DENSITY & TIME
The gravity-train period depends ONLY on density, 2π√(3/4πGρ) — a whole planet or a marble of the same stuff take identical time.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
The gravity-train period seems like it should depend on how big the planet is. It does not. Take Earth’s period, then carve a marble out of the same rock — the marble gives the identical answer. This module shows why, and how far it goes.
R cancels
Write the period 2π√(R³/GM) and replace the mass by four-thirds π R³ ρ. The R³ on top and the R³ hidden in the mass cancel completely. What survives has no size in it at all — only the density ρ remains.
Only ρ
A denser world pulls harder but also packs its matter closer to your path, and the two effects fold into ρ alone: T = 2π√(3/4πGρ). Earth’s 84 minutes come from its density and nothing else — a boulder of Earth-stuff would agree exactly.
Every world
Line the materials up by density: water is the slow end near 198 minutes, the Sun is surprisingly loose near 167, and lead is the fast end near 59. A large body and a marble of the same density keep the same clock.
Audit
Audited: Earth’s density reproduces the gravity-train’s 84.3 minutes to twelve digits, and the result is radius-free by construction — planet and pebble agree to the bit. The honest caveat: real worlds are denser at the core, so this is the clean uniform idealisation.