INTERACTIVE COMPANIONVOL 3, ISSUE 1FALL 2026

The Hilltop Bench

A live model accompanying “Five Places to Park a Spacecraft, Two of Which Are Stable for a Strange Reason”

Study: Computational study
Review: Peer-edited by the club review board
Article: 12 sectionsabout 31 min
Files: LaTeX source22 kB
Analysis code37 kB
Raw output18 kB

Two models. The first draws the effective potential of the rotating frame as a contour map, marks all five Lagrange points live as you move the mass ratio, and drops a test particle wherever you click so you can watch it stay, wander, or leave. The second runs the article's own stability experiment: perturb a chosen point by a controllable amount and watch how long it stays close, then drag the mass ratio slider past the critical value and watch the triangular points stop being safe.

Model 1. The potential and a dropped particle

Darker green is lower effective potential (a valley); paler is higher (a hill). All five Lagrange points sit exactly where the gradient is zero, marked below as they move with the mass ratio slider. Click or tap anywhere on the map to drop a test particle there at rest in the rotating frame and watch its path unfold.

mu = 0.012151 (Earth-Moon), click the map to drop a particle

Mass ratio μ0.012151Slider position is a squashed scale so small mass ratios are not crushed into one pixel; the label always shows real μ. Start at the default (Earth-Moon). Drag past about 80% and watch L4/L5 slide toward the barycentre as the secondary gets heavier. At exactly μ=0.5 (equal masses), L4, L5 and L1 all land directly on top of the symmetry point.Trail length4000stepsHow long a dropped particle is integrated for (step = 0.01 time units). Drop a particle near a filled Lagrange point marker at the default mass ratio and it barely moves even at the maximum trail length; drop one near an open marker (L1/L2/L3, always unstable) and watch it leave within a few hundred steps.

dropped at: –distance from drop point now: –Jacobi drift so far: –

The five markers are computed the same two ways the article uses: Newton's method for the three collinear points (small circles, on the horizontal line through both masses) and the exact equilateral formula for the triangular pair (larger circles, off the line). At the default mass ratio you should see the two triangular markers sitting on an honest local rise of the potential, not in a valley – the whole point of the article is that they stay put anyway.

Model 2. Push it and see

Pick a point, pick how hard to nudge it, pick a mass ratio, and run the same fourth-order Runge-Kutta integrator the analysis script uses. The plot shows distance from the point against time, with the containment threshold marked; the verdict line says whether the particle stayed inside it for the whole run.

PointL1, L2, L3 escape no matter what you set below. L4 and L5 are the interesting ones: try L4 at the default settings first, then push the mass ratio slider up.Mass ratio μ / μ_crit0.31532 on this slider is Earth-Moon's real ratio, 0.315× μ_crit. Push it past 100 (μ>μ_crit=0.038521) and watch L4/L5 flip from contained to escaped with nothing else changed. Pluto-Charon's real ratio is 2.823×, at slider position 282.Perturbation size1.0e-5Log scale, 10⁻⁶ to 10⁻². The article's own single-trajectory demonstration uses 10⁻⁵, which is the default here.

outcome: –max distance reached: –Jacobi drift up to escape: –

Press Run.

What to try

Three checks against the article's own printed numbers, all reproducible by hand from what the readouts show:

Model 2 at its defaults (L4, Earth-Moon's real ratio, perturbation 10⁻⁵) should read contained with a max distance under 10⁻³, matching the article's 1.99×10⁻⁴.
Switch the point to L1 and press Run again with everything else unchanged: it escapes in a handful of time units, matching the article's L1 escape time of 2.69.
Move the mass ratio slider to 282 (Pluto-Charon's real ratio) with the point back on L4: it escapes, matching the article's t=15.0 result for that system.