The shadow that also switches off the satellites

After sunset the sky overhead is lit only above the Earth’s own shadow, and the shadow climbs as the Sun sinks. By the time the Sun is eighteen degrees down, the shadow overhead has risen past the top of the scattering air, nothing above the observer is still in sunlight, and the sky is as dark as it will get. That is the end of astronomical twilight, and for a century it was the definition of when observing can begin.

The definition assumed that nothing above the air reflects sunlight. Satellites do, and there are now many thousands of them. A satellite five hundred kilometres up is fifty times higher than the air that makes twilight, and the shadow that switched off the sky has a long way further to climb before it switches off the satellite.

How high the shadow is overhead #

The geometry is the same as for twilight itself, with the scattering layer replaced by a shell of satellites. The Earth’s shadow is a cylinder behind the Earth, pointed away from the Sun. An observer on the night side looking straight up sees through the shadow to some height and then out of it. With the Sun a depression angle $\delta$ below the horizon, the height of the shadow’s edge directly overhead is

$$h_{\rm shadow} = R\left(\sec\delta - 1\right),$$

with $R$ the Earth’s radius. For a small depression this is approximately $\tfrac12 R\delta^2$, so the shadow climbs slowly at first, as the square of the depression, and then steeply.

The numbers fall straight out of the formula. A shell at 550 kilometres goes dark overhead when $\sec\delta = 1 + 550/6371$, which is a depression of 23.0 degrees. At 340 kilometres it is 18.3 degrees — almost exactly the end of astronomical twilight, which is not a coincidence: it is the reason the lowest shells were proposed as the least intrusive on the night. At 1200 kilometres it is 32.7 degrees, deep into the night at any latitude.

The scattering air that makes twilight is twelve kilometres high, effectively; the ratio of that height to the Earth’s radius sets twilight’s length, about nine degrees of the Sun’s descent for the sky overhead to fade. Satellites are fifty to a hundred times higher, and the depression angle needed to shade them grows as the square root of the height, so they stay lit two and a half to three and a half times as long.

Lit after dark, mostly in the west #

Overhead is not the worst direction. A satellite seen low in the sky is far away along the line of sight, and the point where the line of sight meets the satellite shell is hundreds or thousands of kilometres from the observer, displaced toward wherever the observer is looking. If the observer looks west, toward the Sun, that point is closer to the day side, and the shadow above it is lower. Satellites in the western sky stay lit long after those overhead have gone dark.

That is the asymmetry in the first figure. With the Sun 20 degrees below the western horizon, satellites overhead at 550 kilometres are still lit — the zenith goes dark at 23 — and so is the whole western half of the sky down to the horizon. The eastern sky, where the line of sight meets the shell farther into the night side, is dark. As the Sun sinks further, the boundary sweeps west across the zenith and down toward the western horizon, and the lit region shrinks to a crescent.

The restriction to altitudes above 30 degrees matters. Telescopes rarely observe lower, because the air they look through grows thick and the blurring of seeing grows with it. Within that observable cap, the fraction lit at the end of astronomical twilight is ninety-five per cent for the 550-kilometre shell. The definition of “night” that astronomers have used for a century is, for satellites at that height, the definition of “still almost entirely lit”.

How many at once #

A fraction of the sky is not yet a number of satellites, and the number depends on how many there are in the shell and how they are distributed.

The count is a product of three things: the number of satellites per unit area of the shell, the area of the shell visible above 30 degrees, and the fraction of that area in sunlight. The second is the largest lever and the least intuitive. A shell at 550 kilometres seen above 30 degrees altitude covers a patch of about two thousand kilometres across, a few per cent of the whole shell, so of forty thousand satellites spread evenly, about a hundred and fifty are above 30 degrees at any moment — and in twilight almost all of them are lit.

The count does not translate directly into lost observations, because a satellite crossing an image ruins a strip of it rather than the whole frame, and how much depends on the field of view and the exposure. A wide-field survey camera with a field of ten square degrees and a thirty-second exposure intercepts trails in a large share of its frames when a hundred sunlit satellites are above 30 degrees; a narrow-field spectrograph intercepts very few. The noise a trail adds is the noise of the trail’s own light where it crosses a faint target, and a bright trail across a deep image is not a source of noise that more exposure removes.

Trails per frame #

The count converts into trails with one more piece of arithmetic. A satellite 550 kilometres overhead crosses the sky at about four-fifths of a degree per second — its orbital speed of 7.6 kilometres a second divided by its distance. During a thirty-second exposure it moves about twenty-four degrees, so each lit satellite sweeps a band twenty-four degrees long. A camera with a field three and a half degrees across catches every satellite whose band passes through its field: the satellites in a strip about three and a half degrees wide and twenty-four degrees long, plus those already inside the field.

With a hundred and forty lit satellites spread over the ten thousand square degrees of sky above 30 degrees altitude — the count for forty thousand satellites at the end of astronomical twilight — there are about 0.014 per square degree, and the strip and field together cover nearly a hundred square degrees. The expected number of trails per exposure is a little over one. For the ten-thousand-satellite shell it is about a third. Halfway through the twilight described above, as the lit region retreats westward, the number falls toward zero, and in the dark hours of a winter night it is zero.

The numbers are rough — satellites near the horizon move more slowly across the sky, the field may point away from the lit region, and the shell is not uniform — but their size is the point. A survey that takes its twilight exposures to catch objects near the Sun, such as asteroids on orbits inside the Earth’s or comets just emerging from solar conjunction, takes them exactly where and when the lit satellites are densest, and should expect a trail in most of its frames.

A summer night with no dark zenith #

The depression angle the Sun reaches at midnight depends on latitude and season, and that turns the geometry into a question about where and when there is any night at all.

The lowest the midnight Sun reaches is ninety degrees minus the latitude minus the Sun’s declination. At the June solstice that is 23.0 degrees below the horizon at a latitude of 43.6 degrees north. North of that, the 550-kilometre shell is never in shadow overhead on midsummer night. Between 43.6 and 48.6 degrees there is still an astronomical night — the sky goes dark — but no moment of it when satellites overhead are dark too. North of 48.6 degrees there is no astronomical night at midsummer at all.

Those latitudes contain a great deal of the world’s astronomy: most of Europe’s observatories, much of North America’s, and a large share of the amateur community. In the southern hemisphere’s summer the same band lies at the same southern latitudes, which contain fewer observatories. The large telescopes of the Atacama, at 24 to 30 degrees south, have midsummer nights in which the zenith is dark for a few hours and twilight satellites cover the beginning and end of each — the pattern of the middle of the figure.

Both ends of every night #

Everything above was drawn for the evening, with the Sun in the west. The morning is the mirror image: the lit region rises from the eastern horizon before dawn, sweeps up past the zenith as the Sun approaches the horizon, and covers most of the sky well before the morning’s astronomical twilight begins. A night therefore has two satellite twilights, one after dusk and one before dawn, each lasting as long as the Sun takes to move from eighteen degrees to the depression at which the shell goes dark — five degrees for a 550-kilometre shell.

How long five degrees takes depends on how steeply the Sun’s path crosses the horizon. Near the equator the Sun descends nearly vertically and five degrees takes twenty minutes. At mid-latitudes in summer it descends at a shallow angle and the same five degrees take forty minutes or more; near the latitude where the Sun just reaches the shell’s critical depression at midnight, it takes the whole night. That is why the middle figure’s curves diverge so sharply with latitude: the geometry of the shadow is the same everywhere, and the rate at which the Sun moves through it is not.

In winter the picture reverses. At mid-latitudes a December night has eleven or twelve hours with the Sun more than eighteen degrees down, and the Sun reaches forty or fifty degrees below the horizon at midnight. Satellites at 550 kilometres go dark overhead twenty to thirty minutes after astronomical twilight ends and stay dark until the same interval before morning twilight. Most of a winter night is clean, and most of the satellite problem at mid-latitudes is a problem of summer, of the hours near twilight, and of the western and eastern skies in which surveys must look to catch the objects just after sunset or just before sunrise.

How bright a lit satellite is #

Being in sunlight is necessary but not sufficient for a satellite to spoil an image. Its brightness depends on how far away it is, how large and reflective it is, and the angle at which it reflects sunlight toward the observer.

Distance favours the satellites overhead. A satellite at the zenith is 550 kilometres away; one at 30 degrees altitude is nearly a thousand; one near the horizon, two and a half thousand. Brightness falls as the inverse square of the distance, so the same satellite is several times fainter low in the sky than overhead. The geometry of the shadow and the geometry of brightness therefore pull in opposite directions: the lit region lingers longest in the western sky, where the satellites are farthest and faintest, and leaves the zenith first, where they would be brightest.

The angle of reflection matters at least as much. A satellite between the observer and the Sun shows its dark side and is faint; one opposite the Sun, near the anti-solar point, shows its lit face and is brightest. In evening twilight the anti-solar direction is the eastern sky — the part of the sky that goes dark first. So the satellites still lit in the western sky late in twilight are the ones seen at unfavourable angles, far away, and the brightest trails come early, when the whole sky is lit.

Telescopes in orbit are not above the problem #

It might seem that a telescope in space escapes all of this. One in low Earth orbit does not. A telescope at 540 kilometres shares the altitude of the constellations, and satellites in higher shells, or passing through its field of view from neighbouring orbits, cross its images too. A search of a decade of one such telescope’s archive found satellite trails in a few per cent of its images, the fraction rising over the period as the constellations grew.

A telescope in a high orbit, far beyond the shells, sees the satellites only when it looks back toward the Earth, which it is designed not to do. The observatories at the Sun–Earth Lagrange point a million and a half kilometres away are effectively immune. That is one more reason, besides thermal stability and an unobstructed view, why the largest space observatories are sent so far out.

Why the lower shell is gentler #

The figures say the same thing three ways: a shell at 340 kilometres is far less intrusive than one at 550, and one at 1200 is far more so. The 340-kilometre shell goes dark overhead almost exactly at the end of astronomical twilight; the 1200-kilometre shell is lit for an hour or more into the dark.

That geometry, together with a satellite’s brightness, which falls as the inverse square of its distance, led the astronomical community to recommend that large constellations be placed below about 600 kilometres and that individual satellites be made fainter than about seventh magnitude — too faint for the naked eye, faint enough to limit the damage to a trail in a deep image. Low orbits have their own cost: atmospheric drag brings a dead satellite down in years rather than centuries, which is good for debris and means the constellation must be continually replaced. The orbit that is kindest to the night sky is also the one that cleans itself.

The shadow as a single picture #

The surprising connection is that this is not a new phenomenon at all. It is twilight, measured at a different height. The shadow that climbs the zenith after sunset is the same cylinder whether the thing being shaded is a molecule of air twelve kilometres up or a satellite five hundred kilometres up. The sky darkens when the shadow passes the air; the satellites darken when it passes them. The difference is only the height of the reflecting layer, and the time each takes to go dark follows from that height alone by the same square root.

Seen that way, a sky full of low-orbit satellites is a planet that has acquired a second, much taller twilight layer — sparse, made of discrete points rather than a continuous haze, and lasting into the hours that were previously dark. The definitions of civil, nautical and astronomical twilight were written for one layer. A fourth, for when the satellites at a given height go dark, is now a practical part of planning an observation, and it varies with the height of the shell the observer cares about.

What the figures leave out #

The shadow is a sharp cylinder. The real shadow has a penumbra, since the Sun is half a degree across, and the atmosphere refracts sunlight into the shadow’s edge, both of which move every threshold by a fraction of a degree. A satellite crossing the penumbra fades over a few seconds rather than switching off.

The shell is uniform. Real constellations are made of orbital planes at particular inclinations, and their satellites spend more time near the latitudes of those inclinations, so the density over an observer at mid-latitudes is higher than an even shell implies, and lower at the equator.

Brightness is not modelled. A satellite’s brightness depends on its size, its reflectivity, the angle between the Sun, the satellite and the observer, and its distance, and satellites near the zenith are closest and brightest while those near the western horizon are far and faint. Low in the west, many of the lit satellites in the first figure are too faint to matter for some instruments and not for others.

Only the Sun’s depression is varied. The figures do not follow a satellite along its orbit; they ask which points of a fixed shell are lit at an instant. A real survey sees satellites moving through the lit region and leaving trails across it.

Still open: what the satellites do to the darkest sky #

The sky’s brightness on a moonless night is set by airglow, zodiacal light and the integrated light of faint stars, and it is the background against which every faint measurement is made. Individually resolved satellites add trails to that background. Collectively, the light scattered from tens of thousands of satellites and pieces of debris too faint to be seen individually adds a diffuse glow, which estimates put at about a tenth of the natural background at present and rising with the number of objects in orbit.

That diffuse component is not a trail that can be masked or an image that can be discarded. It brightens the whole sky, uniformly, at every hour when the satellites are lit — which, by the geometry here, is much of the night at mid-latitudes and all of a summer night near the observatories of Europe. How large it becomes depends on how many satellites are launched and how high, and it is the part of the problem the shadow does not solve.