Period of a planet at 2.4 AU

Problem

A new hypothetical planet orbits the Sun at a mean distance $a = 2{,}4\ \text{AU}$ (astronomical units). Using Kepler’s third law in the form $T^2/a^3 = \text{const}$, calculate the planet’s period of revolution expressed in Earth years.

Solution

Kepler’s third law. The constant is the same for all bodies orbiting the Sun. Compare the planet with the Earth: $\frac{T^2}{a^3} = \frac{T_\text{Earth}^2}{a_\text{Earth}^3}$

Choice of convenient units. Measuring time in years and distance in AU, for the Earth $T_\text{Earth}=1\ \text{year}$ and $a_\text{Earth}=1\ \text{AU}$, so the constant equals $1$: $T^2 = a^3 \;\Rightarrow\; T = a^{3/2}$

Numerical substitution. $T = (2{,}4)^{3/2} = 2{,}4\cdot\sqrt{2{,}4} = 2{,}4\cdot 1{,}55 \approx 3{,}72$

$\ev{T \approx 3{,}7\ \text{years}}$

The farther the planet, the more slowly it travels its orbit: at $2{,}4\ \text{AU}$ it takes almost four Earth years for one revolution.

Links

Topics: Gravitation Concepts: Kepler’s laws · Orbits Skills: Symbolic setup