The atmosphere on a planet is possible only if [ where `v_(rms)` is root mean square speed of gas molecules on planet and `v_(e)` is escape speed on its surface ]

Options

A. `v_(rms) = v_(e)`
B. `v_(rms) gt v_(e)`
C. `v_(rms) le v_(e)`
D. `v_(rms) lt v_(e)`

Answer

To determine the condition for the presence of an atmosphere on a planet in terms of the root mean square speed of gas molecules (\(v_{rms}\)) and the escape speed (\(v_e\)), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- **Root Mean Square Speed (\(v_{rms}\))**: This is a measure of the average speed of gas molecules in a gas. It is given by the formula:
\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]
where \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(m\) is the mass of a gas molecule.
- **Escape Speed (\(v_e\))**: This is the minimum speed required for an object to break free from the gravitational attraction of a planet without any further propulsion. It is given by:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where \(G\) is the gravitational constant, \(M\) is the mass of the planet, and \(R\) is the radius of the planet.

2. **Establish the Condition for an Atmosphere**:
- For a planet to retain its atmosphere, the gas molecules must not have enough energy to escape the gravitational pull of the planet. This means that the average speed of the gas molecules (\(v_{rms}\)) must be significantly less than the escape speed (\(v_e\)).
- Mathematically, this can be expressed as:
\[ v_{rms} < v_e \]

3. **Interpret the Condition**:
- If \(v_{rms}\) is greater than or equal to \(v_e\), the gas molecules will have enough kinetic energy to escape the planet's gravitational field, leading to the loss of the atmosphere.
- Therefore, for a stable atmosphere, the condition must be:
\[ v_e \gg v_{rms} \]
- This means that the escape speed must be much larger than the root mean square speed of the gas molecules.

4. **Conclusion**:
- The atmosphere on a planet is possible only if the escape speed is much greater than the root mean square speed of the gas molecules. Thus, the correct statement is:
\[ v_e \gg v_{rms} \]