Kepler's Laws of Planetary Motion/Third Law/Examples

Theorem
Let $P$ be a planet orbiting the sun $S$

Let $P$ be:
 * $\text{(a)}: \quad$ Twice as far away from $S$ as the Earth;
 * $\text{(b)}: \quad$ $3$ times as far away from $S$ as the Earth;
 * $\text{(c)}: \quad$ $25$ times as far away from $S$ as the Earth.

Then the orbital period of $P$ is:
 * $\text{(a)}: \quad$ approximately $2.8$ years;
 * $\text{(b)}: \quad$ approximately $5.2$ years;
 * $\text{(c)}: \quad$ $125$ years.

Proof
Let the orbital period of Earth be $T'$ years.

Let the mean distance of Earth from $S$ be $A$.

Let the orbital period of $P$ be $T$ years.

Let the mean distance of $P$ from $S$ be $a$.

By Kepler's Third Law of Planetary Motion:

Thus the required orbital periods are:


 * $\text{(a)}: \quad 2^{3/2} = 2 \sqrt 2 \approx 2.8$ years


 * $\text{(b)}: \quad 3^{3/2} = 3 \sqrt 3 \approx 5.2$ years;


 * $\text{(c)}: \quad 25^{3/2} = 125$ years.