02 · Shapes

Ellipse

Earth at one focus. Fast at periapsis, lazy at apoapsis. Burn at periapsis to raise apoapsis — every transfer in disguise.

In English

Most orbits are ovals with Earth at one focus. Near Earth (periapsis) you run. Far away (apoapsis) you crawl. A burn at the low point raises the high point. That is how transfers start.

Try this: Raise apoapsis. Slide true anomaly from 0 to 180. Watch speed drop as you climb.
Keep this: Fast at periapsis. Burn there to raise apoapsis.

Lab · fast at periapsis, lazy at apoapsis

Kepler: an ellipse with Earth at one focus. Vis-viva: v² = μ(2/r − 1/a). Close in, r is small, v is large. Raise apoapsis with a burn at periapsis — that is every transfer in disguise.

Apoapsis 2000 km

True anomaly 0°

r − R = 400 km · 8.07 km/s

e = 0.11

Vis-viva v² = μ(2/r − 1/a). Periapsis rp = a(1−e), apoapsis ra = a(1+e). The lab pins periapsis at 400 km and stretches apoapsis so the burn-at-peri story is visible.

Equal areas in equal times is Kepler II: the satellite is lazy at apoapsis because the radius arm is long. That laziness is where plane changes are cheapest.