# Design a Space Elvator

TV series Foundation shows a space elevator 911 event. How should a space elevator be designed and what will its collapse look alike?

Consider the space elevator to be a link with linear mass \hat{m}(l) and a total length L.

To have a straight link, everything below the Geosynchronous orbit, or GSO would be rotating too slow to support its own mass. To balance the effect one can choose from 2 sources of support: base reaction from the ground, or tension from above GSO.

The tension of link at height x

T(x) = - F_{r} - \int_0^x{\cfrac{GM\hat{m}}{(R_{e}+l)^2}}dl + \int_0^L{\hat{m}\omega_{e}^2(R_{e}+l)}dl

where R_e is earth radius, \omega_e is earth rotating angular speed, F_r is base reaction force.

The boundry condition

T(L) = 0

Intuitively, a reasonable model will have the ground part in compression, but not that much of the entire gravity of things below synchronous orbit. The link at synchronous orbit will be in tension, so somewhere below there would be a zero force point.

The major payload should be deployed right at GSO, such that it doesn't pose any extra load into the link. Another benefit of the design is that the major payload can always survive any 911 attacks not targeted at it directly by simply cutting itself off the link. In such design, cut the link anywhere will have the lower part fall to Earth and upper part go into space. The TV series made a reasonable assumption.

Though seemingly unavoidable, it is by design inevitable for a (common civillian) building to fail a 911 attack due to economic concern, even a space elevator.

Axial force diagram should look like moment diagram of a cantilever beam under distributed load, and so should a reasonable section design be alike.