Grand Lunar Canals

In one of his novels, H. G. Wells describes an encounter of amateur earthling astronauts with a lunar civilization living in very deep caverns beneath the surface of the Moon. The caverns are connected to the surface by long channels filled with air. The channel is dug between points A and B on the surface of the Moon so that the angle AOB = 90^o (see Figure 1.1). Assume that the air pressure in the middle of a channel is P₀ = 1 atm. Estimate the air pressure in the channel near the surface of the Moon. The radius of the Moon a ~ 1750 kg. The acceleration due to gravity on the surface of the Moon g_m ~ g/6, where g is the acceleration due to gravity on the surface of the Earth.

Figure 1.1

SOLUTION

Consider the atmosphere to be isothermal inside the channel. The pressure depends only on the distance from the center of the moon r (see Figure 1.2), we have

(1)

So

(2)

Figure 1.2

The acceleration of gravity

(3)

where M is the mass of the Moon and ρ₀ is the average density of the Moon (which we consider to be uniform). Therefore,

(4)

where we have set g_m(a) = g/6. Now, from (2) and (4), we have

where P₁ is the pressure on the surface of the Moon.

(5)

(6)

which implies that it is not impossible to have such cavities inside the Moon filled with gas (to say nothing of the presence of lunars).