Chapter 7.1, Problem 3PP

To determine

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

Answer to Problem 3PP

  $0.724\times$ (average distance of earth from sun)

Explanation of Solution

Given:

Venus takes 225 Earth days to circle 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 revolve around the Sun, thus $T_E=365\text{days = 1 year}$ .

And, Venus takes 225 Earth days to revolve around the Sun, thus $T_V=\text{225 Earth days}$

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

  $\begin{array}{c}{\left(\frac{T_V}{T_E}\right)}^2={\left(\frac{r_V}{r_E}\right)}^3\\ {\left(\frac{225}{365}\right)}^2={\left(\frac{r_V}{r_E}\right)}^3\\ 0.724=\frac{r_V}{r_E}\\ r_V=0.724r_E\end{array}$

Conclusion:

Thus, the Venus’s average distance from the Sun is 0.724 times of the average distance of Earth from the Sun.