Astronomy & Celestial Mechanics
the sky, computed
The observed sky

Phases are not shadows, and eclipses are

Half the Moon is lit at every instant of every month. The phase is which part of the lit half faces the Earth — and confusing that with a shadow is the commonest error in astronomy.
12 min read 6 figures Light is the only messengerThe sky as a clock

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.

Phases are a viewing angle, not a shadowA satellite at eight points of its orbit. Exactly half of it is lit at every one of them; what changes is how much of the lit half faces the centre. Nothing is in shadow except at an eclipse.sunlightnewwaxing crescentfirst quarterwaxing gibbousfullwaning gibbouslast quarterwaning crescentas seen from the centrehalf lit, always
Fig. 1 A satellite at eight points of its orbit, with the lit hemisphere computed from the illumination angle. Half of it is lit at each position; the column on the right shows the same body as it appears from the centre.

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⁡θ)/2(1 + \cos\theta)/2 where θ\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 sky from latitude 52°The celestial sphere seen from latitude 52 degrees. The pole stands 52 degrees above the horizon, the celestial equator meets the horizon due east and west, and a star at declination -18 degrees traces the drawn circle once a day. Everything below the horizon is drawn faint.celestial poleobserverzenithnorthsouthcelestial equatorthe daily circle of a star at δ = -18°the pole sits 52° up, because the observer is at latitude 52°faint arcs are below the horizon
Fig. 2 The daily circle of a body at declination −18°, which is roughly where the full moon sits in June. It stays low and its path above the horizon is short — the mirror of the Sun’s own summer track.

The shadow, when there is one

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

Umbra and penumbraThe shadow of a body lit by a source larger than itself. The umbra is a cone of finite length, computed from the two radii and the separation; the penumbra spreads outward and is the region that sees only part of the source.the umbra ends hereumbrapenumbranot to scale: the source is 400 times further away than drawnumbra length = D·r/(R − r) = 150 units for these radii
Fig. 3 The shadow of a body lit by a larger source. The umbra is a cone of finite length, set by the two radii and the separation; the penumbra spreads outward and is the region that sees only part of the 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 Dr/(R−r)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.

Umbra and penumbraThe shadow of a body lit by a source larger than itself. The umbra is a cone of finite length, computed from the two radii and the separation; the penumbra spreads outward and is the region that sees only part of the source.the umbra ends hereumbrapenumbranot to scale: the source is 400 times further away than drawnumbra length = D·r/(R − r) = 254 units for these radii
Fig. 4 A larger body, or a nearer one. The umbra now extends much further and the penumbra opens more slowly — which is the lunar-eclipse case, where the shadow comfortably engulfs the target.

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.

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 Sun's altitude through the day at latitude 52°Solar altitude against the hour of the day, at one latitude, for the solstices and the equinox. Where a curve crosses zero is sunrise or sunset, and the width between the crossings is the length of the day.05101520-40-200204060hour (local solar time)high — where the December full moon rides — 61° at noon, 16.5 h of daylow — where the June full moon skims — 15° at noon, 7.5 h of daybelow the horizon
Fig. 5 Two daily tracks at latitude 52°. Because the full moon sits opposite the Sun on the same tilted circle, its altitude is roughly the mirror of the Sun’s: high in winter, low in summer.

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.

An orbit at eccentricity 0.055An orbit of eccentricity 0.055. The primary sits at a focus, offset from the centre by 0.055 of the semi-major axis, and the closest and furthest points differ by a factor of 1.12.empty focusrperiapsisapoapsis
Fig. 6 The Moon’s orbit at its true eccentricity. The 11% swing in distance changes the apparent diameter by the same fraction, which is what decides whether a central eclipse is total or annular.

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.

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 ladder from here

Later rungs: the lit fraction as a function of elongation, and the brightness curve that does not follow it. 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.

Where this goes next

celestial poleobserverzenithnorthsouthcelestial equatorthe daily circle of a star at δ = 20°the pole sits 52° up, because the observer is at latitude 52°faint arcs are below the horizon The sphere that is not there, and why it is still the right model The stars are at wildly different distances and the celestial sphere is a fiction. It is also the most useful fiction in observational astronomy, because for pointing at things, distance is exactly the information to throw away. toward the sourcethe near side is pulled harder than the centrethe far side is pulled less — and so falls behindnot to scale: the source is far outside this frame The tide is a difference, which is why there are two of them The Moon pulls the ocean toward it. That explains one bulge. The second one, on the far side, is the whole of the physics — and it comes from subtracting. 05101520-40-200204060hour (local solar time)June solstice — 61° at noon, 16.5 h of dayequinox — 38° at noon, 12.0 h of dayDecember solstice — 15° at noon, 7.5 h of daybelow the horizon The Sun's path, and the tilt that makes the seasons Summer is not when the Earth is closest to the Sun — that happens in January. It is when the Sun climbs higher and stays up longer, and both come from a 23.4° tilt.