A full Moon that stops rising later

The arithmetic of moonrise is one of those facts that everybody can derive. The Moon circles the sky once in about 27.3 days, so each day it has moved 13.2 degrees east among the stars. The Earth has to turn that much further to bring it back to the horizon, which at fifteen degrees an hour takes about fifty-three minutes. Allow for the mean Sun and the figure comes down to fifty. So the Moon rises fifty minutes later each day, and this is in every almanac and popular account.

Anyone who lives at a latitude of fifty degrees and pays attention to it during autumn knows that this is wrong. Late in September the full Moon rises close to sunset, and the next night it rises close to sunset again, and the next, and it goes on for days. For the several evenings around the full Moon there is no dark interval between the end of twilight and moonrise. In an agricultural society before artificial light, that stretch of the calendar had practical value, since a farmer could go on bringing in the crop by moonlight, and it was given its own name.

The explanation is not an effect of the season on the Moon. The Moon’s motion among the stars is the same as it ever is. It is an effect of the geometry of the horizon, and the same geometry is at work every month. Only in September is the Moon also full when it happens.

Fifty minutes, on average #

Compute the delay for a Moon on its true path, tilted 5.14 degrees to the ecliptic, at latitude 52° north, against where it stands on that path.

So a Moon near one point of its orbit is rising some ten minutes later than the night before, and two weeks later it is rising almost an hour and a half later. The average is close to fifty and almost no night has it. A person who keeps records of moonrise once a month has no reason to notice this, because each month’s delay varies through this whole range in an order that changes little from month to month. What is different in each season is only which phase of the Moon falls where.

That distinction is the central fact. The month is a cycle of phases, while the orbital position is a cycle against the stars. They are slightly out of step, and the phase at a given orbital position slips by about a day per sidereal month. Near longitude 0°, in the constellation Pisces at the moment, the moon’s rising delay is smallest. A new crescent may stand there in March, a half Moon in June, a full Moon in September.

The orbital position at which the delay is smallest is the same every month of every year, apart from the slow movement of the Moon’s node. Every month, for a few days, the Moon’s rising time scarcely changes. It simply goes unnoticed when the Moon then is a thin crescent in the morning or a half Moon rising in the middle of the night. It is only the full Moon that everybody sees rising at dusk.

The angle between the ecliptic and the horizon #

Why that orbital position? The Moon is moving nearly along the ecliptic, and the delay depends on how much further below the horizon its daily 13-degree move carries it. If the ecliptic crosses the horizon steeply, a day’s motion along it moves the Moon far below the horizon, and it takes a long time to come up. If the ecliptic lies nearly along the horizon, the same motion moves it mainly sideways, and it hardly sinks at all.

The angle between ecliptic and horizon changes through each day as the sky turns, because the ecliptic is tilted against the equator.

The value of the extremes is easy to derive. The equator makes an angle of $90^\circ-\varphi$ with the horizon at the east point, at latitude $\varphi$. The ecliptic’s tilt to the equator adds or subtracts up to 23.4 degrees. When the First Point of Aries is on the eastern horizon, the ecliptic runs from there up into the north at the shallowest possible angle, and it makes $90^\circ-\varphi-\varepsilon$ with the horizon. Half a sidereal day later, when the opposite point is rising, it makes $90^\circ-\varphi+\varepsilon$.

At 52° north that is the range between 14.6° and 61.4°. The delay in rising is essentially the extra arc on the sky divided by the rate of rotation, and its size goes as the sine of that angle. Hence a ratio of about three, from this geometry alone, between the shallowest and the steepest. The further reduction to a ratio of eight in the first figure comes from the combination with the Moon’s motion and from the change in the Moon’s declination, and that needs the full calculation.

It is the same geometric fact that governs a separate phenomenon, the seasons of the young crescent Moon. A crescent stands higher after sunset in the northern spring, because then the evening ecliptic stands steeply up from the western horizon, and the same crescent in autumn hugs the horizon and is hard to see. These two effects are the same tilt seen from opposite sides of the sky, half a year apart.

A week of evenings #

The observable consequence comes when a full Moon reaches the shallow point. At the September equinox, the Sun stands at longitude 180°, so the full Moon stands near longitude 0°, the shallow point rising at sunset.

Compare the two lines. On the seven evenings around March’s full Moon, moonrise goes from four hours before sunset to more than three hours after it. In September it moves by less than an hour and a half. The word for this is harvest moon, and country people knew it. The name is a feature of the English year, in use well before most people who used it could have said anything about the ecliptic.

The geometrical explanation had reached popular print by the eighteenth century. James Ferguson, the Scottish lecturer in astronomy, set it out in 1756 in Astronomy Explained upon Sir Isaac Newton’s Principles, along with a mechanical orrery to show it, and his account, in which the Moon’s orbit makes the least angle with the horizon when it rises, is essentially that given here. He knew that it was a matter of phase coinciding with geometry, and he noted the fact for the Southern Hemisphere as well: there it happens in March, which has the same effect at the opposite equinox. The northern and southern harvest moons are not two separate phenomena.

There is a mild further effect in the word “full”. A full Moon is full only for a moment, and in practice it looks round for about three nights, so the nights with a short delay and those with a large and bright Moon overlap for quite a long time. The next full Moon, in October, is also close to the shallow point, and the effect repeats; it was given the separate name hunter’s moon.

An effect with a nineteen-year rhythm #

So far the Moon’s orbit has been taken to be in the ecliptic. It is tilted 5.14 degrees, and its nodes slip westward round the ecliptic once in 18.6 years. The Moon’s orbit therefore makes an angle with the equator that swings between $18.3^\circ$ and $28.6^\circ$, a slow nodding that has its own monument history. When the Moon’s orbit has its ascending node at the First Point of Aries, the orbit’s tilt against the equator is largest, and its angle at the horizon at the shallow point is at its lowest. When the descending node is there, nearly ten degrees is removed from the tilt.

So harvest moons vary from year to year on a nineteen-year rhythm. At 52°, the best of them, near the major standstills of 2006 and 2025, has a delay of about ten minutes a night. The worst, near the minor standstill of 2015, has about twenty-five. The difference is noticeable, and it would be noticed by anyone who kept records on those timescales. Nothing else about the Moon’s appearance changes in the same way. Its brightness and its size vary with the distance on a different rhythm, a cycle of about fourteen lunations, and they do not correlate with the delay in rising.

There is a further subtlety to do with distance. The rate of the Moon’s motion along its orbit varies by about twelve per cent over each anomalistic month, from fifteen degrees a day at perigee to twelve at apogee. A full Moon that falls near apogee moves more slowly, and its delay is smaller at every point of the ecliptic. So a so-called micromoon gives a slightly stronger harvest moon than a so-called supermoon, a coincidence that happens at various intervals. The figures omit this effect, which amounts to a tenth or so of the delay.

North of a certain latitude, the sign changes #

There is a strange consequence at high latitude. The minimum angle between the ecliptic and the horizon falls as latitude increases and becomes zero at the Arctic Circle. Before that, the Moon’s motion along its orbit can carry it north at a rate that advances its rising time as fast as its daily motion retards it. Beyond that, for part of each month, the Moon rises earlier each night.

An equatorial observer sees almost no such effect. The ecliptic stands near vertical to the horizon at all times of year, and the Moon’s delay stays between 40 and 60 minutes throughout the month. The effect has a very simple dependence on latitude, because the only thing that matters is how much of the Moon’s eastward motion is along the horizon. That means the rule of fifty minutes is a statement about a low-latitude observer, and the harvest moon is the first symptom of its failure further north.

There is a further strange point. A delay can go negative. So in one sense, the Moon then rises twice in one calendar day, at about fifteen minutes before midnight and again just after, because the rising time is moving backward through the clock. Conversely, at some times of the month there is a whole calendar day with no moonrise at all, for the simple reason that the delay carries the rise past midnight. The latter is not a rare event, and it happens about once a month at any middle latitude, as every almanac’s blank line shows.

What the picture leaves out #

The calculation has one mean rate and one plane. It uses a circular lunar orbit at 13.18° a day. That omits the variable speed described above, the equation of centre and the evection. Perturbations can shift the time of rising by a few minutes on any one day, and the true almanac for a given date differs from these curves by about that much.

Moonrise means an upper limb on an ideal horizon. It uses a fixed altitude of $+0.125^\circ$, from a parallax of about 57 arcminutes less a refraction of 34 and a semi-diameter of 16. Refraction near the horizon is variable and much larger at low altitudes. A real horizon has hills, and a modest skyline moves every moonrise later by several minutes, and by a different amount for each azimuth; the azimuth of moonrise is itself changing quickly from night to night in the same week the delay is small.

The dates are not fixed. The full Moon nearest the equinox can fall anywhere within a fortnight of it, and the effect varies accordingly. The two curves are drawn for a full Moon exactly at the equinox.

It cannot say how anyone used it. Whether the extra light mattered in practice is a matter of social history. English harvests came in August in many years. In some medieval calendars the term is plainly of later use. The name, as far as it can be traced, goes back only to the early modern period, and the figures have nothing to say about the literary record.

Apparent size, and the wrong explanation #

The harvest moon has attracted a set of other beliefs. People report that it looks larger and more orange than other full Moons. The colour is what any Moon near the horizon has, from the same scattering that reddens a sunset, but the harvest moon spends unusually long at low altitude in the evening, since its path is at a shallow angle, and the observer has time to see it. And the apparent size is the Moon illusion, which applies to any Moon near the horizon.

It is worth being exact about the limit of the geometry here, because a plausible error lurks in it. The shallow angle belongs to the ecliptic and not to the Moon’s climb. On any one night the Moon rises along its daily circle, which meets the horizon at $90^\circ-\varphi$ whatever the season. A harvest moon does not stay low for longer than any other, and it is no more orange. What changes is only where tomorrow’s rising will fall, not how tonight’s proceeds. Size and colour have nothing to do with it, and all that can be said is that people are out of doors and looking east at dusk that week.

The same slope has its counterpart at the summer solstice, a less familiar one. When the Sun is furthest north and low in the south at midnight, the full Moon is opposite it, deep in the southern sky at low altitude during a short night. A summer full Moon at 52° north gets no higher than about fifteen degrees and rises and sets close together in the south. A midwinter full Moon stands nearly overhead, the reverse of the Sun’s path, for most of the long night. Thus the full Moon in each season does the opposite of what the Sun did six months earlier. It lights the long nights more than the short ones, without any consideration for human need; and the harvest moon adds the only case in which it seems to oblige.

A check any observer can make #

A useful aspect of this effect is how easily it can be tested. A wristwatch and a clear eastern horizon are enough, and over a week it gives a direct measure of the latitude. Ask how many minutes separate successive moonrises around a September full Moon. For a very rough estimate, the delay goes as $\sin(90^\circ-\varphi-\varepsilon)$ at that moment. Measuring ten minutes rather than twenty would mean being considerably further north. It may be the only method for finding latitude that works by comparing the time of two moonrises without a star catalogue or sextant.

There is a good classroom demonstration in the same result. Students who are told about the fifty-minute delay and then asked to record the Moon over a whole month see it vary by a factor of eight, which is a much bigger surprise than most things in a first astronomy course, and they can see for themselves that the only variable is where the Moon is on the sky. The Moon’s own speed does not change enough to explain it. So the answer has to be in the horizon, in the same way as with the zones the Sun can reach.

The other bright evening in the calendar #

Calendars that depend on the first sighting of the Moon feel this geometry from the opposite side. A month that starts with a visible crescent is at the mercy of how high the young Moon stands after sunset, and that is the shallow or steep ecliptic again, this time at the western horizon in the evening. In the northern spring the evening ecliptic stands steeply and the crescent can be seen early. In autumn it lies low and a crescent a day older may be lost in the twilight. The same 46-degree swing in the angle of a line on the sky therefore moves the starts of months as well as the risings of a full Moon, and the Sun’s own path has no comparable effect, because the Sun defines the line and has no motion along it worth counting in a single day.

Still open: how a word followed the sky #

The name is the part that cannot be computed. “Harvest moon” appears in English from the early eighteenth century at least, and related terms appear earlier in other European languages. Its use in Old and Middle English is less sure. Was the phenomenon noticed first, and given a name, or was it one of the many months’ moon names that later had a geometrical explanation fitted to it? Traditional names for the full Moons, some of which appear in almanacs today, have complicated and partly invented histories, and it is unclear how old any given one is.

The astronomical side raises a question that can be studied: whether societies at lower latitudes, where the delay varies less, have noted anything similar. At 30° north, a delay falls to about 30 minutes at most. Southern latitudes should have a March equivalent in Chile and New Zealand, but those societies had a separate calendar of harvest and received the name by import. Comparing farming calendars, and traditions about moonlight, with the latitude-dependence of this curve would make a modest and unusual test of how much an obscure bit of celestial geometry has actually been noticed by people who had reason to look.