solar-system terms. A star whose inner regions are being fed at a much higher rate than that is not being fed by drag from an outer belt, and something else has to be delivering the material.

The mark the floor leaves on everything above it

A collisional cascade in equilibrium has a size distribution with a specific exponent. Dohnanyi derived it in 1969 from a single requirement — that the mass flowing through every size be the same — and got n(s) ds∝s−3.5 dsn(s)\,ds \propto s^{-3.5}\,ds, with no property of the material in it at all.

Real discs do not show a clean power law, and the reason is the floor.

A power law with a wave printed on it by its own bottom edge. The number of grains per size interval in a collisional cascade, against size, both axes logarithmic. The straight line is the classical result: if every size is losing mass to smaller sizes at the same rate, the exponent is −3.5 and nothing about the material enters. The drawn curve is what that becomes when the cascade is given a bottom. Below 0.46 microns radiation pressure removes grains faster than collisions can make them, so the population ends; the grains just above the edge then have nothing small left to be broken by, survive longer than the steady state allows and pile up; the pile then over-grinds the sizes above it, which deplete; and the alternation propagates upward, damping as it goes. The wave is a few tenths of a decade in amplitude and about a decade of size in period, and it matters because the size distribution is what sets a disc's colour: a survey that fits a straight line to a disc's spectrum is fitting a curve that is not straight for a reason nothing about the grains themselves would suggest.
Fig. 5 What the cascade becomes when it is given a bottom edge. Below the blowout size the population ends. The grains just above the edge then have nothing smaller left to break them, survive longer than the steady state allows, and pile up; the pile over-grinds the sizes above it, which deplete; and the alternation propagates upward, damping as it goes. The wave is a few tenths of a decade in amplitude with a period of about a decade in size, and it is an entirely predictable consequence of removing the small end of a cascade.

The wave matters observationally because the size distribution is what sets a disc’s colour and its total emitting area. A survey that fits a straight power law to a disc’s infrared spectrum is fitting a curve that is not straight, for a reason that has nothing to do with the grains’ composition — and the residuals of such a fit look like a compositional signature.

A power law with a wave printed on it by its own bottom edge. The number of grains per size interval in a collisional cascade, against size, both axes logarithmic. The straight line is the classical result: if every size is losing mass to smaller sizes at the same rate, the exponent is −3.5 and nothing about the material enters. The drawn curve is what that becomes when the cascade is given a bottom. Below 1.15 microns radiation pressure removes grains faster than collisions can make them, so the population ends; the grains just above the edge then have nothing small left to be broken by, survive longer than the steady state allows and pile up; the pile then over-grinds the sizes above it, which deplete; and the alternation propagates upward, damping as it goes. The wave is a few tenths of a decade in amplitude and about a decade of size in period, and it matters because the size distribution is what sets a disc's colour: a survey that fits a straight line to a disc's spectrum is fitting a curve that is not straight for a reason nothing about the grains themselves would suggest.
Fig. 6 The same distribution for icy grains, whose lower density puts the blowout size higher — 1.15 microns rather than 0.46 — and therefore moves the whole wave to larger sizes. The exponent far from the edge is unchanged at −3.5, because that number comes from the requirement that mass flow steadily through every size and knows nothing about what the grains are made of. Everything that is a property of the material is in the position of the cut-off and in nothing else.

A further complication, and it is the one that makes real discs interesting rather than merely computable: a planet in the way. Grains spiralling inward under drag pass through the mean-motion resonances of any planet they encounter, and a resonance can halt the drift — the resonant torque balances the drag and the grain stops migrating, accumulating in a clump that co-orbits with the planet. The same resonant capture that locks a moon’s rotation or clears a gap turns a smooth inward flow into a structured one, and the structure is a map of a body that may be far too faint to see. The Earth has such a ring of trapped zodiacal dust, detected as a slight enhancement of the infrared background at the Earth’s own orbital longitude and trailing it.

What was actually measured

Three measurements anchor the picture, and they are of three different kinds.

The blowout population, detected directly. Dust detectors on Pioneer, Ulysses and Helios have recorded grains on unbound orbits moving outward through the inner solar system, at fluxes consistent with the cascade’s production rate. These are the grains the cascade cannot keep, caught in the act of leaving.

The inward flux, weighed at the top of the atmosphere. The rate at which interplanetary dust arrives at the Earth is measured several ways — from the accumulation of extraterrestrial helium and iridium in deep-sea sediments and polar ice, from radar meteors, and from spacecraft impact counters. The methods agree on tens of tonnes a day, which matches the flux the drag calculation requires for a cloud of the observed optical depth.

The absence of an inner disc where collisions win. Resolved images of debris discs around young stars show rings with sharp inner edges and clear interiors. If drag were transporting material inward, the interior would be filled at a level set by the drag time; it is not, which is the observational statement that those discs are collision-dominated. The measurement is of something that is not there, which makes it a stronger constraint than a detection would be, because it cannot be explained by a sensitivity limit.

Three fates, and the grain only chooses one of the boundaries. What becomes of a grain at 3 astronomical units, as a function of its size and of the disc's optical depth. The vertical boundary at 0.46 microns is blowout: everything to the left of it leaves on a hyperbola within one orbit, and the line is fixed by the star's luminosity and the grain's density with nothing about the disc in it. The sloping boundary is the equality of the two clocks, and it runs at exactly −1 because the inspiral time rises in proportion to the size while the collision time falls in proportion to the density of material. Below the slope a grain spirals into the star; above it, it is hit and broken into pieces that face the same map one step to the left. The solar system sits near the bottom of this plot and its dust is a flow; a young star's debris disc sits near the top and its dust is a mill, which is why a disc that is obviously bright in the infrared can be delivering nothing at all to the star it surrounds.
Fig. 7 The whole argument on one map: what becomes of a grain, as a function of its own size and of the density of the disc it is in. The vertical boundary is blowout and belongs to the grain. The sloping boundary is the equality of the two clocks and runs at exactly minus one, because the drag time rises with size while the collision time falls with density. The solar system sits near the bottom of this plot and its dust flows; a young star’s disc sits near the top and its dust is milled. Same star, same physics, two completely different systems.

Where the picture stops

Three of them, and the third is the one that keeps the subject alive.

The efficiency is not one. QprQ_{\rm pr} is treated above as a constant and it is not: for grains comparable to or smaller than the wavelength of the starlight it depends on size and on composition in a complicated way that requires a scattering calculation. That matters exactly at the blowout size, which is where the grains are smallest, so the sharp edge in the drawings is in reality a smeared one whose position depends on what the dust is made of.

Grains are charged, and the star has a magnetic field. A small grain in a stellar wind picks up a charge of a few volts and feels a Lorentz force from the wind’s magnetic field. For grains near the blowout size this is comparable to the radiation forces, and it makes their trajectories depend on the phase of the stellar cycle — which is measurable in the solar case and is one of the reasons the beta-meteoroid flux varies.

And the whole picture assumes a steady state that a real disc may not have. A single large collision injects a burst of dust that decays over a collision time, and if such events are rare and large the disc’s brightness is dominated by the aftermath of the last one. Several bright discs are best explained that way, and for those the machinery in this essay describes the decay rather than the equilibrium.

There is a fourth limit worth separating out, because it is about the observation rather than the theory. Every optical depth quoted above is inferred from a brightness, and converting a brightness into an area requires knowing how efficiently the grains scatter or emit — which is the same QQ that was set to one two sections ago. Dust between the observer and a source removes light and re-radiates it, and the two halves of that energy budget are measured at completely different wavelengths with completely different systematics. A disc’s optical depth measured in scattered light and the same disc’s optical depth measured in thermal emission routinely disagree by a factor of a few, and the disagreement is a statement about the grains rather than about the disc.

Why this belongs with the cascade rather than with radiation

The reason to file this argument with the collisional cascade rather than with the forces is that it is what makes a cascade a closed system with a definite output. Without radiation the grinding would continue indefinitely and the distribution would have no bottom. With it, the cascade has a well-defined product — grains at the blowout size, leaving — and a well-defined lifetime, and both are computable from the star and the disc.

There is also a pleasing symmetry with the way the argument runs in the solar system’s small bodies. A crater count on an old surface is a measurement of a cascade’s product at one particular size, and the contamination that count suffers comes from a second population created by the same grinding. A rubble pile held together by almost nothing is the cascade seen at its top end, where fragments are large enough for gravity to reassemble them. The cascade is one process, and which of its features is visible depends entirely on the size at which a particular instrument happens to look.

That closure is what makes a debris disc a measurable object rather than a description. The dust seen in the infrared is a tiny fraction of the mass, and the mass is inferred by running the cascade backwards from the observed emitting area to the reservoir of large bodies that must be feeding it. Every step of that inference uses one of the numbers in this essay, and the answer is the mass of a planetesimal belt that no telescope can see directly.

The same idea appears twice more in the collection with different clothes. A collision rate in Earth orbit that needs no collision to measure is the same competition of timescales with drag from an atmosphere rather than from starlight. And the size-dependent drift that spreads an asteroid family is the same statement about a force that scales with area acting on a body whose inertia scales with volume — sorting a population by size because it must.

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. 8 Size sorting in the one place it can be seen directly. An asteroid family spreads in semi-major axis at a rate inversely proportional to the diameter of each member, so a plot of inverse diameter against semi-major axis has a V whose opening angle is the age of the collision. It is the same physics as this essay’s drag — a surface force acting on a volume of matter — and it is the case where the sorting is slow enough and the bodies large enough to be catalogued one by one.

One observation to end on, and it is about why a quantity like the blowout size is worth naming at all. It is not a threshold anybody can measure directly — no experiment weighs a grain and watches it leave. It is a boundary in a parameter space, inferred from the shape of a distribution that stops there, from a population of unbound grains flowing outward past it, and from the wave it prints on the sizes above it. All three of those are indirect, and all three agree. That is the ordinary condition of a number in this subject, and the agreement of three indirect measurements is a stronger statement than one direct one would be.

Where the ladder goes next

The rung directly above is the reservoir itself: how a disc’s observed infrared brightness is turned into a mass of unseen parent bodies, and what that inference assumes. The one above that is the case this essay has kept setting aside — a disc with a planet in it, where resonances trap the drifting grains and the smooth inward flow becomes a structure with gaps and clumps that maps a body nobody can see.

Where this goes next

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. 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. Every model curve has slope −1, and four measurements agree on κ to 1.5×. Semi-major-axis drift against body diameter, for a thermal recoil in which a fraction κ = 0.085 of the absorbed sunlight comes back out along-track. The three curves are the same expression at 1, 1.6, 2.5 astronomical units, and each has a slope of exactly −1: the acceleration is the absorbed power divided by the mass, which is a cross-section over a volume, so it falls as one over the size and nothing else on this axis changes it. A kilometre-wide body drifts a few metres a year; a ten-metre one drifts hundreds. The four filled marks are the bodies whose drift has actually been measured as a fitted parameter in an orbit solution, and they do not lie on any single curve because each carries its own density, distance and obliquity. What they agree about is the number beside each: solve every measured drift for the efficiency that would produce it and the four answers are 0.084, 0.085, 0.089, 0.129 — a factor of 1.5 apart, for a quantity that could in principle have been anything from zero to a fifth. That agreement is the evidence that the mechanism is understood, and it is the only evidence there is, because the thermal conductivity that sets κ has never been measured for any of them. An orbit moved by heat A rotating body re-radiates absorbed sunlight from the hemisphere that has had time to warm, so the recoil is not aimed at the Sun. The resulting force is a few parts in ten billion of gravity, it is the only orbital force whose sign depends on which way the body spins, and it has been measured to four figures. 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.

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.

Beta meteoroidBlowout sizeCollision timescaleCollisional cascadeDebris discOptical depthPoynting robertson dragRadiation pressureSize distributionZodiacal light