Question

2. The diameter of Saturn’s largest moon Titan is about 5.5 percent greater than the diameter of the planet Mercury. The mass of Mercury is 2.4 times as great as the mass of Titan. Because of these differences in size and mass, the escape velocity for Mercury (4.3 km/s) is greater than the escape velocity for Titan (2.6 km/s).

   

     Atoms and molecules for which 10 times their average speed is greater than the escape velocity will not remain in the atmosphere of a planet or moon. How can it be that Titan, with the lower escape velocity has an atmosphere, while Mercury, with the higher escape velocity has almost no atmosphere?

Answer 1

the average velocity of a gas molecule (Vavg) is given by

therefore , average velocity of a gas varies directly with the square root of temperature.

now since escape velocity has been defined as 10 times the average speed of gas molecule ,

therefore, it is easier for gas molecules to achive escape velocity at higher temperature (in kelvin scale).

Now since mercury recieves a large amount of heat from the sun and its temperature reaches to a level of 700 kelvin during day time , due to which the gas molecules easily achieve an average velocity which when multiplied by a factor of 10 easily exceeds the escape velocity of the planet , this is why the mercury does not have an atmosphere.

on the other hand, the surface temperature is quite very low on the surface of Titan ( about 100 kelvin) and thus the average velocity of gas molecules is quite low even when multiplied by a factor of 10, as compared to the escape velocity of the Titan and thus the Titan has an atmosphere because gases does not escape out of the surface of the Titan.

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