Chapter 8, Problem 3P

Look up the masses and radii of Mercury and Jupiter and calculate their escape velocities, using the equation in chapter 3. Does this help you see why the one body has an atmosphere but the other doesn’t? (Note: Be sure to convert kilometers to meters as necessary.)

To determine

The escape velocities of Mercury and Jupiter. And explain why the one body has an atmosphere but the other doesn’t.

Answer to Problem 3P

The escape velocity of Jupiter is $60\text{km}/\text{s}$ and the escape velocity of Mercury is $4.25\text{km}/\text{s}$. The escape velocity of Mercury is much lesser than the escape velocity of Jupiter. Hence Mercury does not have an atmosphere.

Explanation of Solution

The mass of Jupiter is $M_{Jup}=1.90\times {10}^{27}\text{ kg}$ and its radius is $R_{Jup}=7.1492\times {10}^7\text{m}$. The mass of Mercury is $M_{Mer}=3.30\times {10}^{23}\text{ kg}$ and its radius is $R_{Mer}=2.44\times {10}^6\text{ m}$.

Write the formula to find the escape velocity

    $v_{esc}=\sqrt{\frac{2GM}{R}}$        (I)

Here, $v_{esc}$ is the escape velocity, $G$ is the Universal gravitational constant, $M$ is the mass of the planet and $R$ is the radius of the planet.

Substitute $6.67\times {10}^{-11} {\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$ for $G$, $1.90\times {10}^{27}\text{ kg}$ for $M_{Jup}$ and $7.1492\times {10}^7\text{m}$ for $R_{Jup}$ in equation (I) to find the value of $v_{esc}\left(Jup\right)$

$\begin{array}{c}{v}_{esc}\left(Jup\right)=\sqrt{\frac{2\left(6.67\times {10}^{-11} {\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}\right)\left(1.90\times {10}^{27}\text{ kg}\right)}{7.1492\times {10}^7\text{m}}}\\ =\sqrt{3.5453\times {10}^9{\text{ m}}^2{s}^{-2}}\\ =5.95\times {10}^4\text{m}/\text{s}\\ \approx 6\times {10}^4\text{m}/\text{s}\end{array}$

The escape velocity of Jupiter is $60\text{km}/\text{s}$.

Substitute $6.67\times {10}^{-11} {\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$ for $G$, $3.30\times {10}^{23}\text{ kg}$ for $M_{Mer}$ and $2.44\times {10}^6\text{ m}$ for $R_{Mer}$ in equation (I) to find the value of $v_{esc}\left(Mer\right)$

$\begin{array}{c}{v}_{esc}\left(Mer\right)=\sqrt{\frac{2\left(6.67\times {10}^{-11} {\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}\right)\left(3.30\times {10}^{23}\text{ kg}\right)}{2.44\times {10}^6\text{ m}}}\\ =\sqrt{1.80\times {10}^7{\text{ m}}^2{s}^{-2}}\\ =4.25\times {10}^3\text{m}/\text{s}\end{array}$

The escape velocity of Mercury is $4.25\text{km}/\text{s}$.

Mercury is closer to the Sun and therefore, its gas molecule in its surface will have high temperature and the molecules will have higher velocity. Jupiter is far away from the Sun and hence, it is cold and has gas molecules with lower velocity. Also the escape velocity of Mercury is must lesser than Jupiter, the gases will escape from Mercury easily. Therefore, Mercury does not have an atmosphere.

Conclusion:

Thus, the escape velocity of Jupiter is $60\text{km}/\text{s}$ and the escape velocity of Mercury is $4.25\text{km}/\text{s}$. The escape velocity of Mercury is much lesser than the escape velocity of Jupiter. Hence Mercury does not have an atmosphere.