Chapter 12, Problem 21Q

(a)

To determine

The increase in temperature during a descent in the atmosphere of Jupiter from an altitude at 100-millibar to an altitude which is 100 km below that level. A diagram which shows the relation between altitude and temperature in Jupiter’s atmosphere is given below.

Answer to Problem 21Q

Solution:

$-1.3°\text{C/}\text{km}$

Explanation of Solution

Given data:

Temperature at 100-millibar level (0 on the scale of altitude) is $160°\text{C}$

Temperature at 100 km below the 100-millibar level ($-100$ on the scale of altitude) is $30°\text{C}$

Formula used:

Write the expression for lapse rate.

$\Delta T=\left(T_2-T_1\right)/d$

Here, $\Delta T$ is the change in temperature, $T_2$ is the temperature at final location (100 km below the 100-millibar level), $T_1$ is the temperature at initial location (100-millibar level) and $d$ is the distance of separation between the two locations.

Explanation:

In the diagram, the zero altitude level in the atmosphere represents the point where the pressure is 100 milibar or one-tenth the atmospheric pressure on Earth.

The rate at which the temperature changes as we move higher or lower in terms of altitude is called lapse rate. With the help of lapse rate, we can determine the relation between the change in temperature and the change in altitude for a given case, usually a planet. Refer to the expression for lapse rate.

$\Delta T=\frac{\left(T_2-T_1\right)}{d}$

Substitute $30°\text{C}$ for $T_2$, $160°\text{C}$ for $T_1$ and $100\text{ km}$ for $d$.

$\begin{array}{c}\Delta T=\frac{\left(30°\text{C}-\text{160}°\text{C}\right)}{\text{100 km}}\\ \text{=}-\text{130}°\text{C/100}\text{km}\\ \text{=}-\text{1}\text{.3}°\text{C/km}\end{array}$

Conclusion:

Hence, the temperature increases at the rate of $1.3°\text{C/km}$ in Jupiter’s atmosphere.

(b)

To determine

The increase in temperature during a descent from an altitude at 100-millibar in Saturn’s atmosphere to an altitude which is 100 km below that level from the given diagram. It is given in the diagram that the zero altitude in the atmosphere is chosen as the point where the pressure is 100 millibar or one-tenth of Earth’s atmospheric pressure.

Answer to Problem 21Q

Solution:

$-0.6°\text{C/km}$

Explanation of Solution

Given data:

Temperature at 100-millibar level (0 on the scale of altitude) is $180°\text{C}$

Temperature at 100 km below the 100-millibar level ($-100$ on the scale of altitude) is $120°\text{C}$

Formula used:

Write the expression for lapse rate.

$\Delta T=\frac{\left(T_2-T_1\right)}{d}$

Here, $\Delta T$ is the change in temperature, $T_2$ is the temperature at final location (temperature at 100 km below the 100-millibar level), $T_1$ is the temperature at initial location (temperature at 100-millibar level) and $d$ is the distance between the two locations.

Explanation:

The rate at which the temperature changes as we move higher or lower in terms of altitude is called lapse rate. With the help of lapse rate, we can determine the relation between the change in altitude and the change in temperature for a given case, usually a planet. Refer to the expression for lapse rate.

$\Delta T=\frac{\left(T_2-T_1\right)}{d}$

Substitute $120°\text{C}$ for $T_2$, $180°\text{C}$ for $T_1$ and $100\text{km}$ for $d$.

$\begin{array}{c}\Delta T=\frac{\left(120°\text{C}-\text{180}°\text{C}\right)}{100\text{ km}}\\ \text{=}-\text{60}°\text{C/100}\text{km}\\ \text{=}-\text{0}\text{.6}°\text{C/km}\end{array}$

Conclusion:

Hence, the temperature increases at the rate of $0.6°\text{C/km}$ in Saturn’s atmosphere.

(c)

To determine

The planet, out of the three – Earth, Jupiter and Saturn, in whose atmosphere the temperature increases most rapidly with descreasing altitude. It is given that the air temperature increases by $6.4°\text{C}$ for each kilometer covered during a descent.

Answer to Problem 21Q

Solution:

At the Earth’s atmosphere, the temperature decreases rapidly as its lapse rate is $6.4°\text{C/km}$, while in the atmosphere of Jupiter and Saturn, the values of lapse rate are $1.3°\text{C/km and 0}\text{.6}°\text{C/km}$, respectively, which have been evaluated in part (a) and part (b), and are lower than that on Earth.

Explanation of Solution

Given data:

Lapse rate in Earth’s atmosphere is $6.4°\text{C/km}\text{.}$

Explanation:

The rate at which the temperature changes as we move higher or lower in terms of altitude is called lapse rate. With the help of lapse rate, we can determine the relation between the change in altitude and the change in temperature for a given case, usually a planet.

Conclusion:

Hence, out of the atmospheres of Earth, Jupiter and Saturn, Earth’s atmosphere shows the most rapid increase in temperature with decrease in attitude.