On the Stability of the Track of the Space Elevator

Paper number

IAC-06-D4.2.05

Author

Prof. Hans Troger, UT Vienna, Austria

Coauthor

Prof. Alois Steindl, UT Vienna, Austria

Year

2006

Abstract

Since 1991, the time of the discovery of ``carbon nanotubes'', it is
feasible to form macroscopic ropes with a theoretical strength of 100 times
higher than steel, but with only one-sixth of the weight of steel. 
Such a material would allow to build the track of a
space elevator rotating with geostationary angular velocity around the
Earth.

 This idea makes use of the
 fact that a massive string, moving on a circular orbit
 around the Earth, under the action of gravitational and centrifugal
 forces, has a radial relative equilibrium position, in which the
 string is  under tension. For minimum weight design, the shape of the
 string must be tapered being thickest at the
 geosynchronous radius and thinnest at its ends, where one end touches the
 Earth  and the other one extends far beyond the geostationary hight. 

So far for the strength of the macroscopic cable almost the full
theoretical strength of ``carbon nanotubes'' is assumed. However both
theoretical (fracture mechanics applied to defective ``carbon
nanotubes'') and experimental investigations show that for practical
applications of ``carbon nanotubes'' the available strength will be
much smaller and hence the  taper ratio of the string,
that is, the ratio between the cross section at the geosynchronous
orbit to the cross sectional area of the cable at Earth level, which for the
theoretical strength is about 1.5, must be calculated for the defective
case. Depending on the practically occuring reduction of strength a taper
ratio of about 50 seems more realistic.  

The stability of the radial configuration of the
long unperturbed cable
in the  spherical symmetric Newtonian gravitational field is not
guaranteed at all, because  it is well known, that a
dumbell satellite, which is a system of two point masses connected by
a massless rigid rod, has a stable radial relative equilibrium
position only if the distance does not exceed a critical value, which
is  of the order of the radius of the orbit. 

The aim of this paper is to investigate 
for defective ''carbon nanotubes'' whether and under which conditions a
continuous  massive tapered string has a  stable radial
relative equilibrium. The proper theory to answer this question is 
the Reduced Energy Momentum Method. We also give the necessary mass of
a satellite at geostationary height to stabilize the radial
configuration, in case the equilibrium is unstable, as it is the case
for tethers with the large, theoretical strength.

Abstract document

IAC-06-D4.2.05.pdf

Manuscript document

IAC-06-D4.2.05.pdf (🔒 authorized access only).

To get the manuscript, please contact IAF Secretariat.