Choosing a propellant is choosing a molecular weight

Every figure on this anchor so far has taken the exhaust speed as an input. The first rung drew propellant fraction against velocity for three named combinations and read their exhaust speeds off a table; the third divided a budget between two stages whose exhaust speeds differed and never asked why.

The number is not a fact about rocketry. It is a fact about a hot gas expanding through a hole, and it has exactly two inputs — one of which is bounded by chemistry and the other of which is not.

Where the formula comes from #

A rocket chamber is a reservoir of hot gas at high pressure and near-zero velocity. A nozzle converts its enthalpy into directed kinetic energy, and for an ideal gas expanding isentropically the conversion is exact bookkeeping: the enthalpy per unit mass at the chamber is cₚT_c, at the exit it is cₚT_e, and the difference has gone into ½v². Writing cₚ in terms of the ratio of specific heats and the gas constant, and the temperature ratio in terms of the pressure ratio, gives

$$v_e = \sqrt{\frac{2\gamma}{\gamma-1}\,\frac{R_u T_c}{M}\left[1 - \left(\frac{p_e}{p_c}\right)^{(\gamma-1)/\gamma}\right]}$$

with R_u the universal gas constant and M the mean molar mass of the exhaust. Three of the four factors are nearly fixed. The ratio of specific heats for a hot polyatomic exhaust is between 1.15 and 1.25 whatever it is made of. The pressure ratio is set by the nozzle’s area ratio and reaches 99 per cent of its asymptotic contribution by an expansion of about a hundred to one, so it saturates. What is left is T_c/M.

That is why the figure has the shape it has, and why the two axes it could have been drawn against are not equivalent. Temperature is bounded — by the chemistry of what is being burned and by what the chamber wall can survive — and the bound is around 3,700 K for anything chemical. Molecular weight is bounded below by hydrogen and has a factor of twenty of range above it. The lever with room in it is the denominator.

The hottest is not the fastest #

The propellant combinations marked on the figure make the point better than an argument does.

Kerosene and oxygen burn at 3,670 K, hotter than anything else in ordinary use, and deliver about 298 seconds of specific impulse. Hydrogen and oxygen burn cooler, at about 3,500 K in a flight engine, and deliver 441. The ratio of temperatures is 0.95; the ratio of molecular weights is 10 to 23, and the square root of the ratio of those two ratios is 1.48, which is the whole of the difference.

The molecular weight of 10 is itself a choice. Stoichiometric hydrogen–oxygen makes water, molar mass 18, and would run at nearly 4,000 K. Flight engines run at a mixture ratio of about 6:1 by mass rather than the stoichiometric 8:1, deliberately fuel-rich, so the exhaust is water plus a great deal of unburnt hydrogen and its mean molar mass falls to around 10. That trade lowers the flame temperature and raises the specific impulse, which is the formula’s advice taken literally.

There is a limit to how far the trade goes, and it is worth knowing that the fuel-rich mixture is not chosen by this formula alone. Running richer than 6:1 lowers the molar mass further and lowers the temperature further, and the two effects cross: past about 4:1 the temperature is falling faster than the molar mass and the specific impulse turns over. Flight engines sit near the peak of that curve, displaced a little to the rich side because unburnt hydrogen also cools the chamber wall — an engineering constraint arriving at the same answer as a thermodynamic one, which is the usual situation.

The same reasoning is what makes nuclear thermal propulsion interesting on paper. A reactor heating pure hydrogen has M = 2 rather than 10 and no combustion temperature ceiling at all, only a material one — and √(T/M) with the denominator cut by five gives about 900 seconds against a chemical engine’s 450. That is the one route to doubling the exhaust speed that chemistry cannot supply, and it has been demonstrated on the ground and never flown.

The convention that hides the physics #

Exhaust speed is a velocity and specific impulse is quoted in seconds, and the relationship between them is a piece of history rather than of physics: I_sp = v_e/g₀, with g₀ the standard gravity, 9.80665 m/s² exactly.

The division has no physical content at all. It exists because the quantity was originally defined as thrust per unit weight flow rather than per unit mass flow, in an era when propellant was weighed rather than massed, and the weight in question was weight on Earth. A rocket firing on Mars has the same specific impulse; so does one in free space. The seconds are not a duration of anything except in the strained sense that a rocket producing one newton of thrust could do so for I_sp seconds while consuming one newton’s worth of Earth-weight of propellant.

What the convention costs is visibility. Written as an exhaust speed, the number in the rocket equation is manifestly a velocity that the vehicle’s own Δv is compared against, and the ratio Δv/v_e is what the exponential carries. Written in seconds, it is a figure of merit with no obvious relationship to anything, and the comparison that decides a mission — is my Δv one exhaust speed or three? — has to be reconstructed by multiplying by 9.8.

What the number is worth #

Before the complication, it is worth pricing the exhaust speed against everything else in this anchor, because it is not one lever among several.

The rocket equation reads the ratio Δv/vₑ and nothing else. A 48 per cent improvement in exhaust speed therefore reduces that ratio by a third, and since the propellant fraction is 1 − e^(−Δv/vₑ), the effect on the mass ratio is exponential. For the 9.4 km/s of a low orbit: kerosene’s 2.94 km/s of exhaust speed needs a mass ratio of 24.4, and hydrogen’s 4.41 needs 8.4. Three times less mass has to be lifted for the same payload.

Set that against what dividing the budget optimally buys — 21 per cent of the payload — or what a better structural coefficient buys, or what a launch site’s latitude is worth. Nothing else in the subject is a factor of three. The exhaust speed is the only quantity in the rocket equation with that kind of leverage, and it is fixed in a combustion chamber before any trajectory is planned.

What the equation cannot see #

Hydrogen wins on the only axis the rocket equation has. It is not used on first stages, and the reason is a quantity the equation does not contain.

A tank’s mass scales with its surface area for a given wall thickness, and its volume with the propellant it holds, so a low-density propellant costs structure. Liquid hydrogen costs more than that: it boils at 20 K, so the tank needs insulation and the vehicle needs to tolerate boil-off, and the tank cannot be a load-bearing thin-walled shell in quite the way a kerosene tank can. All of it lands in the structural coefficient ε, which the previous rung treated as a property of a stage handed down from outside.

It is not handed down. It is partly a consequence of the propellant choice, which means the two quantities the rocket equation reads — vₑ and ε — are not independent, and optimising one degrades the other. A hydrogen first stage buys 48 per cent more exhaust speed and pays perhaps 40 per cent more structural coefficient, and on a first stage, where the Δv is small in units of the exhaust speed and the mass being lifted is enormous, the structure wins.

On an upper stage it does not, and that is why upper stages are the place hydrogen appears. The Δv is large in units of the exhaust speed, the mass being lifted is small, and the tank has no atmosphere to push through.

The reversal has an exception worth naming, because it shows the trade is about the mission and not about the propellant. A low-thrust electric stage uses xenon, which is heavy — molar mass 131 — and reaches three thousand seconds anyway, because its exhaust is not thermal at all: the ions are accelerated electrostatically and the exhaust speed is set by a voltage rather than by a temperature. On that stage the density is excellent, the specific impulse is enormous, and the constraint is power. The formula above simply does not apply, and the figure of merit that replaces it is a different one.

That is the general point about density impulse rather than an aside. It is a crude figure of merit that captures a real coupling, and the coupling it captures is between the propellant and the tank. Anything that changes what a tank has to be — a solid grain that is its own structure, a xenon tank at three hundred atmospheres holding a hundredth of the mass — moves the comparison somewhere the two axes cannot express.

The nozzle, and why the number quoted is two numbers #

One further complication is worth stating because it is why a single stage has two specific impulses.

The pressure-ratio term in the formula reaches its asymptote only if the exhaust is expanded all the way to vacuum, which needs an infinite nozzle. A real nozzle has an area ratio, and it is sized so that the exit pressure matches the ambient pressure at some chosen altitude. Below that altitude the flow is over-expanded — the exit pressure is below ambient and the atmosphere pushes back — and above it the flow is under-expanded and some of the available energy is left in the gas.

A sea-level engine therefore has a modest area ratio and a specific impulse that rises through the ascent as the back-pressure falls; a vacuum engine has an enormous bell and a specific impulse quoted for vacuum alone. The difference is around ten per cent for the same propellant. That is why the marked numbers in the figures here are vacuum values, why a first stage’s effective exhaust speed over its whole burn is an integral rather than a number, and why the “specific impulse” of a stage is a shorthand for a quantity that changes by a tenth while the stage is using it.

Two engines with the same propellant and different numbers #

A last complication is worth a paragraph because it explains why published figures for the same combination disagree.

Chamber pressure enters the formula only through the pressure ratio, and that term saturates — but it saturates at a ratio, and the exit pressure is fixed by the ambient. So a higher chamber pressure allows a larger expansion ratio at the same nozzle exit pressure, and therefore a higher exhaust speed, for the same propellant and the same flame temperature. It also allows a smaller engine for the same thrust, which reduces the mass the stage carries.

That is why chamber pressures have climbed from about 70 bar on a Saturn V’s F-1 to over 300 on a modern staged-combustion engine, and it is most of the difference between two kerosene engines quoted at 300 and 330 seconds. The propellant did not change. What changed is how far it was expanded, and how far it could be expanded was set by how hard it could be pushed into the chamber.

What the picture cannot show #

The formula drawn is the ideal one-dimensional isentropic expansion of a calorically perfect gas, and three things in a real engine depart from it.

The exhaust is not calorically perfect. It is a mixture whose composition changes as it cools and expands — recombination of dissociated species in the nozzle releases energy and raises the exhaust speed by a few per cent over the frozen-composition answer, and how much depends on how fast the flow is going relative to the chemistry, which is not a thermodynamic question. Published specific impulses lie between the frozen and equilibrium limits and are computed with a chemistry code rather than with this formula.

The flow is not one-dimensional. A real nozzle’s exhaust has a radial velocity component that contributes nothing to thrust, and the divergence loss costs a per cent or two for a conventional bell. And the boundary layer on the nozzle wall is neither isentropic nor at the core temperature, which costs another fraction.

Taken together those corrections are of order five per cent and they do not change the ranking of anything on the figures. What they do mean is that the numbers here are the physics rather than the engineering: they say why hydrogen beats kerosene by about fifty per cent and they should not be used to choose between two combinations that differ by ten.

The habit #

The structure worth extracting is that a performance number often depends on a ratio of two quantities of which only one is usually discussed, and that the neglected one has the wider range.

Chamber temperature is what a reader expects to matter, it is what a rocket looks like it is about, and it has a range of about a factor of three across everything chemical. Molar mass is invisible in the picture and has a range of a factor of sixty between hydrogen and xenon. The lever with room in it is the one nobody looks at.

The same shape recurs across this collection. A star’s luminosity depends on its central temperature to a large power and on its composition weakly, so a thermometer built from a nuclear rate is precise about temperature and blind to everything else. A transit depth is a ratio of radii and carries no dependence on distance at all, which is why the method works on stars whose distances are unknown. In each case the useful question is not what the number depends on but which of its dependences has range.

What is actually measured, and where the numbers come from #

A specific impulse is not read off a formula in practice. It is measured on a test stand, and the measurement is worth describing because it explains why published values carry qualifiers.

The engine is mounted on a thrust frame with load cells, in the same way a laboratory measures any force it cannot compute, the propellant flow rates are measured by turbine or Coriolis meters in both feed lines, and the specific impulse is the thrust divided by the total weight flow. That is a direct measurement of exactly the defined quantity, to a fraction of a per cent, and it is done at sea level — where the nozzle is over-expanded and the number is not the one anybody wants.

Getting the vacuum figure requires either an altitude chamber, which is a large vacuum vessel with a diffuser to swallow the exhaust and is available at only a handful of facilities, or a correction: measure at sea level, add the exit area times the ambient pressure, and quote the result. The correction is exact in principle — the thrust difference between two ambient pressures is A_e Δp and nothing else — and it assumes the flow does not separate from the nozzle wall, which at a large area ratio at sea level it does. An engine with a vacuum bell cannot be fired at sea level at all without destroying itself.

So the specific impulses quoted for upper-stage engines are altitude-chamber measurements or corrected sea-level ones, those for first-stage engines are usually measured directly at both conditions, and the ones for combinations rather than engines — the numbers on the figures here — are computed from a chemical equilibrium code and are theoretical maxima that no engine reaches. The gap between the theoretical and the delivered figure is the nozzle efficiency, and it is 96 to 98 per cent for a good bell. That is worth knowing before comparing a number from a textbook with a number from a data sheet: they are measuring different things, and the difference is larger than most of the distinctions being argued about.

Where the ladder goes next #

Two rungs have now taken the rocket equation’s two inputs — how the budget is divided and where the exhaust speed comes from — and both have treated the budget itself as given. A low Earth orbit costs 9.4 km/s, the essays say, and orbital speed at that altitude is 7.8.

The next rung is the difference. It is not overhead in any accounting sense: it is three integrals along the ascent, and only one of them can be reduced by flying better. Computing them requires integrating a trajectory rather than quoting a budget, and the answer contains a trade — a vehicle that leaves the pad faster fights gravity for less time and pushes air aside harder — whose two halves move in opposite directions and whose sum barely moves at all.

Further rungs: the relativistic rocket equation, where the mass ratio becomes exponential in the rapidity rather than in the speed; nuclear thermal propulsion, which raises the chamber temperature and lowers the molecular weight at once by heating pure hydrogen rather than burning it; and the specific-impulse-versus-power trade for electric propulsion, which is where the exhaust speed stops being a property of a propellant and becomes a design variable.