Galactic Dynamics Simulator

Face-on (top left)

One star orbiting the galactic center (the white dot), seen from above the disk. Galactic mass is spread out instead of concentrated in a point, so the orbit doesn't close into an ellipse. Each radial in-and-out takes a little less time than one trip around, and the path traces a rosette. The dashed circles are the closest and farthest radii the star can reach at its energy E and angular momentum L_z.

Meridional plane (top right)

This plot tracks only the star's distance from the rotation axis (R) and its height above the disk (z), and ignores which way around the galaxy it is. The potential is the same at every azimuth and L_z is conserved, so the 3D orbit reduces to 2D motion in an effective potential Φ + L_z²/2R².

The shaded area is where Φ_eff > E, which the star can never enter. This is the same idea as the zero-velocity curves of the CR3BP Jacobi integral. The star never fills the whole allowed region, only a smaller box inside it, and that is the visible sign of a hidden third integral.

Surface of section (bottom left)

A Poincaré map. Every time the star crosses the disk plane going upward (z = 0, v_z > 0), the plot records its radius R and radial speed v_R, and only that dot is kept. The gold curve is the outer limit set by energy.

If E and L_z were the only conserved quantities, the dots would scatter over the whole area inside it. Instead each regular orbit draws a closed curve, which is a slice through the torus the orbit lives on. Nested curves are different orbits with the same E and L_z. When curves break into a scattered spray, the orbit is chaotic, which is what the Hénon–Heiles mode shows.

Rotation curve (bottom right)

The speed of a circular orbit at each radius, v_c² = R ∂Φ/∂R, split into bulge, disk and dark-halo contributions. Stars alone would give a curve that falls off beyond about 10 kpc. Observed curves stay flat, and the halo term is what keeps them flat. The ring marks your launch speed at R₀.