Calculating Jupiter's Dry Adiabatic Lapse Rate

The dry adiabatic lapse rate ($\Gamma_d$) is the rate at which the temperature of a parcel of dry air decreases as it rises through the atmosphere under adiabatic conditions. It can be calculated using the acceleration due to gravity ($g$) and the specific heat capacity at constant pressure ($c_p$).

Formula for Dry Adiabatic Lapse Rate

The formula used is:

$ \Gamma_d = \frac{g}{c_p} $

Where:

• $g$ = acceleration due to gravity
• $c_p$ = specific heat capacity at constant pressure

This formula yields the lapse rate in Kelvin per meter (K/m).

Applying the Formula to Jupiter

We are given the following information for Jupiter:

• Composition: $\text{H}_2$
• Specific Heat ($c_p$): $13000 \ \text{J/kg/K}$
• Acceleration due to gravity ($g$): $26 \ \text{m/s}^2$

Substitute these values into the formula:

$ \Gamma_d = \frac{26 \ \text{m/s}^2}{13000 \ \text{J/kg/K}} $

To perform the calculation, note the unit relationship: $1 \ \text{J} = 1 \ \text{kg} \cdot \text{m}^2/\text{s}^2$. Therefore, the units of $c_p$ are equivalent to $\text{m}^2/(\text{s}^2 \cdot \text{K})$.

$ \Gamma_d = \frac{26 \ \text{m/s}^2}{13000 \ \text{m}^2/(\text{s}^2 \cdot \text{K})} $

$ \Gamma_d = \frac{26}{13000} \ \text{K/m} $

$ \Gamma_d = 0.002 \ \text{K/m} $

Converting Units to °C/km

The question asks for the answer in degrees Celsius per kilometer (°C/km). We need to convert K/m to °C/km.

Unit Conversion Factors:

• Temperature difference: $1 \ \text{K} = 1 \ \text{°C}$
• Distance: $1 \ \text{km} = 1000 \ \text{m}$

Apply these conversion factors:

$ \Gamma_d = 0.002 \ \frac{\text{K}}{\text{m}} \times \frac{1 \ \text{°C}}{1 \ \text{K}} \times \frac{1000 \ \text{m}}{1 \ \text{km}} $

$ \Gamma_d = 0.002 \times 1000 \ \text{°C/km} $

$ \Gamma_d = 2 \ \text{°C/km} $

Conclusion

The dry adiabatic lapse rate in Jupiter's atmosphere, based on the given values, is 2 °C/km.