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

The celestial sphere as a model of the sky, and how it explains the day, the seasons, the phases of the Moon and why eclipses do not happen every month.

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 Arts).

The celestial sphere as a model

Established Picture a huge sphere centred on you, with the stars fixed to its inside. Extend the Earth's axis and it meets the sphere at the two celestial poles. Project the Earth's equator outwards and you get the celestial equator, a great circle cutting the sphere in half. The horizon, where ground meets sky, is another. 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, so no real shell is needed. The wandering planets need more; see Models of the Heavens.

The celestial pole's altitude above your horizon equals your latitude, so travel north and the pole climbs. Aristotle used the shifting sky 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 changes the horizon markedly, he argues, so the Earth is "a sphere of no great size" (Attested, Stocks's translation).

The daily turn

Established Relative to the stars, the Earth turns once in 23.9345 hours, so the sky seems to wheel about the poles. Meanwhile the Earth moves along its orbit, so the Sun slides east along the ecliptic. That is a full circle a year, or 0.986 degrees a day. Turning that extra degree 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 sky at a fixed clock time repeats 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. So the Sun's declination, its angular distance north or south of the celestial equator, 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).

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 that gives 45 degrees at the equinoxes, 68.44 at the June solstice and 21.56 at the December one. The Sun can stand directly overhead only between 23.44 degrees south and north, a fact Eratosthenes' measurement builds on. A high Sun works like a torch pointed straight down. Tilt the torch and the same light spreads over more ground. A high Sun also stays up longer, and both make summer. With the declination north, the north gets the high Sun and the south the low one: opposite seasons.

Refuted If distance made summer, both hemispheres would have it at once, when the Earth is nearest the Sun. Both parts fail. The hemispheres have opposite seasons. And the Earth is nearest the Sun (perihelion, 147.095 million km) in early January. The Observatory's tables put perihelion between 2 and 5 January (Universal Time) in every year from 2015 to 2030. The Earth is farthest (aphelion, 152.100 million km) in early July. Sunlight at perihelion 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. Network troubleshooting runs on the same move: find the observation the rival explanations disagree on, then look.

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

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), using the cosine from the unit circle in Geometry, Euclid and Trigonometry. H is the angle the sky turns between sunrise and noon, at 15 degrees an hour, so the day lasts 2H/15 hours.

At the equator the day is 12 hours all year. At 45 degrees north the June and December solstice days last 15.4 and 8.6 hours; at 60 degrees north, 18.5 and 5.5. North of the Arctic Circle (90 − 23.44 = 66.56 degrees) the equation has no solution at the solstices. The Sun does not set in June or 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

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.
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.
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.
On clear evenings in the week after a new Moon, sketch the lit side and check that it faces where the Sun set.