by Michael Brunnbauer, 2015-04-04
Kim Stanley Robinsons novel Red Mars tells of a space elevator on Mars that loses its counterweight and starts falling - wrapping itself eastward around Mars within 4 hours and once again in another hour.
This made me suspicious. One turn in 4 hours and another in one hour? In the direction of rotation? Impossible! And this book is described as representative hard science fiction on Wikipedia! ;-)
After some unsuccessful reasoning, I decided to write a simulation. I am no expert on the subject matter and the simulation may be too crude or inappropriate but it confirms two things: The falling space elevator cable wraps itself around Mars faster than I expected and the second turn happens much faster than the first one (If you allow the second turn with an excessive safety margin - otherwise the cable breaks and the upper end flies away).
In my simulation, it takes ca. 14 hours for the cable with excessive safety margin to wrap itself around Mars completely.
Is my simulation too crude and 5 hours for two wraparounds is actually correct while parts of the cable do not break and fly away? Feedback is welcome!
The elevator in Red Mars has a height of 37.000 km, a mass of 6 billion tons and a counterweight of 7.5 billion tons. I simulated it with 231 point masses of 26 million tons connected with weightless springs. Atmosphere is neglected.
The springs have a spring constant of 26000000 N/m. At length 100000m, they exert no force. If they are compressed n meters, they repel with 26000000 * n Newton and if they are extented n meters, they attract with 26000000 * n Newton. The safety margin is 100%. If a spring has to bear two times the maximum force during normal operation, it breaks.
The calculated counterweight is 8.29 billion tons - a bit more than in the book due to the low number of elevator point masses (231). Simulating it with more parts takes too long.
If an elevator part collides with Mars, it stays where it is but rotates with it. The connection to the rest of the elevator also stays.
The simulation uses a fourth order Runge Kutta method to minimize error. This paper helped much (But the correct form of equation 10 is knri+1 = vi + kn-1vi+1 * h/2 for n = 2,3 and knri+1 = vi + kn-1vi+1 * h for n = 4).
Errors still show: If the elevator is not connected to Mars with a spring, it tends to fly away or crash after a half turn. Using a spring seems to delay disaster due to calculation error to much later.
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The Wikipedia Article about (earth based) Space elevator safety mentions "upper sections breaking off and being flung away" in the case of counterweight loss. The reference is a similar simulation by Blaise Gassend.