From Kepler to Ptolemy 7

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The Full Ptolemy

We now start the second part of this series: an in-depth look at the Ptolemaic system.

The last chapter gave us the “Kepler-to-Ptolemy” transformation: Sun-centered orbits turn into deferents plus epicycles, ellipses become eccentric circles equipped with equants.

Now to fill out the story a bit. I’ll start with some basic terminology, before running through the full details of Ptolemy’s models: the Sun, the Moon, and the planets. I’ll also do a “cycle count”. But I’ll leave aside two significant facets of the Almagest: how Ptolemy determined his parameters (e.g., sizes of deferents and epicycles), and his computational techniques.

Ptolemy wrote three major works on astronomy besides the Almagest: the Handy Tables, the Tetrabiblos (astrology), and the Planetary Hypotheses. I’ll explore the Planetary Hypotheses in a later post.

Terminology

This section covers basic terminology, most of it going back to the ancients or medieval astronomers. (However, often not to Ptolemy. For example, he used neither deferent nor equant.)

Coordinates: For the earthly globe, we have latitude and longitude coordinates. Astonomy, even in antiquity, employed several coordinate systems for the celestial sphere; only the ecliptic system will occupy us. Here’s a figure for it:

And here’s table summarizing the three main systems:

System	Equator	Vertical Coord	Horiz Coord
horizontal	horizon	altitude	azimuth
equatorial	celestial equator	declination	right ascension
ecliptic	ecliptic	latitude (β)	longitude (λ)

The ecliptic is the plane of Earth’s orbit. On the celestial sphere, the ecliptic determines a circle, also called the ecliptic. The path of the Sun (viewed from Earth) moves along the ecliptic. The zodiac lies along the ecliptic.

Angular measure above or below the ecliptic is called latitude, β. Angular measure along the ecliptic is called longitude, λ.

The planets stay close to the ecliptic, as we’ve noted earlier. However, the plane of a planet’s orbit is inclined to the ecliptic by an amount called, naturally, the (orbital) inclination, or tilt. Thus the orbit intersects the ecliptic plane in two points, called the nodes.

The celestial equator is the intersection of Earth’s equatorial plane with the celestial sphere. The celestial equator is inclined by about 23° to the ecliptic. The intersections of the ecliptic and celestial equator are also called nodes; these are the vernal equinox and the autumnal equinox.

The nodes move slowly around the ecliptic; Hipparchus discovered this precession of the equinoxes. Nowadays we know the cause: the axis of the Earth’s spin changes because of the tug of the Moon’s gravity on the Earth’s equatorial bulge. Rate: approximately one degree every 72 years, or about 26,000 years for a complete circuit. Mostly I will ignore the precession of the equinoxes.

Longitude is measured from the vernal equinox, eastward. That’s eastward to someone standing outside the celestial sphere, as in the figure above. How it looks in the night sky depends on how you twist your head (or lie in the grass). For example, in the figure in post 5 showing the retrograde motion of Mars, eastward (increasing longitude) is right-to-left.

We won’t really need to keep these conventions straight. But it helps to remember that practically everything in the solar system moves counterclockwise when viewed “from above”, that it, looking down on the Earth’s north pole. The Sun and the Earth both spin counterclockwise, all the planets revolve counterclockwise around the Sun, the Moon revolves counterclockwise around the Earth. Viewed geocentrically (but putting aside the Earth’s daily rotation¹), the Sun revolves counterclockwise around the Earth, and on average so do the planets (that is, ignoring retrogressions).

A planet is closet to the Sun at perihelion, and is farthest at aphelion. Likewise, a body is closest to the Earth at perigee and is farthest at apogee. Perihelion and aphelion are the apsides of a heliocentric orbit, likewise perigee and apogee are the apsides of a geocentric orbit. The line connecting the apsides is called, naturally, the line of apsides, or just the apse.

Opposition and Conjunction: A planet is in opposition when the Sun, the Earth, and the planet lie in a straight line, with the Earth in the middle². Inner planets never appear in opposition. (See why?)

A planet is in conjunction when the Earth, the Sun, and the planet lie in a straight line, with the Earth at one end. Either the Sun or the planet can take the middle position. Outer planets, like Mars, never take the middle. An inner planet, like Venus, can take the middle or not: inferior conjunction happens when the planet is in the middle, superior conjunction when the Sun is in the middle.

Like so:

These terms seem most natural from a geocentic perspective (although the definitions work either way). For Mars to be in opposition means that Mars and the Sun appear in opposite signs of the zodiac, 180° apart. When a planet is in conjunction, it appears in the same zodiac sign, separated from the Sun by 0°. Because we’ve ignored latitude (see previous footnote), these statements are not literally true: they apply rather to the planet’s longitude compared with the Sun’s.

For an outer planet, apogee occurs at conjunction and perigee at opposition. For an inner planet, apogee occurs at superior conjunction and perigee at inferior conjunction. Our figure makes these facts obvious.

Oppositions are directly observable. Not so for conjunctions (the planet is in front of or behind the Sun), but they can be inferred by interpolation.

Periods: Modern astronomy regards the sidereal period of a planet as the most fundamental; this is the time it takes for it to make a complete revolution around the Sun, judged by the fixed stars. This is almost the same as the time for the planet’s longitude to advance 360°, but the precession of the equinoxes introduces a very small correction, which we will ignore. The ancient astronomers laid more stress on the synodic period: the time from one opposition to the next (for an outer planet), or one inferior conjunction to the next (for an inner planet).

We have a relation between the synodic and sidereal periods (again, ignoring the precession of the equinoxes):

outer planet: 1/T_synodic = 1/T_Earth − 1/T_sidereal

inner planet: 1/T_synodic = 1/T_sidereal − 1/T_Earth

Here’s the argument for an outer planet, say Mars. Take a heliocentric perspective. Treat it as a race around a track, starting with Earth and Mars lined up. Earth has a faster angular speed, so it will “lap” Mars. The angular speeds are 360°/T_Earth and 360°/T_sidereal for Earth and Mars, respectively, so Earth gains on Mars at a relative angular speed equal to the difference. But this relative angular speed must be equal to 360°/T_synodic, since it takes one synodic period for Earth to lap Mars. This gives the outer planet formula; with a slight modification, it’s the same for an inner planet.

[1] From a fully geocentric perspective, the Earth does not spin: instead, every celestial object acquires a daily clockwise motion about the Earth.

[2] Strictly speaking, this is not quite correct: we’ve ignored matters of latitude. First project the planet’s position onto the ecliptic plane. Our definitions of opposition and conjunction should refer to this projected position, instead of the actual planet.

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