Chapter 14, Problem 13A

What is the mathematical relationship between how long it takes a planet to orbit the sun and its distance from the sun?

To determine

The mathematical relationship between time taken by a planet to orbit the Sun and its distance from the sun.

Answer to Problem 13A

The mathematical relationship between how long it takes a planet to orbit the sun and its distance from the sun is $T^2\propto r^3$ .

Explanation of Solution

Given:

Planets orbit around the sun.

Formula used:

The expression for the gravitational force between the planet and the sun as follows:

  $F_{\text{G}}=\frac{GMm}{r^2}$

Here, $G$ is the gravitational constant, $M$ is the mass of Sun, $m$ is the mass of the planet, and $r$ is the distance between the planet and Sun.

The expression for the centripetal force as follows:

  $F_{\text{c}}=mr{\left(\frac{2\pi }{T}\right)}^2$

Here, $m$ is the planet’s mass, $r$ is the distance between the planet and Sun, and $T$ is the planet’ revolution period.

Calculation:

Consider the gravitational force between the planet and Sun provides the centripetal force acting on the planet. That is $F_{\text{G}}=F_{\text{c}}$ .

The mathematical relationship between planet’s revolution period to orbit the sun and planet’s distance from the sun is,

  $\begin{array}{l}{F}_{\text{G}}=F_{\text{c}}\\ \frac{GMm}{r^2}=mr{\left(\frac{2\pi }{T}\right)}^2\\ {\left(\frac{2\pi }{T}\right)}^2=\frac{GM}{r^3}\\ \frac{4{\pi }^2}{T^2}=\frac{GM}{r^3}\end{array}$

  $\begin{array}{l}GM{T}^2=4{\pi }^2{r}^3\\ T^2=\frac{4{\pi }^2{r}^3}{GM}\\ T^2\propto r^3\end{array}$

Conclusion:

Thus, the mathematical relationship between how long it takes a planet to orbit the sun and its distance from the sun is $T^2\propto r^3$ .