A year kept by a star seen for one morning

Most calendars are counts of days arranged to keep step with something in the sky. The thing kept is usually the seasons, measured by the Sun’s position among the equinoxes and solstices, or the phases of the Moon, measured by a crescent someone has to see. One of the oldest calendrical markers is neither. It is a single event in the year of a single star: the first morning on which Sirius, the brightest star in the sky, can be seen rising in the east just before the Sun, after weeks in which it has been above the horizon only in daylight.

That morning is the heliacal rising of Sirius, and in Egypt it mattered. It fell in the weeks before the Nile’s annual flood began, and the Egyptians called the star Sopdet, which Greek writers rendered Sothis. The civil calendar of Egypt had twelve months of thirty days and five extra days, 365 in all, with no leap year. Against that calendar the rising of Sothis did not stay put. It arrived a day later in the civil calendar every four years and wandered through every month, returning to its starting point after about fourteen and a half centuries — the Sothic cycle, named by the Roman writer Censorinus, who recorded that a cycle had begun in AD 139.

Two questions are buried in that history. Why should the first morning of a star’s visibility mark a year at all, and why is the year it marks so close to 365¼ days, rather than the 365.2422 days of the seasons or the 365.2564 of a complete orbit against the stars? The second has an answer in precession, and it is not the answer that comes first to mind.

What makes a morning the first #

Sirius sits about forty degrees south of the ecliptic, and each year the Sun passes it in longitude in early summer. For weeks around that conjunction the star rises and sets with the Sun, above the horizon only in daylight, and cannot be seen. As the Sun moves east, the star begins to rise earlier than the Sun each morning, by about four minutes more each day. Soon after conjunction it rises before the Sun, but in a sky already bright with twilight, and remains invisible. Only when it rises with the Sun far enough below the horizon that the sky near the eastern horizon is dark enough to show a bright star does it become visible: low in the east, for a few minutes, before dawn washes it out.

The condition can be stated as an angle. At the moment Sirius rises, the Sun must be at least a certain depth below the horizon — the arcus visionis, about seven to nine degrees for a star as bright as Sirius, larger for fainter stars. The first morning on which that holds is the heliacal rising, and the date depends on the star’s position, the Sun’s, the observer’s latitude, and the clarity of the horizon. It is the same kind of threshold as the one that begins a lunar month: an astronomical configuration plus a judgement about visibility, which is why both are known to vary by a day or two with the observer.

Seventy days in the glare #

The invisible stretch before the rising was as much a part of the Egyptian year as the rising itself. Sirius was said to spend seventy days in the underworld, and the figure was used as the length of the embalming period. Its length is set by the same geometry as the rising: the interval between the last evening on which the star can be seen setting after the Sun, in spring, and the first morning on which it can be seen rising before it, in summer. Both ends are thresholds defined by the Sun’s depression below the horizon, and both are moved by the brightness of the twilight sky near the horizon, which depends on how night is defined — the Sun six, twelve or eighteen degrees down — and by the Earth’s shadow climbing the opposite sky.

Seventy days is about right for Sirius seen from Egypt with an arcus visionis of eight or nine degrees. Further north the period is longer, because a southern star stays closer to the horizon and is lost in the glare for more of the year; further south it is shorter. The number was a fact about the latitude of the Nile, recorded in ritual as faithfully as the length of the star’s year was recorded in the calendar.

Refraction plays a part in both ends of the window. Near the horizon the atmosphere lifts the image of a star by about half a degree, so that the Sun sets after it has geometrically set and a star rises before its geometric rising; it also dims the star by extinction through the long path of air, which is why the arcus visionis is larger than the depression needed to see the same star high in the sky. The single number used in the figures here stands for all of that.

A star that walked through the calendar #

Against the 365-day civil calendar, the heliacal rising moved regularly.

The civil year was shorter than the star’s by a quarter of a day, so the rising arrived one day later in the civil calendar every four years and a month later every hundred and twenty. After about 1460 years it had passed through every day of the calendar and returned to the civil New Year. The figure runs the cycle backward from the coincidence Censorinus recorded for AD 139: the previous coincidences fell around 1320 BC and 2780 BC.

Those dates were for a long time the backbone of Egyptian chronology. A papyrus that records the heliacal rising of Sothis on a given civil date in the reign of a given king fixes the year of that reign, modulo the Sothic cycle, and a handful of such records were used to tie the chronology of the Middle and New Kingdoms to absolute dates. The arithmetic is simple. What makes it uncertain is everything that sets the date of the rising other than the calendar — and the first of those is the star’s own year.

A year that is the Julian one #

The drift through the civil calendar shows that the star’s year is close to 365¼ days. How close is worth asking, because there is no obvious reason it should be.

Against the seasons, the rising moves steadily later, by about a month in three and a half thousand years. That is precession at work: the equinoxes slide westward along the ecliptic as the Earth’s axis turns by fifty arcseconds a year, so a star fixed among the other stars moves eastward relative to them, and the Sun reaches its longitude a little later each year by the seasonal calendar — about a day and a third a century. Against the Julian calendar, which has a year of exactly 365¼ days and so runs ahead of the seasons by about three quarters of a day a century, the rising barely moves at all: two days in three and a half millennia.

The Julian calendar’s drift against the seasons and the star’s drift against the seasons are nearly equal. That is a coincidence, and an inspection of where each comes from shows how much of a coincidence it is.

Two effects of precession that cancel #

If precession did nothing to a star but carry it eastward along the ecliptic, the star’s year would be the sidereal year, the time the Sun takes to return to the same position among the stars: 365.2564 days, twenty minutes longer than the seasonal year. A star’s rising would then drift against the Julian calendar by a day every hundred and fifty years or so, and through three thousand years by three weeks.

But precession also changes a star’s declination, because the equator itself moves while the star stays put among the stars, and a star’s rising time depends on its declination. Over the last five thousand years Sirius has moved south relative to the equator. A star further south, seen from a latitude of thirty degrees north, rises later relative to the Sun at the same longitude, and that effect works the other way from the eastward drift in longitude: it shortens the interval between risings. For Sirius seen from Egypt, the two nearly cancel, and the star’s year has stayed within a minute or two of the Julian year for three millennia, lengthening slowly towards the sidereal value as the geometry changes.

That is why the Sothic cycle is a well-defined thing. Had Sirius been in a different part of the sky, or Egypt at a different latitude, the cancellation would not have happened, and the star’s rising would have drifted against a Julian year as it drifts against the seasons. The 365¼-day year that the Egyptians knew from Sirius — and that the Canopus decree of 238 BC tried, unsuccessfully, to impose by adding a sixth extra day every fourth year — was a property of the star and the place together.

How particular the cancellation is shows up as soon as the observer moves.

At fifty degrees north the declination effect is stronger, because the rising of a southern star changes faster with its declination when it skims a more oblique horizon, and it overcompensates for the eastward drift. The star’s year there falls below the Julian year by several minutes, and a calendar of 365¼ days would have let the rising slide nine days earlier in three thousand years. Near the equator the effect weakens and the year leans towards the sidereal. The Julian year and Sirius agree to within a day in three millennia only in a band of latitudes that happens to include the Nile valley — and the Egyptian astronomers who noticed that Sothis kept a year of 365¼ days were reporting, correctly, a fact about their own place.

A rising seen a week apart along one river #

The heliacal rising is not one date but a set of dates, one for each place the star is watched from.

Going north, a southern star like Sirius rises later and sets earlier and spends less time above the horizon, so the first morning on which it can be seen comes later: by about a day per degree of latitude. Egypt stretches over seven degrees from Aswan to the Mediterranean coast, so the rising of Sothis was observed on different days in Thebes and in Memphis, a week apart at the extremes. And the arcus visionis, the Sun’s depression needed to see the star, depends on the clarity of the eastern horizon, the observer’s eyesight and the dust in the air: each degree of it shifts the date by about a day.

That is the trouble with dating a reign by a recorded rising. A record of the rising on a given civil date fixes the year to within a few years only if the observer’s place and the horizon conditions are known, since four years of Sothic drift equals one day of date; an uncertainty of a week in the date, from not knowing whether the observation was made at Thebes or Memphis, is an uncertainty of nearly thirty years in the reign. The Egyptian chronologies pinned to Sothic dates have shifted by decades as different assumptions about the place of observation were made.

The dependence on place does not go away with time; it is a property of the geometry, present at every epoch.

In AD 139 the rising fell about two days later in the Julian calendar than in 1500 BC at every latitude, as the slow lengthening of the star’s year would predict, but the spread along the Nile is the same. Censorinus does not say where the coincidence was observed, and the choice of Alexandria, Memphis or Thebes moves the start of the cycle by several years.

Why a first rising, not a solstice #

It may seem strange to tie a calendar to an event as hard to observe as the first sighting of a star in twilight, when the solstices — the longest and shortest days — are available every year. But the solstices are hard to time to the day without instruments: the Sun’s noon altitude changes by only a few arcminutes in the days around a solstice, and the day of the turn cannot be seen. A heliacal rising, by contrast, is binary: one morning the star is not there and the next it is. It is a threshold, and thresholds are precise even when the quantity crossing them changes slowly, which is the same reason the length of a year is easier to measure by counting a threshold than by finding the exact moment of an extremum.

Its disadvantage is that the threshold is set by the eye and the air, so its precision is a day or two rather than the minutes a solstice can give with a gnomon and patience. For a calendar meant to announce the flood season, a day or two did not matter. For a chronology meant to fix a king’s reign to a year, it is everything.

How the Julian calendar inherited it #

The connection between the star and the calendar that much of the world used for sixteen centuries is direct. The 365¼-day year was known in Egypt from long observation of Sothis, and when the Ptolemaic priests proposed in the Canopus decree of 238 BC to add a sixth epagomenal day every four years, it was to keep the festivals of the civil calendar fixed against the rising. The reform was not adopted. Two centuries later, when Julius Caesar reformed the Roman calendar on the advice of the Alexandrian astronomer Sosigenes, the year of 365¼ days with a leap day every fourth year was adopted for Rome, and Augustus imposed the same leap rule on the Egyptian calendar in Alexandria a few years later.

The irony is that the Julian year was designed to keep step with the seasons, and it did not: it gains on the seasons by three quarters of a day a century, which is why the Gregorian reform of 1582 had to drop ten days and three leap years in every four centuries. A year that depends on the season counting it is a subtler quantity than a count of days can hold. What the Julian year did keep step with, almost exactly, was the first morning on which Sirius could be seen from Egypt. A calendar inherited from the observation of one star ended up holding that star in place and letting the seasons slip.

A day or two in every computed date #

The figures compute the rising with simplified astronomy: the star’s precession applied to its ecliptic longitude with its latitude held fixed, its proper motion neglected — Sirius moves by about a degree and a half in five thousand years, mostly southward, which shifts the rising by about a day — the Earth’s orbital eccentricity ignored in converting the Sun’s longitude to a date, which moves the absolute dates by a day or two, atmospheric refraction and extinction folded into a single arcus visionis, and the horizon taken as flat. Each changes the absolute dates by a day or so; none changes the drift through the civil calendar, the near-equality with the Julian year, or the dependence on place, which are differences between dates computed the same way.

Still open: whether the records were observed or computed #

The deepest uncertainty in Sothic dating is not astronomical. It is whether a recorded rising was observed on the morning it names, or computed — announced by priests from a rule, as the Canopus decree implies was the practice by the Ptolemaic period. A computed date follows the calendar rule exactly and carries no information about the place or the horizon; an observed one carries both and is scattered by a day or two. The handful of records from the second millennium BC that fix Egyptian chronology are read either way by different scholars, and the decades of difference between their chronologies come largely from that choice. Independent dates — from radiocarbon measurements of short-lived plant material in dated tombs, and from tree rings and volcanic eruptions recorded in ice — now constrain the same reigns to a decade or two. Where they agree with a Sothic date read as an observation, the star was being watched; where they agree only with a computation, the rising of Sothis had become, by then, a calendar rule wearing the name of a star.