GRAVITATION & ORBITS

Shell theorem, drawn: g climbs straight to the surface, then falls as 1/r². Zero at the core.

v_esc = √(2GM/R) — the energy line that just clears the rim. Mass and angle need not apply.

v²r = GM, E = −GMm/2r, and 2KE + PE = 0 to 1e-12 — the virial ledger, audited.

T² = a³, equal areas proven as an identity — Newton’s arithmetic, checked.

Drop a stone through the Earth: it oscillates in 84 min — the exact period of a skimming satellite, and any straight tunnel keeps the same clock.

The gravity-train period depends ONLY on density, 2π√(3/4πGρ) — a whole planet or a marble of the same stuff take identical time.