Cosmos - Simulation of Ptolemy's Saturn

Ptolemy's Saturn

The challenge with Superior planets (Mars, Jupiter, Saturn) is to replicate their apparent periods of retrograde motion as well as their variations in apparent angular velocity.

For Ptolemy, the important periods describing their motion were:

The mean tropical period – return to the same longitude around the ecliptic.
The mean synodic period – from opposition to opposition (this is called the anomalistic period by Ptolemy).

Ptolemy must had have access to a long history of observations because his estimates of these periods was remarkably accurate.

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

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

An Equant circle of radius R is drawn with centre E, distant 2e 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.

A deferent circle of radius R is drawn with centre D, distant e from Earth in the direction λa.

The centre of an epicycle C rotates round the deferent circle such that its motion, as seen from the centre of the Equant circle is uniform, (i.e. in the direction of S as seen from E).
Thus C does not rotate round the deferent circle with uniform motion relative to the deferent centre. This is “breaking the rules” of Greek philosophy but Ptolemy does not seem to worry.
“Bisection of Eccentricity” (using D and E spaced 1e and 2e from Earth ) is introduced in the Almagest, but Ptolemy does not explain how he decided on this construct.
The use of equant and deferent circles seeks to replicate the variation of apparent angular velocity during a complete period (Mars 1.88 tropical years, Jupiter 11.86 tropical years, Saturn 29.49 tropical years).
Remember that the equant describes uniform motion but the deferent carries the epicycle.

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.
Inspection of the geometry concludes that the planet moves such that the vector C->Planet is always parallel to the vector Earth->mean Sun .
The epicycle seeks to replicate the apparent retrograde motion of the planet for a period before and after opposition.
Motion is west-to-east so that the planet is nearest Earth (apparently brighter) at opposition.

For Saturn, R=60, e=3.4166, r=6.4155 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). This is the "epoch" used for Ptolemy's planetary models as they were calibrated from observations beginning near that time.

Ptolemy's Saturn	-
	Ptolemy's Saturn The challenge with Superior planets (Mars, Jupiter, Saturn) is to replicate their apparent periods of retrograde motion as well as their variations in apparent angular velocity. For Ptolemy, the important periods describing their motion were: The mean tropical period – return to the same longitude around the ecliptic. The mean synodic period – from opposition to opposition (this is called the anomalistic period by Ptolemy). Ptolemy must had have access to a long history of observations because his estimates of these periods was remarkably accurate. As usual, the coordinate system is longitude and latitude relative to the ecliptic with the vernal equinox ϒ as the reference point. The mean Sun λm☉ rotates round Earth in one Tropical Year. An Equant circle of radius R is drawn with centre E, distant 2e 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. A deferent circle of radius R is drawn with centre D, distant e from Earth in the direction λa. The centre of an epicycle C rotates round the deferent circle such that its motion, as seen from the centre of the Equant circle is uniform, (i.e. in the direction of S as seen from E). Thus C does not rotate round the deferent circle with uniform motion relative to the deferent centre. This is “breaking the rules” of Greek philosophy but Ptolemy does not seem to worry. “Bisection of Eccentricity” (using D and E spaced 1e and 2e from Earth ) is introduced in the Almagest, but Ptolemy does not explain how he decided on this construct. The use of equant and deferent circles seeks to replicate the variation of apparent angular velocity during a complete period (Mars 1.88 tropical years, Jupiter 11.86 tropical years, Saturn 29.49 tropical years). Remember that the equant describes uniform motion but the deferent carries the epicycle. 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. Inspection of the geometry concludes that the planet moves such that the vector C->Planet is always parallel to the vector Earth->mean Sun . The epicycle seeks to replicate the apparent retrograde motion of the planet for a period before and after opposition. Motion is west-to-east so that the planet is nearest Earth (apparently brighter) at opposition. For Saturn, R=60, e=3.4166, r=6.4155 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). This is the "epoch" used for Ptolemy's planetary models as they were calibrated from observations beginning near that time.
Tony Evans 2022-24