Question

Assume that the Moon and Titan both have the same mass of 10^22 kg and the same radius of 1000km. Titan is about 10x as far away from the Sun as the Moon. Assume for both bodies a reflectivity of zero and assume they have a no greenhouse temperature.

a. Is there a difference in the average no greenhouse temperature of the Moon and Titan? What is it for Titan, and what is it for the Moon?

b. Is there a difference in the escape velocity of a hydrogen atom of mass 10^-27 on the Moon and Titan?

c. Is there a difference in the thermal velocity of a hydrogen atoms on the Moon and Titan?

d. What do you conclude from your calculations? Why does the Moon have no atmosphere but Titan does?

Answer 1

(a) Since both are at different distances from the moon, the radiation energy received from the sun for both of the planets will be different, The energy received by Moon will be more as compared to what is received by the Titan.

The energy emitted by sun, S₀ is,

Now, for whole surface of the sun, the energy will be,

Now, at a distance of r, the energy received would be,

Now, if the planets radius is r_p. The amount of energy absorbed by the planet will be, (this can be equated with the the radiation due to temperature of the planet, because the reflectivity is 0)

Now, for moon the distance is 150 x 10^6 km. and radius of moon is 1000 km, Therefore,

Now for the Titan the distance will be, 150 x 10^7 km, then we can write,

(b) The escape velocity from the surface of the planet is given by,

Since the mass and radius of the planets are same, we can that there is no difference between the escape velocities for both the planets,

The escape velocity will be,

(c) The thermal velocity is given by,

There will be difference in thermal velocities since, the temperature of the planets are different.

Therefore thermal velocity for hydrogen atom in Moon is,

And thermal velocity for hydrogen atom in Titan is,

(d) Since in Titan the thermal velocity of hydrogen is less than the escape velocity, we can say that Titan will be having an atmosphere, whereas since, the thermal velocity of hydrogen atom in moon is more than its escape velocity, therefore moon will not have any atmosphere.