The rocket that leaves its fuel at home

The fuel a starship needs found that relativity does not make a rocket’s task harder; the logarithm does. What a rocket reaches depends on the logarithm of its mass ratio times its exhaust speed, and even with light itself as the exhaust, half the speed of light costs a mass ratio of the square root of three. Every proposal to go further that survives arithmetic, it concluded, leaves something at home.

The simplest is a mirror. A sail pushed by a beam of light from the ground carries no fuel, no engine and no tank; the power plant stays behind, and the only thing that travels is the reflecting surface and whatever it carries. The radiation pressure on such a sail was worked out in light has a pressure, which found that a mirror takes twice the push of a black surface. What changes when the sail is fast is almost everything else, and the changes follow from one ledger.

A photon rocket with the tank left behind #

Treat the beam, the sail and the reflected light as one system and keep its four-momentum, the same bookkeeping that makes a rocket push against nothing. A beam delivering energy $E$ to a sail of rest mass $m$ carries momentum $E/c$. The reflected light leaves backwards with some energy $E'$ and momentum $E'/c$ the other way. The sail ends with energy $\gamma mc^2$ and momentum $\gamma m\beta c$. Conservation of momentum and energy are then two equations, and eliminating $E'$ between them gives

$$\gamma(1 + \beta) = 1 + \frac{2E}{mc^2}.$$

The left side is the exponential of the sail’s rapidity, the quantity that adds exactly where velocities do not. So the sail’s rapidity is $\ln(1 + 2E/mc^2)$. A photon rocket that carries the same energy aboard as fuel, and throws it out behind, has a mass ratio of $1 + E/mc^2$, and its rapidity is $\ln(1 + E/mc^2)$. A mirror is a photon rocket whose fuel stays at home and whose energy counts twice.

The figure checks the closed form by not using it. The dots come from integrating the force on the mirror step by step — each intercepted joule delivering momentum $2/c(1+\beta)$, because the light arrives with momentum $1/c$ per joule and leaves with its energy lowered by the mirror’s recession — and they sit on the curve to a part in $10^{15}$.

Twice has a simple origin. A photon rocket gets momentum from its photons once, when it emits them. A mirror gets momentum from each photon twice, once when it stops it and again when it sends it back. An absorber gets it only once and does worse than either: the energy it keeps becomes internal energy, which is rest mass it then has to accelerate, and its speed is $E/(E + mc^2)$. To reach a fifth of the speed of light a gram of absorber would need a quarter of its rest energy, the photon rocket a little under that, and the mirror 0.1124 — about ten terajoules, the energy of two and a half kilotons of explosive.

The light a slow sail wastes #

The ledger has a second entry that the closed form hides. Most of the energy a beam delivers to a slow sail does not stay in the sail.

A mirror at rest reflects the light with its energy unchanged and gains momentum but, to first order, no energy; everything goes back down the beam. A mirror moving away reflects the light shifted down in frequency by $(1-\beta)/(1+\beta)$ — the double shift a moving mirror gives — and the energy the light lost is exactly what the sail gained. The share of each new joule that stays is $2\beta/(1+\beta)$: two per cent at a hundredth of the speed of light, a third at a fifth, nearly all of it at nine tenths.

Integrated from rest, a sail that reaches $0.2c$ has kept 18.4 per cent of all the beam energy it intercepted, and one that reaches $0.9c$ has kept 77.1 per cent. A slow sail is a poor engine and a fast one an excellent one, which inverts a rocket’s economics. A rocket’s difficulty grows with speed, because each increment of speed has to accelerate all the fuel still aboard. A sail’s difficulty is concentrated at the start, where most of the light is reflected back with its energy intact, and it improves as the sail escapes.

Sixty-eight thousand g for two minutes #

The closed form says how much energy a sail needs and nothing about how quickly it must be delivered. For a laser on the ground, the answer is fixed by something the ledger does not contain.

The sail in the figure is roughly the one proposed for a flight to the nearest stars: a gram, four metres across, pushed by a hundred gigawatts. Its areal density is 80 milligrams per square metre, a hundred times lighter than the sails that have flown in sunlight, which the size the light cannot blow away found come out near ten grams per square metre. On it, a hundred gigawatts of reflected light is a force of 667 newtons and an acceleration of 68,000 g. With every photon caught, the sail would pass a fifth of the speed of light in 113 seconds, 3.7 million kilometres out, and go on accelerating as long as the beam lasted — though never at a constant proper acceleration of the kind the ship that never arrives at c follows, since the Doppler factor weakens the push as the sail speeds up.

It cannot catch every photon for that long. A beam leaving an aperture of diameter $D$ can be focused at a distance $x$ into a spot no smaller than about $\lambda x/D$, the limit that sets how far apart two things have to be for a telescope to separate them. For a one-kilometre aperture and infrared light, the spot is as wide as the four-metre sail at about three million kilometres — eight times the distance to the Moon. The sail reaches that point in under two minutes, moving at 0.181c. Beyond it the sail intercepts a shrinking share of the beam, falling as the square of the distance, and it coasts to 0.267c on the light it catches while the spot outgrows it.

The acceleration is enormous because the runway is short, and the runway is short because of diffraction. Nothing about the sail’s material or the beam’s power sets the three million kilometres. The wavelength and the aperture do.

Diffraction sets the speed #

The estimate behind the dashed line is one line of mechanics. The sail is pushed with a constant force $2P/c$ over a distance $L = \pi D a/2\lambda$, where the spot reaches the sail’s radius $a$, and a constant push over a distance gives $v^2 = 2(2P/mc)L$. The speed therefore grows as the square root of the aperture, the power and the sail’s radius, and falls as the square root of its mass and the wavelength. The integrated flights follow the square root at small apertures, with an exponent of 0.46, and run above the estimate by up to half, because the sail keeps catching some light beyond $L$. At large apertures they fall back towards it, because the sail is relativistic by then and each photon pushes less.

The consequence is a set of exchange rates. Doubling a sail’s final speed takes four times the aperture, or four times the power, or a sail a quarter as heavy. A kilometre-scale array of lasers phased to act as one aperture is therefore not an extravagance of the proposal but the core of it, and the achievable speed is decided more by how well a large array can be phased than by how much power it has.

Why the sail should be small #

The same one-line estimate says something less obvious about the sail itself. The speed grows as the square root of the sail’s radius over its mass, $a/m$, because a wider sail stays inside the spreading spot for longer and a lighter one is pushed harder while it does. But a sail of a given material and thickness has a mass proportional to its area, $\sigma\pi a^2$, so at a fixed areal density $a/m$ falls as $1/a$ and the speed goes as one over the square root of the sail’s radius. Halving the radius quarters the mass and halves the runway, and the sail ends about 1.4 times faster.

That runs against the intuition from sunlight, where a sail’s thrust grows with its area and a bigger sail is simply better. In sunlight the light fills all of space and the sail’s size decides how much of it is caught. In a focused beam the whole beam is caught for as long as the spot is smaller than the sail, and making the sail bigger only lengthens that phase in proportion to its radius while the mass grows as its square.

The shrinking stops where the sail is no longer most of the mass. A spacecraft of a gram carries a payload — a camera, a transmitter, a power source — whose mass does not scale with the sail, and once the sail is lighter than the payload, making it smaller shortens the runway without lightening what is being pushed. The best sail for a given payload is the one whose own mass is comparable with it, which is why a gram-scale probe ends up with a sail a few metres across rather than a few centimetres or a few hundred metres.

A mirror that must not get warm #

The figures so far assume a perfect mirror. Any real one absorbs a little, and in a hundred-gigawatt beam a little is a great deal.

A sail absorbing one part in a hundred thousand of the beam takes in a megawatt over twenty-five square metres of its two faces, and it can lose that only by radiating it. At an emissivity of one it settles at 915 K; at an emissivity of 0.05, typical of a good metallic reflector, it settles at 1936 K. The temperature grows only as the fourth root of the absorbed power, so a mirror a thousand times better is only 5.6 times cooler.

The emissivity is the trap. A surface that reflects well tends to absorb little and, by the law that ties a surface’s glow to its absorption, to emit little at the same wavelength. What a sail needs is to reflect nearly perfectly at the laser’s wavelength and to radiate strongly at the much longer wavelengths of its own heat. Kirchhoff’s law does not forbid that — it binds absorption and emission only wavelength by wavelength — but it does mean that a plain metal film, which reflects well everywhere, is close to the worst choice, and that the sail has to be a structured dielectric designed separately for two parts of the spectrum.

The motion helps slightly. In the sail’s own frame the beam arrives redshifted and weakened by $(1-\beta)/(1+\beta)$, so at 0.2c the absorbed power is two thirds of its value at rest and the temperature 0.904 of it.

Energy that can be gathered slowly #

The ten terajoules a gram needs to reach a fifth of the speed of light is about three gigawatt-hours — a large power station’s output for three hours, or about a week’s output from a hundred-megawatt solar farm. The energy is not the problem; the rate is. A hundred gigawatts is several per cent of the world’s average electrical consumption, and it has to be delivered for a few minutes.

That is the sense in which a beamed sail moves the energy problem somewhere it can be solved slowly. A rocket must carry its energy, and so pays for it exponentially in mass. A sail can have its energy gathered over days, stored, and released in the minutes before the spot outgrows it. What cannot be stored is the aperture: diffraction sets the runway at launch, and no amount of preparation lengthens it.

The trade has a cost at the far end. The laser is at home, so nothing slows the sail at its destination; the rapidity it reaches is spent crossing the gap and it arrives at full speed. A flight to the nearest stars at a fifth of the speed of light takes about twenty years and passes through in hours. Stopping needs another mechanism — a magnetic field dragging on the thin interstellar gas, or a second sail reflecting light from the destination star — and each trades the propellant it saves for a difficulty of its own.

Where the sail stops being this simple #

The beam was a perfect Gaussian from a filled aperture, refocused on the sail continuously. A real array of separate lasers, phased across a kilometre, delivers a beam with side lobes and less power in the central spot, and the atmosphere adds turbulence that phasing has to correct. The coasting speeds in the figures are upper bounds for their apertures.

Light’s travel time was ignored. At three million kilometres the beam takes ten seconds to arrive, so it has to be pointed and focused on where the sail will be, not where it is, with no feedback faster than twenty seconds for the round trip. The steering of a sail being pushed at 68,000 g is a control problem the calculation does not touch.

The sail was held flat and centred. A flat mirror in a beam is unstable: pushed off the beam’s axis, it is pushed further off, and tilted, it is tilted further. A sail that rides a beam has to be shaped — curved, or patterned so that the light it deflects pushes it back towards the centre — and whether a gram-scale shape can be made stable under that acceleration is not yet settled.

The sail was a mirror with no structure. At 68,000 g a four-metre film is under large stress, and at a fifth of the speed of light each grain of interstellar dust arrives with the kinetic energy of an explosive many times its mass. Neither the survival of the film during the push nor its erosion during the cruise appears in any figure.

And the push was purely along the beam. Any reflectance less than perfect or any tilt adds sideways force and heating, and the absorbed momentum of a partly black sail changes both the thrust and the temperature.

What the figures leave out #

The mission figure puts the Moon and one astronomical unit on its distance axis for scale, and a logarithmic axis gives every decade of distance the same width, which is the opposite of how the flight spends its time. The sail is at 0.181c within two minutes, before the dashed line; the slow climb to its coasting speed to the right of that line takes hours, and occupies less of the figure than the first hundred kilometres.

The rapidity figure measures energy in units of the sail’s own rest energy, which is the natural unit and a misleading one: 0.1124 of the rest energy of a gram is ten terajoules, and of a tonne it is ten million times that. The curves are the same for any mass; the beam is not.

Still open: whether a sail can ride its beam #

Every number in the figures assumes a sail that stays in the beam. A flat mirror does not, and the question of how to make one that does, at a gram and several metres, under tens of thousands of g, in a beam whose own pointing wanders, is open. Designs have been proposed that use the diffraction of light by patterned surfaces — gratings and metasurfaces that bend part of the reflected beam sideways — to push a displaced or tilted sail back to the centre, and simulations show restoring forces for small disturbances. Whether such a structure can be made light enough, reflective enough and cool enough at the same time, and whether its stability survives the real beam’s imperfections and the sail’s own flexing, has not been shown in any experiment at the relevant scale.

The habit worth carrying away is to ask what a propulsion scheme is limited by once its energy is paid for. For a rocket, the limit is the logarithm of what it carries; for a sail, it is the distance over which diffraction lets the energy be handed over, and a beam that cannot stay narrow cannot deliver its energy however much of it there is.