Period-Square over Radius-Cubed Matches - Rotational Dynamics and Orbital Mechanics Step by Step

Kepler's third law ties a body's orbital period to its radius through T^2 / r^3 = constant, so two orbits sharing a common central mass share that same ratio. The arithmetic is exact only for the stated periods, radii, and central mass in the ideal circular two-body model. Real orbits add eccentricity, inclination, perturbations, central-mass uncertainty, finite-body effects, and measurement uncertainty.

Two orbit ledgers match the same normalized Kepler ratio. 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 first orbit matches the normalized period law

The first period-square 4 over radius-cubed 8 gives one half.

\frac{T_A^{2}}{r_A^{3}}=\frac{4}{8}=\frac{1}{2}

The second orbit matches the same normalized law

The second period-square 32 over radius-cubed 64 also gives one half.

\frac{T_B^{2}}{r_B^{3}}=\frac{32}{64}=\frac{1}{2}