The fuel a starship needs

A rocket goes forward because its exhaust goes backward, and the arithmetic of that trade gives Tsiolkovsky’s equation: the achievable change in speed is the exhaust speed times the logarithm of the mass ratio. The logarithm is the whole difficulty, because a logarithm grows very slowly and its argument therefore has to grow very fast.

Relativity changes one thing about that. Velocities do not add, so the exhaust’s contribution to the ship’s speed cannot simply be accumulated; what accumulates is the rapidity, and the equation survives with that substitution.

What survives and what changes #

Speeds do not add and rapidities do. The rapidity $\phi$ is defined by $\tanh\phi = v/c$, and a boost followed by another boost has the rapidity of the sum.

Since a rocket’s acceleration is a continuous sequence of small boosts, each in the frame the ship is momentarily in, what accumulates through the burn is the rapidity. The relativistic rocket equation is therefore Tsiolkovsky’s with the speed replaced by a rapidity:

$$\operatorname{artanh}\!\left(\frac{v}{c}\right) = \frac{v_{\text{e}}}{c}\ln\frac{m_0}{m_1}$$

and the mass ratio needed is $\exp\!\left[(c/v_{\text{e}})\operatorname{artanh}(v/c)\right]$.

Rapidity is the quantity that adds under successive boosts, and it runs to infinity as the speed approaches light’s. That is why the rocket equation is best written in it: the awkward composition law for velocities becomes ordinary addition, the exponential in the mass ratio stays an exponential, and a mission budget becomes a sum of legs rather than a sequence of compositions.

Two features of that are worth extracting. Below about a tenth of light speed the artanh is indistinguishable from its argument, so the relativistic and Newtonian answers agree — the dashed and solid curves in the opening figure overlap there, and every rocket ever flown lives in that region.

Above it they part, and they part in the direction that is usually got backwards. The relativistic mass ratio is smaller than the Newtonian one at every speed, because artanh grows more slowly than its argument would need to for the two to agree. Relativity makes the propellant problem slightly easier, not harder.

The number that ends the discussion #

Put the numbers in and the chart’s message is blunt.

A chemical rocket has an exhaust speed of about 4.4 kilometres per second, which is $1.5 \times 10^{-5}$ of light’s. Reaching a tenth of light speed therefore needs a mass ratio of $10^{2968}$.

There are about $10^{50}$ atoms in the Earth and about $10^{80}$ in the observable universe. The requirement is not a difficult engineering problem; it is an arithmetic impossibility by two thousand nine hundred orders of magnitude, and no improvement in structural mass fraction, staging or manufacturing touches it.

It is worth noticing where the impossibility comes from, because it is not relativity and it is not a shortage of engineering. It is the ratio of the required speed to the exhaust speed appearing inside an exponential. A rocket is good at reaching speeds comparable with its exhaust speed and hopeless at reaching speeds many times it, and the transition is savage: a mass ratio of 3 gets one exhaust speed, 20 gets three, 22,000 gets ten, and $10^{43}$ gets a hundred. Every propulsion difficulty in the solar system is about getting a handful of exhaust speeds; every interstellar one is about getting thousands, and no exponential tolerates that.

That is the honest content of the chart. Chemical propulsion is not slow for interstellar purposes, it is not in the discussion. Staging improves matters within the solar system by a factor of a few and does nothing at all here, because it attacks the structural fraction rather than the exponent.

What the good end looks like #

The other end of the chart is more cheerful and is worth reading carefully, because it is where the shape of the answer changes.

A photon rocket — exhaust at exactly the speed of light, which is the best any drive can do — has $v_{\text{e}} = c$, and the mass ratio collapses to

$$\frac{m_0}{m_1} = \sqrt{\frac{1+\beta}{1-\beta}}$$

which is 1.732 at half light speed, exactly the square root of three, and 4.36 at nine tenths. Those are ordinary numbers. A ship that could convert mass into a collimated beam of light with any efficiency would reach a substantial fraction of light speed on a mass ratio a chemical rocket uses to reach orbit.

There is a counterintuitive point in there that is often got wrong. A photon carries momentum $E/c$, which is the least momentum per unit energy of any possible exhaust; a massive particle carries $\gamma m v$, which for the same energy is larger at every sub-light speed. So a photon rocket is the worst drive by momentum-per-joule and the best by mass ratio, and the two measures answer different questions. What the logarithm cares about is how fast the exhaust leaves, because that is what multiplies it.

Between the extremes, an exhaust at a tenth of light speed — which is roughly what fusion products carry — gives a mass ratio of about 200 to reach a tenth of light speed. That is very large and it is not absurd, which is why every serious interstellar proposal is either fusion, antimatter, or something that carries no propellant at all.

Where the exhaust speed comes from #

Since everything turns on the exhaust speed, it is worth asking what sets it, because the answer is the same in every case and explains the ordering of the curves.

The fraction of a fuel’s rest mass that can be released decides how fast its products leave, and that fraction is set by the binding-energy curve. Chemical burning manages a part in $10^9$, fission about a thousandth, fusion about seven parts in a thousand, and annihilation all of it — so the exhaust speed available spans four orders of magnitude and the mass ratio required spans very much more than that.

An exhaust leaves with a kinetic energy supplied by whatever reaction released it, so its speed is set by the energy released per unit mass. That fraction is the whole story.

Chemical. A bond rearrangement releases a few electronvolts per molecule out of a rest energy of tens of GeVs, a fraction of about $10^{-10}$. The exhaust speed follows as roughly $\sqrt{2\times10^{-10}}\,c$, which is four kilometres per second — the number every chemical rocket has, whatever it burns.

Fission. About 0.1 per cent of the rest mass, giving products at a few per cent of light speed. A nuclear thermal rocket does far worse than that because it heats a working fluid rather than expelling the fragments, and the working fluid’s temperature is limited by the reactor’s materials.

Fusion. About 0.4 per cent, giving products near a tenth of light speed. That is the best any process short of annihilation offers, and it is the exhaust speed most serious interstellar designs assume.

Annihilation. All of it, in principle, giving a photon rocket. Nothing beyond it exists, because there is no more mass to convert.

So the chart’s five curves are not five engineering choices but four physical processes and a limit, and the gaps between them are gaps between binding energies rather than between designs. That is why no amount of engineering moves a chemical rocket towards the fusion curve: the exhaust speed is set by chemistry, and chemistry releases what it releases.

The trip nobody costs properly #

Every number above is for accelerating. The rest of the journey is what turns a large number into an impossible one.

Drawn in spacetime, a voyage under constant proper acceleration has four legs — accelerate, coast, decelerate, and the same again to come home — and each costs the same rapidity. That is what makes rapidity the right currency: it adds where velocity does not, so a round trip is four times one leg rather than some awkward composition, and the fuel requirement is an exponential in the total.

Stopping squares the ratio. Arriving at another star at a tenth of light speed is arriving as a very energetic collision, so the ship must decelerate, and that costs the same rapidity again. Since the mass ratio is an exponential of the rapidity, doing it twice squares it.

Coming home squares it again, and then once more for the final stop — so a return mission with all four legs under power needs the fourth power of the one-way accelerating ratio. The fusion drive’s 200 becomes $1.6\times10^9$.

Mass and energy being the same thing is what makes the good end of the chart good, and it is worth being clear that the relation is doing real work rather than decorating the argument. A photon rocket’s mass ratio is small because the propellant’s entire rest energy is available as exhaust kinetic energy; a chemical rocket’s is enormous because a bond rearrangement releases a ten-billionth of it. The equation is the same equation in both cases, and what differs is a number supplied by a quite different part of physics.

The way out of that is not to carry the propellant, which is why every proposal that survives arithmetic uses something external: a light sail pushed by a laser left behind, a magnetic sail braking against the interstellar medium, or a ramjet collecting fuel on the way. Each replaces the mass ratio with a different difficulty, and each of those difficulties is severe — but they are not exponential in the trip length, which is the only property that matters.

The other three problems #

Propellant is the difficulty that can be computed exactly, which is why it dominates the discussion. It is not the worst one.

Three problems sit outside the fuel calculation entirely, and each is capable of ending the design on its own: what the ship runs into on the way, where the waste heat goes when there is nothing to conduct it to, and the deceleration at the far end, which costs the same again and is routinely left out of the arithmetic. The fuel equation is a lower bound on one requirement among several, and it is already the discouraging one.

The power. Reaching a tenth of light speed with a thousand-tonne ship requires about $4.5\times10^{20}$ joules of kinetic energy. Delivering that over ten years is $1.4\times10^{12}$ watts, which is about a tenth of humanity’s total power consumption, sustained, into one vehicle — and that is the energy that ends up in the ship, before any efficiency.

The dust. At a tenth of light speed a one-gram grain arrives with $4.5\times10^{11}$ joules, the energy of a hundred tonnes of TNT. The interstellar medium contains roughly one hydrogen atom per cubic centimetre and dust besides, so a shield is not optional and its mass is part of the payload.

And the time. Even at a tenth of light speed the nearest star is forty years away, and time dilation does not help at that speed — the Lorentz factor is 1.005. Dilation only becomes a useful passenger benefit above about 0.8c, which is where the mass ratios have already become severe.

Those three are why the mass-ratio chart is a lower bound on the difficulty rather than a statement of it.

What a rapidity budget looks like #

There is a bookkeeping advantage to working in rapidity that is worth having, because it makes the four-leg arithmetic trivial instead of awkward.

Adding two speeds requires the composition law and gives an answer smaller than the sum; adding two rapidities is addition. That is why a mission profile is drawn as a rapidity budget — accelerate, coast, decelerate, and home — with the legs simply summed, and why the fuel requirement is an exponential of that sum rather than of anything a velocity budget would produce.

A mission’s total requirement is the sum of the rapidities of its legs, because rapidity is what adds. A four-leg round trip at a tenth of light speed is four times $\operatorname{artanh}(0.1) = 0.1003$, which is 0.4013 — and the mass ratio is the exponential of that divided by the exhaust speed, in one step.

That is the same convenience the Newtonian budget has, where delta-v is added leg by leg and the mass ratio computed once at the end. Relativity does not take it away; it replaces the quantity being added, and the replacement reduces to the old one below a tenth of light speed.

The reason it works is worth stating, because it explains why rapidity is the natural variable and velocity is not. Successive boosts in the same direction form a group, and rapidity is the parameter in which the group law is addition — the same role an angle plays for rotations. A velocity is the hyperbolic tangent of that parameter, which is why velocities saturate at $c$ while the parameter behind them does not, and why every relativistic acceleration problem simplifies when written in it.

The trade the rocket equation does not contain #

The chart says the exhaust speed is everything and implies that a designer should make it as large as possible. Within the solar system that is wrong, and the reason is a quantity the rocket equation has no room for.

Thrust is the mass flow times the exhaust speed, and the power carried away in the jet is half the mass flow times the exhaust speed squared. Eliminating the mass flow between them gives

$$P = \tfrac12 F v_{\text{e}},$$

so at a fixed power the thrust falls in proportion as the exhaust speed rises. Doubling the exhaust speed halves the propellant needed and halves the thrust available.

For a chemical rocket that trade does not bite, because the power comes from the propellant itself: burn twice as fast and twice the power appears, at no cost in hardware. For an electric drive it bites hard, because the energy comes from a solar array or a reactor whose mass has to be carried and whose output is fixed. High exhaust speed then buys a small propellant tank and a large power plant, and there is an optimum somewhere between.

Where the optimum sits depends on how long the mission may take and on how many kilograms of power plant each watt costs. A longer mission tolerates lower thrust and therefore favours a higher exhaust speed; a lighter power plant does the same. The result is that electric propulsion is chosen for missions measured in years and never for a launch from a planet, where the required thrust exceeds the weight and no plausible power plant supplies it.

The numbers on the flown examples make the trade concrete. The ion engine on the Dawn spacecraft produced about ninety millinewtons of thrust — the weight of a postcard — from two and a half kilowatts, at an exhaust speed of thirty kilometres a second, seven times a chemical engine’s. Running it for nearly six years of accumulated thrusting gave a total speed change of some eleven kilometres a second, the largest of any spacecraft, using 425 kilograms of xenon out of a launch mass of 1,240.

A chemical stage delivering the same eleven kilometres a second would have needed a mass ratio of twelve, which is ninety-two per cent propellant. Dawn’s was thirty-four. That factor is what the exponent does when the exhaust speed is multiplied by seven, and it is why every mission that can afford to be slow now flies electric.

The lesson for the interstellar case is the discouraging one. The chart’s fusion curve assumes an exhaust at a tenth of light speed, and the power relation says that a drive at that exhaust speed producing usable thrust must handle power at a level that dwarfs the energy figures in the section above. The mass ratio is the easy half of the problem, and it is the half that can be calculated.

The speed that is borrowed rather than carried #

There is one way of changing a spacecraft’s speed that does not appear anywhere in the rocket equation, because no propellant is expended at all, and it is the reason the outer planets have been visited.

Fly past a planet. In the planet’s own frame the encounter is a deflection: the spacecraft comes in at some speed, follows a hyperbola, and leaves at exactly the same speed in a different direction, because gravity is conservative and nothing has been exchanged.

In the Sun’s frame it is not a deflection. The planet is moving, and the spacecraft’s velocity in the Sun’s frame is its velocity relative to the planet plus the planet’s own. Rotating one vector and adding a fixed one changes the length of the sum, and in the most favourable geometry it changes it by twice the planet’s orbital speed.

Jupiter orbits at 13.1 kilometres a second, so the theoretical ceiling on a single Jovian assist is 26 kilometres a second. Voyager gained about ten, which took it from an orbit that would have fallen back toward the Sun to one that leaves the solar system.

The energy is not free; it is taken from the planet. Jupiter loses exactly the kinetic energy the spacecraft gains and slows in its orbit accordingly — by a fraction of order the mass ratio of the two bodies, which is $10^{-24}$, and which no measurement will ever resolve. It is the same accounting as an elastic ball bouncing off an approaching bat: the ball leaves faster, and the bat is imperceptibly slower.

What it costs instead is timing. An assist requires the planet to be in the right place, and a sequence of them requires several planets to be, which is why the alignment that allowed one spacecraft to visit Jupiter, Saturn, Uranus and Neptune recurs about every 175 years.

A related trick belongs in the same paragraph. Since kinetic energy goes as the square of the speed, a given change of speed adds more energy when the ship is already moving fast — so a burn made at the bottom of a gravity well, where the ship is at its quickest, is worth more than the same burn made far away. That is the Oberth effect, it is why an interplanetary injection burn is done at periapsis rather than in cruise, and it is another gain the rocket equation’s bookkeeping does not display, because the equation gives a change of speed and says nothing about where in an orbit it is spent.

What the equation assumes #

Two idealisations are built into every number here, and both are generous.

The structural mass is zero. The ratio computed is the total starting mass over the final mass, and the final mass includes tanks, engine and structure as well as payload. A real vehicle’s payload fraction is a fraction of the already-tiny inverse mass ratio, and staging exists precisely to shed structure on the way.

And the exhaust is perfectly collimated and perfectly efficient. A real drive throws some of its exhaust sideways, which contributes nothing to the thrust and everything to the mass consumed; a photon drive would need to convert mass to a beam with no waste heat, which no known process approaches. Both idealisations make the chart optimistic.

What the pictures cannot show #

The acceleration is unspecified. The rocket equation contains no time and no acceleration; a burn of one second and a burn of one century give the same mass ratio for the same speed change. Everything about how long the trip takes is a separate calculation.

Nothing carries a payload. Every ratio is the ship’s own mass, so a mission carrying anything at all is worse by the payload fraction.

Nothing radiates. A drive releasing enough energy to accelerate a ship also produces waste heat that has to be radiated away, and at these powers the radiator area is a design driver comparable with everything else. It contributes no thrust and considerable mass, and it is absent from every curve here.

And the vertical axis is decades. A curve that leaves the top of the chart is not merely off-scale, it is off-scale by an amount that grows without bound, and the chemical curve leaves it before the chart begins.

The ladder from here #

Later rungs on this anchor: the relativistic ramjet and why Bussard’s version does not close; the light sail with the accelerator left behind, which moves the energy problem to a place where it can be solved slowly; deceleration against a magnetic field or the interstellar medium, which costs no propellant and a great deal of drag; and antimatter as a propellant, where the mass ratio is manageable and the production energy is not.

The neighbouring ladders are the push that needs nothing to push against, which is the momentum bookkeeping this equation integrates, speeds that refuse to add, which is where the rapidity comes from, and the ship that never arrives at c, which is the same journey seen from the passengers’ clocks.