Hans.Moravec@ROVER.RI.CMU.EDU (12/22/85)
Skyhooks again, eh? The following two old but barely published papers
may be of interest. The second one has a value for the strength of Kevlar
buried in it, the first one talks about delta-v from cables rotating
in free space.
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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 at ends 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
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NON-SYNCHRONOUS ORBITAL SKYHOOKS FOR THE MOON AND MARS
WITH CONVENTIONAL MATERIALS
Abstract. A satellite in low circular orbit has two huge tapered
cables extending outwards and rotating in the orbital plane, touching the
planet each rotation. The tip velocity cancels the orbital velocity at the
contacts, as in a rolling wheel. It can gently lift loads from the surface
and accelerate them to escape velocity, and capture and lower speeding
masses. Taper is minimized when the satellite's radius is one third the
planet's, and for Mars and the moon is reasonable with existing materials
such as fiberglass and Kevlar.
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The idea of a planet to orbit transportation system involving an
enormous tapered cable extending from a synchronous satellite to the ground
has been in the literature for almost two decades (1, 2, 3). It has hitherto
been considered applicable only in the distant future, when materials
stronger than any now available come into existence.
This report points out that the combination of a new material,
Kevlar (4) and a new, less expensive, satellite skyhook configuration (5, 6)
makes skyhook transportation feasible now on bodies as large as Mars. On the
moon, in particular, a Kevlar skyhook has enormous advantages over rockets
for the supply and crew rotation missions envisioned for space
industrialization efforts (7).
A synchronous skyhook is made by lowering a cable from a synchronous
satellite to the surface, balanced by an even longer cable extending
outwards from synchronous orbit. Anchored to the ground and put into
tension by a ballast at its far end, it would be a cosmic elevator cable,
able to deliver mass to high orbit with extreme efficiency, also providing a
means for extracting the rotational energy of the planet. Such a structure
cannot reasonably be built on Earth given existing structural materials. It
would be possible if a cable with 10 times the strength/weight ratio of
steel, or 1/8 the theoretical strength/weight ratio of crystalline graphite
could be fabricated. A graphite cable with a density of 2.2 gram/cm^3 and a
tensile strength of 2.1x10^11 dyne/cm^2 could be fashioned into a
synchronous terrestrial skyhook which had only 100 times ground level cross
section at synchronous orbital height. At any one time it could support one
powered elevator massing 1/6000 of the cable mass (6).
Mars has a much shallower gravity well, and a synchronous skyhook
for it is almost reasonable with conventional materials. Kevlar is a new
superstrong synthetic from the DuPont Co. With a density of 1.44 gram/cm^3
and a tensile strength of 2.76x10^10 dyne/cm^2 it has about 5 times the
strength/weight of steel. Stressed to half this, to build in a safety
factor of two, Kevlar can be used to construct a synchronous martian skyhook
with a taper of 16,000:1, able to support 10^-6 of its own weight at a time.
The numbers for the moon, which has little gravity, but rotates very slowly,
are 17.5:1 and 1/13,000.
In very high orbits the forces on the cable must be integrated over
long distances, resulting in large tapers. For very low orbits, the
satellite must spin rapidly to keep the contact points stationary, and the
quadratic dependence of centrifugal force on rate of spin results in a large
taper in the limit. The taper is minimized between these extremes, when the
radius of the skyhook is about 1/3 the radius of the planet.
An optimum size skyhook of this kind touches down six times per
orbit. It is much smaller than the synchronous variety for the earth, moon
and Mars, but its length is still enormous by conventional standards.
Because of its scale, its motion near the ground during a touchdown is
purely vertical. It appears to descend with a constant upward acceleration,
coming to a gentle momentary stop, then ascending again. This acceleration
is 1.4 gravities on Earth, 0.28 g on the moon and 0.5 g on Mars.
A load attached to the bottom end of such a skyhook during a
touchdown will be accelerated to a maximum of 1.6 times escape velocity at
the highest point of the cable end's trajectory. Launching a mass in this
manner extracts rotational and orbital energy from the skyhook, and lowers
the orbit. Conversely, a high velocity craft which rendezvous with and
attaches itself to the upper end of the cable, and is then decelerated and
lowered to the ground, injects a similar amount of energy. Simultaneous
docking of equal masses at both ends of a skyhook would leave the orbit
essentially unchanged. The most plausible way to operate a device like this
may be to have the cable ends merely approach the surface at a safe
distance. A small rocket could be used to match the relatively tiny
velocity and position differences between the cable tip and the ground. It
would then be possible to borrow and deposit small amounts of orbital energy
without risking collisions of the cable and surface.
TABLE I. Parameters for Optimally Sized Skyhooks
Orbital Liftoff Fiberglass Kevlar
Body Period (hr.) Accel (g) Taper Mass Taper Mass
Mercury 2.37 0.57 2200 23000 49 350
Venus 2.37 1.39 1.2x10^20 3.0x10^21 1.3x10^10 2.3x10^11
Earth 2.16 1.40 7.2x10^21 1.9x10^23 1.0x10^11 1.9x10^12
Moon 2.78 0.28 13 72 3.6 13
Mars 2.62 0.49 17,000 200,000 136 1100
Ganymede 3.41 0.26 35 240 6.0 28
Titan 3.39 0.26 29 190 5.4 24
Table I lists parameters for optimum size fiberglass and Kevlar
skyhooks for some solar system bodies. Fiberglass is assumed to have a
density of 2.5 gram/cm^3 and a tensile strength of 2.41x10^10 dyne/cm^2.
Kevlar has a density of 1.44 gram/cm^3 and a tensile strength of 2.76x10^10
dyne/cm^2. Orbital period is how long it takes the skyhook to make a full
circuit of the body. The liftoff acceleration is the vertical acceleration
experienced by a skyhook payload near the ground, not including the surface
gravity of the planet. It gives an indication of how long the touchdown
lasts. Taper is the ratio in cross sectional area between the center of the
skyhook, where it is thickest, and the tips, where it is thinnest. The Mass
columns give the ratio between the mass of the skyhook and the largest
payload that it can support at one time at each end. Thus a lunar Kevlar
skyhook can lift 1/13 of its own mass. The numbers assume the skyhooks are
stressed to at most half the tensile strength of the material of which they
are made, thus incorporating a safety factor of two.
Evidently Earth and Venus are too large for Kevlar skyhooks. Kevlar
is strong enough for Mars, Mercury and all the moons of the solar system.
Some current plans for space industrialization call for transport of
large quantities of equipment and people to and from the moon. The proposed
linear accelerator mass driver (7) is ideal for launching ore from the moon.
It provides no way of bringing payloads down to the surface, and with its
1000 g accelerations and small mass unit is unsuitable for launching bulkier
and more delicate loads.
A Kevlar lunar skyhook is able to lift and deposit 1/13 of its own
mass every 20 minutes, and subjects payloads to a maximum 0.45 g of
acceleration. It would seem to be a desirable alternative to expensively
fuelled rockets for routine supply and crew rotation missions to the moon's
surface.
The tapers for non-synchronous skyhooks used in this report
were obtained by integrating the forces on the cable between ground
level and satellite center, at the instant of a touchdown. This is
when the stress is at its highest (8).
Define
r[p] the radius of the planet
m[p] the mass of the planet
w[p] the rotation rate of the planet (in radians per unit time)
r[o] the radius of the orbit
w[o] the orbital rate of the satellite's mass center
w[s] the rotation rate of the satellite
D the density of the cable material
T the tensile strength of the cable material
A(r) cable cross section at distance r from the planet center
G the universal gravitational constant
to make contact point stationary,
r[o] W[o] - r[p] W[p]
W[s] = ---------------------
r[o] - r[p]
and for a circular orbit
W[o] = SQRT(G m[p] / r[o]^3)
this last substitution is only an approximation, since the extended
satellite does not orbit and rotate exactly like a point at its mass center.
The stresses in the cables are caused by their weight in the
planet's gravitational field and the accelerations due to the orbital
motion and spin of satellite. They are maximum in the downward hanging
cable. Both cables must be built to take this stress and the satellite is
thus symmetric about its center. If the cables are constructed so as to
make the tension per unit area constant, the cross section of the downward
hanging cable at distance r from the planet center is given by
A(r) = A(r[p]) EXP{ D/T (r-r[p])
(Gm[p]/(r r[p]) - r[o]W[o]^2 + (r[o]-(r+r[p])/2) W[s]^2) }
The mass ratios were found by numerically integrating this
expression over r. Some confidence in the general stability of skyhooks of
this kind has been obtained by observing computer simulations of optimum
size terrestrial graphite versions (6). The only serious problems revealed
were caused by launches not complemented by captures. These lowered the
satellite's orbit and caused collisions with the ground.
Hans P. Moravec, 1977
References and Notes
1. Y. Artsutanov, Komsomolskaya Pravda, July 31, 1960
(contents described in Lvov, Science 158 946 (1967)).
2. J.D. Isaacs, A.C. Vine, H. Bradner, G.E. Bachus,
Science 151 682 (1966) and 152 800 (1966) and 158 946 (1967).
3. J. Pearson, Acta Astronautica 2 785 (1975).
4. J.H. Ross, Astronautics & Aeronautics, 15-12 44 (1977).
5. The central idea in this paper, of a satellite that rolls like a wheel,
was originated and suggested to me by John McCarthy of Stanford.
6. H.P. Moravec, Advances in the Astronautical Sciences, 1977, also
J. Astronautical Sciences 25 (1977).
7. G.K. O'Neill, The High Frontier, Human Colonies in Space
(William Morrow & Co., New York, 1976).
8. The derivations were done using the MACSYMA symbolic mathematics computer
system at MIT.
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