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\C\FBNon-Synchronous Orbital Skyhooks for the Moon and Mars
\C\FBwith Conventional Materials
\J\F0Abstract. \FEA 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.\F0\.
\J 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
(\FE1, 2, 3\F0). 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 (\fE4) and a new, less expensive, satellite skyhook configuration
(\FE5, 6\F0) 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 (\fE7).
\F0A 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 \!jfrx(1,8); the theoretical strength/weight
ratio of crystalline graphite could be fabricated. A graphite cable
with a density of 2.2 \!jfrx(g,(cm\!jsup(3);)); and a tensile strength
of 2.1x10\!jsup(11);\!jfrx(dyne,(cm\!jsup(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 \!jfrx(1,6000); of the cable mass (\fE6).
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 \!jfrx(g,(cm\!jsup(3);)); and a tensile
strength of 2.76x10\!jsup(10); \!jfrx(dyne,(cm\!jsup(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\!jsup(-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 \!jfrx(1,(13,000));.\.
\_=15;\\N
\C\FABCDEF\F0
\_;\-=210;\Y\C\FAGHIJK\F0
\_;\-=200;\Y\_=50;\\N
\!fig(1,(A non-synchronous skyhook's progress around a planet: two spokes
of a giant wheel.));
\J (\fE6) introduced the concept of a non-synchronous skyhook.
Figure 1 illustrates the idea. A satellite in low circular orbit
is elongated enough to just touch the surface in certain positions.
It spins so that, like a rolling wheel, its rotation cancels
its tangential velocity during the contacts with the surface.
Such a structure can be constructed to orbit at any height, and
a synchronous skyhook is a special case.
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 \!jfrx(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.\.
\,
\_=0;\\_=0;\\F0\_L\I\R\-=1184;\/=2;\+I\+=30;\L\_=320;\E\+L\A\+E\B\+E\C
\8TABS(A,B)[\_A\-=90;\.A\_A\+=70;\.B]\N
\8TABT(A,B)[\_B\-=90;\.A\_B\+=70;\.B]\N
\8TABO(A,B)[\_C\-=90;\.A\_C\+=70;\.B]\N
\C\F1TABLE I. Parameters for Optimally Sized Skyhooks\F0
\!TABS((\_.\-=10;\.Orbital),Liftoff);\N
\!TABT( Fiberglass,);\!TABO( Kevlar,);
\_L\+=15;\.Body\!TABS((\_.\-=35;\.Period (hr.)),(\_.\-=15;\.Accel (g)));\N
\!TABT(Taper,Mass);\!TABO(Taper,Mass);
\_=50;\\_;\-=32;\G\N
Mercury \!TABS((2.37),(0.57));\N
\!TABT((2200),(23000));\N
\!TABO(( 49),(350));
Venus \!TABS((2.37),(1.39));\N
\!TABT((\_.\-=20;\.1.2x10\!jsup(20);),(\_.\-=20;\.3.0x10\!jsup(21);));\N
\!TABO((\_.\-=20;\.1.3x10\!jsup(10);),(\_.\-=20;\.2.3x10\!jsup(11);));
Earth \!TABS((2.16),(1.40));\N
\!TABT((\_.\-=20;\.7.2x10\!jsup(21);),(\_.\-=20;\.1.9x10\!jsup(23);));\N
\!TABO((\_.\-=20;\.1.0x10\!jsup(11);),(\_.\-=20;\.1.9x10\!jsup(12);));
Moon \!TABS((2.78),(0.28));\N
\!TABT(( 13),( 72));\N
\!TABO(( 3.6),( 13));
Mars \!TABS((2.62),(0.49));\N
\!TABT((17000),(200000));\N
\!TABO((136),(1100));
Ganymede\!TABS((3.41),(0.26));\N
\!TABT(( 35),( 240));\N
\!TABO(( 6.0),( 28));
Titan\!TABS((3.39),(0.26));\N
\!TABT(( 29),( 190));\N
\!TABO(( 5.4),( 24));
\YG\F2\_=0;\\\_L\-=30;\L
~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~
%$
\ ;\_=0;\\_=50;\\F0\N
\,
\J 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 \!jfrx(g,(cm\!jsup(3);)); and a tensile strength
of 2.41x10\!jsup(10); \!jfrx(dyne,(cm\!jsup(2)););. Kevlar
has a density of 1.44 \!jfrx(g,(cm\!jsup(3);)); and a tensile
strength of 2.76x10\!jsup(10); \!jfrx(dyne,(cm\!jsup(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
\!jfrx(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.\.
\J 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 (\fE7)
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 \!jfrx(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.\.
\,
\J 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 (\fE8).\.
Define\QL\_.\+=100;\L\+=200;
r\!jsub(p);\.the radius of the planet
m\!jsub(p);\.the mass of the planet
\f5w\!jsub(p);\.the rotation rate of the planet (in radians per unit time)
\!jtop( ,r);\!jsub(o);\.the radius of the orbit
\f5w\!jsub(o);\.the orbital rate of the satellite's mass center
\f5w\!jsub(s);\.the rotation rate of the satellite
\!jtop( ,(\f5d));\.the density of the cable material
\!jtop( ,(\f5t));\.the tensile strength of the cable material
A(r)\.cable cross section at distance r from the planet center
\!jtop( ,G);\.the universal gravitational constant
\qL
\Jto make contact point stationary,\.
\C\f5w\!jsub(s); = \!jdiv((r\!jsub(o);\f5w\N
\!jsub(o); - r\!jsub(p);\f5w\!jsub(p);),(r\!jsub(o); - \N
r\!jsub(p);));
\Jand for a circular orbit\.
\C\f5w\!jsub(o); = \N
\!jnrt(,(\!jdiv((Gm\!jsub(p);),(r\!jsab(3,o);));));
\Jthis last substitution is only an approximation, since the
extended satellite does not orbit and rotate exactly like a point at
its mass center.\.
\J 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\.
\CA(r) = A(r\!jsub(p);) \f0e\N
\!jsup((\!jsup((\!jsup((\!jsup((\N
\F0\!jdiv((\f5d),(\f5t));(r - r\!jsub(p);)\!jOP;\N
\!jdiv((Gm\!jsub(p);),(r r\!jsub(p);)); - r\!jsub(o);\f5w\!jsab(2,o); + \N
(r\!jsub(o); - \!jdiv((r + r\!jsub(p);),2);)\f5w\!jsab(2,s);\!jCP;\N
));));));));
\J 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 (\fE6). 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.\.
\,
\_L\A\+=600;\L
Hans P. Moravec
\FEComputer Science Dept.,
Stanford University,
Stanford, Ca. 94305\F0
\_A\L
\C\F1References and Notes\F0
\_L\+=55;\l\_R\r\N
1.\jY. Artsutanov, \FEKomsomolskaya Pravda\F0, July 31, 1960
(contents described in Lvov, \FEScience\F1 158\F0 946 (1967)).\.
2.\jJ.D. Isaacs, A.C. Vine, H. Bradner, G.E. Bachus,
\FEScience\F1 151\F0 682 (1966) and \F1152\F0 800 (1966) and
\F1158\F0 946 (1967).\.
3.\jJ. Pearson, \FEActa Astronautica\F0 \F12\F0 785 (1975).\.
4.\jJ.H. Ross, \FEAstronautics & Aeronautics\F0, \F115-12\F0 44 (1977).\.
5.\jThe 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.\jH.P. Moravec, \FEAdvances in the Astronautical Sciences, 1977\F0, also
\FEJ. Astronautical Sciences \F125\F0 (1977).\.
7.\jG.K. O'Neill, \FEThe High Frontier, Human Colonies in Space\F0
(William Morrow & Co., New York, 1976).\.
8.\jThe derivations were done using the MACSYMA
symbolic mathematics computer system at MIT.\.
\'100;\F0\N