A cable with a rocket equation of its own

Every route to orbit described in this collection pays the exponential that decides what can be flown. A rocket’s ratio of fuelled to empty mass is $e$ raised to the velocity change divided by the exhaust speed, and because the velocity a launch must buy, about 9.4 kilometres a second, is two to three times the exhaust speed any chemical propellant gives, a rocket to orbit is mostly propellant and the payload a few per cent of the whole.

The space elevator was proposed as the way out. Imagine a cable stretched from a point on the equator straight up past the altitude of synchronous orbit, where a satellite goes round once a day and so hangs over one spot. The whole cable turns with the Earth. A vehicle climbs it by gripping it, driven by electric motors powered from the ground by laser or by wire, and when it reaches synchronous orbit it is moving at orbital speed without having burned any propellant at all. The energy needed is the same — it is set by the climb in gravity and the speed of the orbit — but it is supplied as electricity, from the ground, at the price of electricity.

The idea is old: Tsiolkovsky imagined a tower in 1895, and Yuri Artsutanov in 1960 and Jerome Pearson in 1975 worked out the hanging cable. What stopped it then and has stopped it since is not the climber. It is the cable, which has to hold up its own weight, and when that requirement is written down it turns out to have the form of the very equation the elevator was meant to escape.

A cable that must hold itself up #

A cable hanging in a uniform gravitational field, with constant cross-section, breaks at its top once it is longer than the material’s breaking length: its tensile strength divided by its density and by $g$. For steel that is about twenty-five kilometres; for the strongest commercial fibres, two to four hundred. A cable to synchronous orbit is thirty-six thousand kilometres long, so a cable of constant thickness made of anything that exists would snap under its own weight many times over.

The answer is to taper it. If the cable is made thicker towards the top in just the right way, every part of it carries the same stress — the most the material can safely bear — and no part is overloaded. Each slice of the cable must hold up the weight of everything below it, so the slice’s cross-section must grow in proportion to that load. A cross-section that grows in proportion to the load it carries grows exponentially, because the load is itself the integral of the cross-section below.

In a rotating frame the weight of a slice is set not by gravity alone but by gravity minus the outward pull of the rotation, which together form the effective potential $\Phi(r) = -GM/r - \omega^2 r^2/2$. Writing the balance for a slice of cable of density $\rho$ at stress $\sigma$ gives $dA/A = (\rho/\sigma)\,d\Phi$, and so

$$\frac{A(r)}{A(R)} = \exp\!\left[\frac{\Phi(r) - \Phi(R)}{\sigma/\rho}\right].$$

The same exponential, moved into the rope #

Set that beside the rocket equation, $m_0/m_1 = \exp(\Delta v / v_e)$. The ratio of top to bottom cross-section plays the part of the mass ratio. The climb in effective potential, $\Delta\Phi$, plays the part of $\Delta v$, but it is an energy per unit mass, so the natural comparison is with $\Delta v^2/2$. And the material’s specific strength, $\sigma/\rho$ — its breaking stress divided by its density, also an energy per unit mass — plays the part of the exhaust speed. The rocket equation’s essential structure, an exponential of a demand divided by a capability, has reappeared in the rope.

For the Earth the climb from the equator to synchronous orbit is 48.4 megajoules per kilogram. That is the energy needed to lift a kilogram up the potential well and give it synchronous orbital speed, and it corresponds to a velocity change of $\sqrt{2 \times 48.4\ \mathrm{MJ/kg}}$, about 9.8 kilometres a second — the same size as the velocity a launch to orbit must buy, as it should be, since the destination is the same. A rocket has an exhaust speed of three or four kilometres a second, so its mass ratio is $e^{2.5}$ or so: large but buildable. The question for the cable is how its specific strength compares with 48.4 megajoules per kilogram.

The answer is: very badly. Steel’s specific strength is a quarter of a megajoule per kilogram, and its taper ratio, even with no safety margin at all, would be a number with eighty-four digits. Kevlar, at two and a half, gives a cable a hundred million billion times thicker at the top than at the bottom, with a safety factor of two. The best carbon fibres and polymer fibres, near four megajoules per kilogram, still give ratios of tens or hundreds of billions. A cable a millimetre across at the ground would be hundreds of kilometres across at the top.

Only one family of materials comes close. The carbon–carbon bonds in a perfect carbon nanotube or a sheet of graphene have a theoretical strength of around a hundred gigapascals at a density of 1.3 grams per cubic centimetre — a specific strength of fifty megajoules per kilogram or more, comparable to the climb itself. At that strength the taper is a factor of a few, and a cable is a long, thin ribbon thickening gently to the top. The trouble is that a perfect nanotube is microscopic, and every macroscopic yarn or sheet made from nanotubes so far has had a specific strength of a few megajoules per kilogram, because the tubes slip past one another and their defects concentrate stress. The gap between the ideal bond and the best bulk material is a factor of ten, and the taper is an exponential of the reciprocal: ten times weaker is not ten times worse, it is $e^{10}$ times worse.

A cable thickest where it weighs nothing #

The profile of the ideal cable has a shape that follows from the effective potential.

The cable thickens all the way to synchronous orbit and thins again beyond it. Below synchronous orbit each slice is pulled down, because gravity exceeds the rotation’s outward pull, and must hold the slices below; above it each slice is pulled out, and the cable beyond hangs outward from the slices nearer the Earth. The thickest point is the synchronous radius, where a free object would float, because that is where the tension — the accumulated load from both directions — is greatest. A rope hung from a ceiling is thickest at the ceiling; a rope stretched across a balance point is thickest at the balance point.

Even at a working strength of twenty-five megajoules per kilogram — half the ideal nanotube’s — the taper is a factor of seven. That is the number that makes the elevator a matter of engineering rather than of impossibility, and it is the number that a material ten times stronger than anything yet made in bulk would deliver.

Nearer to what can be made, the profile swells far more steeply.

At eight megajoules per kilogram — twice the best fibre yet spun from nanotubes, used with no margin — the cable is four hundred times thicker at synchronous orbit than at the ground, and most of its mass sits in the ten thousand kilometres round the balance point. At twelve the ratio falls to sixty, at twenty to eleven. The cable’s total mass falls faster still, because the thick section is also the long one: each gain in strength shrinks the cable everywhere along it. The whole engineering case for the elevator lies in this stretch of the curve, where each doubling of strength divides the cable’s mass by a factor that is itself growing exponentially. Just as the split between stages is not an equal split because the exponential punishes the stage that carries the most, the cable’s mass is concentrated where the exponential is largest.

Why it needs a weight beyond the orbit #

A cable from the equator to synchronous orbit alone would not stay up. Its whole length is below the balance point, so all of it is pulled downward, and nothing pulls it up.

The cable must extend beyond synchronous orbit, into the region where the net force is outward, far enough that the outward pull on the outer part balances the downward pull on the inner. That means either a counterweight — a captured asteroid or the spent upper stage that built the cable — some tens of thousands of kilometres above synchronous orbit, or a cable that simply continues to about a hundred and fifty thousand kilometres, where its own outer length provides the balance. The outer end is then moving faster than orbital speed for its height: a payload released from it is flung onto an escape trajectory, at no cost in propellant beyond the electricity of the climb. A cable that reaches far enough is a sling as well as a ladder.

That outward pull is also what keeps the cable taut and vertical. Without the counterweight there is no tension at the bottom to hold, and with too much the mooring on the ground must hold the excess. The cable is a pendulum hanging upward, its stability set by the balance between the two parts, and every climber that ascends perturbs it: a climber gains angular momentum as it rises, which it takes from the cable, so the cable leans westward behind it and must be allowed to swing back.

A smaller, faster world is kinder #

The difficulty is entirely in the size of the climb, and the climb is a property of the planet.

Mars has a fifth of the Earth’s climb, because it is smaller and lighter and turns at about the same rate, and the taper for a Kevlar cable falls from a ratio with seventeen digits to one of about two thousand — enormous, but a matter of how much fibre can be made rather than of physics. A cable of carbon fibre on Mars would taper by a few hundred. On Ceres, the largest body in the asteroid belt, small and spinning once every nine hours, the climb is so small that a cable of almost any material barely needs to taper at all. The same equation that rules out an elevator on the Earth with present materials makes one routine on the small, fast-turning bodies of the solar system.

The Moon has no synchronous orbit within its gravitational reach — it turns once a month, so its synchronous radius would be far beyond the region where its gravity dominates the Earth’s — and a lunar elevator would instead hang through one of the Lagrange points between the Earth and the Moon, balanced by the Earth’s tide rather than by rotation. Its climb is modest, and a lunar cable of existing fibres is within the range of the plausible, which is why it has been studied more seriously than the terrestrial one.

The same comparison with a better material shows how sensitive the conclusion is to the last factor of a few.

Carbon fibre is half as strong again as Kevlar, and on Mars that cuts the taper from two thousand to two hundred — a cable whose top is a couple of hundred times the cross-section of its foot, which with a ground section of a square millimetre is a ribbon a few centimetres wide at synchronous orbit. On the Earth the same improvement takes the taper from seventeen digits to twelve: still absurd. Mars sits at the point on the exponential where existing materials work; the Earth sits ten times further along it.

A lunar elevator, hung through the Lagrange point between the Earth and the Moon — the first of the five places that keep station in that pair — has a climb of a few megajoules per kilogram from the lunar surface, and with existing fibres a taper of a few. It would be long, about sixty thousand kilometres, but light, and a study in the 2000s found it within reach of materials already in production. The reason it has not been built is the reason no lunar infrastructure has been built: there has not yet been enough to carry.

Where the rocket equation went #

The rocket equation is often described as a limit on rockets, and the elevator as a way round it. The arithmetic in this essay says something more interesting. Any system that delivers mass to orbit must deliver the energy of the climb, and must hold the payload against gravity on the way. A rocket does both by carrying its reaction mass with it, and the cost of carrying the reaction mass that will be expended later is the exponential. A cable does both by holding the payload on a structure, and the cost of a structure that holds itself is the same exponential, with strength in place of exhaust speed. In both, the decisive number is the ratio of what must be done — 48 megajoules per kilogram — to what one kilogram of the working material can do. Choosing a propellant is choosing a molecular weight because the exhaust speed is fixed by chemistry; choosing a cable material is choosing a bond strength for the same reason.

What the elevator changes is where the exponential is paid. A rocket pays it on every launch, in propellant burned and, for most of history, in hardware thrown away — the stage that has to be thrown away. A cable pays it once, in the mass of the cable, and then carries payloads for the price of electricity, much as a reusable booster pays with the stage above it and then flies again. If a material with fifty megajoules per kilogram were ever made in bulk, the exponential would shrink to a factor of a few, and the cost of reaching orbit would be set by the cost of electricity — about fifteen kilowatt-hours per kilogram for the climb itself.

A cable that only hangs #

The calculations assume a cable at one uniform stress throughout, a perfectly rigid mooring on the equator, and a cable that does nothing but hang. A real design would add the dynamics of climbers and the swinging they cause, the meteoroids and orbital debris that would cross the cable — every object in low orbit crosses the equatorial plane twice a revolution, and the cable sweeps through the region of most satellites — the electrical currents the ionosphere would drive along a conductive cable, and the vibration modes of a structure with a period of hours. Debris is the most serious: a cable is a target that cannot move out of the way, swept by every object whose orbit crosses the equator, and the rate of collisions among objects in orbit grows with the cross-section exposed. The safety factor of two assumed here is lower than engineering practice for structures that cannot be repaired. And the specific strengths quoted for materials are approximate: published values for the same fibre vary by tens of per cent between grades. None of this changes the core result, which is that the taper is an exponential in the ratio of the climb to the strength, and that nothing made in bulk today comes within a factor of ten of the strength the Earth demands.

Still open: whether a bond’s strength can be made into a rope’s #

The question on which the elevator rests is one of materials science, not of orbital mechanics: whether the strength of an individual carbon–carbon bond, demonstrated in single nanotubes a few micrometres long, can be carried into a continuous ribbon tens of thousands of kilometres long. The strongest nanotube fibres have doubled in strength several times over two decades and remain an order of magnitude short. If the gap closes, the exponential in this essay shrinks from impossible to modest, and the economics of spaceflight change more than any engine could change them. If it does not, the elevator remains what it has been since 1895: a correct calculation about the wrong planet, which becomes practical as soon as it is moved to Mars, the Moon or an asteroid.