Real Simulation Ledgers

Orbit Simulation Conserves Energy and Momentum

Every value in this lesson is rounded to four decimal places from a real physim numerical integration of an elliptical Kepler orbit (scenario examples/kepler-orbit-elliptical.json, integrator dp45, rtol 1e-9, atol 1e-11), not derived from a closed form. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The simulated path is a real ellipse that closes

The simulated radius starts at 1.8000 meters, swings out to 4.1988 meters at the far side (t = 10.1177 seconds), and comes back to 1.8000 meters after one full simulated period (t = 20.6484 seconds) — a genuinely elliptical orbit, not the circular idealization of the earlier chapters.

r: 1.8000 m → 4.1988 m → 1.8000 mr:\ 1.8000\ \mathrm{m}\ \to\ 4.1988\ \mathrm{m}\ \to\ 1.8000\ \mathrm{m}
Simulated elliptical orbitGhost trail and sampled positions come from the physim trajectory; the radius labels are the pinned CSV values.t=0.0000 s: r=1.8000 mt=10.1177 s: r=4.1988 m (far side)t=20.6484 s: r=1.8000 m (one period)

A real integrator holds both conservation laws

At all three sampled rows the specific energy rounds to the same -0.4167 joules per kilogram and the specific angular momentum to the same 2.5100 meters squared per second, even though the radius more than doubles along the way — a real numerical integrator, not just the closed-form algebra of the earlier chapters, holds the conservation laws.

e=−0.4167 J/kg,h=2.5100 m2/se=-0.4167\ \mathrm{J/kg},\quad h=2.5100\ \mathrm{m}^{2}/\mathrm{s}
Conserved quantities on the simulated orbitSpecific energy and specific angular momentum stay pinned at the same rounded value at every sampled row.t=0.0000 s: r=1.8000 mt=10.1177 s: r=4.1988 m (far side)t=20.6484 s: r=1.8000 m (one period)specificEnergy=-0.4167 J/kgspecificAngularMomentum=2.5100 m^2/s