Chapter 7.1, Problem 4PP

To determine

Uranus’s average distance from the Sun as a multiple of Earth’s average distance from the Sun.

Answer to Problem 4PP

  $19.18\times$ (average distance of earth from sun)

Explanation of Solution

Given:

Uranus requires 84 years to orbit the Sun.

Formula used:

Kepler’s Third Law,

  ${\left(\frac{T_A}{T_B}\right)}^2={\left(\frac{r_A}{r_B}\right)}^3$

Where $T_A$ and $T_B$ are the periods of the planets A and B around the Sun, and $r_A$ and $r_B$ are their average distances from the Sun.

Calculation:

It is known that Earth takes 365 days or 1 year to orbit the Sun, thus $T_E=365\text{days = 1 year}$ .

And, Uranus takes 84 years to orbit the Sun, thus $T_U=\text{84 years}$

Let, the average distance of Earth from the Sun be $r_E$ , and that of Uranus from the Sun be $r_U$ . Then, using Kepler’s law,

  $\begin{array}{c}{\left(\frac{T_U}{T_E}\right)}^2={\left(\frac{r_U}{r_E}\right)}^3\\ {\left(\frac{84}{1}\right)}^2={\left(\frac{r_U}{r_E}\right)}^3\\ 19.180=\frac{r_U}{r_E}\\ r_U=19.18r_E\end{array}$

Conclusion:

Thus, Uranus’s average distance from the Sun is 19.18 times of the average distance of Earth from the Sun.