கணித வடிவம்The maths behind the orrery
Real two-body mechanics, real J2000.0 elements, deterministic. The orrery is reproducible: give it an era-week seed and it produces the same configuration every time. Nothing here predicts a person's future; it documents how nine bodies actually move under Newtonian gravity, and uses that as a mnemonic for nine structural patterns in the caste record.
The discipline cited is graha-gaṇita — the mathematical-astronomical layer of classical jyotiṣa, documented as history of science by Pingree (1981; 1997) — not phalita (the predictive layer). The orrery is computational astronomy with civilisational labels, not forecast, not horoscopy.
Kepler's third law
T² = (4π² / μ) · a³
For heliocentric orbits, μ = G·M☉ ≈ 1.327 × 10²⁰ m³ s⁻². T is the sidereal period, a is the semi-major axis. The orrery uses periods derived directly from the J2000.0 elements (Standish & Williams, JPL Solar System Dynamics).
Vis-viva
v² = μ · (2/r − 1/a)
Instantaneous speed at radius r along an orbit of semi-major axis a. Used to verify the numerical integrator against the analytic solution at perihelion and aphelion.
Kepler's equation, solved by Newton-Raphson
M = E − e · sin E
M is the mean anomaly (advances linearly with time), E is the eccentric anomaly (the quantity we need to compute the position on the ellipse), e is the eccentricity. The equation is transcendental; we solve it iteratively:
E_(n+1) = E_n − (E_n − e · sin E_n − M) / (1 − e · cos E_n)
Convergence is quadratic for e < 1. We use 8 iterations with a tolerance of 10⁻¹². For the lunar nodes (Rāhu, Ketu) we substitute the 18.612958-year regression period rather than solve a Keplerian orbit — they are points, not bodies.
J2000.0 mean elements used
| Body | Sanskrit | a (AU/km) | e | i (°) | T (yr) | L₀ (°) |
|---|---|---|---|---|---|---|
| Sun | Sūrya | 0 AU | 0 | 0 | 0.000 | 0.00 |
| Moon | Chandra | 384,399 km | 0.0549 | 5.145 | 0.075 | 218.32 |
| Mars | Mangala / Cevvāy | 1.524 AU | 0.0934 | 1.85 | 1.881 | 355.45 |
| Mercury | Budha | 0.387 AU | 0.20563 | 7.005 | 0.241 | 252.25 |
| Jupiter | Bṛhaspati / Guru | 5.203 AU | 0.04839266 | 1.305 | 11.863 | 34.40 |
| Venus | Śukra | 0.723 AU | 0.00677323 | 3.394 | 0.615 | 181.98 |
| Saturn | Śani | 9.537 AU | 0.0541506 | 2.485 | 29.457 | 49.94 |
| Lunar ascending node | Rāhu | 384,399 km | 0.0549 | 5.145 | 18.613 | 125.04 |
| Lunar descending node | Ketu | 384,399 km | 0.0549 | 5.145 | 18.613 | 305.04 |
Determinism & era-week seed
The orrery's animation phase is seeded by ⌊now / (7·86400·1000)⌋ — a monotonically incrementing era-week index. Two visitors loading the page in the same era-week see the same configuration; the next era-week advances all bodies by exactly seven days of simulated motion. This makes the page archivable: the Continuity Changelog can record which era-week's configuration accompanied a publication.
Sources
- NASA JPL, Approximate Positions of the Planets. ssd.jpl.nasa.gov/planets/approx_pos.html
- Standish, E.M. & Williams, J.G., Keplerian Elements for Approximate Positions of the Major Planets. JPL Solar System Dynamics.
- Pingree, D. (1981). Jyotiḥśāstra: Astral and Mathematical Literature. Harrassowitz.
- Pingree, D. (1997). From Astral Omens to Astrology. IsIAO.