The Orbit Bench

Two benches. The first integrates one Kepler orbit three ways at once, live, on the same timestep, so you can watch forward Euler leave the frame while velocity Verlet and RK4 stay close to the dashed exact ellipse underneath them. The second runs the convergence-order fit from §7 of the article live, in your browser, for any eccentricity you choose, and reports the fitted slope beside the textbook value it is supposed to match. Every number below is computed by the same physics as the analysis script, not looked up from a table; the defaults reproduce the article's headline numbers exactly, and you can check that yourself with the "skip to orbit 300" button.

Model 1. Watch the orbit fail

The orbit panel rescales itself as the trajectories spread out, which is why the ellipse visibly shrinks in the frame once Euler starts wandering: the view is zooming out to keep everything on screen, not the orbit itself changing size. The two strips below it are exactly Figure 2 from the article, live: Euler's on a scale wide enough to hold its catastrophe, Verlet and RK4 on a scale twenty times finer, with Verlet drawn as the true min-to-max range within each completed orbit rather than a single sample of it.

Model 2. Does the error actually shrink the way theory says

What's going on here?

Both benches run the identical stepping rules used in the analysis script: kick-drift-kick velocity Verlet, classical four-stage RK4, and plain forward Euler, all checked against the same closed-form Kepler solution via Newton's method on $M = E - e\sin E$. Nothing here is a canned animation; every frame is a real integration step. The convergence bench (Model 2) recomputes seven step counts per method from scratch on every eccentricity change, the same procedure §7 of the article runs once, offline, and prints.

The simulation pauses automatically when this tab is hidden and never autoplays if your system is set to reduce motion; press Play yourself when you are ready to watch it.