How high can we build in a space habitat?

Living in a space habitat is quiet different from living on the surface of a planet. On Earth our head is oriented outwards, i.e. our head is pointed away from the centre of the Earth. But in a space habitat our head is oriented inwards, pointing towards the centre. Therefore we cannot build higher than the distance between the wall of the habitat and its centre, but this is not the only, or even most important, restriction for the height of building in space habitats.

In a space habitat gravity is replaced by the centrifugal force, which is generated by rotating the habit around its axis. This (virtual) force is given by the following equation: Fcent = m(w^2)r (I have used ‘w‘ instead of the small omega, since I don’t know how to type Greek letters in WordPress), with m the mass of the object on which the force acts, w the angular velocity of the habitat and r the distance between the object and the axis. On Earth gravity is given by the equation Fgrav = mg, with g the so-called gravitational acceleration, which is (on average) g = 9.81 m/s^2.

If we want that the centrifugal force acting on the inner wall of the space habitat (which we will refer to as “street level”), we have to solve Fcent = Fgrav. Or

m(w^2)r = mg

We see that we can cancel m on both sides of the equation, and write

(w^2)r = g = 9.81

Since r is in fact nothing else than the radius of our space habitat (which would be in case of an O’Neill cylinder be 3,000 meter), and hence a design parameter, we can only play with the habitat’s rotational speed in order to fix the strength of the centrifugal force.

A consequence of the last equation, the larger the radius of a space habitat how smaller its angular velocity will be. But also that by a given w, the closer you are to the axis of rotation, the smaller the centrifugal force acting on you will be. At the axis of rotation the centrifugal force is zero.

The whole point of substituting gravity with centrifugation is to counteract the health effects of low or zero-gravity. Therefore the height of a building will be restricted by the minimum accepted level of gravity. Recall that in a space habitat we are building towards the axis. So the question of height becomes a question of what is the accepted minimum gravity?

Gerard O’Neill have suggested that 70 % of Earth’s gravity, or 0.7g is an acceptable minimum level of gravity. Our equations show that for a given space habitat the strength of the centrifugal force is linearly dependent of the distance to the axis of rotation. Hence 0.7g is present at 0.7r, or at 0.3r if we are counting from street level. Assuming r = 3,000m than we can erect building up to 900m (counted from street level).

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