Chapter 10.3, Problem 80AYU

In Problems 79-83, use the fact that the orbit of a planet about the Sun is an ellipse, with the Sun at one focus. The aphelion of a planet is its greatest distance from the Sun, and the perihelion is its shortest distance. The mean distance of a planet from the Sun is the length of the semimajor axis of the elliptical orbit. See the illustration.

Mars The mean distance of Mars from the Sun is 142 million miles. If the perihelion of Mars is $128.5$ million miles, what is the aphelion? Write an equation for the orbit of Mars about the Sun.

To determine

To find: The measurement of aphelion, if the perihelion of Mars is $128.5$ million miles. An equation for the orbit of Earth around.

Answer to Problem 80AYU

Solution:

Apehilion $=155.5$ (million miles).

$\frac{x^2}{20,164}+\frac{y^2}{19981.75}=1$ ($x\text{and}y$ in million of miles).

Explanation of Solution

Given:

Use the fact that the orbit of a planet about the Sun is an ellipse, with the Sun at one focus. The aphelion of a planet is its greatest distance from the Sun, and the perihelion is its shortest distance. The mean distance of a planet from the Sun is the length of the semimajor axis of the elliptical orbit.

The mean distance of Mars from the Sun is 142 million miles.

Formula used:

Center	Major axis	Foci	Vertices	Equation
$(0,0)$	$x\text{-axis}$	$(c,0)$	$(a,0)$	$\begin{array}{l}\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\\ a>b>0\text{and}b²=a²-c²\end{array}$
$(0,0)$	$x\text{-axis}$	$(-c,0)$	$(-a,0)$

Calculation:

Mean distance $=142$ million miles, Hence $a=142$ million miles.

The length of the major axis: $2a=284$ million.

From the figure we see that,

$\text{Apehilion }=\text{ Major axis }–\text{ Perihelion}$

Apehilion $=284-128.5$ (million miles)

Apehilion $=155.5$ (million miles)

Distance from center of the ellipse to focus (Sun’s position) is given by $\text{Mean}\text{distance}-\text{perihelion}$.

$c=142\text{million}-128.5\text{million}=13.5\text{million}$

To find $b$,

$b²=a²-c²$

$b²={142}^2-{13.5}^2{\left(\text{in}\text{million}\right)}^2$

$b^2=19981.75{\left(\text{in}\text{millions}\right)}^2$

$b=141.36$ (in million)

Hence the equation is,

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

$\frac{x^2}{{142}^2}+\frac{y^2}{141.36²}=1$ (in million)

$\frac{x^2}{20,164}+\frac{y^2}{19981.75}=1$ ($x\text{and}y$ in million of miles).