Chapter 14, Problem 53Q

(a)

To determine

The comparison between the brightness of sunlight on Earth and the brightness of sunlight on Pluto at perihelion. The distance of Pluto from the Sun at perihelion is $29.649\text{ au}$.

Answer to Problem 53Q

Solution:

The ratio of Sun’s brightness on Pluto at perihelion and the brightness on Earth is $1.14\times {10}^{-3}$.

Explanation of Solution

Given data:

The distance between Pluto and Sun at perihelion is $29.649\text{ au}$.

Formula used:

According to the Inverse square law, the brightness of sunlight is inversely proportional to the square of the distance between the sun and the object.

$\text{brightness}\propto \frac{1}{{\left(\text{square of distance between the sun and the planet}\right)}^2}$

Explanation:

The distance between the Sun and the Earth is about $\text{1 au}$. The comparison is made with the brightness of the Sun on the Earth.

Brightness of the Sun on the Earth is:

${\text{brightness}}_1\propto \frac{1}{{\left(\text{square of distance between the sun and the earth}\right)}^2}$

Similarly,

Brightness of the Sun on the Pluto at perihelion is:

${\text{brightness}}_2\propto \frac{1}{{\left(\text{square of distance between the sun and Pluto at perihelion}\right)}^2}$

Consider the brightness of the Sun on the Pluto at perihelion compared to the brightness on Earth be $b_1$.

Combining these two above expressions to get,

$\begin{array}{c}{b}_1=\frac{{\left(1\text{ au}\right)}^2}{{\left(29.649\text{ au}\right)}^2}\\ =1.14\times {10}^{-3}\text{ bright}\end{array}$

Conclusion:

The brightness of the Sun on Pluto at perihelion is $1.14\times {10}^{-3}$ times as compared to that on the Earth.

(b)

To determine

The comparison between the brightness of sunlight on Earth and the brightness of sunlight on Pluto at aphelion. The distance of Pluto from the Sun at aphelion is $\text{49}\text{.425 au}$.

Answer to Problem 53Q

Solution:

The ratio of Sun’s brightness on Pluto at aphelion and the brightness on Earth is $4.09\times {10}^{-4}$.

Explanation of Solution

Given data:

The distance of Pluto and Sun at aphelion is $\text{49}\text{.425 au}$.

Formula used:

According to Inverse square law, the brightness of sunlight is inversely proportional to the square of the distance between the sun and the object.

$\text{brightness}\propto \frac{1}{{\left(\text{square of distance between the sun and the planet}\right)}^2}$

Explanation:

The distance between the Sun and the Earth is near about $\text{1 au}$. The comparison is made with the brightness of the Sun on the Earth.

Brightness of the Sun on the Earth is:

${\text{brightness}}_1\propto \frac{1}{{\left(\text{square of distance between the sun and the earth}\right)}^2}$

Similarly,

Brightness of the Sun on the Pluto at aphelion is:

${\text{brightness}}_2\propto \frac{1}{{\left(\text{square of distance between the sun and Pluto at aphelion}\right)}^2}$

Consider the brightness of the Sun on the Pluto at aphelion compared to the brightness on Earth be $b_2$.

Combinine these two above expressions to get,

$\begin{array}{c}{b}_2=\frac{{\left(1\text{ au}\right)}^2}{{\left(\text{49}\text{.425 au}\right)}^2}\\ =4.09\times {10}^{-4}\text{ bright}\end{array}$

Conclusion:

The brightness of the Sun on Pluto at aphelion is $4.09\times {10}^{-4}$ times as compared to that on the Earth.

(c)

To determine

The comparison between the brightness of sunlight on Pluto at perihelion and the brightness of sunlight on Pluto at aphelion.

Answer to Problem 53Q

Solution:

The ratio of brightness of the Sun on Pluto at perihelion and that at aphelion of Pluto is $2.787$.

Explanation of Solution

Given data:

The distance of Pluto and Sun at aphelion is $\text{49}\text{.425 au}$ and the distance of Pluto and Sun at perihelion is $29.649\text{ au}$.

Formula used:

According to Inverse square law, the brightness of sunlight is inversely proportional to the square of the distance between the sun and the object.

$\text{brightness}\propto \frac{1}{{\left(\text{square of distance between the sun and the planet}\right)}^2}$

Explanation:

By referring the part (a) and (b),

Brightness of the sunlight at perihelion as compared to the brightness on Earth is:

$\begin{array}{c}{b}_1=\frac{{\left(1\text{ au}\right)}^2}{{\left(29.649\text{ au}\right)}^2}\\ =1.14\times {10}^{-3}\text{ bright}\end{array}$

Brightness of the sunlight at aphelion as compared to the brightness on Earth is:

$\begin{array}{c}{b}_2=\frac{{\left(1\text{ au}\right)}^2}{{\left(\text{49}\text{.425 au}\right)}^2}\\ =4.09\times {10}^{-4}\text{ bright}\end{array}$

The brightness of sunlight at perihelion compared to the brightness at aphelion is:

$b=\frac{b_1}{b_2}$

Substitute $1.14\times {10}^{-3}$ for $b_1$ and $4.09\times {10}^{-4}$ for $b_2$.

$\begin{array}{c}b=\frac{1.14\times {10}^{-3}}{4.09\times {10}^{-4}}\\ =2.787\text{ bright}\end{array}$

Conclusion:

The brightness of the Sun on Pluto at perihelion is $2.787$ times as compared to that on Pluto at aphelion.