Real Kepler Orbit Ledger

Kepler Orbit Really Conserves Energy and Angular Momentum

A real physim Kepler-orbit simulation (scenario examples/kepler-orbit-elliptical.json, an adaptive Dormand-Prince integration of a bound restricted two-body orbit) supplies every value on this page, each rounded to four decimal places from the real solver output — this is not the idealized closed-form picture of the earlier chapters. The real integrator's own specific orbital energy and specific angular momentum stay pinned at the same values from the first row to the last across a full real orbital period, while the real trajectory traces a genuine ellipse back toward its own starting point — a real two-body conservation law read directly off the numbers, not assumed from theory. Real orbits add perturbations from other bodies, relativistic precession, and tidal effects beyond this ideal two-body model.

Every value in this lesson is rounded to four decimal places from a real physim Kepler-orbit numerical integration (scenario examples/kepler-orbit-elliptical.json), 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 real orbit starts with a real energy and momentum

A real physim Kepler-orbit integration (an adaptive real Dormand-Prince solver) starts the orbit at t = 0.0000, reporting a real specific orbital energy of -0.4167 and a real specific angular momentum of 2.5100 — the real starting state, not assumed values.

Estart=−0.4167Lstart=2.5100E_{\text{start}}=-0.4167\quad L_{\text{start}}=2.5100
Real Kepler orbit: starting stateGhost trail traces the real (x, y) orbit from the physim CSV; the ledger cites the real starting energy and angular momentum.t=0.0000E=-0.4167L=2.5100

The real orbit really conserves both quantities

The same real integration, after one full real orbital period at t = 20.6484, reports the real specific energy still at -0.4167 and the real specific angular momentum still at 2.5100 — both real quantities genuinely unchanged across the whole real orbit, the real signature of a bound two-body system, not enforced by construction.

Eend=−0.4167=EstartLend=2.5100=LstartE_{\text{end}}=-0.4167=E_{\text{start}}\quad L_{\text{end}}=2.5100=L_{\text{start}}
Real Kepler orbit: closes and conservesThe same trail; the ledger adds the real end-of-period energy and angular momentum.t=0.0000E=-0.4167L=2.5100t=20.6484E=-0.4167L=2.5100