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 under which an atmosphere can exist on a planet, we need to compare the root mean square speed of gas molecules (\(v_{rms}\)) with the escape speed from the planet's surface (\(v_e\)).

### Step-by-Step Solution:

1. **Understanding Escape Velocity**:
Escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a celestial body without any additional propulsion. For a planet, this speed depends on its mass and radius.

2. **Understanding Root Mean Square Speed**:
The root mean square speed (\(v_{rms}\)) of gas molecules is a measure of the average speed of particles in a gas. It is given by the formula:
\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]
where \(k\) is the Boltzmann constant, \(T\) is the absolute temperature, and \(m\) is the mass of a gas molecule.

3. **Condition for Atmospheric Retention**:
For an atmosphere to be retained by a planet, the gas molecules must not have enough energy to escape the planet's gravitational pull. This means that the average speed of the gas molecules must be less than the escape speed. Mathematically, this can be expressed as:
\[ v_{rms} < v_e \]

4. **Conclusion**:
Therefore, the atmosphere on a planet is possible only if the root mean square speed of gas molecules is less than the escape speed of the planet. If \(v_{rms}\) is equal to or greater than \(v_e\), the gas molecules will escape into space, and the atmosphere will not be able to sustain itself.

### Final Answer:
The atmosphere on a planet is possible only if:
\[ v_{rms} < v_e \]