The solar system, derived from a single force law — and the eclipses it casts.

The orrery above holds one rule in its loop: Newton’s law of universal gravitation. Between every pair of bodies the force is F = G·m₁m₂ / r², directed along the line that joins them. There is no table of orbits, no pre-baked ellipse — only mass, position, and that inverse-square attraction.

We seed the state vectors from the standard J2000 orbital elements (Standish’s reduction): each planet’s heliocentric position and velocity at the epoch, then a shift to the barycentre so total momentum is ~0 and the system doesn’t drift off-screen. From that single snapshot, the future is integrated, not looked up.

The integrator is symplectic — velocity-Verlet (leapfrog): a half-kick on the velocities, a full drift of the positions, then a second half-kick. Unlike a naive Euler step, a symplectic scheme conserves the system’s energy over long runs (watch the live drift in ppm badge stay tiny even at high speed), which is exactly why the ellipses stay closed instead of spiralling.

And then the payoff. Recover each planet’s orbit from its current state vector and compute T² / a³ — orbital period squared over semi-major axis cubed. For every planet it lands on ~1.000. That constant is Kepler’s third law, and we never wrote it down. It fell out of the force law alone. (Jupiter reads ~0.999 because it is massive enough that the real (M + m) term matters — physics, not integrator error.)

A solar eclipse is the Moon’s shadow falling on Earth, so it can only happen at New Moon — when the Moon sits between us and the Sun. But the Moon’s orbit is tilted ~5° to the ecliptic, so most New Moons the shadow sails harmlessly above or below us. An eclipse needs that New Moon to land near one of the two nodes, where the lunar orbit crosses the ecliptic plane.

How close is close enough is captured by γ (gamma): the least distance of the shadow axis from Earth’s centre, measured in Earth radii. When |γ| < 0.9972 the axis actually strikes the globe and the eclipse is central (total, annular, or hybrid). Larger than that and only the penumbra grazes us — a partial.

Because the geometry repeats, eclipses arrive in a rhythm: eclipse seasons come roughly every ~173 days (~5.8 months), so each calendar year carries at least two and at most five solar eclipses. The predictor above walks new-moon index by index and keeps only the ones whose shadow reaches us. And the same alignment at Full Moon runs in reverse — the Moon crosses Earth’s shadow, a lunar eclipse — which is why solar and lunar eclipses arrive paired inside one season, about a fortnight apart.

Total or annular is then purely a question of distance. The Moon’s orbit is an ellipse: near perigee it looks slightly larger than the Sun and its disk covers it completely — total, the corona blazes out. Near apogee it looks slightly smaller, leaving a bright rim — annular, the “ring of fire.” That is the whole story in one number: the Moon-to-Sun apparent size ratio. Read it live in the disk view.

γ

Signed least distance of the shadow axis from Earth’s centre, in Earth radii. Sign marks north (+) or south (−) of centre.

u

The antumbral radius parameter (Meeus). Its sign separates a total cone from an annular one at greatest eclipse.

size ratio

Moon’s apparent diameter ÷ the Sun’s. > 1 ⇒ total; < 1 ⇒ annular ring. For a partial it is not a coverage.

umbral mag

Lunar eclipses: the fraction of the Moon’s diameter inside Earth’s umbra at greatest eclipse. ≥ 1 ⇒ total (the copper “blood moon”); ≤ 0 ⇒ penumbral only.