THE KEPLER THIRD LAW the period squared is the size cubed
For every body orbiting the same mass, T² ∝ a³ — the period squared is proportional to the semi-major axis cubed, with the same constant for all of them. Kepler found it in Tycho’s data in 1619, decades before Newton explained it as T² = 4π²a³/GM. The ratio T²/a³ is a fingerprint of the central mass.
THE TECHNIQUE T² / a³ = constant
In units where 4π²/GM = 1, the demo shows T²/a³ is the same for a = 1, 2, 3: live demo
HISTORY & CREDIT Kepler · Harmonices Mundi 1619
“Kepler derived his laws from theory.” — he found them empirically in Tycho Brahe’s observations; Newton derived them from gravity 68 years later. cited
the harmony · T²/a³ is identical for every satellite of the same body. the constant · = 4π²/GM — it weighs the central mass. 1619 · Johannes Kepler, Harmonices Mundi; from Tycho Brahe’s data.
Distance and time, locked by a cube. physics
RECOMMEND FOR I-13 the ratio, on the compiler
On i-13 (units where 4π²/GM=1), T²/a³ = 1 for a = 1, 2, 3:
$ i13 run gw_kepler.i13 # T^2 / a^3
RUN OK · 56 step(s)
T1sq=1 T2sq=8 T3sq=27 -- a^3 for a=1,2,3
ratio1=1 ratio2=1 ratio3=1
same_ratio = 1
Recommend as a NULL — a theorem (B39). T²/a³ = 4π²/GM is an exact law of the 1/r potential; any correct mechanism reproduces the same ratio. NULL — the harmony of the worlds.