An equation that does not break at the speed of light

The first rung of this anchor established the exponential and its consequence: every extra kilometre a second costs a larger share of what remains, so a mission’s Δv measured in units of the exhaust speed is the only quantity the equation reads. Every rung since has stayed within a few units of that ratio — a low orbit is 2.1 exhaust speeds on hydrogen, a Mars transfer about 3.

At speeds where velocities no longer add, the equation changes form. It does not change conclusion.

Why the rapidity is the right variable #

The Newtonian derivation is momentum bookkeeping: expel dm of propellant at relative speed vₑ, gain dv = vₑ dm/m, integrate. The relativistic derivation is the same bookkeeping with the correct momentum, and the reason it produces a different variable is that velocities do not add.

If a rocket at speed v expels propellant at vₑ relative to itself, the propellant’s speed in the original frame is not v − vₑ but the relativistic composition of the two. Successive small boosts therefore do not sum to a velocity. What they do sum to is the rapidity,

$$\varphi = \operatorname{artanh}(\beta), \qquad \beta = v/c$$

which is additive under composition by construction: two boosts of rapidity φ₁ and φ₂ in the same direction give a boost of φ₁ + φ₂, exactly, at any speed. Rapidity is to velocity what an angle is to a slope, and Lorentz boosts are hyperbolic rotations in the same sense that ordinary rotations are rotations — the same hyperbolic geometry an unbound orbit is drawn on, where the eccentric anomaly is also replaced by its hyperbolic analogue and the equations keep their shape.

Redo the bookkeeping in rapidity and the derivation is Newtonian again — dφ = (vₑ/c) dm/m, integrate — and the answer is

$$\frac{m_0}{m_1} = \exp\!\left(\frac{c\,\varphi}{v_e}\right) = \left(\frac{1+\beta}{1-\beta}\right)^{c/2v_e}$$

The exponential is unchanged. What changed is what is in the exponent, and since artanh(β) diverges as β → 1, so does the mass ratio.

It is worth checking the limit rather than asserting it. For small β, artanh(β) = β + β³/3 + …, so the relativistic mass ratio exceeds the Newtonian one by exp(cβ³/3vₑ) — third order, which at the speeds every other rung of this anchor works in is a correction of order 10⁻¹⁵. The two forms are not merely close there; they are indistinguishable by any measurement that will ever be made of a chemical rocket. That is why the Newtonian equation was derived three times independently before anybody needed the other one.

Nothing forbids a speed #

That divergence is worth stating carefully, because it is not a prohibition.

For every β strictly below one there is a finite mass ratio that achieves it. The equation does not say that light speed is a barrier and it does not say anything about a maximum velocity; the constraint that light speed cannot be reached comes from the kinematics, not from the propulsion. What the equation says is that the cost of a speed rises without bound as the speed approaches c, and that the rise is exponential in a quantity that itself rises without bound — a double exponential in the ordinary sense of the word.

That structure is worse than a wall. A wall can be moved by better engineering; a divergence cannot be. Doubling the exhaust speed halves the exponent, which at low β is a large improvement and at β = 0.99 takes the mass ratio from 10⁵ to 10²·⁵, and nothing in the family ever becomes comfortable.

What an exhaust speed of a tenth of c means #

Every number on these figures is quoted as a fraction of c, and it is worth converting one of them.

An exhaust speed of 0.03c is 9,000 km/s, or a specific impulse of 917,000 seconds. The best chemical engine reaches 450. The best ion thruster flown reaches about 4,000, and the exhaust is xenon at perhaps 60 km/s. A fusion drive expelling helium ash at a realistic fraction of the reaction energy might reach a few per cent of c — the same reaction, at a millionth of the density, that runs a star — and that is the optimistic end of what a nuclear reaction can deliver: the energy per unit mass available from fusing hydrogen is 0.7 per cent of mc², so even perfect conversion into directed kinetic energy caps the exhaust speed at about 0.12c.

The only propellant that does better is antimatter, whose annihilation converts the entire rest mass, and even there the exhaust is gamma rays and charged pions that are difficult to direct. A photon rocket is the limit: exhaust at exactly c, mass converted entirely to directed radiation, and no propellant of any kind does better.

The last sentence of that caption is the honest summary of interstellar flight by rocket. Four times the mass ratio is not four times the difficulty; it is the fourth power, because each leg multiplies.

The number that a chemical rocket would need #

It is worth doing the arithmetic once for the case everybody’s intuition starts from, because the answer is the reason the subject exists.

A hydrogen–oxygen engine has an exhaust speed of 4.41 km/s, which is 1.47 × 10⁻⁵ c. Reaching a tenth of light speed needs a rapidity of artanh(0.1) = 0.1003, so the mass ratio is exp(0.1003/1.472 × 10⁻⁵) = e^6,816. That number has 2,961 digits. The mass of the observable universe is about 10⁵³ kilograms, and the mass of a proton about 10⁻²⁷, so the entire universe expressed in proton masses is 10⁸⁰ — a number with eighty-one digits.

The exponential is not a difficulty of degree. It is the reason chemical propulsion is not a candidate for interstellar flight at any scale of investment whatever, and the reason every serious proposal either changes the exhaust speed by five orders of magnitude or abandons the rocket.

The one assumption that is doing all the work #

Every figure here assumes the vehicle carries its propellant. That is what a rocket is, and it is the assumption the whole anchor has been built on since the first rung.

Remove it and the arithmetic changes completely. A solar sail carries no propellant and has no mass ratio, so no rocket equation applies to it — its acceleration falls as the inverse square of the distance from the source and its final speed is an integral rather than an exponential. A laser-pushed sail moves the energy source off the vehicle entirely, and its limit is the laser’s power and the sail’s ability to survive being pushed, neither of which is a mass ratio. A ramjet that scoops interstellar hydrogen carries no propellant either, and its problem is that the scooped material has to be accelerated to the vehicle’s own speed before it can be used, which at high β costs more than the fusion of it delivers.

None of those is easy and none of them is priced by anything on this page. What they have in common is that they escape the exponential rather than improving its exponent, which is the only kind of escape available.

The trip that is possible and the one that is not #

The figures price a final speed. What a mission cares about is a journey, and the two are related by an arithmetic worth doing.

Proxima Centauri is 4.24 light years away. A probe that accelerates to 0.1c, coasts, and does not stop takes 42 years of coordinate time plus the acceleration phases — a mission within a human career, and the reason 0.1c is the number every interstellar study picks. At 0.01c it is 420 years, which is not a mission but a civilisation-scale project. At 0.5c it is 8.5 years, and the mass ratio has gone from 1.11 to 1.73 for a photon rocket and from 28 to 9 × 10⁷ for anything at 0.03c.

Stopping doubles the exponent, which squares the mass ratio. A flyby at 0.1c on a fusion drive at vₑ = 0.05c needs a ratio of 7.4; stopping there needs 55. That single factor is why most serious proposals are flybys, and why the most-studied laser-sail concept does not decelerate at all: it passes through the system in a few hours, takes what data it can, and continues.

Coming back squares it again. There is no version of the arithmetic in which a crewed round trip at a useful speed is priced by a rocket, and the numbers on this rung are the reason rather than a failure of ambition.

What the divergence is, exactly #

The mass ratio diverges as β → 1, and it is worth knowing how fast, because “diverges” covers a range of behaviours.

Write the ratio as ((1+β)/(1−β))^(c/2vₑ). As β approaches one the numerator approaches two and the denominator approaches zero linearly, so the ratio behaves as (2/(1−β))^(c/2vₑ) — a power law in the deficit 1 − β, with exponent c/2vₑ. For a photon rocket that exponent is a half, so halving the deficit multiplies the mass ratio by √2; for a fusion drive at vₑ = 0.05c it is ten, so halving the deficit multiplies it by a thousand.

That is a milder divergence than the exponential-in-rapidity form suggests, and it is the reason the photon-rocket numbers in the figures are as small as they are. A vehicle at 0.99c is only fourteen times its dry mass in antimatter; at 0.9999c it is 141. The wall is not near c, and there is not really a wall — there is a power law whose exponent is set by how good the exhaust is, and every improvement in the exhaust flattens it.

Which is why the honest way to state the interstellar problem is not that light speed is unreachable. It is that the exponent c/2vₑ is 10⁴ for anything anybody can build, and a power law with an exponent of ten thousand is indistinguishable from a wall at any deficit worth naming.

What the picture cannot show #

Three things about a relativistic rocket are outside the equation and two of them make it worse.

The equation is one-dimensional and impulsive in spirit: it says what mass ratio buys what final speed and says nothing about how long it takes or what the crew experiences. A vehicle accelerating at one gravity in its own frame reaches 0.77c in one year of ship time — proper acceleration and coordinate acceleration part company in the same way proper and comoving distance do in cosmology — and 0.99993c in five, while the elapsed time outside runs to decades — the ship-time and coordinate-time accounts diverge, and the mass ratio in the figures is the coordinate-frame one.

The propellant has to be stored, and antimatter has to be stored without touching anything. Nothing in the mass ratio accounts for a containment system, and the containment system for kilogram quantities of antimatter is not a small fraction of the vehicle by anybody’s estimate.

And the exhaust has to be directed. A photon rocket radiating isotropically produces no thrust at all; one radiating into a hemisphere produces half of what the equation assumes; annihilation products that are charged pions decaying to muons decaying to neutrinos lose most of the energy to particles that cannot be steered. The “exhaust speed of c” in the figure is an idealisation of an idealisation.

The one thing that makes it better is that the vehicle’s mass falls as it accelerates, so the same thrust produces more acceleration — which is already in the equation and is the only reason the mass ratios are as small as they are.

That last observation deserves emphasis because it is the standard hope and it does not work. Staging helps in the Newtonian launch case because the structural fraction is what limits a single stage, not the exponential — a stage cannot carry a mass ratio of 25 because its tank weighs more than that. It buys nothing against a large exponent as such: the product of the per-stage mass ratios is the total mass ratio, and dividing the Δv into pieces does not change it. What staging removes is dead structure, and at a mass ratio of 10²⁶ the structure is not the problem.

Two things that are true at every speed #

Two features of the rocket equation survive the change of variable, and they are the features that make it a general statement rather than a piece of rocketry.

It has no time in it. The mass ratio for a given Δv is the same whether the burn takes a second or a decade, which is why an impulsive transfer and a low-thrust spiral can be compared on the same axis at all — and why the difference between them, when there is one, comes from something outside the equation: gravity acting during a long burn, or a trajectory that is no longer a conic. The relativistic form inherits the property exactly.

And it has no force in it. Thrust does not appear; only the exhaust speed and the mass expelled. A gramme of propellant leaving at a given speed buys the same rapidity whether it leaves in a microsecond or a year. That is why the thrust-to-weight trade of an ascent is a trade about losses rather than about the equation, and why a low-thrust interstellar mission and a high-thrust one need the same mass ratio and differ only in how long they take to spend it.

Both properties are consequences of the derivation being momentum conservation and nothing else. Anything a rocket does that the equation does not price is something other than momentum conservation acting on it.

Why this rung is in a collection about celestial mechanics #

The relativistic rocket equation has never been used to design anything, and it is worth being clear about why it belongs here anyway.

It is the one place in this collection where an engineering equation and a piece of special relativity meet without either being an approximation to the other. The Newtonian rocket equation is not a low-speed limit of a different equation; it is the same equation, written in a variable that happens to equal the velocity when velocities are small. Every other Newtonian result in this collection that has a relativistic correction — Mercury’s perihelion, the light deflection, the orbital decay of a binary pulsar — has one because a term was missing. Here nothing was missing. The variable was wrong.

That distinction is worth having because it changes what a correction means. A missing term is an approximation that fails when the term becomes large, and the failure is graded. A wrong variable is exact wherever the two variables agree and exact everywhere else too, once the right one is used — the equation was never approximate, and the Newtonian form is not an approximation to it but a special case written in a coordinate that happens to be additive at low speed.

There are very few results in physics with that character, and it is why the derivation is a paragraph rather than a chapter. Momentum conservation is momentum conservation; the only question is what adds.

Where the ladder goes next #

Six rungs of this anchor have taken one equation — a mass ratio, an exhaust speed, and an exponential between them — and asked what each of its parts is. The derivation from momentum. The wall that is structural rather than exponential. How a budget divides between unequal stages. Where an exhaust speed comes from. What a launch spends on top of orbital speed. And what happens when the speeds are large enough that velocities stop adding.

What is left divides into things that change the exponent and things that leave the framing. Nuclear thermal propulsion and nuclear pulse change it, by factors of two and thirty; the specific-impulse-against-power trade for electric propulsion changes it continuously and turns the exhaust speed into a design variable rather than a propellant property. Aerobraking, solar and laser sails, and a tether leave the framing entirely, because none of them carries what it accelerates against.

And one leaves it in a way that is available now rather than in principle. A vehicle departing a filled depot in orbit resets its mass ratio, so the inequality that this whole anchor is about — that the propellant is on board from the beginning and every kilogram of it has to be lifted by the kilogram before it — simply does not arise. Orbital refuelling is not a better rocket. It is a different arithmetic.