What are two factors, which determine why some members of solar system have atmosphere and others don’t?

The existence or nonexistence of atmosphere on any members of solar system depends on two factors. One is the temperature of the planet. The root mean square velocity of gaseous molecules in any environment is, \[c_\text{rms}=\sqrt{\frac{3RT}{M_m}}\]

If the temperature is high, the rms velocity will be more and vice versa.

The next factor is the escape velocity of the planet. If the planet has less escape velocity, then the rms velocity will easily exceed the escape velocity. So, the atmosphere will escape even at low temperature. If the planet has higher escape velocity, then the rms velocity can exceed the escape velocity only at high temperature. This means that the atmosphere will escape only when the planet is extremely hot.

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