Chapter 11.2, Problem 52E

To determine

To determine the distance between Pluto and the sun at perihelion and at aphelion.

Answer to Problem 52E

The distance between Pluto and the sun at perihelion is $3,870,000,000\text{ kilometers}$ and at aphelion is $6,450,000,000\text{ kilometers}$ .

Explanation of Solution

Given information:

The eccentricity of Pluto’s orbit is $\text{0}\text{.25}$

The length of the minor axis is approximately is $\text{10,000,000,000 km}$ .

Formula used:

The following formulas are used:

  $\begin{array}{l}\text{Length of the minor axis}=\text{2b}\\ \text{e}=\frac{\text{c}}{\text{a}}\\ {\text{c}}^2={\text{a}}^2-{\text{b}}^2\end{array}$

Calculation:

The sun lies in the foci as shown:

  

The value of $\text{b}$ can be calculated by:

  $\begin{array}{l}\text{Length of the minor axis}=\text{2b}\\ \text{2b}=10,000,000,000\\ \text{b}=5,000,000,000\end{array}$

The eccentricity is $\text{0}\text{.25}$ , so

  $\begin{array}{l}\text{e}=\text{0}\text{.25}\\ \frac{\text{c}}{\text{a}}\text{=}\frac{25}{100}=\frac{1}{4}\left[\text{e}=\frac{\text{c}}{\text{a}}\right]\end{array}$

Let $\text{a}$ and $\text{c}$ be $\text{4x}$ and $\text{x}$ respectively.

  $\begin{array}{l}{\text{c}}^2={\text{a}}^2-{\text{b}}^2\\ {\text{b}}^2={\text{a}}^2-{\text{c}}^2\\ {\text{b}}^2={\left(\text{4x}\right)}^2-{\left(\text{x}\right)}^2\\ {\text{b}}^2=16{\text{x}}^2-{\text{x}}^2\\ {\text{b}}^2=15{\text{x}}^2\end{array}$

Now, according to the question:

  $\begin{array}{l}{\text{b}}^2={\left(5,000,000,000\right)}^2\\ 15{\text{x}}^2=25\times {10}^{18}\\ {\text{x}}^2=\frac{25\times {10}^{18}}{15}\\ \text{x}=\sqrt{\frac{25\times {10}^{18}}{15}}=1.29\times {10}^9\end{array}$

The value of $\text{a}$ can be calculated by:

  $\begin{array}{l}\text{a}=4\text{x}\\ \text{a}=4\times 1.29\times {10}^9\\ \text{a}=5.16\times {10}^9\text{ or 5,160,000,000}\end{array}$

The value of $\text{c}$ can be calculated by:

  $\begin{array}{l}\text{c}=\text{x}\\ \text{c}=1.29\times {10}^9\text{ or 1,290,000,000}\end{array}$

The distance at perihelion can be calculated by:

  $\begin{array}{l}\text{a}-\text{c}\\ \text{=5,160,000,000}-1,290,000,000\\ =3,870,000,000\text{ kilometers}\end{array}$

The distance at aphelion can be calculated by;

  $\begin{array}{l}\text{2a}-\text{perihelion}\\ \text{=10,320,000,000}-3,870,000,000\\ =6,450,000,000\text{ kilometers}\end{array}$

Hence, the distance between Pluto and the sun at perihelion is $3,870,000,000\text{ kilometers}$ and at aphelion is $6,450,000,000\text{ kilometers}$ .