Ptolemy's Mercury

The main feature of the Inferior planets (Venus and Mercury) is that they never appear at opposition to the Sun. They oscillate to the east and west of the Sun, appearing as morning or evening "stars".

Their periods are:

• The mean tropical period – return to the same longitude around the ecliptic.
• The mean synodic (anomalistic) period – return to same (east or west) elongation.

The periods are not fixed but we can establish mean values from long-term observations. These planets may be east or west of the Sun but their average position is the same as the Sun so their mean motion in longitude (tropical period) is the same as the mean Sun. This lead some early astronomers to suspect that Mercury and Venus rotated around the Sun (which in turn rotated round Earth). However, Ptolemy does adopt this scheme and sticks to his constructs of equants, deferents and epicycles.

The coordinate system is longitude and latitude relative to the ecliptic with the vernal equinox ϒ as the reference point.

The Earth is in the centre of the Universe and consists of the “spheres” of earth, water, air and fire, as described by Aristotle.

The mean Sun λm☉ rotates round Earth in one Tropical Year.

Mercury is a particularly difficult case with high eccentricty and inclination, and a short period. Inconsistencies and erroneous observations lead Ptolemy to believe Mercury has two perigees and a unique construction is used to create this condition.

An Equant circle of radius R is drawn with centre E, distant e from the Earth in the direction of λa.

• λa is believed to be fixed relative the the stars so increases by 1 degree per century relative to the vernal equinox.
• The point S rotates round the Equant circle with uniform motion, west-to-east, with a period equal to the mean tropical period of the planet,(in this case, the same as the Sun).

The point F is located distant 2e from earth in the direction of λa.

• The vector F->D, length e, rotates round F, east-to-west, at the uniform angular velocity of the mean Sun such that its angle with the line of apsides is equal and opposite to that of the vector E->S. This causes D, the centre of the deferent circle, to describe a small circle around point F, thus varying the distance and direction of D from Earth.

A deferent circle of radius R is drawn centred on the moving D.

• The centre of an epicycle C rotates round the deferent circle such that its motion, as seen from the centre of the Equant circle E, is uniform.
• Thus C does not travel around the deferent circle with uniform motion relative to the deferent centre. This goes against Greek philosophy but Ptolemy avoids the issue.
• This construct enables Ptolemy to create an orbit with two perigees.

An epicycle is drawn with centre C and radius r.

• The planet moves round the epicycle, west-to-east, at a uniform rate of the mean anomalistic period of the planet relative to the vector E->C.
• This results in the planet appearing to the east and west of the Sun as viewed from Earth.

To account for variations in latitude:

• The deferent circle is tilted by 0.166 degrees and the epicycle is tilted by a further 6.5 degrees.
• The tilt is along the line of the ascending node, Ω, which is deemed to be fixed relative to the stars.

For Mercury, R=60, e=3.0, r=22.5 partes.

The animation starts at noon on the 1st day of the month of Thoth in the first year of the reign of Babylonian King Nabonassar.(Feb 26th 747BC, JD 1448273.0).