Free Space Skyhooks
A non-planetary Kevlar skyhook with tip velocity wrt its center of
1/4 earth escape velocity, which is just enough to catch a Venus-Earth
Hohmann and accelerate it to Earth-Mars Hohmann is able to support 1/425 of
its mass at each end, building in a safety factor of two. If the skyhook
masses 21x10^6 Kg., it can support 50,000 Kg (about the mass of Skylab), at
each end.
Considering different lengths (mass ratio is unaffected by
geometry):
Skyhook Rotational Acceleration Area of cable Area of cable
radius period ends middle
100 Km 3.74 min 8 g 28 cm^2 1700 cm^2
1000 Km 37 min 0.8 g 2.8 cm^2 170 cm^2
2000 Km 1.25 hrs 0.4 g 1.4 cm^2 84 cm^2
5000 Km 3 hrs 0.16 g 0.56 cm^2 34 cm^2
10,000 Km 6.25 hrs 0.08 g 0.28 cm^2 17 cm^2
20,000 Km 12.5 hrs 0.04 g 0.14 cm^2 8.4 cm^2
The cable cross section as a function of radius is a perfect
EXP(-r^2) normal curve. Macsyma was able to integrate it symbolically, to
get an expression for the mass ratio. The integral naturally contains the
error function. The taper ratio and the mass ratio go up exponentially as
the square of the tip velocity (and simply exponentially with the
weight/strength ratio).
A Hohmann catch/Hohmann boost removes or adds orbital energy to
the cable, but does not affect its rotation.
The formula for cable cross section:
M v^2 EXP(D/T v^2/2 (1-(r/r[e])^2))
Area(r) = -----------------------------------
T r[e]
r is distance from cable center
r[e] is cable radius (i.e. 1/2 its length)
v is tip velocity wrt. center
D density of cable material
T design tensile strength of cable
M mass to be supported at each end
this, integrated and multiplied by two and by density, divided by M
gives the mass ratio:
let dtv2 = D/T v^2/2
Mass Ratio = 2 SQRT(P dtv2) EXP(dtv2) ERF(sqrt(dtv2))
Hans Moravec
November, 1978