Real Kepler Orbit Ledger

Kepler Orbit Really Conserves Energy and Angular Momentum

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