Chapter 8.2, Problem 52E

Solution

To Prove: The object with the most eccentric orbit sometimes is closer to the Sun than the planet with the least eccentric orbit.

It has been shown that the object with the most eccentric orbit sometimes is closer to the Sun than the planet with the least eccentric orbit.

Given:

The eccentricities and semi-major axis of the planets and Pluto.

Concept used:

The eccentricity of an elliptic orbit is given as $e=\frac{c}{a}$ and the perihelion distance from the Sun is $a-c$ .

Calculation:

It can be seen from the given table that the object with the most eccentric orbit is Pluto and the planet with the least eccentric orbit is Neptune.

It can also be found from the given table that the eccentricity of Pluto is $0.2484$ and its semi-major axis is $5900$ Gm.

Then, for Pluto, $e=0.2484$ and $a=5900$ .

Put these values in $e=\frac{c}{a}$ to get,

  $0.2484=\frac{c}{5900}$

Simplifying,

  $c=1465.56$

Then, the perihelion distance of Pluto from the Sun is given as,

  $a-c=5900-1465.56=4434.44$

So, the perihelion distance of Pluto from the Sun is $4434.44$ Gm.

Similarly, the eccentricity of Neptune is $0.0050$ and its semi-major axis is $4497$ Gm.

Then, for Neptune, $e=0.0050$ and $a=4497$ .

Put these values in $e=\frac{c}{a}$ to get,

  $0.0050=\frac{c}{4497}$

Simplifying,

  $c=22.485$

Then, the perihelion distance of Neptune from the Sun is given as,

  $a-c=4497-22.485=4474.515$

So, the perihelion distance of Neptune from the Sun is $4474.515$ Gm.

Clearly, the perihelion distance of Pluto from the Sun is less than the perihelion distance of Neptune from the Sun.

This implies that Pluto sometimes is closer to the Sun than Neptune.

This shows that the object with the most eccentric orbit sometimes is closer to the Sun than the planet with the least eccentric orbit.

Conclusion:

It has been shown that the object with the most eccentric orbit sometimes is closer to the Sun than the planet with the least eccentric orbit.