Chapter 9.7, Problem 63E

(a)

To determine

To find: the polar equation of the orbit of a planet

Answer to Problem 63E

The polar equation of the orbit of a planet Neptune is $\frac{4.4977\times {10}^9}{1-0.0086\cos \theta }$

The planet Pluto is $\frac{5.5404\times {10}^9}{1-0.2488\cos \theta }$ .

Explanation of Solution

Given:

  $\text{Neptune }a\approx 4.498\times {10}^9\text{kilometers, }e\approx 0.0086$

  $Plutoa\approx 5.906\times {10}^9\text{kilometers, }e\approx 0.2488$

Calculation:

For Neptune: Use the equation given values.

  $\begin{array}{l}r=\frac{\left(1-e^2\right)a}{1-e\cos \theta }\\ r=\frac{\left(1-{0.0086}^2\right)\left(4.498\times {10}^9\right)}{1-0.0086\cos \theta }=\frac{4.4977\times {10}^9}{1-0.0086\cos \theta }\end{array}$

For Pluto: Use the equation given values.

  $\begin{array}{l}r=\frac{\left(1-e^2\right)a}{1-e\cos \theta }\\ r=\frac{\left(1-{0.2488}^2\right)\left(5.906\times {10}^9\right)}{1-0.2488\cos \theta }=\frac{5.5404\times {10}^9}{1-0.2488\cos \theta }\end{array}$

Conclusion:

Therefore, the polar equation of the orbit of a planet Neptune is $\frac{4.4977\times {10}^9}{1-0.0086\cos \theta }$ and the planet Pluto is $\frac{5.5404\times {10}^9}{1-0.2488\cos \theta }$ .

(b)

To determine

To find: the perihelion and aphelion distances for each planet.

Answer to Problem 63E

The perihelion and aphelion distances of Neptune planet is $4.4593\times {10}^9$ and $4.5367\times {10}^9$ . And perihelion and aphelion distances of Pluto planet is $4.4366\times {10}^9$ and $7.3754\times {10}^9$ .

Explanation of Solution

Calculation:

For Neptune: Use the formula from exercise 49 to find the perihelion.

  $\begin{array}{l}r=a\left(1-e\right)\\ r=\left(4.498\times {10}^9\right)\left(1-0.0086\right)=4.4593\times {10}^9\end{array}$

  $\begin{array}{l}r=a\left(1+e\right)\\ r=\left(4.498\times {10}^7\right)\left(1+0.0086\right)=4.5367\times {10}^9\end{array}$

For Pluto: Use the formula from exercise 49 to find the perihelion.

  $\begin{array}{l}r=a\left(1-e\right)\\ r=\left(5.906\times {10}^9\right)\left(1-0.2488\right)=4.4366\times {10}^9\\ r=a\left(1+e\right)\\ r=\left(5.906\times {10}^7\right)\left(1+0.2488\right)=7.3754\times {10}^9\end{array}$

Conclusion:

Therefore, the perihelion and aphelion distances of Neptune planet is $4.4593\times {10}^9$ and $4.5367\times {10}^9$ . And perihelion and aphelion distances of Pluto planet is $4.4366\times {10}^9$ and $7.3754\times {10}^9$ .

(c)

To determine

To graph: Neptune’s and Pluto’s equations of orbit

Answer to Problem 63E

  

Explanation of Solution

Calculation:

Graph the equations for the orbits of both planets.

Red: Neptune

Blue: Pluto

  

Conclusion:

Therefore, the equation is drawn.

(d)

To determine

To prove: the Pluto called the ninth planet and Neptune the eighth planet.

Answer to Problem 63E

On average, Pluto is farther from the sun than Neptune is.

Explanation of Solution

Pluto's perihelion is less than Neptune's perihelion and aphelion, so there is a point at which Pluto is closer to the sun than Neptune is. On average, Pluto is farther from the sun than Neptune is.

Conclusion:

Therefore, On average, Pluto is farther from the sun than Neptune is.

(e)

To determine

To explain: whether the orbits of Neptune and Pluto intersect and collide.

Answer to Problem 63E

Pluto and Neptune will not collide.

Explanation of Solution

Pluto and Neptune will not collide. Even though the graph in part c shows that their orbits intersect, it does not show what happens in three-dimensional space.

Conclusion:

Therefore, Pluto and Neptune will not collide.