Linear-scale view of a theoretical comet reservoir: a sparse spherical outer cloud surrounding a warped two-arm inner Oort structure.
Galactic tidal torque winds Neptune-scattered comets into a two-arm spiral — drag to rotate
Every comet orbit has a closest point to the Sun called its perihelion — its pointed end. The angle of that pointed end relative to the galactic plane is ω (omega). At the start, all comets scattered by Neptune share a similar orientation: ω ≈ 0°, meaning their pointed ends lie near the galactic equator. That's the flat disc you see at t = 0 in the simulation.
Every comet orbit has a closest point to the Sun called its perihelion — like the sharp tip of an egg shape. The angle ω (say "omega") tells us which direction that tip is pointing. When Neptune flings comets outward, all their tips point roughly the same way: toward the flat middle of the galaxy. You can see this as a thin disc of comets at the very start of the simulation.
The galactic tide doesn't randomly spin orbits — think of it as a landscape with two valleys at ω = 90° and ω = 270° (the galactic poles), and two hills at 0° and 180°. Every comet orbit inclined more than ~39° to the galactic plane sits on this landscape — and all our comets start on the hilltop at ω = 0°. The tide then rolls each one toward the nearest valley.
Think of ω as a ball's position on a hilly landscape. The Milky Way's gravity creates two valleys (low-energy resting spots) at ω = 90° and ω = 270°, with hilltops at 0° and 180°. All our comets start at a hilltop. Just like a ball placed on a hilltop rolls to the bottom, the galaxy's gravity slowly pushes each comet's tip downhill toward the nearest valley. This is the Kozai-Lidov effect.
The roll-down rate depends on orbit size: bigger orbits (larger semi-major axis) roll faster. So at any snapshot, each radial shell is at a different stage of its journey. Inner orbits have barely moved from ω = 0°; outer orbits have rolled further toward 90°. Plot angle vs. orbit size in polar coordinates and you see an Archimedean spiral — like a stadium wave frozen mid-wave where the outer rows are further along.
Bigger orbits roll downhill faster than smaller ones — that's the key. So at any moment, each ring of the cloud is at a different stage of rolling. The inner ring has moved 10°, the middle ring 40°, the outer ring 90°. Connect those dots and you get a spiral. It's exactly like a stadium wave where the outer seats are further ahead — freeze it mid-wave and the hand positions trace a coil.
Each dot below is one comet's perihelion direction — the spot in the sky where its closest approach to the Sun lies. At first they're scattered everywhere. As Kozai locking takes hold, each dot slides to the nearest valley: half stream toward galactic north, half toward galactic south. The result is two bright clusters — one above the galactic plane, one below. That is the two-arm structure: not two ribbons stretching outward, but two opposite concentrations of perihelion directions locked at the poles.
Each dot is one comet's tip direction in the sky. Half the comets roll to the valley at galactic north, half to the one at galactic south — whichever is closer. When they all arrive and lock in place, you get two bright clusters: one above the galaxy's midplane, one below. Those clusters are the two arms. They're not arms stretching outward like a pinwheel — they're two opposite spots in the sky where all the comet tips end up pointing.
Four billion years ago, comets scattered by the giant planets were slowly lifted by galactic tidal forces into a vast spherical shell — stretching a quarter of the way to the nearest star.
The Milky Way's gravity is not uniform. Its disk exerts a tidal pull perpendicular to the galactic plane — a force that slowly precesses a comet's perihelion outward over millions of years. Once the perihelion rises above ~15 AU, Neptune can no longer recapture it, and the comet is permanently stored in the cloud. The outer cloud's near-perfect sphere is a direct imprint of this tidal symmetry.
A comet in a highly inclined orbit experiences a resonance with the galactic plane: eccentricity and inclination trade off periodically, conserving their combined angular momentum. When eccentricity peaks, the perihelion plunges inward — letting Jupiter scatter the comet to a larger semi-major axis, where the galactic tide can then fully detach it from the planetary zone.