An eclipse held in place to a millimetre

The solar corona is a million times fainter than the Sun’s visible surface and extends from the limb outward by several solar radii. It is invisible in daylight because the sky scattered by the atmosphere is far brighter than it is, and for most of history it was seen only during total solar eclipses, when the Moon blocks the photosphere and the sky darkens. Totality lasts a few minutes at most, happens somewhere on Earth once every year or two, and requires clear weather at a narrow track of the surface. Everything learned about the inner corona before the space age came from a few hours of observation in total, spread over a century.

A coronagraph, invented in the 1930s, makes an artificial eclipse inside a telescope with an occulting disc in its optics. It works, but the disc is close to the telescope’s aperture, and light diffracted around it and scattered inside the instrument swamps the faint corona near the limb. A coronagraph in space escapes the sky’s brightness but not its own diffraction, and the innermost region — the first few tenths of a solar radius above the surface, where the corona is heated and the solar wind accelerated — has remained largely an eclipse-only domain.

The Proba-3 mission, launched in 2024, attacks the problem with geometry. It puts the occulting disc on one spacecraft and the coronagraph on another, 144 metres behind it, and flies the two in formation so precisely that the coronagraph sits in the disc’s shadow for hours at a time. It is an eclipse whose occulter has been moved from the Moon’s distance to the length of a football pitch, and every number in its design follows from the shape of a shadow.

A shadow that narrows because the Sun is not a point #

A point source would cast a shadow of the disc’s own size at any distance. The Sun is not a point. It subtends about half a degree, 959.6 arcseconds in radius from the Earth’s average distance, and light from opposite limbs crosses the disc’s edge at different angles. Behind the disc there is therefore an umbra, from which no part of the Sun is visible, surrounded by a penumbra, from which part of it is. The same two regions make every eclipse, on the scale of the Moon and the Earth.

The umbra’s radius at a distance $z$ behind a disc of radius $R_o$ is $R_o - z\tan\theta_\odot$: the disc’s radius, less the distance over which the Sun’s angular radius closes the cone. For a 1.42-metre disc the cone closes 153 metres behind it. At 144.3 metres, where the coronagraph sits, the umbra is 77 millimetres across. The coronagraph’s aperture is 50 millimetres. The penumbra, by contrast, widens with distance and is nearly three metres across at the same place.

The Moon’s shadow on the Earth works the same way, with the numbers in a remarkable coincidence: the Moon’s umbra closes about 375,000 kilometres behind it, and the Earth’s surface is, on average, about 378,000 kilometres away. The Moon’s umbra therefore just reaches the Earth’s surface at some eclipses and falls short at others, which is the difference between total and annular eclipses. Proba-3 chose a separation that puts it, like the Earth at a total eclipse, just inside the tip of the cone.

Why the Moon is the best occulter there is #

The Moon makes a better occulter than any instrument has, for reasons that the formation is trying to reproduce. It is far away, so its angular size is matched to the Sun’s while its physical size is enormous, and the diffraction around its edge is negligible. The Fresnel number of the Moon’s limb for an observer on the Earth, the ratio of its radius squared to the wavelength times its distance, is about ten billion. Diffracted light falls off rapidly with that number, so almost none of it reaches the shadow. Its edge is also outside any telescope, so no light scattered from the occulter enters the optics. And the sky behind it, during totality, is darkened over hundreds of kilometres, so the atmosphere’s own scattered light falls by a factor of a million.

Bernard Lyot’s coronagraph of the 1930s moved the occulter inside the telescope, where it could be used every clear day, at the cost of all three advantages. The occulting disc is centimetres from the aperture, its Fresnel number small, and the light diffracted around it and scattered by the optics sits on top of the faint inner corona. Ground-based coronagraphs at high-altitude sites, and space coronagraphs above the atmosphere, regain the third advantage but not the first two. An external occulter tens of metres in front of the telescope regains most of the first and all of the second. The formation’s 144 metres is the furthest the occulter can go while staying in a shadow the coronagraph can be held in.

The Moon also has a disadvantage that an artificial occulter avoids. Its edge is a horizon with mountains on it, irregular by a few kilometres, so the innermost corona is revealed unevenly around the limb as sunlight slips through the valleys. And its size relative to the Sun changes from eclipse to eclipse with both distances, so the edge of the corona it reveals is never the same twice. A manufactured disc has a known, smooth edge, and a geometry that can be repeated orbit after orbit, which is what turns eclipse snapshots into a time series. Natural eclipses keep their own schedule, predictable for centuries and impossible to move.

How close to the limb, against how much room to move #

The reason for putting the coronagraph so near the cone’s tip is what it can then see.

From the coronagraph’s position the occulter’s edge hides the sky out to an angle $\arctan(R_o/z)$, which for the design separation is 1.058 times the Sun’s angular radius. Corona beyond 1.06 solar radii is in view, and since the Sun’s own radius has been measured to a precision where its edge is itself a question, the instrument is looking a few tens of thousands of kilometres above the photosphere. A shorter formation would see less: at sixty metres the occulter would block everything inside two and a half solar radii, and the coronagraph would be no better than its ground-based predecessors.

The price is the shrinking umbra. The further back the coronagraph sits, the closer the occulter’s angular size comes to the Sun’s and the narrower the umbra. At 147.2 metres it would be exactly the width of the aperture, with no margin at all. At 144.3 metres the margin is 14 millimetres on each side: the coronagraph can wander that far from the line joining the Sun and the occulter’s centre before part of its aperture sees the photosphere. Choosing the separation is choosing how close to the Sun to look against how precisely two spacecraft can be kept in line.

A million times brighter, a few millimetres away #

The penumbra’s light shows why the margin matters.

Inside the umbra no part of the Sun’s disc is visible. At its edge the first sliver of the limb appears, and the visible fraction rises smoothly across the penumbra. The first sliver of photosphere is dim as a fraction of the Sun, but the corona being observed is a million times fainter than the disc, so even a thousandth of the Sun’s light falling on the aperture would overwhelm the observation. The umbra’s edge is effectively a cliff. On one side the instrument works and on the other it is blinded.

That is why the formation is held to millimetres rather than centimetres, and why it is a demonstration as much as a solar observatory. The two spacecraft fly in a highly elongated orbit around the Earth, reaching more than 60,000 kilometres at its far end, where the Earth’s gravity gradient and atmospheric drag are weak and the relative motion of two nearby orbits changes slowly. For about six hours of each 19.7-hour orbit they hold position actively, measuring their separation with lasers and a camera that sees the occulter’s edge, firing small thrusters to stay aligned. Between those intervals they drift apart and are brought back.

What pulls the formation apart #

The difficulty of holding the alignment is easiest to see in the size of the forces trying to break it. Two spacecraft in nearby orbits feel slightly different gravity, and for a separation of 144 metres along the radial direction the difference in the Earth’s pull, at the far end of an orbit 67,000 kilometres from the Earth’s centre, is about $4\times10^{-7}$ metres per second squared. That is a tiny acceleration. Left uncorrected for the six hours of an observing interval it would move one spacecraft relative to the other by about ninety metres, thousands of times the margin. Near the near end of the orbit, 7,000 kilometres from the Earth’s centre, the same separation would feel a gravity gradient a thousand times stronger, which is why the formation is held only around the far end of the orbit and allowed to drift elsewhere.

Sunlight pushes too. The occulter presents a disc of about one and a half square metres to the Sun, and radiation pressure on it gives a spacecraft of a couple of hundred kilograms an acceleration of a few times $10^{-8}$ metres per second squared, enough to move it metres over an observing interval unless the other spacecraft feels a matched push or the thrusters cancel the difference. Neither force is large. Both are many orders of magnitude larger than the millimetre tolerance allows, so the formation is held by continuous measurement and correction rather than by any passive arrangement. Cold-gas thrusters fire in small pulses, guided by a laser that measures the separation and by sensors that detect the penumbra’s light at the coronagraph’s edges — the shadow itself used as the alignment signal.

A year that narrows the margin by a factor of ten #

The Sun’s angular size is not fixed. The Earth’s orbit is eccentric, and the Sun’s angular radius varies by 1.7 per cent either way over the year: largest in early January at perihelion, smallest in early July at aphelion. A 1.7 per cent change in the Sun’s size moves the umbra’s edge by 1.7 per cent of $z\tan\theta_\odot$, which is the large number in the subtraction that gives the margin.

The margin swings from about 25 millimetres in July to about 2 millimetres in January — a factor of ten from a 3.4 per cent change in the Sun’s size, because the margin is a small difference between two large numbers. In January the formation, in this geometry, has to stay aligned to a couple of millimetres over a baseline of 144 metres, an angle of three arcseconds. The separation can be adjusted to compensate, moving the coronagraph a little closer when the Sun is large, at the price of seeing a little less of the innermost corona. The figure shows why some such adjustment, or a margin chosen for January rather than for the average, is a necessity rather than a refinement.

The same seasonal effect operates on natural eclipses. Total eclipses are longer and annular ones rarer when the Sun is small and the Moon near its closest approach, and the Earth’s position in its orbit is one of the factors that decides, eclipse by eclipse, whether the Moon’s umbra reaches the ground. An artificial eclipse inherits the dependence along with the geometry.

The spot in the middle of every shadow #

There is one more property of a disc’s shadow, and it was once the most famous result in optics.

In 1818 Augustin Fresnel submitted a memoir on the wave theory of light to a prize competition of the French Academy. One of the judges, Siméon Denis Poisson, a committed adherent of the particle theory, worked through Fresnel’s equations for the shadow of an opaque disc and found that they predicted something absurd: a bright spot at the exact centre of the shadow, as bright as if the disc were not there. François Arago, another judge, performed the experiment, and found the spot. The spot is now called Arago’s spot, or Poisson’s, and it settled the debate in Fresnel’s favour.

The spot arises because every point on the disc’s circular edge is the same distance from a point on the axis behind it, so the waves diffracted from the whole rim arrive there in phase and add up. Babinet’s principle gives its brightness exactly: the field behind a disc is the unobstructed field minus the field through a hole of the same size, and on the axis those two combine to an intensity equal to the unobstructed one. For a disc lit by a point source the centre of the shadow is as bright as open sky.

For Proba-3 this would be a disaster if the Sun were a point. It is not. The spot’s width is about $0.38\,\lambda z/R_o$, which at visible wavelengths and the formation’s geometry is about 0.04 millimetres, and each point on the Sun’s disc produces its own spot displaced by the angle to that point times the separation. Across the Sun’s half-degree those displaced spots are spread over more than a metre, and they average away. What does not average away is a faint general glow of light diffracted from the occulter’s rim into the umbra. That glow is why the occulter’s edge is not a sharp circle but a carefully shaped profile, designed so that the waves diffracted from different parts of the edge do not reinforce one another, and why the instrument’s quoted inner limit, at about 1.1 solar radii, is slightly outside the purely geometric value of 1.06.

The same diffraction arises in a different guise when an asteroid or the Moon passes in front of a star, where the shadow’s edge is fringed rather than sharp, and in the proposals to image planets around other stars by flying a starshade tens of thousands of kilometres in front of a telescope. There the problem is the same as Proba-3’s on a vastly larger scale, and the starshade’s petals are the shaped edge taken to its extreme.

What an artificial eclipse buys #

The point of all this precision is the inner corona, the region from the limb out to about two solar radii, where the magnetic field shapes the gas into loops and streamers and the slow solar wind is thought to be released. It is the region natural eclipses show best and least often, and the one coronagraphs have seen worst. A natural eclipse gives a few minutes of the inner corona from one point on the Earth, at whatever resolution the observing team brought, through whatever air was overhead. An artificial one gives six hours per orbit, every orbit, from above the atmosphere, with the same instrument each time.

Duration matters because the inner corona changes. Loops brighten and fade over hours, material falls back along field lines, eruptions begin in the low corona long before they are visible further out. A sequence of images of the same region taken hours apart can follow those changes. Snapshots from eclipses months apart cannot. The first images from the formation, released in 2025, showed the corona to within a small fraction of a solar radius of the limb, outside an eclipse, and at a quality that had previously required being in the right place at the right time.

What the figures simplify #

The Sun is drawn as a uniform disc of fixed angular radius, although real sunlight is limb-darkened, so the first sliver of photosphere at the umbra’s edge is dimmer than a uniform disc would make it. The occulter is a flat, sharp-edged disc in the geometric figures, where the real one has a shaped edge designed against diffraction. The diffraction figure uses a Fresnel number of twenty to resolve the pattern, three hundred times smaller than the formation’s own, so the spot and rings it shows are qualitatively right and far wider than the real ones. The station-keeping margin is computed for lateral displacement alone, at a fixed separation, ignoring the coupling between lateral and longitudinal errors and the pointing of the coronagraph itself. The orbit’s own variation of the distance to the Sun, a fraction of a per cent over the Earth’s orbit added to the Earth’s annual change, is ignored.

Still open: how much the formation can be trusted with #

Proba-3 is a demonstration, and the question it was built to answer is not only about the corona. Precise formation flying is the enabling technology for a family of instruments that need structures larger than any single spacecraft can carry: occulters for the corona, starshades for exoplanets, interferometers whose baselines are the separations between free-flying telescopes. Each requires holding spacecraft in a fixed geometry to precisions that scale with the size of the structure being replaced. Proba-3’s millimetres over 144 metres is a ratio of about one part in a hundred thousand. A starshade for imaging planets around Sun-like stars needs metres over tens of thousands of kilometres, a similar ratio. A space interferometer needs nanometres over hundreds of metres, a ratio ten thousand times finer, held by laser metrology rather than by thrusters alone.

Whether the control achieved in orbit — through the varying gravity gradient of an eccentric orbit, the solar radiation pressure on two spacecraft of different shapes, the drift between active intervals — holds to the design margins month after month, including the January weeks when the margin is two millimetres, is what the mission’s long-term operation will establish. Its images of the corona are the direct return. The record of how well two spacecraft stayed in line is the result that later missions will be designed around.