Chapter 18, Problem 4P

To determine

The mass of Saturn, Uranus, Neptune, Earth, and Mercury in terms of mass of Jupiter and distinct them into categories and whether the determination is in agreement with the text.

Answer to Problem 4P

The mass of Saturn is $2.99\times {10}^{-1}{M}_{\text{Jup}}$, Uranus is $4.56\times {10}^{-2}{M}_{\text{Jup}}$, Neptune is $5.37\times {10}^{-2}{M}_{\text{Jup}}$, Earth is $3.14\times {10}^{-3}{M}_{\text{Jup}}$, and Mercury is $1.73\times {10}^{-4}{M}_{\text{Jup}}$ which are divided into three categories and the determination is in agreement with the text.

Explanation of Solution

Refer to Table A-10, “Properties of the Planets” to obtain the value of mass of the Earth as $3.14\times {10}^{-3}{M}_{\text{Jup}}$.

Write the expression for mass of Saturn.

    $M_{\text{Sat}}=95.2M_{\text{Ear}}$        (I)

Here, $M_{\text{Sat}}$ is the mass of Saturn and $M_{\text{Ear}}$ is the mass of the Earth.

Write the expression for mass of Uranus.

    $M_{\text{Ura}}=14.5M_{\text{Ear}}$        (II)

Here, $M_{\text{Ura}}$ is the mass of Uranus.

Write the expression for mass of Neptune.

    $M_{\text{Nep}}=17.1M_{\text{Ear}}$        (III)

Here, $M_{\text{Nep}}$ is the mass of Neptune.

Write the expression for mass of Mars.

    $M_{\text{Mar}}=0.055M_{\text{Ear}}$        (IV)

Here, $M_{\text{Sat}}$ is the mass of Mars.

Conclusion:

Substitute $3.14\times {10}^{-3}{M}_{\text{Jup}}$ for $M_{\text{Ear}}$  in equation (I) to find the mass of Saturn.

    $\begin{array}{c}{M}_{\text{Sat}}=95.2\left(3.14\times {10}^{-3}{M}_{\text{Jup}}\right)\\ =2.99\times {10}^{-1}{M}_{\text{Jup}}\end{array}$

Substitute $3.14\times {10}^{-3}{M}_{\text{Jup}}$ for $M_{\text{Ear}}$  in equation (II) to find the mass of Uranus.

    $\begin{array}{c}{M}_{\text{Ura}}=14.5\left(3.14\times {10}^{-3}{M}_{\text{Jup}}\right)\\ =4.56\times {10}^{-2}{M}_{\text{Jup}}\end{array}$

Substitute $3.14\times {10}^{-3}{M}_{\text{Jup}}$ for $M_{\text{Ear}}$  in equation (III) to find the mass of Neptune.

    $\begin{array}{c}{M}_{\text{Nep}}=17.1\left(3.14\times {10}^{-3}{M}_{\text{Jup}}\right)\\ =5.37\times {10}^{-2}{M}_{\text{Jup}}\end{array}$

Substitute $3.14\times {10}^{-3}{M}_{\text{Jup}}$ for $M_{\text{Ear}}$  in equation (IV) to find the mass of Mars.

    $\begin{array}{c}{M}_{\text{Mar}}=0.055\left(3.14\times {10}^{-3}{M}_{\text{Jup}}\right)\\ =1.73\times {10}^{-4}{M}_{\text{Jup}}\end{array}$

The masses can be categorized in three categories on the basis of the exponential power.

1. 1) Jupiter and Saturn
2. 2) Uranus and Neptune
3. 3) Earth and Mars

This determination is in agreement with the text.

Therefore, the mass of Saturn is $2.99\times {10}^{-1}{M}_{\text{Jup}}$, Uranus is $4.56\times {10}^{-2}{M}_{\text{Jup}}$, Neptune is $5.37\times {10}^{-2}{M}_{\text{Jup}}$, Earth is $3.14\times {10}^{-3}{M}_{\text{Jup}}$, and Mercury is $1.73\times {10}^{-4}{M}_{\text{Jup}}$ which are divided into three categories and the determination is in agreement with the text.