The sunlit area in orbit is almost all in things already tracked

Satellites too faint to see one at a time still reflect sunlight, and spread across the sky their light adds a glow counted in square metres of sunlit metal. The glow does not depend on how far away the objects are, because a shell twice as high holds four times the volume at a quarter of the brightness per object; it depends only on how much reflecting area is up there and whether the Earth’s shadow has reached it. That essay found that a tenth of the natural darkness of the zenith needs about a hundred and seventy thousand square metres of effective reflecting area, and that the published estimate of a sky already brighter by about that much rested on a quantity nobody had measured: the reflecting area of the debris too small to track.

The area of the untracked debris is not measured, but it is constrained, and the constraint turns out to be strong. The pieces in orbit come in a distribution of sizes, and how that distribution’s cross-section is divided among the sizes depends on a single number, the slope of the distribution. Above a certain slope the smallest pieces hold most of the area and the unknown dominates; below it the largest pieces do, and the unknown hardly matters. This essay finds which side of that line the orbital population is on, adds up where the sunlit area actually is, and finds it is almost entirely in objects already in the catalogue.

Which sizes hold the area #

A population of fragments is described by how many pieces are larger than each size, and for debris, as for asteroids and for the dust of collisions, that count usually falls as a power of the size: $N(>L) \propto L^{-b}$. The number of pieces is dominated by the smallest, whatever the slope — there are always more small pieces than large. The area is another matter. A piece of size $L$ has a cross-section proportional to $L^2$, and the number of pieces per decade of size goes as $L^{-b}$, so the area per decade goes as $L^{2-b}$.

That exponent decides everything. If the slope $b$ is less than two, the area per decade falls towards small sizes, each decade holding a fixed fraction of the one above it, and the total converges to a value dominated by the largest pieces. If $b$ is greater than two, the area per decade rises towards small sizes, and the total is dominated by the smallest pieces, growing without limit as smaller ones are counted. At exactly two every decade holds the same area, and the total grows as the logarithm of the size range.

The collisional fragments of satellites and rocket stages follow the standard breakup model, which was fitted to ground tests of hardware being shot at and to the tracked debris of break-ups in orbit, and whose slope is 1.71. Each decade of size down holds about half the area of the one above. Everything below a millimetre together holds less area than the single decade from ten centimetres to a metre. For a population on that slope, the large pieces own the sky.

How much of a breakup’s area the catalogue cannot see #

The space-surveillance radars catalogue objects down to about ten centimetres in low orbit. For a single breakup on the model’s slope, running from metre-sized pieces down to millimetres, the catalogue sees a little more than half the cross-section; the rest is in centimetre and millimetre fragments. That share hardly changes if the distribution is followed down to ten micrometres, because each further decade adds half the area of the one before and the series converges.

For a steeper slope the picture changes. At a slope of exactly two, two-thirds of the area is hidden down to a millimetre and four-fifths down to ten micrometres, and the fraction keeps growing with every decade counted. At 2.5 the catalogue sees almost nothing. The question of whether the uncounted debris could hold the glow is therefore a question about the slope of the distribution below ten centimetres — not a question about how many pieces there are, which no one doubts is very large.

The cross-section in orbit, by kind of object #

The slope of the real population below ten centimetres is known roughly from the debris-environment models, which are fitted to radar campaigns that count centimetre objects statistically by staring at a fixed patch of sky, and to the impact craters on surfaces that spent years in orbit and were brought back — the panels of the Hubble Space Telescope and of returned satellites among them. Those models put about 1.2 million objects between one and ten centimetres in orbit and about 130 million between a millimetre and a centimetre. Between ten centimetres and one the cumulative slope is about 1.5; between one centimetre and one millimetre it is a little steeper than 2.

Adding up the cross-section class by class gives the budget. The operating satellites — about ten thousand now, most of them in the large broadband constellations — hold around a hundred and twenty thousand square metres, mostly in solar arrays. Spent rocket stages, about 2,300 of them, hold nearly sixty thousand; dead satellites another twenty thousand. The tracked fragments larger than ten centimetres, numerous as they are, hold under a thousand. And the untracked fragments — 1.2 million of one to ten centimetres and 130 million of one millimetre to one centimetre — hold about four hundred and three hundred square metres. All the debris below the catalogue limit together holds less than half of one per cent of the cross-section in orbit.

The reason is the slope. The fragments’ distribution is shallow enough that their area sits in their largest members, and even their largest members are tiny next to a single spent upper stage, which presents tens of square metres to the Sun. The objects that matter for the glow are the ones that matter for the debris of the future — the big intact ones — and every one of them is tracked, its orbit known, its size measured by its radar cross-section, its brightness measurable by any telescope that cares to look.

Why the end of twilight is the moment that counts #

The glow is computed for one moment of the night on purpose. Night has three definitions, by how far the Sun is below the horizon, and astronomical twilight ends when it is eighteen degrees down, the point at which the scattered sunlight in the upper atmosphere stops contributing to the sky’s brightness. That is the moment at which the natural sky is first fully dark, and the moment at which the shells of satellites in low orbit are still sunlit overhead: at eighteen degrees of solar depression the Earth’s shadow at the zenith reaches only about three hundred kilometres up, well below the constellations.

So the end of twilight is when the artificial glow is largest relative to the natural sky, and the hours on either side of midnight are when it is smallest, because the shadow has climbed past the shells. At high latitudes in summer the shadow never climbs high enough and the shells stay lit all night. The figures here are for the worst hour of an ordinary night, which is also the hour in which many surveys do their twilight searches for asteroids close to the Sun.

How much reflecting area a tenth of the sky needs #

The glow depends on effective reflecting area — the physical area times the fraction of sunlight it sends towards the ground at the relevant angles. A tumbling dead object of ordinary materials sends something like a tenth; an operated satellite flown so that its arrays face away from the ground at night sends less, a few per cent. With those factors, today’s active satellites add about three-tenths of a per cent to the zenith sky at the end of astronomical twilight, and the dead satellites and rocket bodies about half a per cent. The untracked fragments add four thousandths of a per cent.

A tenth of the natural sky would need about a hundred and seventy thousand square metres of effective reflecting area in low orbit. That is roughly twenty times what the whole catalogued population supplies with these reflectivities, and two thousand times what the untracked fragments supply. A hundred thousand constellation satellites as bright as today’s would bring the total to a few per cent. The arithmetic does not say the sky is unaffected — a few per cent of the natural brightness is a real loss for the faintest observations, and individual satellites leave streaks that this calculation does not include — but it does say where the diffuse part is coming from and will come from: from the large objects people put up and leave up, not from the dust of their destruction.

That also makes it a quantity that can be measured rather than estimated. The brightness of every catalogued object can be predicted from its orbit, its size and its attitude, and checked against wide-field photometry, which is already done for the constellation satellites to monitor their streaks. The sum over the catalogue, at each hour of the night and each latitude, is the glow, to within the reflectivity factors; the untracked debris adds a correction smaller than the uncertainty in those factors.

A collision multiplies the sunlit area #

Breaking an object up does add area, because the same mass divided into many pieces exposes more surface. A thousand-kilogram satellite with a cross-section of about ten square metres, shattered in a collision, becomes about fifty square metres of fragments by the breakup model; a ten-tonne rocket stage, nearly three hundred. A collision is therefore a multiplier of reflecting area by a factor of about five, and the rate of such collisions in the crowded shells is a rate of growth of the sky’s glow as well as of the hazard.

But the multiplication does not move the area out of the catalogue. Even for the fragments of a breakup, the slope of 1.71 keeps more than half the new area in pieces above ten centimetres, which the radars will track within weeks of the event. The fragments nobody can see and nobody can shield against — the centimetre pieces that are the lethal hazard to spacecraft — are exactly the ones that matter least to the sky, because the same shallow slope that makes them numerous makes their total area small. The hazard and the glow are owned by different ends of one distribution: the number by the small end, the area by the large.

Air removes the small pieces first, where there is air #

The population in orbit is not the population produced, because drag removes pieces at rates that depend on their size. A fragment’s orbital lifetime is set by its mass per unit of cross-section, and for fragments of one material that grows in proportion to size: a millimetre flake decays ten times faster than a centimetre fragment of the same shape. In the lower shells, where a centimetre fragment lasts months, the small pieces are removed almost as fast as they are made, and the standing population is far shallower than the produced one — the millimetre decade’s share of the area is a ten-thousandth of the metre decade’s. Higher up, at eight hundred and fifty kilometres, where centimetre fragments last centuries, the standing population keeps almost the shape it was made with.

Both effects push the same way. Drag makes the low-altitude debris even more dominated by its largest pieces than the breakup model already makes it, and at high altitude the breakup model’s own slope keeps the small pieces’ share modest. Nowhere in low orbit does the size distribution of debris become steep enough for the untracked pieces to hold the area.

The same rule in the asteroid belt #

The rule about slopes is not specific to debris. A collisional population of any kind has its number held by its small members and its area held by whichever end the slope favours, and the same two-sided question arises wherever area rather than number is what is observed. The dust of a planetary system is where the light is, while its large bodies are where the mass is, because a collisional cascade grinding down to the size at which radiation removes grains settles at a slope of about 2.5, steep enough that the smallest grains dominate the area. The asteroid belt’s own size distribution, used to count how often the belt breaks, is close to two over the kilometre sizes surveys reach and shallower among its large members, which is why a few hundred large asteroids hold so much of its reflected light.

Orbital debris sits on the shallow side for a reason. It is young: most of it was made in a few dozen break-ups over sixty years, and it has not had time to grind itself down into a steady cascade, which needs pieces to collide with each other many times over. A shell where that grinding had gone on for a long time — where collisions between fragments had become more frequent than collisions involving intact objects — would steepen towards the cascade slope, and its smallest pieces would begin to dominate the area. That is the regime the Kessler syndrome describes, and the calculation here says one more thing about it: a cascade in orbit would show itself in the sky’s glow only late, after the number of fragments had grown by orders of magnitude, because the area follows the slope and the slope changes slowly.

What the arithmetic leaves out #

The population counts are rounded from published environment models and the cross-sections of the intact objects are typical values, not a sum over the catalogue. The effective reflecting fractions are assumptions, uncertain by factors of two or three and different for every satellite design and attitude; the operated satellites’ value depends on how carefully they are flown at night. The fragments’ area uses the breakup model’s average relation between size and area, and its slope is assumed to hold from a metre down; the measured slope steepens slightly below a centimetre, which is included in the budget but not in the single-breakup figures. All the glow is computed for a single shell at 550 kilometres at the end of astronomical twilight at the zenith; the geostationary belt, the higher shells and other hours of the night change the absolute numbers but not the division of the area among sizes. Streaks from individual bright satellites, which are the main harm to most observations, are a different problem and are not part of the diffuse glow.

Still open: whether the reflectivities are what they are assumed to be #

The area is now accounted for, to within a factor of a few, and almost all of it is in catalogued objects. What is not accounted for is how much of that area reflects towards the ground at night. The operated satellites are designed with that in mind — dark coatings, sunshades, attitude modes that turn the arrays edge-on — and their brightnesses are monitored. The dead satellites and spent stages are not designed for anything any longer, tumble freely and present whatever face the tumble gives them, and their reflectivities are known from a modest number of photometric campaigns.

A glow survey would settle it directly: a sensitive photometer measuring the zenith brightness through the night, through the moment the Earth’s shadow climbs past each shell, and subtracting the natural sky by its known behaviour. The step in brightness as each shell enters the shadow is the glow from that shell, and comparing it with the sum over the catalogue’s predicted brightnesses measures the effective reflectivity of everything up there at once. If the step is a few tenths of a per cent, the catalogue is the whole story and the sky’s future brightness is in the hands of the operators who decide what to launch and how to fly it. If it is several per cent, something in orbit reflects far more than its catalogued size suggests, and finding out what would be the first measurement of the sky made by looking at the glow rather than at the objects.