Chapter 10.3, Problem 81AYU

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 perihelion.

The mean distance.

An equation for the orbit of Jupiter around the Sun.

Answer to Problem 81AYU

Solution:

Perihelion $=460.6$ million miles.

Mean distance $=483.8$ million miles.

$\frac{x^2}{234062.44}+\frac{y^2}{233,524.2}=1$ ($x\text{and}y$ in million).

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

Aphelion of Jupiter is 507 million miles.

The distance from center to sun is $23.2$ million miles.

a. $\text{Mean}\text{distance}=\text{Aphelion}–\text{Distance}\text{from}\text{center}\text{to}\text{Sun}$

$\text{Mean}\text{distance}=507-23.2=483.8\text{million}\text{miles}$

b. $\text{The perihelion }=\text{ Mean distance }–\text{ distance from center to Sun}$

Perihelion $=483.8-23.2$ (in million)

Perihelion $=460.6$ (in million)

c. $a=483.8\text{ million and }c=23.2\text{ million}$

$b^2=a^2-c^2$

$b^2={483.8}^2-{23.2}^2$ (in million).

$b²=2.335242$ (in million).

$b=483.2434$ (in million).

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

$\frac{x^2}{{483.8}^2}+\frac{y^2}{{483.2}^2}=1$ (in million).

$\frac{x^2}{234062.44}+\frac{y^2}{233,524.2}=1$ ($x\text{and}y$ in million).