This is derived from the POX source for
H. P. Moravec
A Non-Synchronous Orbital Skyhook
23rd AIAA Meeting, The Industrialization of Space
San Francisco, Ca., October 18-20, 1977
Published in
The Journal of the Astronautical Sciences
v25#4, Oct-Dec 1977, pp. 307-322
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A Non-Synchronous Orbital Skyhook
Hans Moravec
AI Lab, Computer Science Dept.
Stanford University, Stanford, Ca. 94305
Several authors have examined space transportation via an enormously
elongated planet contacting synchronous satellite. To support itself
over such distances the structure must have an exponential
taper. Conventional materials require impractical tapers, but perfect
crystals offer hope. Given 1/8 the strength of crystalline graphite,
such a skyhook for Earth requires a 100:1 area taper. It would be a
cosmic elevator cable able to support 1/6000 of its mass at a time.
We investigate a cheaper system. A satellite in low circular
equatorial orbit has two long cables extending in opposite
directions. It rotates in the orbital plane, and the cables touch the
planet each rotation, with the rotational velocity canceling the
orbital velocity. The system acts like two spokes of a giant wheel
rolling on the equator.
The orbit is stable, and the taper is minimized when the satellite's
diameter is one third the planet's. On Earth it is 4000 km long and
touches down every 20 minutes, every 2 hours at six points. Cable
motion near the ground is vertical and uniformly accelerated at 1.4
g. The maximum velocity in the atmosphere is 2 km/sec. One eighth the
strength of graphite gives it a taper of 10:1, and it can lift 1/54 of
its own mass at each contact.
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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. He also encouraged the work and provided many of the
resources for it. The symbolic mathematics was done with the MACSYMA
system being developed at MIT. This program behaves like a
programmable desk calculator that deals with algebraic expressions
instead of simply numbers. It is capable of solving equations,
integrating formulas, taking limits and much more.
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INTRODUCTION
Rockets are the ferry boats of orbital travel. This paper suggests
that bridges to orbit may also be possible. Chemical rockets require
the most energetic reactions and large mass ratios to attain escape
velocity. Analogously, structures spanning Earth's gravity well can
be built using the very strongest materials and large tapers in the
structural members. The two endeavors are comparably difficult
because both depend on the ratio of the energy in an atom's outer
electron shell to the energy required to raise the atom out of Earth's
potential well.
Moving between the surface and orbit, like travel along the surface,
involves covering an intervening distance. More importantly, and
unlike surface travel, it also involves an enormous change in
velocity. The portion of an orbital bridge near the ground should be
nearly stationary with respect to the ground. The part of the bridge
at orbital altitude should be moving with at least circular orbital
velocity for that height. Depending on its dynamics, such a structure
could be climbed, or would carry passive loads to and from
orbit. Energy would be borrowed from or injected into the bridge by
such trips. To keep the bridge orbit stable, these flows must be made
to cancel in the long run.
We will consider a simple structure: a long cable whose center of mass
is in a circular orbit in the plane of a planet's equator, and which
rotates in the same plane. The cable is long enough that the tips
touch the planet's atmosphere, or even its surface, each rotation, and
spins just fast enough to cancel the horizontal orbital motion during
the contacts. The cable acts like two spokes of a giant wheel that
rolls around the equator [Fig 1].
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Figure 1: A skyhook's progress around a planet: two spokes of a giant
wheel.
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THE MATERIAL
On bodies the size of Mars and smaller, structures of this kind can be
constructed with conventional materials such as steel. For planets
the size of Earth the taper of the structural members is reasonable
only when substances approaching the theoretical strength/weight
ratios for conventional matter are used. For Jupiter, no known
material suffices.
The strength to weight ratio of potential materials will be
characterized, as in [4], by the characteristic length, the length of
the material shaped into a cable of constant cross section that can
just support itself in a uniform 1 gravity field. This is the tensile
strength of the material divided by its density times one earth
gravity.
Table I gives parameters for some candidate substances. It shows the
tensile strength, density and characteristic length for each of these
materials. It also gives the area taper ratio for optimum (minimum
taper) skyhooks built of these materials for the earth, Mars and
Earth's moon. The tapers do not include a safety factor (i.e. the
materials are assumed stressed to their limits). To include a safety
factor of two, the tapers must be squared. Kevlar [1] is a new
superstrength synthetic polymer recently introduced by the du Pont Co.
TABLE I. Material Parameters
Material Tensile str. Density Ch. Len. Earth Mars Lunar
dyne/cm^2 g/cm^3 km taper taper taper
Stainless 1.72x10^10 8.0 22 9.3x10^48 3.0x10^9 298
Steel 3.17x10^10 7.5 43 8.9x10^24 6.7x10^4 18.2
Nylon 9.86x10^9 1.14 88 1.6x10^12 229 4.14
Fiberglass 2.41x10^10 2.5 98 8.5x10^10 130 3.57
Kevlar 2.76x10^10 1.44 195 3.2x10^5 11.65 1.90
Silica 9.0x10^10 2.6 353 1117 3.89 1.43
Graphite 4.2x10^11 2.0 2147 3.17 1.25 1.06
It can be seen that only the strongest and most exotic substances
suffice for building terrestrial skyhooks. At present, neither silica
fibers nor graphite whiskers have been fashioned into large structures
that retain most the basic strength of the microscopic components. On
the other hand, skyhooks for the moon and Mars are possible with
existing materials.
Graphite whiskers embedded in epoxy matrices have been used as
stronger, lighter replacements for metal in golf clubs, airframes,
pressure vessels and rocket engines. The strength/weight ratios of
these composites fall far short of those of individual whiskers. The
nature of the matrix-whisker interface seems to be the major
controlling factor, as interface irregularities cause stress
concentrations and premature breakage [3]. Constructing large members
with durable strengths near that of individual whiskers is the single
most important, and probably difficult, technical hurdle to be
overcome if terrestrial orbiting skyhooks are to become a reality.
Such materials would have many other important applications.
Measured characteristic lengths for graphite whiskers [2] range from
900 to 3200 km. Because it produces some round numbers for structures
built for Earth, we will arbitrarily choose 2147 km from this range as
the material strength assumed in most of the numerical examples. The
strength actually plugged into the formulas will be half that, to
build in an automatic safety factor of two.
This substance will be referred to in the rest of the paper as derated
carbon. It has an actual characteristic length of 2147 km, and a
design length of 1073.5 km, or equivalently a density of 2.2 g/cm^3
and a design tensile strength of 2.1x10^11 dyne/cm^2. The modulus of
elasticity, needed in the simulations but unused in the analytic
calculations, is 1.1x10^13 dyne/cm^2. This means that a piece of the
cable material stretches about 1/52 its original length when a design
maximum load is applied.
The taper ratio varies exponentially with the weight/strength ratio of
the material. For example, halving the strength/weight ratio squares
the taper ratio. The characteristic length of steel is one twentieth
that of derated graphite. Where a graphite cable needs a taper ratio
of 10, one of steel would require a taper of 10^20.
In principle a rolling satellite can be built to orbit at any height,
but three cases are of special interest.
CASE 1: SYNCHRONOUS ORBIT
Variants of this have been examined in [4][5][6][7]. The hub is put
into a synchronous orbit about the planet, and the cable, extending
outward in both directions from synchronous height, stands unmoving,
rotating with the planet. The lower end is anchored to the ground.
Gravity pulls the portion of the cable between the surface and
synchronous height towards the center of the planet. The part beyond
synchronous orbit is pulled away from the center by centrifugal force,
and is made long enough to just cancel the pull on the downward
segment. The outermost tip of the cable is ballasted to put the cable
at its design tension. The force exerted by the ballast shows up as
pull on the anchor.
A weight attached to the cable near its base causes a decrease in the
force on the anchor. The pull on the anchor gradually returns as such
a load travels up the cable, because on the way up gravity decreases
and centrifugal force increases. When the load reaches synchronous
height, it also has synchronous orbital velocity and ceases to exert
any forces on the cable. The energy to do the lifting could be
supplied from orbit, or from the ground, via a superconductor attached
to the cable, or by microwave beam. The energy to accelerate the load
to orbital velocity comes from the planet's rotational energy, through
the anchor. The rising mass drags the part of the cable below it a few
degrees away from the perpendicular, producing a decelerating torque
on the planet, and an equal and opposite acceleration in the load.
Masses moving down the cable reverse this process, and could be used
to inject energy into the system. If inward and outward traffic were
equal, the energy cost of leaving a planet could be made negligible.
Because the forces below and above synchronous orbit have different
gradients, it is best to not make the cable symmetrical about
synchronous height. The cable starts out at ground level with a given
cross sectional area. This increases as we move up the cable until
synchronous height, and then begins to taper down again. At some point
beyond synchronous orbit, the area equals that at ground level. We end
the cable there because at that length the upward and downward forces
on it exactly balance. If the cable is then anchored to the ground, it
exerts no force on the anchor.
A ballast is attached to the outward end. Since the far end of the
cable is moving faster than orbital velocity at that distance, the
ballast pulls outward. Its mass is adjusted to put the anchored cable
at design tension. Fig. 2 shows the configuration for a synchronous
cable built for the earth. The ballast is located 150,290 km from the
center of the Earth, and is 12.57 times as massive as the largest
weight that can be lifted by the cable at one time. This is because
the net outward force on the ballast is about 1/13 Earth's surface
gravity.
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Figure 2: A view of a synchronous derated graphite skyhook for the
earth. The diagram is to scale, except that the thickness of the
cable has been greatly magnified.
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It is evident that for the earth such a skyhook, though elegant, is
very large. Table II shows that it masses about 6000 times as much as
it can lift at one time. On the other hand, a version for Mars is
almost reasonable, massing only 42 times what it can lift. Mars is the
best planet in the solar system for a synchronous cable, having both a
shallow gravity well and a high rotation rate.
CASE 2: MINIMUM TAPER RATIO
It is likely that shorter, rolling, skyhooks will be less massive, and
require less taper, than the stationary, synchronous variety.
Evaluating tapers and masses for different sizes of graphite skyhooks
for Earth gives us the graphs in Fig. 3. Max g force is the peak
force experienced by a payload attached to the end of the rotating
skyhook. It happens at the moment of touchdown. Taper ratio is the
cross sectional area of the cable at its thickest (at the hub) divided
by the area at the cable ends, where it is thinnest. Mass ratio is
the total mass of the cable divided by the largest mass that it can
lift at one time. Note that the maximum load the cable can lift is
reduced by the strain imposed by the takeoff acceleration.
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Figure 3: Maximum g force and taper and mass ratio for derated
graphite cables for Earth, as a function of orbital radius.)); The
graph shows a minimum taper when the radius of the satellite is about
one third the radius of the earth. The formulas found in the last
section of this paper show that the cable size at which this minimum
occurs is independent of the nature of the cable material. They
furthermore show that if the cable's planet is non-rotating the
minimum occurs at exactly 1/3 planetary radius, independent of
everything else.
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CASE 3: ONE THIRD RADIUS ORBIT
The rotation rates of most of the bodies of the solar system are low
enough that a cable exactly one third the planet's size is not
substantially worse than an optimal one. Fig. 4 shows a terrestrial
1/3 size skyhook, but the difference between it and an optimal one is
unobservable at this scale. The 1/3 size has the advantage that if
its orbit is not perturbed it touches down repeatedly at exactly six
points around the planet, instead of slowly precessing around the
equator. This might make possible the establishment of some kind of
fixed transfer points on the surface.
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Figure 4: A view of a 1/3 radius skyhook at the moment of a touchdown.
The thickness has been greatly magnified.
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The velocity of the outer tip of a terrestrial 1/3 radius skyhook is
13.24 km/sec. My colleague Donald Gennery has calculated that a
payload released with this velocity could arrive at the orbit of Mars
in 72 days and reach Venus in 41 days. It is just short of the
velocity needed to reach Saturn.
OTHER SIZES
Fig. 3 suggests other convenient skyhook orbits. At 12/3 earth radii,
the cable touches down at 3 points on the equator. The taper ratio is
still a reasonable 11.9, the mass ratio is 75, and the maximum g load
has been reduced to 1.4 g. When the orbital radius is twice Earth's
radius, the skyhook touches down at two places around the equator. The
taper is then 15.4, the mass ratio is 120 and the maximum force is
1.17 g. At 3 earth radii there is only one touchdown point. The taper
ratio is 31.3, the mass ratio is 425 and the maximum force is 1.02 g.
Orbits smaller than 11/3 radii seem impractical because of the high g
forces involved. Not only do large g loads stress the cable's payload,
but they imply a very quick entry and exit. This makes rendezvous
tricky, and increases drag if the cable enters the atmosphere. The
takeoff acceleration can be found by subtracting 1 gravity from the
maximum g force. The larger orbits are much better by this criterion.
The one and two touchdowns per orbit sizes are interesting for another
reason. They are suitable for non-equatorial orbits. By selecting the
proper rotation rate, they can be made to touch down, with velocity
match, at high latitudes, or even at the poles.
SUMMARY
Table II tabulates taper and mass ratios for synchronous, 1/3 radius
and optimum skyhooks for the planets and some moons of the solar
system. Taper ratio, in the upper left of each box, is the cross
sectional area of the cable at its thickest (at the hub) divided by
the area at the cable ends, where it is thinnest. Mass ratio, in the
lower right, is the total mass of the cable divided by the largest
mass that it can lift at one time. Synchronous cables for slowly
rotating bodies are very long, and thus massive, even if the gravity
well is shallow.
TABLE II. Taper and Mass Ratios for Three Lengths
of Derated Graphite Skyhooks for Solar System Bodies
Synchronous 1/3 radius Optimum
Mercury 2.223 , 2755. 1.424 , 1.843 1.424 , 1.842
Venus 123.3 , 1.4x10^6 8.315 , 42.43 8.315 , 42.36
Earth 100.0 , 5751. 10.05 , 54.19 10.03 , 53.71
Moon 1.297 , 163.8 1.124 , .5164 1.124 , .5162
Mars 2.414 , 41.73 1.563 , 2.498 1.562 , 2.478
Jupiter 2.8x10^26 , 1.8x10^28 7x10^15 , 1.6x10^17 2x10^15 , 4.5x10^16
Ganymede 1.420 , 52.7 1.177 , .7464 1.177 , .7455
Saturn 3.3x10^6 , 7.3x10^7 17430 , 2.1x10^5 7995. , 86330
Titan 1.403 , 109.8 1.166 , .6992 1.166 , .6989
Uranus 2350. , 47500 101.3 , 820.6 88.12 , 671.9
Titania 1.049 , 12.87 1.022 , .0911 1.022 , .0911
Neptune 1x10^6 , 4.1x10^7 2092. , 21580 1962. , 19650
Triton 1.506 , 71.91 1.209 , .8835 1.209 , .8826
Key: (Taper , Mass)
PAYLOAD ACCELERATIONS
The acceleration experienced by a payload at the cable ends is the sum
of three accelerations. These are the gravity of the planet, the
centrifugal force due to the orbital motion of the satellite and the
centrifugal force of the satellite's spin. The directions of these
forces change continuously, and they depend on the mass and size of
the planet, and the radius of the orbit.
Table III gives the accelerations experienced at the end of a 1/3 size
terrestrial cable. 1.96 g is the minimum ever experienced and 2.4 g
is the maximum. At ground level the cable appears to descend and then
lift off with a constant 1.4 g of vertical acceleration.
TABLE III. Payload Accelerations in a 1/3 Size Terrestrial Skyhook
(in Earth Gravities)
Gravity Orbital Rotational Total
at touchdown 1 down 0.56 up 1.96 down 2.4 down
at orbital height 0.56 down 0.56 up 1.96 sideways 1.96 sideways
at apex 0.36 down 0.56 up 1.96 up 2.16 up
This acceleration takes the cable end from a stationary start on the
earth's surface to the fringes of the atmosphere 150 km above the
surface in 150 seconds. The velocity at that height is 2 km/sec.
STABILITY
The dynamic behavior of a rotating skyhook involves interactions with
the planet's non-uniform gravitational field, with intermittent
external loads and possibly with the planet's atmosphere. Many
possible combinations of initial conditions and load and force
sequences need to be considered. My analysis has been superficial.
Digital simulations of terrestrial 1/3 size derated graphite cables,
with the filaments modeled as a chain of point masses and springs,
showed no gross instabilities. The initial conditions were 1) one end
in contact with Earth's surface and stationary with respect to it 2)
central point in the cable 1/3 earth radius above the surface and
moving at circular orbital velocity for that height 3) cable far end
2/3 earth radius above the surface and moving with a velocity equal to
the difference between the velocity of the cable's center and the
ground end, and 4) cable stretched as it would be under design maximum
loads.
If undisturbed the cable orbits at nearly the correct height (i.e. the
touchdowns are close to the surface), but travels farther than
expected between rotations. Instead of touching down repeatedly at the
six expected places, it oversteps the points by increasing amounts. By
the end of the first orbit it touches down too far by about half the
distance between touchdown points. Removing the loads at both ends of
the cable does not significantly affect this. Cables with a higher
taper ratio than the 10:1 for derated graphite touch down more nearly
in the correct places. It is clear that the orbit and rotation of the
extended cable is different (and much more complicated) than of a
small object at the cable's middle. Simulations provide one way in
which the skyhook's length and and velocity could be tweaked to make
the touchdowns come out right. To complicate the issue, the cable
stretches a little during maximum loading at the vertical contacts,
and contracts when the cable is horizontal.
Suddenly removing or adding design maximum loads, besides altering the
orbit a little, creates tension/compression waves that travel the
length of the skyhook. If they are not damped along the way, these
waves reflect and re-inforce themselves temporarily during the
reflection. Often these re-inforced stresses cause pieces of the
cable to break off and fly away. This effect can be greatly reduced by
putting shock absorbers along the cable to gobble passing waves. Such
dampers involve an energy loss and diminish the rotational and orbital
velocity of the skyhook slightly. Picking up and then launching a
large load also removes orbital velocity from the cable. Such energy
losses can cause later trauma since they lower the orbital height,
resulting in high velocity impacts of the cable with the ground, more
waves and possibly fragmentation.
Energy losses could be replaced by catching high velocity loads
launched from some convenient place (such as the moon), or by rockets,
and possibly even by interaction with the atmosphere during re-entry
and touchdown, with active devices such as jets or propellers.
The whole problem of energy loss and sharp waves could be eliminated
by attaching an equivalent mass whenever a load was dropped, and
dropping a mass whenever one had to be picked up. This would require
careful scheduling, and a net mass flow of zero. Since there are
plenty of rocks both down here and up there, this is not completely
implausible. The simulations of this case show it to be very stable
and simple. The tension slowly rises to a maximum during vertical
contacts and ebbs to a minimum halfway between touchdowns. Even in
undamped cables there are no disturbing high frequency oscillations.
The fine adjustments needed for precise touchdowns could be made in
space by sliding small masses up and down the cable, and by "flying"
the end of the skyhook in the atmosphere, either as a glider with
passive control surfaces, or actively with jets and propellers. It is
quite clear that a large computer system will be needed to predict and
control the dynamics.
Additional problems arise if we want atmospheric entry of the skyhook.
Short skyhooks have high entry and exit velocities. A terrestrial 1/3
size cable enters the atmosphere at a supersonic 2 km/sec It
decelerates to a standstill in 150 seconds, and then accelerates again
to an exit velocity of 2 km/sec. There will be heating and energy loss
due to drag, and wind shear effects. This could be avoided by having
the skyhook touch only the fringes of the atmosphere rather than the
surface. It takes 43 times as much energy to accelerate to earth
orbital velocity as it does to merely climb to 150 km. The climb could
probably be accomplished by a conventional jet with rocket assist. The
bulk of the work would still be done by the skyhook.
NON-PLANETARY APPLICATIONS
A giant cable rotating in free space could be used as a velocity bank
where spacecraft would deposit and borrow energy. Any velocity in the
plane of the cable's rotation up to the cable's tip speed could be
matched by entering on tangents at various distances from the cable's
center.
A large number of such structures in successive circular orbits around
the sun, rotating and orbiting in the plane of the ecliptic, could
greatly cut the energy cost of planetary travel. Each would provide
or absorb enough energy to boost spacecraft to the next. They would
provide many of the advantages of gravity assisted spaceflight, where
and when needed.
STRONGER MATERIALS
A terrestrial version of this kind of structure hinges on the
existence of extremely strong materials. This section contains some
speculations about the existence of substances stronger than any now
known.
Material strength comes from interatomic forces. Thermal agitation
reduces the average strength, so cooling usually strengthens the
materials. Unfortunately there is no data available to me on how much
stronger cryogenically cooled graphite is.
It may be possible to re-inforce structural members by application of
external forces. Current flowing in a superconducting solenoid wrapped
around and attached to a structural member in tension would increase
its effective tensile strength. The magnetic field pulls the ends of
the solenoid, and thus the structural member, towards one another. At
the same time, it tries to burst the solenoid radially with
approximately equal force. In a material such as graphite, which is
equally strong in two dimensions (at least in principle), the strength
in the radial or circumferential direction could be used to perhaps
double the longitudinal strength. Diamond's strength might be tripled
this way.
Much farther afield, there are perhaps exotic forms of matter with
intrinsically higher strength/weight ratios. The nuclei of atoms are
the bulk of the mass of conventional materials, but contribute very
little to their strength. If they could be replaced by something much
lighter, higher strength/weight ratios would result. A simple
possibility is single crystal frozen hydrogen.
If a lightweight, stable, replacement for the proton existed, even
lighter matter might be possible. The fact that such a low energy
particle has not already been observed in accelerators makes this
unlikely. Heavyweight replacements for the electron might also
improve the s/w ratio, by shrinking atom size, which would increase
the material's density, but also the strength of the chemical bonds,
which obey roughly an inverse square law. There are no known electron
substitutes. Muons, all of them very short lived, come the closest.
Some gauge theories predict the existence of stable magnetic monopoles
weighing about 1000 proton masses. If there were at least two stable
types, analogous to electrons and protons, we could build monopole
matter. Monopoles offer advantages in addition to the density effects
we expect with heavy electron atoms. The quantum of magnetic charge
is at least 68.5 times as strong as the electric quantum. This means
that two magnetically charged particles mutually attract 68.5^2 = 4700
times as strongly as electric ones. Magnetic matter would have its
s/w ratio increased 4700 times by this fact alone. The energy in
monopole "electron" orbitals would be so high that terrestrial
temperatures would leave virtually all of them in ground state. This
should make effects such as superfluidity and superconductivity (but
of magnetic currents!) possible at room temperature. This would make
solenoid re-inforcement convenient.
Even more exotic, consider the possibility of hybrid electric/magnetic
matter. A magnetically charged particle can be held in a circular
orbit by an electric current flowing through the center of the
orbit. Analogously, an electric particle will orbit a monopole
current. If we take the straight line current in either of these
examples and bend it into a circle, closing the loop outside the path
of the orbiting particle, we have an interlocked electric and magnetic
current, each perfectly happy with the other. Doing this at the
quantum level, we could have a single monopole and a single electron,
interlocked like two links of a chain. If the chain could be extended
to more links, alternately electric and magnetic, a very strong
material should result. Net charge could be cancelled by using both
electrons and protons (and the monopole analogs). A single type of
particle with both magnetic and electric charge might also be used in
this way.
DERIVATIONS
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Figure 5: A view from high above one pole of a planet equipped with an
orbital skyhook, shown at the instant of touchdown.
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A satellite is in a circular equatorial orbit around a planet. The
satellite rotates in the plane of its orbit, along with two very long
filaments or cables which extend radially outward from it. The length
of the filaments is such that their tips brush the planet's surface.
If the rotation rate of the satellite is low, these contacts occur at
almost orbital velocity. It is possible, however, to spin the
satellite so that its rotation cancels the the orbital motion at the
contact point. The filaments then appear to be two spokes of a giant
wheel which rolls around the equator.
We will investigate this device, and determine the structural
strengths required for different orbital radii.
Define
r[p] the radius 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
w[s] the rotation rate of the satellite
d the density of the filament material
t the tensile strength of the filament material
G the universal gravitational constant
Taper Ratio
Suppose the satellite is oriented as shown in Fig. 5. The stresses in
the filaments are caused by their weight in the planet's gravitational
field, and the accelerations due to the orbital motion and the spin of
structure. The maximum stress occurs in the downward hanging filament,
on which gravity and spin pull in the same direction. This is the case
we will analyze.
We assume that the filaments are constructed with the cross section
varying so as to make the tension per unit area constant along their
length.
Scanning up the satellite from ground level, we observe that the force
in the cable above any slice of it is equal to the force below that
slice plus the weight of the slice and its mass times its
acceleration.
dF = { Gm[p]/r^2 - r[o] w[o]^2 + (r[o]-r) w[s]^2 } dm
Equation 1
Let A(r) represent the cross sectional area of the cable at a distance
r from the center of the planet. We can now substitute
dm = dA(r) dr dF = t dA(r)
Equation 2
into Eq. (1) such that
t/d dA(r)/A(r) = { Gm[p]/r^2 - r[o] w[o]^2 + (r[o]-r) w[s]^2 } dr
Equation 3
Eq. (3) may be integrated between the planet's surface and the general
position, r
t/d Integral[A(r[p]:A(r)] dA(r)/A(r) =
Integral[r[p]:r] { Gm[p]/r^2 - r[o] w[o]^2 + (r[o]-r) w[s]^2 } dr
Equation 4
Giving
A(r)/A(r[p] =
exp( d/t { Gm[p](1,/r[p] - 1/r) + r[o]( w[o]^2 -
w[s]^2 )(r[p] - r) + ( w[s]^2 (r[p]^2 - r^2))/2 } )
Equation 5
to make contact point stationary,
r[o] w[o] - (r[o] -
r[p]) w[s] = r[p] w
[p]
or
w[s] = (r[o] w[o] - r[p] w[p])/(r[o] - r[p])
Equation 6
and for a circular orbit
w[o] = sqrt(Gm[p]/r[o]^3)
Equation 7
the maximum area is found at r = r[o]. Substituting Eqs.(6,7) into
Eq. (5), we get
A[max]/A(r[p] = exp( d/t {
(Gm[p](2r[p]^2 - 3r[o]r[p] + 2r[o]^2) - 2r[o]^(3/2)
r[p]^2 w[p] sqrt(Gm[p]) + r[o]^2r\!jsab(3,p); w[p]^2 )
/ (2 r[o]^2 r[p]) } )
Equation 8
To calculate the total mass of the cable, it is necessary to integrate
its cross section over its length. It appears to be impossible to do
this analytically with the usual functions. The mass ratios in this
paper were obtained by numeric integration.
Surface Accelerations
Liftoff accelerations of payloads attached to rotating skyhooks at the
moment of touchdown are the sum of the accelerations due to orbital
motion and satellite rotation.
Liftoff Acceleration = (r[o] - r[p]) w[s]^2 - r[o] w[o]^2
Equation 9
Substituting Eqs. (6,7) into Eq. (9)
and dividing by the surface gravity of the planet gives us
Liftoff Acceleration/Surface Gravity = r[p]^2
{sqrt(Gm[p]/r[o]) - r[p] w[p] }^2)/(Gm[p](r[o] - r[p])) - r[p]^2/r[o]^2
Equation 10
In the case of a non-rotating planet, w[p] = 0 and this simplifies to
Liftoff Acceleration/Surface Gravity = (r[p]^3)/(r[o]^3 - r[o]^2r[p])
Equation 11
Minimum Taper Ratio
There is a minimum area ratio for a certain cable size. This can be
found by setting the derivative of the maximum area ratio with respect
to the orbital radius to zero and solving for the orbital radius.
Equivalently, we consider only the variable part of the exponent of
the maximum area ratio. This is
(Gm[p](2r[p]^2 - 3r[o]r[p] + 2r[o]^2) - 2r[o]^(3/2)
r[p]^2 w[p] sqrt(Gm[p]) + r[o]^2r[p]^3 w[p]^2 ) / (2r[o]^2 r[p])
Equation 12
the problem is simplified by inventing the variables
t = sqrt(r[p]^3 w[p]^2 / Gm[p])
u = sqrt(r[o]/r[p])
Equation 13
substituting Eq. (13) in Eq. (12) gives us
(r[p]^2 w[p]^2 (u^4 (t^2 + 2) - 2u^3 t - 3u^2 + 2))/(2u^4 t^2)
Equation 14
Now, differentiating Eq. (14) with respect to u we get
d/du (r[p]^2 w[p]^2 )/t^2 (t^2 + 2) u^4 - 2tu^3 - 3u^2 + 2)/(2u^4)
= (r[p]^2 w[p]^2 )/(t^2) (t u^3 + 3 u^2 - 4)/(u^5)
Equation 15
setting Eq. (15) to 0, we can cancel the denominator
t u^3 + 3 u^2 - 4 = 0
Equation 16
note that when w[r] = 0, t = 0 and Eq. (16) becomes
3 u^2 - 4 = 0 or r[o] = 4/3 r[p]
Equation 17
thus for a non-rotating planet the minimum area ratio occurs when
the orbital radius of the skyhook is 11/3 planetary radii,
i.e. when the radius of the cable is 1/3 the radius of the
planet.
a simple form for the general solution when w[r] /= 0 is
u = (2 cos { acos(2 t^2 - 1)/3 } - 1) / t
or
r[o] = { (2 cos { acos(2 t^2 - 1)/3 } - 1)/t} ^2 r[p]
Equation 18
Synchronous Orbit
Another interesting case occurs when the satellite is placed in a
synchronous orbit. In that case
r[o] = (Gm[p]/w[p]^2 )^(1/3)
and
w[p] = w[o] = w[s]
Equation 19,20
and the filaments are perpetually stationary with respect to the planet's surface.
Substituting Eqs. (19,20) into Eqs. (5,8) gives us
A(r)/A(r[p] = exp( d/t { Gm[p] (1/r[p] - 1/r) + ( w[p]^2 (r[p]^2 - r^2))/2 } )
A[max]/A(r[p] =
exp( d/t { (Gm[p]/r[p]) - 3 (Gm[p] w[p])^(2/3)/2 + (r[p]^2 w[p]^2)/2 } )
Equation 21
REFERENCES
1. ROSS, Jack H., "Superstrength-Fiber Applications", Astronautics &
Aeronautics, Vol. 15, No. 12, December 1977, pp 44-65.
2. HARMAN, Cameron G., "Non-Glassy Inorganic Fibers and Composites",
NASA Technology Utilization Report SP-5055, August 1966.
3. CHAMIS, Christos C., "Mechanics of Load Transfer at the
Fiber/Matrix Interface", NASA TN-D-6588, 1973
4. PEARSON, Jerome, "The Orbital tower: a spacecraft launcher using
the Earth's rotational energy", Acta Astronautica Vol 2, pp. 785-799,
Pergamon Press, 1975.
5. ISAACS, John D., VINE, Allyn C., BRADNER, Hugh and BACHUS, George
E., "Satellite Elongation into a True "Sky-Hook"", Science Vol. 151,
February 11, 1966, pp 682-683 and Science Vol. 152, May 6, 1966, p
800. and Science Vol 158, November 17, 1967, pp 946-947.
6. ARTSUTANOV, Y., "V kosmos na elektrovoze.", Komsomolskaya Pravda,
July 31, 1960. (contents described in LVOV, Science Vol 158, pp
946-947).
7. TSIOLKOVSKY, K.E., "Grezi o zemle i nebe (in Russian) (Speculations
between earth and sky)", Moscow, Izd-vo AN SSSR, 1959, p.35.