A glow counted in square metres of sunlit metal

A satellite crossing a telescope’s field during an exposure leaves a trail, and trails can be dealt with: predicted and avoided, masked out of the image, or caught by taking several short exposures instead of one long one. The Earth’s shadow helps, because a satellite in shadow reflects nothing, and for a shell at five hundred and fifty kilometres the zenith goes dark when the Sun is twenty-three degrees below the horizon. The trails are a nuisance with a geometry, and astronomers have learned to schedule around geometry.

There is a second effect that has no trail. Every object in orbit that is in sunlight reflects some of it, and most objects in orbit are far too faint to leave a visible trail — fragments of broken rockets and satellites, centimetres to tens of centimetres across, numbering in the hundreds of thousands. Their light, and the light of satellites too faint or too far from a given exposure to be noticed one at a time, adds to the sky’s background. It is a glow rather than a streak, it covers every exposure, and no software can subtract it without knowing exactly how bright it is.

How bright it is turns out to be a simple calculation with one surprising feature.

A brightness that does not depend on distance #

Take a shell of objects at altitude $h$, spread evenly over the sphere, with a number per unit area $n_s$ on the shell. Each object reflects sunlight with some effective area $A_p$ — its cross-section times its reflectivity — and a dependence on the angle between the Sun and the observer as seen from the object, which for a rough sphere is a smooth function falling from full brightness when lit face on to zero when lit from behind. An object at distance $\rho$ delivers a flux proportional to $A_p/\rho^2$.

Now look along a line of sight. A patch of sky of solid angle $\Omega$ contains the objects on a patch of shell of area $\Omega\rho^2/\cos\theta$, where $\theta$ is the angle at which the line of sight crosses the shell. The number of objects in it grows as $\rho^2$ and the flux from each falls as $1/\rho^2$, and the two cancel. The sky’s brightness from the shell is

$$B \propto \frac{n_s\, A_p\, \Phi(\alpha)}{\cos\theta},$$

the surface density on the shell times the reflecting area per object times the phase function, divided by the slant. The distance has dropped out. It is the same cancellation that makes a uniform wall of stars equally bright at every distance, Olbers’s argument applied to a shell of metal, and it means that moving a constellation higher does not make its glow fainter. Fewer objects are needed to cover the Earth from a higher shell, but each square degree of sky contains the same reflecting area.

The opening figure shows the consequence. The two shells, at 550 and 1,200 kilometres, add almost the same glow at the zenith in twilight, three per cent of the natural sky for the illustrative hundred thousand objects. What differs is when the glow stops.

The same physics as the zodiacal light #

The cancellation of distance has a natural precedent, and it is already in the sky. Interplanetary space is filled with dust — grains shed by comets and ground off asteroids, spread in a thick disc around the Sun’s equator — and the sunlight those grains reflect is the zodiacal light, the faint cone that rises from the horizon after evening twilight in spring and before dawn in autumn. Away from the Milky Way it is one of the three largest contributions to the natural night sky, beside the glow of the upper atmosphere’s own chemistry and the integrated light of faint stars.

The zodiacal light obeys the same arithmetic as the glow from a shell of satellites. Its brightness along a line of sight depends on how much reflecting area the line passes through, not on how far away the grains are, and it is brightest where the grains are lit at the angles that reflect most — toward the Sun, for dust grains, which scatter forward, unlike the rough metal surfaces of satellites, which reflect most when lit face on. A sky that already carries a glow of reflected sunlight from a natural cloud of dust is acquiring a second one from an artificial cloud of metal, and the two can be told apart only by their geometry: the dust glow follows the ecliptic and does not care about the Earth’s shadow; the metal glow switches off when the shadow passes it.

When the shadow reaches the shell #

Objects in the Earth’s shadow reflect nothing, so the glow at each point of the sky switches off when the shadow, climbing as the Sun sinks, reaches the shell along that line of sight. At the zenith the shell at 550 kilometres goes dark when the Sun is twenty-three degrees down, the shell at 1,200 kilometres at thirty-three. Astronomical twilight ends, by definition, at eighteen degrees, so both shells are lit at the zenith for the first part of every dark night and the higher one for much longer.

The glow is not uniform across the sky. Toward the horizon each line of sight crosses the shell at a slant and passes more objects, which brightens the sky there; opposite the Sun the objects are seen nearly face on, which brightens it further, so the brightest part of the glow is low in the sky away from the Sun — until the shadow, rising from that horizon, reaches it. Toward the Sun the objects are backlit, and the phase function dims them more than the slant brightens them. As the night deepens the shadow sweeps over the sky from the anti-solar horizon, past the zenith, toward the Sun, and the last glow left is a band low in the sky above where the Sun went down.

Area, not number #

Because the glow is proportional to the reflecting area per unit of shell, what matters is the total reflecting area in orbit — the number of objects times the area of each.

Satellites of the large constellations now being launched have effective reflecting areas — area times reflectivity at typical viewing angles — of a few tenths of a square metre, which is what makes them fifth or sixth magnitude at the zenith. A hundred thousand of them, the scale of the constellations that have been proposed or licensed, would add a few per cent to the zenith sky in twilight. For the glow to reach a tenth of the natural sky from satellites alone needs several hundred thousand at that brightness, or a hundred thousand that are several times brighter.

That is lower than the figure often quoted. Published estimates put the present diffuse brightening at around a tenth of the natural sky already, with today’s few thousand operational satellites a small part of the contribution. The rest is debris: the fragments of exploded rocket stages, collisions and anti-satellite tests, from centimetres to metres, in numbers that are known only above about ten centimetres, where radar tracks them. The fragments nobody can see are estimated from models of break-ups and a handful of returned surfaces, and their total reflecting area — the only quantity the glow depends on — is uncertain by a factor of several. The figure says which number would have to be right for the tenth to be right: around a hundred thousand square metres of sunlit reflecting area in low orbit.

That figure is not implausible. The large constellations’ satellites have panels and arrays of tens to a hundred square metres each, most of which faces away from the ground when they are operated with care, and the debris population’s total cross-section, extrapolated from the objects radar can track down to the centimetre fragments it cannot, runs to tens of thousands of square metres. The uncertainty is in the reflectivity and in how much of each object’s area is presented to the ground at a given moment, and those are exactly the quantities a brightness monitor watching the glow switch on and off would measure.

What can be seen and what cannot #

The difference between a trail and a glow is a threshold. An object brighter than the faintest trail a survey’s software detects can be found and masked; anything fainter becomes part of the background.

How the glow splits depends on how the reflecting area is distributed over sizes. Debris populations follow power laws, with the number above a given size falling as the size to about the minus two, and the slope decides whether the area sits mostly in the few large objects or the many small ones. For a shallow distribution almost all of the glow comes from objects bright enough to leave trails, which can in principle be masked one by one; for the steepest distributions measured, more than half comes from objects that will never be seen individually. Large satellites add a separate population at the large end, which pushes the balance toward trails.

The glow from the unresolved objects is the part that matters most for the faintest work, because it cannot be removed. A faint star is measured against the sky, and the precision of the measurement is set by the photon noise of the background; a sky ten per cent brighter makes every faint measurement take ten per cent longer to reach the same precision. A few per cent of extra background is a few per cent of telescope time, across every observatory, every night the glow is on.

What a few per cent costs #

The cost of a brighter sky is easy to state for the kind of observation most affected by it. A measurement of a source much fainter than the sky is limited by the photon noise of the sky in the aperture, which grows as the square root of the sky’s brightness; to reach the same signal-to-noise ratio, the exposure has to grow in proportion to the sky. A sky three per cent brighter is three per cent more telescope time for the same result, or a limiting depth three hundredths of a magnitude shallower in the same time.

That sounds small, and for one observation it is. Natural sky brightness varies by more than that from night to night with the solar cycle’s effect on the upper atmosphere, and by a factor of ten with the Moon. But the glow is not a fluctuation that averages out; it is a systematic addition to every exposure, at every observatory, on every night it is on, and it grows with every satellite launched and every fragment made. For a survey designed to reach a fixed depth over a decade, a few per cent of extra background is a few per cent of the survey, every year, with nothing to show for it. For the surveys of low surface-brightness light — the faint outskirts of galaxies, measured at a thousandth of the sky’s level — a background that changes with the shadow’s geometry through each night is not just more noise but a structure that has to be modelled before anything fainter can be seen.

How much of the night the glow is on #

The glow is on only while the shell above the observer is sunlit, and that depends on the latitude and the season through the length of the night.

At the latitudes of most large observatories, around thirty degrees, the zenith of a 550-kilometre shell is lit for less than an hour of each equinox night after astronomical twilight ends, at the start and at the end, and dark through the middle. That is the geometry that makes low shells tolerable. Further from the equator and nearer midsummer the nights are shorter and the shadow lower, and the lit fraction grows: at forty degrees in June more than a third of the short dark night has the glow on. Further north still, a June night never becomes astronomically dark at all, and the question of the glow becomes moot only because the sky is already bright.

A high shell at midsummer #

The altitude of the shell does not change the glow’s brightness, but it changes how long the glow lasts, and at midsummer that matters a great deal.

At midnight on the June solstice at forty-five degrees north, the Sun is only twenty-two degrees below the horizon, and the Earth’s shadow over the zenith reaches only to about four hundred and eighty kilometres. Every shell above that is lit at the zenith all night. At fifty-five degrees, every shell above three hundred kilometres is. A constellation placed at twelve hundred kilometres to reduce the number of satellites it needs adds its glow to the whole of every short summer night at the latitudes of the European observatories, and to much of every summer night further south. A constellation at five hundred and fifty kilometres is dark at the zenith at midnight from thirty-five degrees south to thirty-five north, and lit in summer beyond.

Why higher was supposed to help #

The independence from altitude cuts against an intuition that has shaped how constellations are discussed. A satellite twice as far away is four times fainter, and a constellation at twelve hundred kilometres needs fewer satellites than one at five hundred and fifty to give the same coverage of the ground, so a high constellation seems better for the sky on both counts. For trails, part of that is true: each satellite is fainter, and its trail is fainter in proportion — though it also crosses the field more slowly, so the trail’s surface brightness falls by less than the satellite’s.

For the glow, neither advantage survives. Fainter satellites at a greater distance are exactly compensated by more of them per square degree of sky, and the shell’s glow is set by its reflecting area per unit of shell area. Fewer satellites over a larger shell reduce that only if the total reflecting area falls — which it does if the satellites are no larger, and does not if a higher orbit needs larger satellites with bigger antennas and solar arrays to reach the ground. What altitude certainly changes is the time the glow lasts, and it changes it in the wrong direction: the higher the shell, the deeper the Sun must sink before the shadow covers it, and at midsummer the higher shells never go dark at all.

What the calculation leaves out #

The objects are spread evenly over a sphere, which is not how constellations are arranged: they fly in orbital planes at chosen inclinations, which concentrates them at the latitudes their inclinations reach and leaves the poles and the highest latitudes almost empty. A shell inclined at fifty-three degrees covers the latitudes of most observatories more densely than the uniform average. The reflecting area is a single number per object, while real satellites reflect specularly from flat panels, producing flares and a brightness that depends strongly on the angle. And the atmosphere is ignored: extinction dims the glow near the horizon, and the glow itself is scattered by the air into a faint extra component the calculation omits.

None of those changes the result’s structure — the glow is proportional to the reflecting area and independent of altitude, and the shadow decides when it is on. They change its numbers by tens of per cent in either direction.

Still open: how much sunlit area is already up there #

The calculation reduces the glow to one quantity per shell — the sunlit reflecting area in orbit — and that quantity is known for satellites and not for the debris. For the satellites it is in the operators’ hands: darker coatings, sunshades and attitudes chosen to turn reflective surfaces away from the ground have already reduced the brightness of the newest constellation satellites by factors of several. For the debris it is in nobody’s hands. Fragments are made by collisions and break-ups, each of which adds reflecting area that stays in orbit for years to centuries depending on altitude, and the rate of break-ups is a collision rate that needs no collision to grow.

The glow can be measured directly, in principle: its brightness should follow the shadow’s geometry, rising and falling with the depression of the Sun in a way the natural sky’s airglow and zodiacal light do not. A sky-brightness monitor that tracked the zenith through each night, compared with the shadow’s passage over the main shells, would see the glow switch off. That measurement has not yet been made to the precision needed, and until it is, the share of the night sky’s brightness that is sunlight reflected from metal is an estimate that no one has checked against the sky.