91Ó°ÊÓ

The given statement is:

The aphelion of Jupiter is 507 million miles. If the distance from the center of its elliptical orbit to the Sun is 23.2 million miles.

We can conclude from it that the greatest distance of Jupiter from the sun is 507 million miles.

We have to find the perihelion, the mean distance, and an equation for the orbit of Jupiter around the Sun.

If we subtract the distance from the center of the elliptical orbit to the Sun from the aphelion, we can find the distance from the center of the elliptical orbit to one of the vertex.

Therefore the distance will be:

$507-23.2=483.8$million miles.

As we know that the length of the major axis is twice of the distance from the center of the elliptical orbit to one of the vertex.

Therefore the length of the major axis is:

$2\left(483.8\right)=967.6$million miles

If we subtract aphelion from the major axis we will get the distance of perihelion.

$967.6-507=460.6$

Therefore, the perihelion is $460.6$million miles.

The mean distance is equal to half of the major axis of the ellipse describing the trajectory of a planet around the sun.

Therefore the mean distance is $483.8$million miles.

The center of the elliptical orbit is at the origin and the major axis is parallel to x-axis.

Then the general form of the equation of the ellipse will become:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

The distance from the center to the vertex is $483.8$, it is the distance a.

And the distance from the center to the focus is $23.2$, it is our distance c.

role="math" localid="1647022495952" $$\begin{gathered} b^2={\left(483.8\right)}^2-{\left(23.2\right)}^2 \\ =234062.44-538.24 \\ =233524.2 \\ b=\sqrt{233524.2} \end{gathered}$$

Substitute the values of aand bin the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$,

$\frac{x^2}{{\left(483.8\right)}^2}+\frac{y^2}{233,524.2}=1$

Therefore, $\frac{x^2}{{\left(483.8\right)}^2}+\frac{y^2}{233,524.2}=1$is the equation for the orbit of Jupiter around the Sun.