Non-Synchronous Orbital Skyhooks for the Moon and Mars with Conventional Materials Hans Moravec 1978 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. --- 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 g/cm^3 and a tensile strength of 2.1e11 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 g/cm^3 and a tensile strength of 2.76e10 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 one millionth 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. Figure 1: A non-synchronous skyhook's progress around a planet: two spokes of a giant wheel. [6] 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 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 1.2e^20 3.0e^21 1.3e10 2.3e11 Earth 2.16 1.40 7.2e21 1.9e23 1.0e11 1.9e12 Moon 2.78 0.28 13 72 3.6 13 Mars 2.62 0.49 17000 200000 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 g/cm^3 and a tensile strength of 2.41e10 dyne/cm^2. Kevlar has a density of 1.44 g/cm^3 and a tensile strength of 2.76e10 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, w[s] = (r[o] w[o] - r[p] w[p]) / (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]) ( G m[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. 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.