A booster that flies home pays with the stage above it

The rocket equation charges for everything a rocket carries by the same rule. Every kilogram that has to reach a given speed costs propellant in proportion to the exponential of that speed divided by the exhaust speed, and every kilogram of that propellant has to be lifted too. It is the exponential that decides what can be flown, and it is the reason rockets have stages: a stage whose tanks are empty is dead mass, and dropping it lets the rest go on without paying to accelerate it further.

For sixty years the dropped stages fell into the ocean or onto the steppe, and the rocket equation said nothing about whether they should. Since 2015 first stages have routinely flown back and landed, and several tens of them have flown ten times or more. A stage that is coming back is no longer dead mass at separation. It carries legs and fins, a structure strong enough to survive re-entry, and propellant in its tanks for the burns that bring it home — and all of that is mass the stage itself has to accelerate on the way up. The rocket equation charges for it at the rate it charges for everything else, and the equation has no special case for reuse — it prices every velocity change exponentially, whatever the kilograms are for.

What it charges is a question with a precise answer, and the answer depends less on the recovery hardware than on where the rocket separates its stages.

An old idea, flown late #

Recovering rockets is as old as the large ones. The Space Shuttle recovered its two solid boosters from the Atlantic after every flight, towed them back, took them apart and refilled them, and it recovered its orbiter — the upper stage and the payload bay together — by flying it back as a glider. It was the most reusable launch system ever built for thirty years, and one of the most expensive per kilogram. The boosters’ refurbishment cost nearly as much as new ones, and the orbiter carried wings, a heat shield, landing gear and a crew compartment all the way to orbit, which is the upper-stage penalty this essay ends on, paid in full.

In the 1990s a small experimental vehicle demonstrated vertical take-off and powered vertical landing at low altitude, and showed that a rocket could descend on its engine and touch down under control. It never went higher than a few kilometres. The first orbital-class booster to land after sending a payload toward orbit did so in December 2015, at the launch site, after a boost-back; a landing on a ship downrange followed in 2016. By the early 2020s boosters were flying tens of times each, and landing had become routine enough that a launch which lost its booster was news.

The difference between the Shuttle and the boosters that now land is the subject of this essay in miniature. The Shuttle recovered the stage the rocket equation charges at the full rate and refurbished at great cost the stage it charges at the reduced one; the modern boosters recover only the stage that is cheap to recover, and make it cheap to refly.

The split that moves #

A two-stage rocket must supply about 9.4 kilometres a second of ideal velocity change to reach low orbit — the orbital speed plus the losses to gravity and air on the way up — and how that total is divided between the stages decides how much payload it carries. The best division is not an equal one. It hands more of the work to whichever stage has the better exhaust speed, and for a rocket whose first stage burns kerosene at sea level and whose upper stage burns the same propellants in vacuum, the best expendable split asks the first stage for a little over four kilometres a second.

The opening figure’s black curve is that optimisation. Its peak is broad, which is why rockets built by different designers for different reasons all sit near it, and the rocket at the peak carries three and a half per cent of its lift-off mass to orbit — close to what real rockets of this kind achieve.

Recovering the first stage changes the curve’s shape. A stage that lands downrange, on a ship or a pad hundreds of kilometres out, keeps propellant for two burns: one that slows it as it re-enters the atmosphere, so that it does not overheat, and one that brings it to rest at the landing site. A stage that flies back to the launch site adds a third, larger burn first, the boost-back, which cancels most of the horizontal speed it had at separation and turns it around. The faster the stage was going when it separated, the more of all three it needs.

So recovery makes a fast first stage expensive in a way an expendable rocket never sees. The rocket responds by asking the first stage for less. The best split for a downrange landing moves from 4.1 to 3.1 kilometres a second, and for a return to the launch site to 2.5. The upper stage takes the difference.

What the toll depends on #

The second figure follows the toll as the staging speed changes.

The two curves diverge because of the boost-back. A downrange landing needs propellant mainly to slow the stage during re-entry and to land it, and those budgets grow gently with the staging speed. A return to the launch site needs, on top, a burn that cancels most of the horizontal speed — and a stage separating at three kilometres a second has to spend most of three kilometres a second to come back. That propellant multiplies the stage’s dry mass by the exponential of its own velocity change over the exhaust speed, and at three kilometres a second that is a factor of nearly three. The payload is nearly gone.

The figure also shows why a rocket designed for expendable flight is a poor rocket to recover. Its stages separate at the speed that is best for throwing the first stage away, near two and three-quarter kilometres a second, and recovering it there costs a fifth of the payload downrange and almost three-quarters flying back. A rocket designed around recovery separates earlier, and the same recovery costs a seventh and a quarter. The toll quoted for landing a booster depends on which rocket is doing the landing.

That difference is part of why the penalty figures quoted for recovery vary so widely. A booster recovered from a rocket that was designed to be thrown away, and later adapted, pays the larger toll; a rocket designed from the outset around recovery pays the smaller one, because its upper stage was sized to take on more of the work. Comparisons between launch vehicles that quote a single percentage for the cost of landing are comparing vehicles that were optimised for different questions.

The propellant costs more than the legs #

Recovery adds two things to the stage, and they are worth separating.

With no hardware at all — a stage that could somehow land on its existing structure — the return propellant alone costs seven per cent of the payload for a downrange landing and sixteen for a return to the launch site. That part of the toll is set by physics: the velocity change of the return, the exhaust speed, the exponential. The hardware adds to it, about three per cent of payload for each per cent of the stage’s mass that goes into legs, fins, thicker skins and heat protection. At the two per cent assumed in the opening figure, the hardware and the propellant cost about the same for a downrange landing; for a return to the launch site the propellant dominates.

The hardware is the part engineering can shrink. Lighter legs, fins that double as control surfaces, a structure that is reused because it was overbuilt rather than strengthened afterwards — each has been pursued, and each moves the toll toward the floor the return propellant sets. The floor itself moves only if the return needs less velocity change, which means separating slower or landing further away.

Or it moves if the air does the braking instead of the engine. A stage that sheds its speed by drag rather than by burning propellant trades the exponential cost of the return for the linear cost of whatever surfaces do the braking — the same trade that aerocapture offers an arriving spacecraft, and with the same difficulty, that the atmosphere is a less predictable brake than an engine. Some smaller launchers recover their boosters under parachutes and catch them, which removes the landing burn and most of the entry burn and replaces them with a parachute’s mass and a recovery operation at sea. In the rocket equation’s terms, that moves the toll from the exponential column to the linear one. Whether it is cheaper depends on how much the parachutes weigh against how much propellant they save, and for a stage separating slowly — at the speeds a recovery-designed rocket chooses — the propellant saved is not large.

The better the top, the cheaper the bottom #

The way the split moves suggests an unexpected connection between the two stages. A recovered first stage is cheapest when it separates slowly, and a rocket can only afford a slow first stage if its upper stage is good enough to supply most of the velocity change.

With a weak upper stage, an exhaust speed of three kilometres a second, the first stage has to do most of the work, separates fast, and recovering it costs a quarter of the payload downrange and two-fifths flying back. With an upper stage burning hydrogen and oxygen, at four and a half kilometres a second, the first stage can be asked for little more than it needs to clear the dense atmosphere, separates slowly, and comes home for six to ten per cent. Choosing a propellant is choosing a molecular weight, and the choice made for the upper stage turns out to decide what the lower stage costs to keep.

This runs against the usual way of thinking about recovery, which is as a property of the booster: its engines, its legs, its guidance. The rocket equation says that the stage most responsible for the booster’s recovery cost is the one that never comes back.

How many flights repay the toll #

Whether paying the toll is worth it is not a question the rocket equation answers. It depends on what the hardware costs and how many times it flies, and a crude version of that arithmetic shows the shape of the answer.

A recovered booster flown once is a loss: the rocket carried less, and the booster was not reused. Flown twice, it already repays the lost payload under these assumptions. After twenty flights the cost per kilogram has fallen to a little over half the expendable rocket’s, and it approaches a floor set by the parts that are still thrown away — the upper stage, here forty per cent of the rocket — and by the cost of refurbishment between flights. The economics are dominated by the expendable remainder long before the toll stops mattering.

That is the shape of the industry’s experience. Boosters that fly many times have made the first stage a small part of the marginal cost of a launch, and the upper stage, thrown away every time, has become most of it. Which leads to the obvious next step.

The stage above it #

Recovering the upper stage is a different problem in the rocket equation, not merely a harder one.

The first stage’s dry mass is accelerated only to the staging speed and then dropped, so the exponential of the first stage’s velocity change multiplies it, and the upper stage’s own exponential does not. The upper stage’s dry mass goes all the way to orbit with the payload. Every kilogram added to it — a heat shield to survive re-entry from orbital speed, through an entry corridor far narrower than a booster’s, propellant for landing, wings or legs — is a kilogram the payload loses, almost one for one. Doubling the upper stage’s structural fraction from six to twelve per cent, which a heat shield and landing system could easily do, cuts the payload from three per cent of lift-off mass to one and a half.

That is why the bottom of rockets was made reusable first, and why the vehicles designed to recover their upper stages are very large: they make up for the upper stage’s dry mass with sheer size and with propellant refilled in orbit, rather than by evading the arithmetic. The rocket equation lets the first stage come home for a modest toll and makes the upper stage pay at the full rate.

A single stage that comes home #

The limit of the argument is a rocket with only one stage, recovered whole. The stage that has to be thrown away is the reason a single stage barely reaches orbit at all: with kerosene and oxygen its structural fraction would have to be a few per cent to carry anything, and the margin it leaves is smaller than the uncertainty in its own dry mass. Adding a heat shield, legs and landing propellant to a vehicle that had no payload margin to begin with turns a marginal vehicle into an impossible one.

Hydrogen and oxygen make a single stage less hopeless, and several were designed in the 1990s on that basis; none was completed, because the hydrogen tank’s volume and insulation made the structural fraction grow faster than the exhaust speed helped. The two-stage arrangement survives in the reusable era for exactly the reason it survived in the expendable one. Splitting the velocity change lets one stage’s dry mass be dropped, and if that stage is the one that comes home, it comes home at the reduced rate.

What the model leaves out #

The model treats the ascent’s losses as a fixed 1.3 kilometres a second charged to the first stage, which is roughly right for rockets of this kind but changes with the split: a slower first stage that separates lower in the atmosphere leaves the upper stage to fight more gravity loss at low thrust, which raises the cost of moving the split down. The return budgets are simple linear rules in the staging speed, standing in for trajectories that are optimised flight by flight. And the structural fractions are held fixed, although a recovered stage must be built to fly many times and is heavier for it in ways no fraction captures.

None of these changes the main conclusions: the best split moves down when the booster is recovered, the toll is set more by the return propellant than by the hardware, and a better upper stage makes the booster cheaper to keep. Each correction moves the numbers by a few per cent of payload.

Still open: the price of a returning upper stage #

The rocket equation has answered the first-stage question with unusual clarity. Recovery costs a seventh of the payload for a downrange landing on a rocket built for it, a quarter for a return to the launch site, and more on rockets built to be thrown away; the toll is repaid in a couple of flights and then becomes a minor cost beside the upper stage.

The upper stage is where the question now sits, and the rocket equation offers no comparable bargain. A returning upper stage pays for its heat shield and its landing propellant out of the payload directly, and the vehicles that attempt it do so by being large enough that a few per cent of their lift-off mass is still a useful cargo. Whether that is the cheapest way to put a kilogram into orbit depends on how many times the upper stage flies and how much its refurbishment costs after each re-entry from eight kilometres a second — questions about heat shields and engines rather than about the equation, which has already said exactly what the attempt costs before the first flight. The equation’s verdict on the booster took sixty years to be acted on, because nobody built the rocket around it until the economics demanded it; its verdict on the upper stage is just as clear, and the vehicles now being flown to test it are, in effect, measuring how far engineering can push the structural fraction that the arithmetic says decides everything.