The Sky as a Sphere: Days, Seasons and the Phases of the Moon

Astronomy is the art of magnitude in motion. Nicomachus gives geometry the part of size that "abides and is at rest" and astronomy "that which moves and revolves" (Introduction to Arithmetic I.3, D'Ooge's translation, Attested; see The Order of the Seven Liberal Arts: Substance, Quantity and the Mind). This lesson builds the celestial sphere, a model of the sky as it appears, and uses it to explain days, seasons, phases and eclipses.

The celestial sphere as a model

Established Picture a sphere of enormous radius centred on you, with the stars fixed to its inside. The Earth's axis, extended, meets it at the two celestial poles, and the Earth's equator, projected outwards, makes the celestial equator. The horizon is the circle where ground meets sky. Like the celestial equator, it is a great circle, cutting the sphere in half. A third is the ecliptic, the Sun's apparent yearly path through the stars. It is inclined to the celestial equator by the obliquity, 23.44 degrees (NASA).

The sphere maps directions, not distances; nothing below needs the stars to lie on a real shell. The planets, which wander across the sphere, need a different account: Models of the Heavens: Ptolemy, Copernicus, Tycho and Kepler.

The altitude of the celestial pole above your horizon equals your latitude, so a traveller going north sees the pole rise. Aristotle used this effect as evidence (De Caelo II.14, 297b30 to 298a9): some stars seen in Egypt and near Cyprus are not seen in the northern regions. A small change of position, he argues, changes the horizon markedly, so the Earth is "a sphere of no great size" (Attested, Stocks's translation).

The daily turn

Established The Earth turns once in 23.9345 hours relative to the stars, so the sphere seems to turn about the poles in that time. Because the Earth also moves along its orbit, the Sun slides eastwards along the ecliptic, 360 degrees in about 365.25 days, or 0.986 degrees a day. Turning that extra degree to bring the Sun back to noon takes the Earth 3 minutes 56 seconds, so the mean solar day is 24 hours. A star therefore rises about four minutes earlier each night, and the stars seen at a fixed clock time repeat yearly.

The seasons come from the tilt

Established The Earth's axis leans 23.44 degrees from the perpendicular to its orbit, the same angle as the obliquity. The Sun's declination, its angular distance north or south of the celestial equator, therefore swings between plus and minus 23.44 degrees. The solstices are the two extremes and the equinoxes the two crossings of the celestial equator. In 2026 they fell on 20 March, 21 June, 23 September and 21 December (US Naval Observatory, Universal Time; local dates can differ by a day).

The Sun's noon altitude is 90 degrees minus the size of the difference between latitude and declination, counting south as negative. At 45 degrees north it is 45 degrees at the equinoxes, 68.44 at the June solstice and 21.56 at the December solstice. The Sun can stand directly overhead only between 23.44 degrees south and north, a fact Eratosthenes' measurement builds on. A higher Sun's light strikes the ground more directly, concentrating each beam on less ground, and a higher Sun stays up longer. Both make summer. When the declination is north, the north has the high Sun and the south the low one, so the hemispheres have opposite seasons.

Refuted The rival explanation, that summer comes when the Earth is nearest the Sun, predicts summer in both hemispheres at once, at the nearest point of the orbit. Both parts fail. The hemispheres have opposite seasons, and the Earth is nearest the Sun (perihelion), 147.095 million km away, between 2 and 5 January (Universal Time) in every year from 2015 to 2030, by the Observatory's tables. It is farthest (aphelion), 152.100 million km, in early July. At perihelion sunlight is about 6.9 per cent stronger than at aphelion, since light weakens with the square of distance, yet the north is then in winter.

Distance does leave a trace in the calendar. The Earth moves faster near perihelion, as Kepler's second law says: the line to the Sun sweeps equal areas in equal times (Kepler's laws in the lab). So the quarter containing perihelion (December 2025 solstice to March 2026 equinox) lasted 88.99 days, and the quarter containing aphelion (June to September 2026) 93.65 days.

Day length depends on latitude

Demonstrated (by spherical trigonometry, not shown here; for the Sun's centre on the horizon). The sunrise equation is cos H = −tan(latitude) × tan(declination). H is the angle the sky turns between sunrise and noon, at 15 degrees an hour, so with H in degrees the day lasts 2H/15 hours. It uses the cosine from the unit circle in Geometry, Euclid and Trigonometry.

At the equator the day is 12 hours all year. At the June and December solstices it is 15.4 and 8.6 hours at 45 degrees north, and 18.5 and 5.5 hours at 60 degrees north. North of latitude 90 − 23.44 = 66.56 degrees, the Arctic Circle, the equation has no solution at the solstices: the Sun does not set in June and does not rise in December.

Almanacs instead time sunrise with the Sun's centre 0.833 degrees below the horizon, for refraction (the air bends sunlight) and the Sun's width. Real days are therefore longer, by about 7 minutes at the equator and over 20 at 60 degrees at the solstices, and the limits of the midnight Sun and polar night shift about 0.8 degrees.

The phases of the Moon

Established The Sun always lights half of the Moon, and the phase is how much of that lit half we can see (NASA). It depends on the elongation, the angle between the Sun and the Moon as seen from Earth. At 0 degrees the lit half faces away: new Moon. At 90 degrees half the disc is lit: a quarter. At 180 degrees the lit half faces us: full Moon. The lit fraction is very nearly (1 − cos E)/2, because the Sun is about 390 times farther away than the Moon, so its light reaches both from almost the same direction.

The cycle repeats on average every 29.53 days, the synodic month. The Moon circles the Earth in 27.32 days relative to the stars, but the Sun's direction moves on by about 27 degrees in that time, and the Moon needs a little over two more days to catch up. The Moon gains on the Sun 1/27.32166 − 1/365.256 = 1/29.5306 of a turn a day. Single months vary because the Moon's orbit is elliptical: in 2025 and 2026, new to full Moon took 14.0 to 15.6 days (US Naval Observatory).

The phases are not caused by the Earth's shadow, which falls on the Moon only in a lunar eclipse, at full Moon. Aristotle already relied on the difference (De Caelo II.14, 297b23 to 30): the Moon's monthly shapes are "straight, gibbous, and concave", but in eclipses "the outline is always curved". Because "it is the interposition of the earth that makes the eclipse", he takes the curve to show that the Earth's surface is spherical (Attested).

Why eclipses are not monthly

Established The Moon's orbit is inclined 5.145 degrees to the ecliptic. It crosses the ecliptic at two nodes and otherwise lies north or south of the Sun's path. The Moon's disc is about 0.52 degrees wide and the Sun's about 0.53 (NASA data). At new Moon, a Moon 5 degrees off the ecliptic misses the Sun by about ten Moon-widths. At full Moon it misses the Earth's shadow likewise.

An eclipse needs a new Moon (solar) or a full Moon (lunar) near a node; for a solar eclipse, within about 17 degrees of it. The Sun reaches a node about every 173.3 days, so eclipses cluster in eclipse seasons, usually two a year. Each season lasts about 34.5 days, longer than a synodic month, so each brings at least one solar eclipse somewhere on Earth (NASA).

Try this

1. Choose any city and look up its latitude. Predict the Sun's noon altitude there at the equinoxes and both solstices, as Hypotheses, Predictions and Tests advises, then check in the sky lab.
2. For January in both hemispheres, write what the distance and tilt explanations each predict. Then say whether any perihelion date could have saved the distance explanation.
3. Predict the phase at an elongation of 90 degrees and check it in the phases lab. Then, from a published new Moon date and the average month, predict the next first quarter and full Moon, and check the table.
4. On clear evenings in the week after a new Moon, sketch the lit side and check that it faces where the Sun set.

Further reading

• Aristotle, On the Heavens II.14, translated by J. L. Stocks: MIT Classics text.
• NASA, the seasons and Moon phases.
• U.S. Naval Observatory, Earth's seasons and apsides.
• Fred Espenak, Eclipses and the Moon's orbit.