A burn that cannot happen at one point

The cost of moving between orbits is computed, almost everywhere, as a sum of instantaneous changes of velocity. The cheapest transfer between two circular orbits is two such impulses, one at each end of an ellipse; an escape from a parking orbit is one impulse at the right point, sized so the vehicle leaves with the hyperbolic excess speed the mission needs. The impulses are what the vis-viva relation gives, and they are exact for an engine of infinite thrust.

No engine has infinite thrust. A departure stage pushing a spacecraft out of low Earth orbit accelerates at somewhere between a few hundredths and about one gravity, and to deliver three or four kilometres a second it has to run for minutes at the strong end and for hours at the weak. During that time the vehicle is not at the point where the impulse was supposed to be applied. It has travelled round the orbit, climbed, and slowed — and the velocity it gains there is worth less than velocity gained at the low point.

The difference is the gravity loss: the velocity the rocket actually spends, by the rocket equation, minus the velocity an impulse would have needed. It is a real cost, paid in propellant, and it is the price of the engine’s finite strength.

Why the low point is the best point #

The impulsive calculation chooses the low point for a reason. A change of velocity delivered when the vehicle is moving fast changes its kinetic energy more than the same change delivered when it is moving slowly — the energy gained is the velocity change times the speed already present — and at the bottom of a gravity well the vehicle is moving fastest. That is the Oberth effect, and it is why departure burns are made from low orbit at the moment of closest approach.

A finite burn gives up part of that advantage. Its first seconds are delivered at the low point, but as the engine runs the vehicle gains energy and climbs; it is soon higher than the parking orbit and moving more slowly than it would at perigee of its new, larger orbit. Every metre per second delivered there buys less energy than one delivered at the low point. The loss is therefore not wasted propellant in any obvious sense — the engine is pushing in the right direction all the time — but propellant spent where it is less effective.

There is a second, smaller contribution. A burn held along the velocity direction is not quite along the direction that would most efficiently add energy once the vehicle is climbing; the velocity has a component away from the Earth, and thrust along it partly fights gravity rather than adding speed. That is the meaning of the name. During a launch from the ground it is the dominant loss, because the rocket must climb vertically at first; in a departure from orbit it is a small part of a loss that is mostly geometric.

How long the engine runs #

The arc follows from a burn time, and the burn time follows from the rocket equation. A stage of constant thrust starting at an acceleration $a_0$ and exhaust speed $c$ needs a time

$$t_b = \frac{c}{a_0}\left(1 - e^{-\Delta v/c}\right)$$

to deliver a velocity change $\Delta v$, because the propellant burned is the fraction $1-e^{-\Delta v/c}$ of the initial mass and it is consumed at a constant rate. For the 3 km/s departure, $\Delta v$ of about 3.6 km/s, an exhaust speed of 4.41 km/s and a thrust-to-weight of a tenth, that is about two and a half thousand seconds — forty-two minutes, against a parking orbit that takes ninety to go round. The vehicle covers more than a third of an orbit with the engine running, and because it speeds up and climbs as it goes, the arc it sweeps is more than a third of a circle.

At a thrust-to-weight of one the same burn takes four minutes and sweeps about fifteen degrees, which is why it is effectively an impulse. At a thrust-to-weight of a hundredth it would take seven hours: several orbits, and a burn that is no longer a burn at all but the beginning of a spiral.

The arc the burn sweeps #

The size of the loss is set by how far round the orbit the burn extends.

The figure is the problem drawn. A stage that accelerates at less than a tenth of a gravity runs its engine for half an orbit, spiralling outward as it goes, and by the time it is done it has delivered most of its thrust far from where an impulse would have been applied. The loss is a fifth of the impulsive cost.

The arc grows as the thrust falls, nearly in inverse proportion for strong stages: half the thrust, twice the burn time, twice the arc. The loss, in turn, grows as the square of the arc while the arc is small, because the penalty for being away from perigee by an angle θ is proportional to $1-\cos\theta$, and for small angles that is $\theta^2/2$. Put together, the loss of a strong stage falls as the inverse square of its thrust-to-weight. The opening figure’s high-thrust end shows that slope, almost exactly two.

That square law is what makes the impulsive approximation so good for the stages that usually fly departures. A cryogenic upper stage with a thrust-to-weight near one loses a few metres a second on a three-kilometre-per-second departure, a tenth of a per cent, and the impulsive calculation is as good as the navigation that executes it. The same law makes the approximation fail abruptly for weak stages: a thrust-to-weight of a tenth, which sounds modest rather than extreme, loses over a tenth of the impulsive cost, and a thrust-to-weight of a few hundredths loses as much again as the impulse itself.

A lower exhaust speed loses less #

One feature of the losses is counter-intuitive enough to state separately. A stage with a lower exhaust speed — a less efficient engine — has a smaller gravity loss for the same initial thrust-to-weight.

The reason is the mass history. A rocket’s acceleration rises during the burn as it consumes propellant, because the thrust is constant and the mass falls. An engine with a lower exhaust speed has to consume more propellant for the same change of velocity, so its mass falls further and its acceleration rises more steeply towards the end of the burn. It finishes the burn sooner and over a shorter arc than an engine of the same initial thrust-to-weight but higher efficiency.

This does not make the less efficient engine better. The total propellant it burns is still larger, by the rocket equation, and that dwarfs the difference in gravity loss. It does mean that the thrust-to-weight ratio at ignition is not the whole description of a stage’s finite-burn penalty, and that the comparisons of propulsion systems made from impulsive costs alone are slightly unfair to the efficient ones.

Splitting the departure #

The square law suggests the remedy. If the loss grows as the square of the arc, a burn divided into several shorter ones, each centred on the low point, loses much less — provided each piece can be delivered at perigee.

That is possible because each burn changes the orbit without changing where its perigee is. A first burn centred on perigee raises the apogee and leaves the vehicle on an elliptical orbit with the same low point. The vehicle coasts round the ellipse and returns to perigee, where a second burn raises the apogee further. After several such burns the orbit is very elongated, and the last burn, still at perigee, completes the escape. Each burn sweeps only a fraction of the arc a single burn would.

The recovery is dramatic. A weak stage that would lose more than a third of its impulsive cost in a single burn loses barely more than one per cent in eight. The cost is paid in time. Each intermediate orbit is larger than the last, and the coasts between perigees lengthen from an hour and a half to days; an eight-burn departure takes a week or more where a single burn takes an hour. Each perigee passage also carries the spacecraft through the inner radiation belt twice, and an elongated orbit spends a long time in the outer belt near apogee. For a robotic spacecraft that can tolerate the dose, and whose mission timeline has room, the trade is often worth making.

Lunar and interplanetary missions launched on rockets whose upper stages cannot complete the departure have used exactly this scheme. A spacecraft is left in a highly elliptical orbit by the launcher and raises its apogee with its own small engine at a sequence of perigees, and the final burn sends it to the Moon or beyond. India’s first lunar and Mars missions did it in several phasing orbits over weeks; the method is standard for small spacecraft that ride as secondary payloads and must use whatever orbit the launcher delivers them to.

What the loss costs in payload #

A gravity loss is a velocity, and velocity is paid for through the rocket equation’s exponential. The useful way to read the opening figure is therefore not in metres a second but in what fraction of the departing mass reaches the destination.

For the 3 km/s departure with an exhaust speed of 4.41 km/s, an impulsive burn of 3.60 km/s leaves 44.2 per cent of the vehicle’s initial mass as stage and payload when it is done. A stage with a thrust-to-weight of a tenth spends another four hundred metres a second, and leaves 40.4 per cent: nearly nine per cent less mass delivered, for the same departure. Since the stage’s own dry mass is part of what remains and does not shrink, the payload loses a larger share still — for a stage whose dry mass is a tenth of its loaded mass, the payload falls by about an eighth.

That is the reason departure stages are built with thrust-to-weight ratios near one even though the engine needed for that thrust is heavy. An engine twice as powerful weighs more, and its mass counts against the payload throughout; a weaker engine is lighter but loses more to gravity. The optimum sits where the marginal kilogram of engine saves a kilogram of propellant, and for chemical departures from low orbit it lands at a thrust-to-weight of a few tenths to one, which is where the opening figure’s curves have flattened to a few per cent.

The launch, where the loss is largest #

A departure from orbit is the gentle case. The vehicle starts on a circular orbit, already moving at orbital speed, and its burn mostly adds speed along the direction it is already going. A launch from the ground starts with no horizontal speed at all, and for the first minutes the rocket must climb almost vertically to get out of the dense atmosphere, holding itself up against gravity with every second of thrust it spends.

That is the purest form of the loss, and it is large: a launch to low orbit spends about 9.4 kilometres a second to reach an orbital speed of 7.8, and most of the difference is gravity loss — roughly 1.2 kilometres a second spent holding the rocket up while it climbs. Launch vehicles reduce it the same way departure stages do, by being strong: first stages lift off at thrust-to-weight ratios of 1.2 to 1.5 and climb out of the atmosphere in two or three minutes, because every extra minute of near-vertical climb costs another half a kilometre a second. A rocket that lifted off at exactly one gravity would hover until it ran out of propellant.

The two cases share the square law’s spirit. In each, the loss is what thrust delivered away from the ideal point costs, and the remedy is to deliver the thrust faster, or in pieces closer to the ideal point. The difference is only in where the ideal point is — at perigee for a departure, and for a launch, nowhere, because the rocket must climb before it can turn.

Where the impulse is still a good approximation #

Most of orbital mechanics is safely impulsive, and it is worth being clear about when. The loss is negligible when the burn time is short compared with the orbital period — when the arc is a few degrees. For a departure from low orbit, with its ninety-minute period, that requires a thrust-to-weight of about one. For a manoeuvre far from the Earth, where the period is days or years, a much weaker engine qualifies: a mid-course correction of a few metres a second on an interplanetary trajectory is impulsive by any measure, because the trajectory barely curves while the engine runs. The relevant ratio is always burn time to the time the trajectory takes to turn.

That ratio also governs the arrival. A spacecraft braking into orbit around a planet faces the mirror image of the departure problem: it must slow down at the low point of its approach hyperbola, and a long burn delivers much of its braking after the spacecraft has passed periapsis and started to climb away. The cheapest orbit to brake into is set by the Oberth effect for an impulse; for a real engine the capture burn is centred on periapsis and accepts a loss, and planetary orbiters with modest engines routinely spend several per cent of their capture budget on it.

Electric propulsion, where the burn never stops #

At the far extreme, the burn is never short. An electric thruster produces thrust measured in millinewtons, accelerations of a millionth of a gravity, and to deliver kilometres a second it runs for months. Its arc is not a fraction of an orbit but hundreds of orbits, and the trajectory is not a burn followed by a coast but a slow spiral, steered so that one continuous thrust both raises and turns the orbit. The gravity loss in the sense used here becomes enormous: a spiral escape from low orbit costs nearly the full circular orbital speed, about 7.7 kilometres a second, against an impulsive 3.2 for a parabolic escape.

The loss is accepted because the exhaust speed is ten times higher. The rocket equation’s exponential penalises velocity in units of exhaust speed, and an electric engine can spend twice the velocity of a chemical one and still use a small fraction of the propellant. The finite-burn penalty that is ruinous for a chemical stage is a footnote for an electric one, because its currency is cheaper. Which is better for a given mission depends on whether time or propellant is scarcer — the same trade as splitting a chemical burn, taken to its limit.

What the calculation leaves out #

The figures hold the thrust along the velocity throughout the burn, which is simple and slightly suboptimal: the best steering pitches the thrust a little below the velocity during the second half of the burn, so that more of it adds speed and less of it adds altitude, and that recovers a modest fraction of the loss for weak stages. The departure is planar and from a circular orbit; a real departure must also turn the plane to the outgoing asymptote’s direction, and the timing of the burn relative to the asymptote adds a further penalty when the arc is long. The engine’s thrust and exhaust speed are constant, and the vehicle’s mass is all propellant and stage, with no account taken of the stage’s dry mass, which slows the late rise in acceleration and increases the loss slightly.

None of these changes the scaling. The loss grows as the square of the arc for short burns and dominates the budget for long ones, and splitting the burn at perigee recovers it in proportion to the square of the number of pieces.

Still open: how to steer a long burn best #

The optimal steering of a finite burn — the direction history that minimises the propellant needed to reach a given departure — is a problem in optimal control with a known general structure and no closed-form solution, and for the long burns of weak stages the optimal history departs substantially from thrust along the velocity. Missions compute it numerically for each departure. What remains open is less the computation than the trade around it: for small spacecraft with weak engines and tight propellant budgets, how many perigee burns to plan, how much radiation dose and schedule to accept for each metre per second saved, and how much margin to hold against a missed perigee that turns a multi-burn departure into a costly recovery. Those choices are made mission by mission, and the arithmetic of the square law is the starting point every one of them shares.