Kepler's Third Law: A Planet's Year from Its Orbit

A planet twice as far from the Sun doesn't take twice as long to go around: it takes about 2.8 times as long. Kepler found that the orbital period grows like the distance to the 3/2 power, a rational exponent: square the period and you get the distance cubed.

How it works

Rational exponents

In one sentence

For $a > 0$ and integers $m$ and $n > 0$ , $a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$ . Kepler's third law states that the square of a planet's orbital period is proportional to the cube of its orbit's semi-major axis; in years and astronomical units, $T^2 = a^3$ , so $T = a^{3/2}$ .

General formula

{\htmlData{sym=0}{T}} = {\htmlData{sym=1}{a}}^{3/2} = \sqrt{{\htmlData{sym=1}{a}}^3}, \qquad {\htmlData{sym=0}{T}}^2 = {\htmlData{sym=1}{a}}^3

Cube the distance and take the square root: the exponent 3/2 does both in one step.

What the symbols mean

$T$
Orbital period
How long one trip around the Sun takes, in Earth years.
$a$
Orbit size
The planet's average distance from the Sun, in astronomical units (AU).

A rational exponent combines a power and a root: $a^{3/2} = \left(\sqrt{a}\right)^3 = \sqrt{a^3}$ .
Measure distance in astronomical units (Earth's distance from the Sun is 1 AU) and time in Earth years. Then Kepler's third law reads $T = a^{3/2}$ .
Squaring both sides clears the fraction: $T^2 = a^3$ . For Jupiter at 5.2 AU, $T = 5.2^{3/2} \approx 11.9$ years.
The exponent sits between 1 and 2: periods grow faster than the distance (outer planets also move more slowly) but slower than the distance squared.

Learn the math

Rational ExponentsAlgebra · Lesson 2, Preliminaries RadicalsAlgebra · Lesson 3, Preliminaries