A Moon lit by every sunset at once

A shadow of something opaque has an inside that is dark. The Earth’s shadow at the distance of the Moon is about 9,200 kilometres across, more than two and a half times the Moon, and a totally eclipsed Moon is deep inside it for up to about an hour and forty minutes. No direct sunlight reaches any part of it. By rights it ought to vanish, leaving a hole in the stars.

It does not. On most total eclipses the Moon stays plainly visible as a dim red or orange disc, bright enough to see from a city. Kepler had the right explanation in 1604: light must be coming through the Earth’s air. Eclipse brightness varies from one eclipse to another by about a factor of a hundred and more, a much larger variation than anything the geometry could cause, and it is not the Moon that varies. The eclipsed Moon is a screen, a white card a quarter of a million miles away, onto which everything happening in the Earth’s atmosphere round the terminator is being projected. Its appearance is an atmospheric measurement.

How the light gets in #

A ray of sunlight skimming the Earth is bent towards the ground by the atmosphere’s gradient in density. The bending in passing horizontally through the whole of the atmosphere is about twice the refraction familiar at a sunset, once on the way in and once on the way out, so about 1.1 degrees for a ray at the surface. At a height $h$ the bending falls with the density:

$$\delta(h) = \delta_0\, e^{-h/H},\qquad \delta_0\approx 0.0196\ \text{radian},\quad H\approx 7.6\ \text{km}.$$

The geometric shadow’s edge at the Moon comes in by the Moon’s distance times the Sun’s angular radius. So the shadow has a radius of about 4,580 kilometres at the Moon. The ray that grazed the Earth has been bent inward by more than this.

Rays passing through the air all round the Earth, at every position angle, are being bent into the shadow, so that seen from the Moon the Earth is a black disc surrounded by a thin bright ring: a ring of every sunrise and sunset on Earth at that moment. An observer on the Moon during a lunar eclipse would see a solar eclipse by the Earth, with that glowing ring around it. The bending near the limb of the atmosphere is itself an instrument studied in other ways, but it has no better showcase than this.

The effect is not small. A ring of about a thirty-kilometre depth around the circumference of the Earth has an area of a little over a million square kilometres, and all of its light is spread over a shadow about seventy million square kilometres in cross-section at the Moon’s distance.

Each height lights one place #

The geometry of a bent ray gives a simple rule. From the Sun’s centre, a ray with impact height $h$ reaches the Moon’s distance $D$ at a distance from the axis of

$$\rho(h) = R + h - D\,\delta_0\, e^{-h/H}.$$

That function has a single value for each height, and the height determines the place.

So the centre of the shadow is lit by rays that passed through the lowest two kilometres of the atmosphere, through the densest air and the thickest clouds. Most of the umbra is lit by rays from the troposphere. Only the outer umbra gets rays that passed above about ten kilometres. The Sun’s disc spreads this somewhat, because light from the Sun’s near limb already leans inward, and needs less bending. With that included, the outer parts of the umbra can be reached by rays coming through up to about twenty kilometres.

This single relationship does a lot of the work. It explains why the umbra is usually darkest at its centre. It says that the shadow’s middle depends mainly on tropospheric cloud cover along the terminator at that moment, and it says that the rim depends on the stratosphere. So if a person watching the eclipse sees a bright rim and dark centre, or a rather uniform copper, they are effectively reading off where in altitude the atmosphere is clear.

There is a concentration effect as well, the atmosphere acting as a lens of a sort. Since the deflection changes rapidly with height, a range of impact heights is spread over a large range of distances. Rays near the surface are spread out, and each kilometre of height is smeared across hundreds of kilometres. In the other direction there is cylindrical focusing: the whole circumference of the ring is converging on the axis of the shadow, so that the region near the centre receives light from all sides. A small area on the axis gets rays from every azimuth at once. This brightening on the axis ought to be visible in the Moon if the air were clean, and it is observed in some records of very clear eclipses.

What survives the path #

Bending carries the light in. What colour it has when it arrives depends on what the air does on the way. A ray at grazing incidence passes through a far greater column of air than a ray going straight up, by a factor of about

$$\sqrt{\frac{2\pi R}{H}}\approx 73,$$

for a full passage through at the surface. Air scatters light with a strength going as the inverse fourth power of the wavelength, and with a vertical optical depth at green of about 0.1, the grazing column has an optical depth of seven or so, which leaves about a thousandth of the light in green, about a hundredth or a tenth in the red, and essentially nothing in the blue.

That is a sunset, in effect taken twice, and it depends on the same air that sets the depth of three definitions of night. An observer at the terminator sees a Sun through half of the slant column, as an orange or red disc. The ray that gets to the Moon has passed through the same air, and gone on through the other half. This doubling is why eclipsed Moons are a good deal redder than sunsets.

The figure has a second feature that would be missed if only the low rays were considered. Higher up, where the air is thin and the scattering weak, the colours are not reddened. And there is a contribution from ozone: its Chappuis bands absorb broadly around 600 nanometres, in the yellow and orange, which passes the red and the blue. At a height of about 20 to 30 kilometres, where most of the column of ozone lies, the rays can come through with their blue intact and their orange taken out. This creates the turquoise band that observers sometimes see along the edge of the umbra at the beginning and end of totality, and that photographers now capture routinely. The same ozone is what keeps the twilight sky blue overhead, a fact that was established for the sky at dusk by Chappuis and Hulburt.

That single effect shows the map operating. The blue band has its position on the Moon because a particular height of air reaches that point, and its existence requires ozone at just that altitude. An astronomer in 1900 who photographed that colour would have been looking at the stratospheric ozone layer, unknown at the time.

The disc from edge to centre #

Combining bending, transmission, focusing and the Sun’s finite disc gives the brightness across the shadow, in each colour.

The behaviour across the disc is characteristic. The blue falls by many orders of magnitude from edge to centre. The red hardly changes by more than ten times. In the outer umbra the colours are nearly equal, which makes that area relatively pale and grey, and the centre is deep red. A real total eclipse shows just this structure, particularly when the Moon’s path takes it through only part of the shadow, because one limb is then near the edge and is visibly brighter and less red than the other.

Measurements of the actual brightness were first made systematically in the twentieth century. An eclipsed Moon has a typical surface brightness about ten to a hundred thousand times less than a full Moon at the centre of the umbra, and the difference between eclipses can be a hundred times or more. That is many magnitudes, which is much the largest thing in the picture. All geometric factors, the distance of the Moon and the depth of the track in the shadow, explain only a small part of this range.

The eclipse as a record of volcanoes #

The variation was given a scale by André Danjon in 1921, running in five steps from $L=0$, where the Moon is nearly invisible at mid-totality, to $L=4$, where it is a bright copper. Danjon noticed that L varies with the solar cycle, an idea later abandoned. What later emerged was a much clearer signal, connecting dark eclipses with the years after major volcanic eruptions.

The link is physical and comes from the same map. A strong eruption can put sulphur dioxide into the stratosphere, where it forms droplets of sulphuric acid that last a year or two, spreading round the globe. A layer whose vertical optical depth is about 0.1 will have a slant depth along a grazing ray of several, since the slant column at that height is about 38 times the vertical one.

The eclipse of 9 December 1992, eighteen months after the eruption of Pinatubo, was remarkably dark: at mid-totality the Moon was very hard to see, and photographs show it barely distinguishable from the sky. After Agung in 1963 there was a similar dark eclipse, in December 1963, reported at $L$ near 0, and after Tambora in 1815 eclipses of the following years were recorded as so dark that the Moon could scarcely be found. The eclipse brightness has been used as a retrospective proxy of volcanic loading, for instance by Richard Keen, who has used eclipse data from about 1960 to estimate optical depths for the stratosphere, and concluded that the stratosphere has been nearly clear since the mid-1990s.

The figure shows an honest weakness of the simple model. A stratospheric layer in this model darkens the outer umbra by more than two magnitudes but hardly touches the centre, because the rays reaching the centre grazed below it. The real dark eclipses are dark at the centre too. So real volcanic aerosol reaches down lower than a single thin layer does, and in the months after a big eruption the upper troposphere and lowest stratosphere are also full of it. Together with the normal low-level cloud they can cut off both of the main routes. That has been modelled more fully, but it is a good reminder that the map from height to position is not a clean filter for one layer.

A shadow that is slightly too big #

The same bending seems to have an odd signature in a much older measurement. Since at least the seventeenth century it has been noticed that the Earth’s shadow on the Moon is larger than the Earth’s size and distance predict, by about a fiftieth. Philippe de La Hire noted the excess in 1707, and it has been measured at about 2 per cent from timing the shadow’s edge across lunar craters, using observations collected during eclipses for three centuries.

The rays that graze at about seventy to a hundred kilometres are not bent much, yet they are substantially absorbed by ozone and scattered by aerosols. So a dark layer around the Earth effectively raises its opaque radius. The apparent edge of the umbra is not sharp in any case, since the transition is in a gradient of brightness, and the eye’s location for that edge moves according to the contrast. The eclipse maps of the almanacs add a conventional enlargement of the shadow for this reason. It is an accepted correction, and the enlargement has been shown to vary by a small amount from eclipse to eclipse.

This offers a small example of the same limb method used on another planet. Measuring an atmosphere by the shadow it casts, or by what it does to light passing by it, is how the atmospheres of Pluto and Titan were found from stellar occultations. A lunar eclipse is the same kind of occultation in which the Sun is the source and the Moon the detector. So the same physics carries through to the transmission spectra of exoplanets, for which a planet’s atmosphere is a ring of light around a black disc, and a colour-dependent radius is the only signature. It has been noted that a lunar eclipse could provide a benchmark: the spectrum of an eclipsed Moon gives what the Earth’s own transit spectrum would look like to a distant observer. Spectra of this kind have been taken and they do show water, oxygen, ozone and methane, in the amounts that a distant observer might see.

What the model cannot show #

The atmosphere here is a uniform shell. The real terminator has cloud systems and clear skies, mountains, polar and tropical air at different heights. Around the limb at the time of an eclipse there are thousands of kilometres of different weather. The disc of an eclipsed Moon is sometimes visibly asymmetrical for just this reason. A cloud deck of eight kilometres with a quarter of its paths clear is a crude stand-in for that.

The refraction is that of a simple exponential. Temperature inversions and the tropopause change the bending by amounts that can matter at the low altitude end, and that may shift the location at which light lands by hundreds of kilometres. The effects of water vapour bands and the near-infrared are not computed at all.

The Moon’s own colour and reflectance have been ignored. The Moon is a reddish surface with a reflectance of about ten per cent that increases with wavelength. Its dark seas and highlands make a much bigger difference to the appearance than the small brightness changes being discussed, and they must be accounted for in any real photometry of the disc.

The focusing factor is capped near the axis. Rays within fifty kilometres of the axis are treated at that distance. Atmospheric turbulence and departures from spherical symmetry would break a perfect focus anyway.

A sky with no dark in it #

The view from the Moon’s surface would be something quite unlike anything seen from Earth, and the calculation gives what it would be. An observer in the umbra sees a black Earth nearly four times the Sun’s width, with the Sun behind it. Around the Earth’s edge there is a thin, intensely coloured ring about a thirtieth of a degree thick, bright orange and red for most of its length. Its colour varies around the circle, because the weather does, and because the air’s height against the planet sets the twilight differently at different latitudes.

No one has seen this, although the Apollo missions came close. The camera on Japan’s Kaguya orbiter photographed a diamond-ring effect from lunar orbit in 2009, as the Earth covered the Sun. A spacecraft in orbit round the Moon has since recorded the ring’s spectrum, and it shows much the same absorption bands as the spectra of the eclipsed Moon itself. All of the light on the surface at mid-eclipse comes from that ring, and its total is what a person sees from Earth as the Moon’s dim copper glow.

The ring also explains a persistent detail of the shadow’s edge. At the boundary of a shadow on Earth, the same effects can be seen on a clear evening, as the dark band that rises in the east after sunset, with a pink belt above it. That band is the Earth’s shadow on its own atmosphere. It has an edge, and that edge is soft and coloured for the same reason, seen from inside the shadow rather than from a hundred times further out. An observer in the right place can watch the same ring at work twice, with the same air, by looking first east at dusk and then, a few months later, at the eclipsed Moon. The one is a demonstration that requires no equipment for what the other measures.

Still open: how clean the stratosphere has been #

A hundred years of eclipse observations might seem an easy way to find a long record of the atmosphere, and they are one. Danjon values go back to the seventeenth century in a small number of cases, and Keen’s analysis is thought to detect clean stratospheres over recent decades. But the observers are not standard instruments. Estimates of brightness are affected by the Moon’s position relative to the shadow, by how long the observer’s eyes have been dark-adapted, and by local sky conditions. A Danjon estimate from 1816 cannot have the same meaning as one from the present.

There is also a clean question to be asked about what the record shows beyond the largest eruptions. It has been claimed that air pollution, forest-fire smoke and the exhaust from a growing number of rocket launches and burning-up satellites might raise the aerosol load in the stratosphere. Those contributions are now under active study, because the metals from re-entering spacecraft have been detected in stratospheric droplets. A long record of the umbra’s brightness, kept with calibrated cameras, could register that change in a very direct way. It would be an unusual measurement, needing only a clear night a few times a decade, and a screen that is already in place.