Chapter 12.7, Problem 63STP

a)

To determine

To find :Write an equation for orbit of each planet. The centre of the orbit is the origin and the center of the sun is a focus that lies on x-axis.

Answer to Problem 63STP

1

Explanation of Solution

Given Information :

We know that, Closest and farthest distance of Venus and Jupiter from the center of sun in millions of miles.

Calculation:

  $\begin{array}{ccc}Planet& Closest& Farthest\\ Venus& 66.8& 67.7\\ Jupiter& 460.1& 507.4\end{array}$

  

AB = the closest distance

DE = the farthest distance

BD = $\text{0}\text{.87}$ (million miles)

  $\begin{array}{l}a=\frac{AE}{2}=\frac{AB+BD+DE}{2}\\ a_v=\frac{66.8+0.87+67.7}{2}=67.685\\ a_j=\frac{460.1+0.87+507.4}{2}=484.185\end{array}$

The equation of orbits are ellipse.

  $\begin{array}{l}C=OD=OA-AB-BC=OA-AB-\frac{BD}{2}\\ C_v=OA-AB-\frac{BD}{2}=67.685-66.8-\frac{0.87}{2}=0.45\\ C_j=OA-AB-\frac{BD}{2}=484.185-460.1-\frac{0.87}{2}=23.65\end{array}$

The distance from the center of ellipse to the focus,

  $\begin{array}{l}{b}^2=a^2-b^2\\ b_v=\sqrt{{67.485}^2-{0.45}^2}=67.6835\\ b_j=\sqrt{{484.185}^2-{23.65}^2}=483.6071\\ \frac{x^2}{{67.685}^2}+\frac{y^2}{{67.6835}^2}=1\\ \frac{x^2}{4581.259}+\frac{y^2}{4581.057}=1\end{array}$

For venues,

  $\begin{array}{l}\frac{x^2}{{484.185}^2}+\frac{y^2}{{483.6071}^2}=1\\ \frac{x^2}{234435.1}+\frac{y^2}{233875.8}=1\end{array}$

b)

To determine

To find :Which planet has an orbit that is closer to a circle ?

Answer to Problem 63STP

Venus is closer.

Explanation of Solution

Given Information :

We know that, Closest and farthest distance of Venus and Jupiter from the center of sun in millions of miles.

Calculation:

  $\begin{array}{l}{a}_v=67.685,a_j=484.185\\ c_v=0.45,c_j=23.65\\ e=\frac{c}{a},e_v=\frac{c_v}{a_v}=\frac{0.45}{67.685}=0.006648\\ e_j=\frac{c_j}{a_j}=\frac{23.65}{484.185}=0.0488\end{array}$

  $e_v>e_j$ , it means that the orbit of venues is closer.