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If the atmosphere on Mars was once much thicker, how was it likely lost? Was it due to interaction with the solar wind, the small size of the planet, both, or something else, and approximately how long did it take to reach its current thickness?

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The loss of the Martian atmosphere can be mostly attributed to its mass. The reason why Earth still has an atmosphere made of lighter elements is because with larger mass comes larger escape velocity, which is the speed at which an atom's kinetic energy overcomes the gravitational potential energy of its planet.

The distribution of speeds of most gasses can be described by the Maxwell–Boltzmann distribution.

Maxwell-Boltzmann

This curve represents the probability of finding a particle with a given speed at a given temperature. Holding temperature constant, the above chart illustrates that lighter molecules will have higher probabilities of being found with higher speeds. $$P(\epsilon) = \frac{2}{\sqrt{\pi}k_{B}T} \left( \frac{\epsilon}{k_{B}T} \right)^{1/2} \exp{-\left(\frac{\epsilon}{k_{B}T}\right)} $$ where $\epsilon$ is the energy of the particle, $\frac{p^{2}}{2m}$ in the absence of a gravitational field (see this lecture for information regarding how to include the gravitational term).

Integrated over time, the lighter gasses tend to exceed the escape velocity more often than their heavier counterparts. This is why the larger planets like Jupiter and Saturn still have atmospheres dominated by hydrogen and helium.

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