Applying the Equation of an Ellipse to the Orbit of a Planet

Example

The orbit of the planet Mars is an ellipse with the sun at one focus. The eccentricity of the ellipse is 0.0935, and the closest distance that Mars comes to the sun is 128.5 million mi. Find the maximum distance of Mars from the sun.

Solution

The figure shows the orbit of Mars with the origin at the center of the ellipse and the sun at one focus.

Mars is closest to the sun when Mars is at the right endpoint of the major axis and farthest from the sun when Mars is at the left endpoint. Therefore, the least distance is a − c, and the greatest distance is a + c.

Since $a-c=128.5,\:c=a-128.5$. Use $e=\frac{c}{a}$.

$\frac{a-128.5}{a}=0.0935$

Multiply by $a$.

$a-128.5=0.0935a$

$a\approx141.8$

Then, $c=141.8-128.5=13.3\:$and $a+c=141.8+13.3=155.1$

The maximum distance of Mars from the sun is about 155.1 million mi.