Orbits to scale of feeling, not distance — radii are logarithmic
No. I — The Claim
Nothing on this instrument is drawn. Everything is solved.
Most diagrams of the solar system are pictures — planets parked where a designer liked them. This one is an instrument. Set it to any date between 1600 and 2400 and each jewel is placed by the same celestial mechanics an almanac uses: six orbital elements per planet, aged by the centuries, passed through Kepler's equation, projected onto the plane of the ecliptic.
Spin the flywheel and you are not playing an animation. You are re-deriving the sky, sixty times a second — Mercury's hurry, Neptune's patience, the Moon turning through her synodic month, retrogrades blooming and unwinding as Earth overtakes the slower worlds.
- MethodKeplerian elements → Newton–Raphson → ecliptic
- EpochJ2000.0 · JD 2451545.0
- Bodies8 planets · 1 moon · 1 sun
- Range1 Jan 1600 — 31 Dec 2400
- Convergence|ΔE| < 10⁻¹² rad, ≤ 24 iterations
- Verifiedreconstructs 21 Dec 2020 conjunction to 0°06′
No. II — The Mathematics
One equation, iterated to silence.
Kepler, 1609 · Astronomia Nova
The equation with no algebraic answer
A planet sweeps equal areas in equal times, so its clock-angle — the mean anomaly M — grows perfectly uniformly while the planet itself does not. The angle that locates it on the ellipse, the eccentric anomaly E, is bound to M by one short line that cannot be inverted in closed form. It has been solved by iteration for four centuries. It is solved by iteration here, at every frame.
M = E − e·sin E
Kepler's equation · transcendental in E
I · Wind the clock
T = (JD − 2451545)/36525
M = L − ϖ
Julian centuries since epoch J2000 age each element table — semi-major axis a, eccentricity e, inclination i, mean longitude L, perihelion ϖ, node Ω. The mean anomaly is the planet's uniform fiction: where it would stand if orbits were circles.
II · Newton, four strokes
Eₙ₊₁ = Eₙ − Δₙ
Δₙ = (Eₙ − e·sin Eₙ − M)
÷ (1 − e·cos Eₙ)
Seeded with E₀ = M + e·sin M, Newton–Raphson converges quadratically; for every planetary eccentricity a handful of strokes reaches 10⁻¹² radians — far below the width of any engraved line on this plate.
III · Onto the ecliptic
x′ = a(cos E − e)
y′ = a√(1−e²)·sin E
λ = atan2(y, x)
Coordinates in the orbital plane are rotated through the argument of perihelion, the inclination, and the ascending node onto the plane of Earth's orbit. The longitude λ is what the instrument engraves: a bearing in the zodiac.
ELEMENTS — E. M. Standish, JPL Solar System Dynamics: “Keplerian Elements for Approximate Positions of the Major Planets,” Table 1 (J2000 mean elements with centennial rates, fit 1800–2050). Run here across 1600–2400, where the giants stray by arcminutes: honest enough for brass. The lunar dial keeps the mean synodic month of 29.530589 days from the new moon of 6 January 2000.
No. III — The Long Watch
Great conjunctions, found — not remembered.
Every 19.86 years Jupiter laps Saturn and the two brightest wanderers stand together in the evening sky — the great conjunction, the slow bell of the solar system. The instrument does not look these up. When this page loads it walks its own Kepler pipeline forward day by day, watching the angular separation as seen from Earth, and lists the meetings it finds.
Jupiter ☌ Saturn · scanned live from today's date
- SCANNING…
CALIBRATION — asked to reconstruct the last great conjunction, this pipeline returns 21–22 December 2020 at 0°06′ separation; the recorded minimum was 0°06′. The scan above is computed in your browser, from the same table of elements, in a few milliseconds.
No. IV — Honesties & Provenance
What is true
Every longitude, every zodiac reading, every retrograde flag, the lunar phase, and every conjunction date is computed from the element table — nothing is keyframed, cached, or drawn by hand. Bearings are true to a fraction of a degree across the instrument's range.
What is feeling
The orbit rings are spaced logarithmically, or Mercury would huddle invisibly against the Sun while Neptune left the room. Jewel sizes flatter the small worlds. Orbits are engraved as circles; the planets ride them at their true longitudes, eccentricity and all.
How it is made
One HTML file, one table of eighteenth-century-grade numbers, and vector engraving — no images, no data files, no lookup services. Set the wheel spinning and leave the tab; the instrument rests when unwatched, as any good movement should.