Phases are not shadows, and eclipses are

The Moon is a sphere lit from one side by a source 400 times further away than the Earth. At every instant of every month, exactly half of it is in sunlight and exactly half is not. That never changes, and nothing the Earth does affects it.

What changes is how much of the lit half is turned toward the Earth. A phase is a viewing angle. The Earth’s shadow has nothing to do with it, and the belief that it does — persistent, widespread, and shared by a substantial fraction of any audience — makes the eclipse impossible to understand, because an eclipse is the one occasion when a shadow really is involved.

The angle, and the shape it makes #

Take the angle at the Moon between the direction to the Sun and the direction to the Earth — the phase angle. When it is near 180°, the Earth is looking at the lit face and the Moon is full. When it is near 0°, the Earth is looking at the unlit face and the Moon is new. In between, some fraction shows.

The lit fraction is $(1 + \cos\theta)/2$ where $\theta$ is the elongation from the Sun. That expression is why the phases are not evenly spaced in appearance: a cosine is flat near its extremes, so the Moon looks nearly full for several days either side of full and changes fastest at the quarters.

The terminator — the boundary between lit and unlit — is a great circle on the sphere, always. What varies is how that circle is projected. Seen face-on it appears as a straight line, which is the quarter moon. Seen obliquely it projects to an ellipse of varying width, which is the curved inner edge of a crescent or gibbous moon. The outer edge is always a circular arc because it is the limb of the sphere itself.

That distinction gives a clean test. A crescent moon has one edge that is a circular arc and one that is elliptical. The Earth’s shadow, being cast by a sphere much larger than the Moon, would produce a circular arc on both — and that is exactly what is seen during a lunar eclipse, which is how Aristotle argued the Earth was round.

A phase is a position #

Because the phase is fixed by the geometry, it also fixes where in the sky the Moon is and when. The two pieces of information are the same information.

A full moon is opposite the Sun, so it rises at sunset, is highest at midnight, and sets at sunrise. A new moon is in the same direction as the Sun and shares its timings — which is why it is not seen. First quarter is 90° east of the Sun, rising at noon and setting at midnight; last quarter is 90° west, rising at midnight.

Two consequences follow that surprise people who have not noticed them. A crescent moon is always near the Sun in the sky, so it is only ever visible near dawn or dusk — a crescent high in the sky at midnight is an impossibility, which does not stop it appearing in illustrations. And the full moon’s altitude is opposite to the Sun’s: in midwinter it rides high, in midsummer it skims the horizon, exactly because it sits at the opposite point of the same tilted circle the Sun runs along.

The shadow, when there is one #

An eclipse is a shadow, and the geometry is the geometry of an extended source.

Because the Sun is a disc rather than a point, the shadow has two parts. Inside the umbra the source is completely hidden; inside the penumbra it is partly hidden. The umbra is a cone that narrows to a point at a distance $D r/(R-r)$ from the body, and everything about eclipses follows from how that length compares with the distance between the Earth and the Moon.

For the Earth, the umbra reaches about 1.4 million km — far past the Moon’s 384,000 — so a lunar eclipse is total whenever the Moon enters it, and totality can last well over an hour.

For the Moon, the umbra is about 373,000 km long, and the Moon’s distance varies between 356,000 and 406,000. The shadow tip therefore falls near the Earth’s surface, sometimes just reaching it and sometimes falling short. When it reaches, there is a total solar eclipse along a track a hundred kilometres wide. When it falls short, the Moon’s disc is too small to cover the Sun and a ring remains: an annular eclipse.

That coincidence — a shadow cone whose length is within 5% of the distance to the target — is a genuine accident. The Sun is 400 times wider than the Moon and 390 times further away, so the two look nearly the same size from here, and there is no reason for it. It is also temporary: tidal recession is moving the Moon outward at 3.8 cm a year, and in about 600 million years the umbra will never reach the Earth again.

Why not every month #

If the Moon passes between Earth and Sun every month, the obvious question is why there is not a solar eclipse every month.

The answer is a 5.1° tilt. The Moon’s orbit is inclined to the ecliptic, so at most new moons it passes above or below the Sun rather than across it. Five degrees is about ten solar diameters — a wide miss.

Eclipses happen only when a new or full moon coincides with the Moon being near one of the two nodes, the points where its orbit crosses the ecliptic. Those alignments come round in eclipse seasons about every 173 days, and each season yields at least one solar and usually one lunar eclipse.

The pattern repeats because three periods happen to be nearly commensurate: 223 synodic months is 6585.32 days, and 242 draconic months is 6585.36. That agreement to a part in 150,000 means eclipses recur in a cycle of 18 years and 11 days — the saros — with almost identical geometry. The Babylonians found it from records alone, without any model of what an eclipse was, and it is the earliest example of prediction from a repeating pattern in the sky.

The extra third of a day in the saros shifts each repetition a third of the way round the Earth, so a given saros series returns to the same longitude only every three cycles, which is 54 years. That is why total eclipses are common globally and rare locally: any given place waits about 375 years on average.

What was actually observed #

Nearly everything above was established before anyone knew what the Moon was.

The phase cycle gave the month, and the month gave the calendar — the earliest astronomical instrument there is, and one that needed no theory of what was going round what. The saros gave eclipse prediction from a table of past events. Aristotle’s argument for a spherical Earth came from the circular shadow edge, always circular from every angle, which only a sphere produces.

Aristarchus, in the third century BC, used the geometry to get distances — the first attempt at measuring something unreachable by angle alone. Timing how long the Moon took to cross the Earth’s shadow gives the shadow’s width in lunar diameters, and from that the Earth–Moon distance in Earth radii — he got about 60, which is right. His attempt at the Sun’s distance, from the angle between Moon and Sun at exactly half phase, needed an angle of 89.85° and he measured 87°, so his answer was twenty times too small. But the method was sound, and it established that the Sun was much further and therefore much bigger, which is the argument that eventually put it at the centre.

And the total solar eclipse is still an instrument. The corona is only visible when the disc is covered, so the Sun’s outer atmosphere was known solely from eclipses until the coronagraph. Helium was found in an eclipse spectrum in 1868, in the Sun, before it was found on Earth. And in 1919 the deflection of starlight passing the eclipsed Sun was measured at twice the Newtonian value, which is the observation that made Einstein famous.

The brightness that refuses to follow the geometry #

The lit fraction is a computable number and the brightness is a measured one, and they do not agree. The disagreement is large, it is easy to check, and it is not a small correction.

At first quarter, exactly half the visible disc is lit. The full moon shows all of it. The brightness ratio should therefore be about two, allowing for the fact that the lit half at quarter is illuminated obliquely and returns less light — which would make the honest prediction rather more than two, perhaps three.

The measured ratio is about eleven. A half moon is not half as bright as a full moon; it is a ninth.

Almost all of the discrepancy appears in the last few degrees before opposition. Plotting brightness against phase angle gives a curve that declines smoothly from full and then, within about 5° of exact opposition, spikes upward. That spike is the opposition surge, and it has two causes that took a long time to disentangle.

The first is shadow-hiding. The lunar surface is not smooth; it is a deep, porous, fluffy layer of impact-shattered dust, and every grain casts a shadow. At any phase angle other than zero, some of those shadows are visible from the Earth and darken the surface. At exactly zero — the Sun directly behind the observer — every grain hides its own shadow perfectly, and the surface appears at its full intrinsic brightness. The effect is the same one that makes dewy grass unusually bright directly opposite the Sun, where it is called the heiligenschein.

The second is coherent backscatter, and it is a wave effect rather than a geometric one. Light that scatters through a sequence of grains and light that traverses the same sequence in reverse arrive back along the exact backward direction with identical path lengths, so they interfere constructively. Away from the exact backward direction the path difference destroys the coherence. The result is a narrow peak of about a degree wide, sitting on top of the broader shadow-hiding surge.

The practical consequence is that the Moon’s brightness cannot be predicted from its geometry. It has to be measured, and the measured curve is a probe of the regolith’s structure — its porosity, its grain size distribution, its degree of compaction. The same analysis applied to asteroids gives their surface texture from a light curve, without any image, which is a considerable amount of information to extract from a point of light that got slightly brighter than expected.

Where the Moon is in the sky #

A phase and a position are the same fact, and putting the Moon on the celestial sphere makes the connection concrete. The rule extends to every phase. A first-quarter moon is 90° east of the Sun, so it sits where the Sun will be in three months and is highest at sunset. A waning crescent is just west of the Sun and rises shortly before it. The phase fixes the elongation, the elongation fixes the position, and the position fixes the timing — three descriptions of one geometric fact. That eccentricity is small enough to be invisible in the curve and large enough to matter completely. It is also what drives the monthly variation in tidal range, through a cube law that turns 11% in distance into a third in force.

The light on the dark part #

A thin crescent moon is not merely a crescent. The rest of the disc is faintly visible, a dull grey against the sky, and the explanation is a piece of symmetry worth having.

The dark part is being lit by the Earth. From the Moon’s surface, when the Moon is a thin crescent as seen from here, the Earth is nearly full as seen from there — the phases are complementary, because the two bodies are looking at each other’s illuminated hemispheres from opposite sides of the same geometry. A nearly full Earth is a bright object: about fifty times brighter in the lunar sky than a full Moon is in the terrestrial one, being four times the diameter and rather more reflective.

Earthshine is that light, reflected once more and arriving here. Leonardo da Vinci worked the explanation out around 1510, correctly, and drew the geometry — though he attributed the Earth’s brightness to its oceans rather than its clouds.

It is also a measurement. The ratio of earthshine to direct sunlight on the Moon gives the Earth’s albedo, averaged over an entire hemisphere at once, which is a quantity satellites measure with difficulty and a telescope pointed at the Moon measures easily.

The side that is not dark #

There is a second confusion in the same family and it is embedded in a phrase everybody uses. The far side of the Moon is not the dark side.

The Moon rotates once per orbit, which is what keeps one face turned towards the Earth. It does not keep one face turned towards the Sun. So every point on the Moon experiences a day and a night, each about fourteen Earth days long, and the far side receives exactly as much sunlight over a month as the near side does.

At new moon — when the near side is unlit and invisible from here — the far side is in full daylight. At full moon the reverse. The two hemispheres are lit in antiphase, and neither is ever dark for longer than the other.

What the far side is, is unseen from the Earth, and that is a statement about geometry rather than about illumination. It was unseen until a spacecraft went round in 1959, and the images it returned showed a hemisphere strikingly different from the near one: heavily cratered, with almost none of the dark basaltic plains that make the familiar pattern on the near side.

That asymmetry is a real and unexplained fact about the Moon rather than a lighting effect, and it is one of the more interesting things the wrong phrase obscures.

Where the model stops #

The phase figure is not to scale, at all. The Moon is drawn far too large and far too close. At true scale, with the Earth a centimetre across, the Moon would be 2.5 mm at 30 cm — and the Sun would be over a metre wide at 1.2 kilometres away. The orbit’s curvature toward the Sun is so slight that the Moon’s path around the Sun is everywhere concave toward it: the Moon never actually loops backwards.

The sunlight is drawn as parallel rays, which is right for the phases and wrong for the shadows. If the rays really were parallel there would be no penumbra and no annular eclipses.

Both orbits are drawn circular. The Moon’s eccentricity of 0.055 is what decides total against annular, and it cannot be omitted from any quantitative account.

The tilt is not drawn. Every phase diagram in existence, including the one here, is drawn in a plane — which is precisely the drawing that makes an eclipse look inevitable every month. The 5.1° that prevents it is invisible in a coplanar figure, and that invisibility is the reason the question keeps being asked.

The shadow’s geometry depends on two ratios, and each is worth drawing at a second value.

The ladder from here #

The same geometry seen from outside a system altogether is a transit, where the shadow is never resolved and the whole event is a number that dips by one part in a hundred.

Later rungs: the lit fraction as a function of elongation, derived. Earthshine, and Leonardo’s explanation of it. The saros derived from the three lunar months. Eclipse limits and the geometry of the nodes. Baily’s beads and the diamond ring, which are lunar topography seen in silhouette. The Moon’s libration, which shows 59% of its surface rather than 50%. Occultations as a measuring tool. Transits of Mercury and Venus, and the distance to the Sun. And eclipse timing as a record of the Earth’s slowing rotation, where a Babylonian tablet constrains the length of the day two and a half millennia ago.

The Moon has shown the same face throughout recorded history and its far side was unseen until 1959. Half of the nearest object in the sky was unknown territory until sixty-six years ago.