A real physim numerical integration of an elliptical Kepler orbit (scenario examples/kepler-orbit-elliptical.json, integrator dp45, rtol 1e-9, atol 1e-11) supplies every value on this page, each rounded to four decimal places from solver output — these are not closed-form idealizations like the earlier circular-orbit chapters. The radius really swings and really closes after one simulated period, while the specific energy and specific angular momentum stay pinned at the same rounded value at every sampled row. Real missions add perturbations, extended bodies, reference-frame choices, finite solver tolerance, and measurement uncertainty beyond this two-body model.
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:1.8000m→4.1988m→1.8000m
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.