Period Ratios

Larger Radius Takes More Period-Square

Because T^2 / r^3 stays constant for a shared central mass, growing the radius raises the required period-square, so a larger orbit takes a longer period. The trend 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.

Radius-cubed and period-square scale together in this normalized law. 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 larger orbit has eight times radius-cubed

The radius-cubed grows from 8 to 64, a factor of 8.

rB3rA3=648=8\frac{r_B^{3}}{r_A^{3}}=\frac{64}{8}=8
Radius-cube scaleThe period law uses radius-cubed, not raw radius.keplerConstant=1/2 s^2/m^3leftRadiusCubed=8 m^3leftPeriodSquared=4 s^2leftRatio=1/2 s^2/m^3rightRadiusCubed=64 m^3rightPeriodSquared=32 s^2rightRatio=1/2 s^2/m^3acceptedBit=1 bit

The period-square grows by the same factor

The period-square grows from 4 to 32, also a factor of 8.

TB2TA2=324=8\frac{T_B^{2}}{T_A^{2}}=\frac{32}{4}=8
Period-square scaleEqual ratios keep the same normalized K.keplerConstant=1/2 s^2/m^3leftRadiusCubed=8 m^3leftPeriodSquared=4 s^2leftRatio=1/2 s^2/m^3rightRadiusCubed=64 m^3rightPeriodSquared=32 s^2rightRatio=1/2 s^2/m^3acceptedBit=1 bit