Question

Earth The mean distance of Earth from the Sun is $93$million miles. If the aphelion of Earth is $94.5$million miles, what is the perihelion? Write an equation for the orbit of Earth around the Sun.

Short answer

The equation for the orbit of the Earth as $\frac{x^2}{{93}^2}+\frac{y^2}{8646.75}=1$.

Step-by-step solution

. Given information.

The length of the major axis is twice the length of the mean distance of Earth from the sun.

Length of the major axis is twice the length of the mean distance of the Earth from the Sun.

Length of the major axis =$2(93millionmiles)=186millionmiles$

The major axis is sum of the aphelion and the perihelion.

So, Aphelion$+$Perihelion$=186millionmiles$

The aphelion of earth is $94.5$million miles.

We get ,

role="math" localid="1647100683737" $$\begin{gathered} 94.5+perihelion=186 \\ perihelion=186-94.5 \\ perihelion=91.5 \end{gathered}$$

Step 2. To find equation for the orbit of the Earth.

Equation of the ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

We have semi major axis $a=93$

The distance $c$is the distance of the sun from the center of the ellipse is $=MeanDis\tan ce-perihelion$

$$\begin{gathered} c=93-91.5 \\ c=1.5 \end{gathered}$$

Substitute $c=1.5,a=93$for $b^2=a^2-c^2$to get $b^2$.

$$\begin{gathered} b^2={93}^2-1.5^2 \\ {b}^2=8649-2.25 \\ {b}^2=8646.75 \end{gathered}$$

Now, the equation for the orbit of the Earth as$\frac{x^2}{{93}^2}+\frac{y^2}{8646.75}=1$