Synodic Arc

Definition

The synodic arc is the difference in ecliptic longitude between two consecutive synodic phenomena of the same kind for a given planet — for example, between successive first visibilities, successive stations, or successive last visibilities. It is the central predicted quantity of the Babylonian arithmetical planetary theory: each phase is treated as if it were an independent celestial object, so synodic arcs are always positive and retrogradation is removed to a secondary position.

In Tradition

Neugebauer, Rochberg, and Hunger-Pingree concur in treating the synodic arc as the organizing concept of Babylonian planetary theory. Babylonian astronomy took the synodic appearances (risings, settings, stations, retrogradations) as primary observational data, deriving positions at intermediate times by interpolation — the opposite procedure from Greek geometric astronomy, which models continuous motion and derives synodic phenomena as secondary consequences.

In Practice

For each planet the central task of the Babylonian theory is to compute how the synodic arc varies as a function of where on the ecliptic the phenomenon occurs. The variation is captured arithmetically: System A uses a step function (zones of constant synodic arc — two zones for Saturn, six zones for Mars), and System B uses a linear zigzag function bounded by fixed maximum and minimum. Once the synodic arcs are known, the corresponding synodic times follow directly: the synodic time is the interval the sun takes to traverse the same arc. Rochberg notes that the period relations underpinning both systems are framed as "x synodic phenomena = y revolutions of the ecliptic" — for Jupiter, 391 synodic phenomena occur in 36 revolutions of the 360° ecliptic, fixing the mean synodic arc as Δλ ≈ 33;8,45°.

Historical Origin

Attested across the Seleucid-period ACT planetary procedure texts and ephemerides from Babylon and Uruk (c. 250-50 BCE); the Venus synodic-arc value 3,35;30° survives in cuneiform and is recovered by Neugebauer in Varahamihira's *Pañcasiddhāntikā* as evidence of transmission to Indian astronomy. Modern critical treatments: Neugebauer, *The Exact Sciences in Antiquity* (1957) Ch. V-VI; Rochberg, *The Heavenly Writing* (2004) pp. 54-55 and 158-159; Hunger-Pingree, *Astral Sciences in Mesopotamia* (1999) §C5.

Further Reading

• Otto Neugebauer, The Exact Sciences in Antiquity
• Francesca Rochberg, The Heavenly Writing
• Otto Neugebauer, Astronomy and History: Selected Essays
• Hermann Hunger & David Pingree, Astral Sciences in Mesopotamia