Hollow Earth

Today I heard about an hypothesis from the 18th century called Hollow Earth, which supposes the Earth to be hollow. I figured such a space vessel would be possible to construct, with air on the inside, and a hard shell around it. Here is what I discovered...

Constants:

G is the gravitational constant.
P is Earth-normal Atmospheric pressure.
ρ_shell is the mass density of the shell, assume solid iron.
ρ_air is the mass density of the inner, Earth-like atmosphere.

Variables:

r is the radius of the world.
s is the thickness of the iron shell.

The Air pressure is balanced by weight of shell is:

(1) P = (m * g) / 2

Why did I divide by two, you ask? Well, the force of gravity falls off, linear, from g gravity at the surface, to 0 on the inside of the shell, so the average weight is half g.

Mass of a square meter slice of the shell is:

(2) m = ρ_shell * V_slice

Assume that one meter is small compared to the curvature of the shell.
The volume of a square meter slice of the shell is:

(3) V_slice = (1 meter)² * (s)

Surface gravity on Hollow World is:

(4) g = (G * M) / r²

The total mass of the planet, shell plus atmosphere, is:

(5) M = ρ_shell * V_shell + ρ_air * V_air

The volume of the metal shell is a difference of two spheres:

(6) V_shell = V_world - V_air

The volume of the world is:

(7) V_world = (4π/3) * r³

The volume of the atmosphere is:

(8) V_air = (4π/3) * (r - s)³

Putting it all together, we have an equation with only two variables:

(9) P = (ρ_shell * s * ((G * ((4π/3) * ((ρ_shell * (r³ - (r - s)³)) + (ρ_air * (r - s)³)))) / r²)) / 2

Graphing this last equation shows us the possible configurations of Hollow World.

654000=1.1006*10^-6*y*x^-2*(7874*x^3+(x-y)^3*-7872)
The horizontal axis is the r, the radius of the world, in meters.

The vertical access is s, the thickness of the iron shell, also in meters.

Notice that upper left is desaturated, because it's meaningless for the thickness to be larger than the radius. The function quickly drops down to around 5 km thick shell around a world with radius between 100 km (that's no moon) and 10,000 km (super-Earth).

For an object as large as the sun, the thickness would be 0.684 km. It keeps getting thinner because the surface gravity gets stronger, pulling down more on the shell, while the inside pressure stays the same. As a result, less and less thickness is required to do the job.

It's mostly the air inside that's responsible for the increased gravity, too!

Due to the Shell Theorem, people on the inside would experience free fall, no matter the size. Cities could be designed like rafts, lashed together.

Just don't let your world spring a leak. Too much decrease in air pressure would put too much load on the shell. Enough inward force could cause the metal to buckle, and the whole world to collapse in on its self.

You could set up a fusion reactor to act as a miniature sun in the center. It could use exhaust to maintain it's position in the center. At night, it could power down, and condense water on its cooling surfaces to acquire hydrogen for the next day's sunshine.

Sounds like a fun science fiction world to explore!