A month that begins when someone sees it

Every calendar has to decide what to do about the fact that a year is not a whole number of days and a year is not a whole number of months. The solar calendars keep the year and let the months drift free of the Moon. The lunisolar calendars keep both, at the price of an intercalated month every two or three years and a table that replaces the sky. There is a third choice, and it is the one made by the calendar used by the largest number of people for religious purposes: keep the Moon, abandon the Sun, and let the year be whatever twelve months add up to.

A calendar that keeps the Moon still has to say when a month begins. The astronomical new Moon, when the Moon passes the Sun’s longitude, cannot be seen — the Moon is lost in the Sun’s glare — so the traditional answer is the first evening on which the thin crescent is seen in the western twilight after sunset. That is a definition by observation, and an observation depends on where it is made, at what time of year, through how much air and by whom. The month that begins with a sighting inherits all of that.

What makes a crescent visible #

The crescent is the sunlit limb of the Moon seen nearly edge-on. Just after new Moon only a sliver is lit, and its width grows as one minus the cosine of the Moon’s angular distance from the Sun — slowly at first, then faster. A crescent ten degrees from the Sun is a quarter of an arcminute wide at its centre, about the resolution of the eye, and it has to be picked out against a twilight sky that is still bright where the Moon is low. Whether it can be seen therefore depends on two things: how wide it is, which is set by its distance from the Sun, and how dark the sky behind it is, which is set by how far the Sun is below the horizon when the Moon is still above it.

Astronomers have tried to turn centuries of recorded sightings and failures into a rule, and the most widely used one is Yallop’s, published in 1997 from about three hundred observations. It uses exactly those two quantities. The arc of light is the Moon’s angular distance from the Sun; the arc of vision is the Moon’s altitude above the Sun at the moment of sunset. A thin crescent must stand high, a thicker one need not.

The strip on the left is not a gap in the data. Below an elongation of about seven degrees no crescent has ever been reliably seen, by eye or telescope, and the reason was worked out by André Danjon in the 1930s: the lunar limb near the Sun is lit at such a grazing angle that the shadows of mountains and crater walls along it swallow the light entirely, and the crescent’s horns shorten until there is nothing left. Danjon’s limit is a property of the Moon’s rough surface, not of the eye or the sky, and it means that no amount of telescope or clear air can make a month begin less than about thirteen hours after the astronomical new Moon. The youngest crescents ever recorded by eye are around fifteen to sixteen hours old.

The season decides how high the Moon stands #

The criterion is in terms of the Moon’s height above the Sun at sunset, and the Moon’s height depends on more than its age. A Moon of a given age is a given number of degrees east of the Sun along the ecliptic — about twelve degrees a day — and how much of that distance is height depends on the angle at which the ecliptic leaves the horizon at sunset.

In the northern spring, after sunset, the ecliptic rises almost vertically out of the western horizon, and a Moon one day old stands nearly twelve degrees above the Sun. In the autumn the evening ecliptic lies low and slanted, and the same Moon is carried along the horizon rather than above it, standing only a few degrees high. The same geometry makes the full Moon near the autumn equinox — the harvest Moon — rise at nearly the same time on successive evenings, and it makes the young spring crescent the easy one and the young autumn crescent the hard one. At thirty degrees north the difference is eight hours — a third of a day — and at fifty it is most of a day.

The Moon’s own latitude adds to it. Its orbit is tilted five degrees to the ecliptic, so at a given new Moon it may be up to five degrees above or below the Sun’s path, and above is a gift of height while below is a loss. The dashed curves show that this alone moves the first visibility by several hours either way. And none of the figures includes the weather, which in practice moves it by days.

A window of a few tens of minutes #

The criterion is evaluated at a single moment, and the choice of moment is itself a piece of the astronomy. At sunset the sky behind the Moon is too bright; an hour later the Moon has set. In between, the sky darkens as the Sun sinks while the Moon sinks after it, and the crescent’s contrast against the twilight first rises and then falls as the Moon reaches the thick, hazy air near the horizon. Yallop’s best time is four-ninths of the way from sunset to moonset, and the useful window around it lasts only a quarter of an hour or so for a Moon that sets forty minutes after the Sun.

How fast the twilight darkens depends on how steeply the Sun descends, which is again a matter of latitude and season: near the equator the Sun drops vertically and twilight is short; at high latitudes it slants down and the bright twilight lingers. The darkening is set by how high the lit air stands above the observer, so the crescent is a faint object seen through the brightest part of the sky at the worst altitude, and a few minutes either way, or a little dust in the western sky, decides whether a sighting is made. That is why the records the criteria are fitted to scatter as they do.

The oldest prediction of all #

Predicting the first crescent is one of the oldest computational problems in astronomy, because the calendars of the ancient Near East depended on it. Babylonian astronomers of the last centuries before the common era kept nightly diaries of the Moon’s positions and recorded six intervals around each new and full Moon — the times between moonset and sunset on the evening of first visibility among them — and by the third century before the common era they computed those intervals in advance, with arithmetic schemes that tracked the Moon’s varying speed and the changing angle of the ecliptic. The schemes predicted whether the new month would begin on the thirtieth day of the old one or the day after, which is exactly the question the first figure answers.

The Babylonian solution was to predict the observation rather than replace it, and it worked well enough that their lunar tables are among the most accurate astronomical computations of antiquity. The arithmetic of the variable angle of the ecliptic that their scribes tabulated is the same seasonal effect drawn in the third figure, discovered empirically two thousand years before anybody wrote down why it happens.

A line that sweeps west #

The first figure combines the age a crescent needs with the time at which it is evening in each place. Sunset arrives later in universal time the further west an observer stands, and the Moon is older by the same amount, so if the crescent is too young to see at sunset in one longitude it may be old enough a few time zones further west. At each latitude there is a longitude west of which the crescent can first be seen on a given evening, and joined up they make a curve: the visibility line, which on the next evening has moved round the world.

The shape of the line follows from the figure before it. Where the evening ecliptic is steepest the crescent needs the least age and the line reaches furthest east; in the hemisphere where it is autumn, the line falls away to the west. For a new Moon at midnight universal time near the spring equinox, the first evening’s line runs through western Africa and Europe in the north and swings out across the Atlantic towards South America in the south. Everyone west of it can see the crescent that evening; everyone east of it must wait a day.

An observed lunar calendar therefore begins the month on two different dates across the inhabited Earth in most months. The line is not a fixed boundary between peoples; it moves every month, with the time of new Moon, the season and the Moon’s latitude, so that a city east of the line one month may be west of it the next.

How often two cities disagree #

The question a community actually faces is not where the line is but whether two of its places fall on the same side of it.

The answer is a fraction of months that grows with longitude difference and with latitude difference. Two cities three hours of sunset apart disagree whenever the line falls between them, which with nothing else varying would be about an eighth of the time; differences of latitude add the seasonal effect, because the two cities need differently aged Moons in the same month. Those numbers are for perfect skies and one criterion applied identically in both places. Real disagreements are more frequent, because skies are not perfect, because communities use different criteria — some require a sighting by eye, some accept a telescope, some accept a sighting anywhere — and because the visibility line on a given evening passes through regions where some observers will report the crescent and others will not.

The arguments over whether a sighting anywhere should begin the month everywhere, or only one’s own, or the one in a particular city, are therefore not arguments about astronomy. The astronomy says that the calendar defined by sight is a family of local calendars, and the disputes are about which member of the family to keep.

A line the Moon does not know about #

The visibility line runs roughly from east to west across the Earth in successive evenings, and the calendar date runs from west to east, starting each day at the International Date Line in the Pacific. The two do not agree on where a day begins. A crescent first seen in the Americas on the evening of a given date is seen when it is already the next morning in East Asia, so a rule that one sighting anywhere begins the month everywhere makes the month begin in Asia on a day when the Moon was not visible there that evening, and the date assigned to it depends on which side of the Pacific the line falls.

For a calendar whose day itself begins at sunset, as the lunar calendars’ days do, the problem is sharper. The first day of the month is the day that begins at the evening of sighting, and a sighting thousands of kilometres to the west, made hours later in universal time, can begin the month after sunset has already passed in the east. A single worldwide month therefore requires a convention about which sunset counts — and every convention proposed puts the line in a different place and draws the boundary of disagreement somewhere else rather than nowhere. The Earth turns under a Moon that becomes visible progressively, and any rule that gives the whole planet one date has to cut that progression at an arbitrary longitude.

Replacing the sky with arithmetic #

The other lunar calendars of history ran into the same problem and some resolved it the same way. The Hebrew calendar began months by sighting until the fourth century, when the dispersal of the community made a central declaration impossible to communicate in time and a calculated calendar was adopted in its place — the tabulated lunisolar arithmetic whose cycles are the convergents of one continued fraction. An arithmetic lunar calendar exists alongside the sighted one: thirty-year cycles of 354- and 355-day years, eleven of them long, which average a month of 29.530556 days against the true 29.530589. That rule drifts against the real Moon by a day in about 2,500 years, which makes it better than the Julian calendar was against the Sun, and it agrees with the sighted month to within a day or two almost always.

Several countries now use calculated criteria for civil purposes — a month begins if the Moon sets after the Sun on the evening after the astronomical new Moon at a particular place, for instance — while keeping sighting for religious ones. A calculated criterion removes the weather and the observer, and keeps the dependence on place, because it still asks about one location’s sunset.

A year that walks through the seasons #

The calendar that keeps the Moon pays for it in the year. Twelve synodic months are 354.37 days, eleven fewer than the solar year, and with no intercalation to make up the difference the lunar year slides backwards through the seasons.

The consequence is not abstract. A month of fasting from dawn to sunset is a month of short days when it falls in the winter of a high-latitude country and long days when it falls in the summer — at fifty degrees north the difference between the two is about eight hours of daylight — and over a lifetime every observer meets it in every season. The calendar trades a fixed relation to the seasons for a fixed relation to the Moon, and the eleven days a year are the price written in the solar calendar.

Where the crescent cannot be seen at all #

At high latitudes in summer the criterion can fail for a different reason. North of about fifty degrees around the summer solstice the Sun does not go far enough below the horizon for the sky to darken properly, and north of about sixty-six it does not set at all. A crescent close to the Sun is then never seen against a dark enough sky, and for some weeks of the year a strictly sighting-based calendar cannot begin a month by local observation. The communities of northern Europe and Canada follow a sighting or a calculation made somewhere further south, which is a practical answer and an admission that the definition was written for the latitudes where it arose.

The same latitudes make the fasting day long when the lunar month comes round to summer, and short in winter; at the extremes, scholars have ruled that the hours of a city further south be followed. The astronomy that sets the start of the month and the astronomy that sets the length of its days are both the geometry of the Sun’s path at a latitude, and both bend away from their intended meaning in the same places.

What the picture leaves out #

The visibility criterion is an empirical fit, and sightings near its boundary are uncertain by a few hours in either direction; other criteria fitted to other records differ from it by about that much. The geometry used here reduces the problem to the Moon’s distance from the Sun along the ecliptic and its latitude off it, with the ecliptic’s angle to the horizon taken from the season and latitude alone. A full calculation uses the true positions of the Sun and Moon for a given evening, includes the Moon’s parallax — which lowers it by up to a degree as seen from the Earth’s surface — and the atmospheric refraction that lifts everything near the horizon, and uses the Moon’s real speed, which varies by more than ten per cent around its eccentric orbit. Each moves the line by a fraction of a day. None changes its shape, its westward sweep or its dependence on season.

The disagreement fractions are for clear skies everywhere, which is the most favourable case; in practice the weather is the dominant source of disagreement between nearby places, and no figure of geometry can include it.

Still open: whether one sighting should start the month everywhere #

The astronomy of first visibility is now settled to within the scatter of the observations. What is not settled — and is not an astronomical question — is which observation counts: a sighting by eye or by instrument, at one’s own location or anywhere, or a calculation in place of either. The proposals for a single worldwide lunar calendar all have to choose a reference line and accept that on some evenings it will declare a month begun where nobody could have seen the crescent, or not yet begun where many already have. The visibility line drawn above is the reason no choice can avoid that: the Moon does not become visible on one date on a round, rotating Earth.