No single way to point a long burn

A rocket gains most from its propellant where it is moving fastest, deep in a gravity well, because the same burn is worth more when moving fast: the work done by a given thrust over a given time grows with the speed at which it is applied. An impulsive calculation puts the whole of a departure burn at the lowest point of the orbit and takes the full advantage. A real engine needs time, and a burn that cannot happen at one point spends part of its thrust away from the lowest point, climbing, where each kilogram of propellant buys less. The difference is the gravity loss, and for a stage that accelerates at a tenth of a gravity it can be a few hundred metres per second of a three-and-a-half-kilometre-per-second departure.

That loss was computed with the simplest steering rule there is: thrust held along the velocity throughout. Whether a better rule exists, how much better it is, and how far the simplest of all — holding the engine pointed in one direction in space — falls short, are questions about the shape of the loss rather than its size, and they have answers that depend on how long the burn lasts.

Why the loss grows as a square #

The size of the loss follows from a simple picture. Escaping from a low orbit to a given excess speed requires reaching a given energy, and the energy a kilogram of propellant adds is its velocity change multiplied by the vehicle’s speed at the moment — which is the vis-viva relation read as bookkeeping. The speed is greatest at the lowest point of the orbit. A burn centred there, of duration τ, spends some of its propellant a little before and a little after, where the vehicle is a little higher and a little slower. The shortfall in speed at a time t from the centre grows in proportion to $t^2$ for small $t$, because the orbit is curved and the speed is at a maximum; averaged over the burn it grows as $\tau^2$.

That is why the loss falls as the inverse square of the thrust for strong stages: the burn’s duration, and with it the arc it sweeps, falls in inverse proportion to the thrust, and the loss as its square. It is also why the loss is so sensitive to the weakest stages. Halve the thrust and the loss quadruples; halve it again and the burn begins to sweep a large fraction of the orbit, the quadratic picture breaks down, and the loss grows faster still.

Steering cannot change that geometry. It can only choose the direction of the thrust at each point of the arc, and the arc itself, set by the thrust and the energy needed, is fixed before the steering is chosen.

Three rules a stage can fly #

The rules are chosen because each corresponds to something real. Thrust along the velocity is what a simple guidance loop does when it points the engine where the vehicle is going, and it is what the gravity-loss estimate assumed. Thrust fixed in space is what a spinning stage does: a solid motor on a spin table, stabilised by its own angular momentum, holds its axis for the whole burn because it cannot do anything else. Thrust turned at a constant rate is the simplest rule that lets the direction change with time independently of the velocity, and it is close to the form the theory of optimal steering predicts for short burns.

That theory, worked out in the 1960s, says that the thrust in an optimal trajectory should point along a vector called the primer, which obeys a simple equation along the orbit and whose direction, over a short burn near the bottom of a well, rotates almost uniformly. A constant-rate turn is the primer’s direction approximated by its first two terms, and for a burn short compared with the orbit it captures nearly all of the optimum. For long burns the primer’s rotation is not uniform, and no two-parameter rule follows it.

The calculation here integrates the departure under each rule, at constant thrust with the vehicle growing lighter as it burns, until the energy reaches what a three-kilometre-per-second escape needs, and searches over each rule’s free parameters for the one that spends least.

Short burns turn, long burns follow #

For strong stages the three curves are nearly parallel on the logarithmic plot, falling as the inverse square of the thrust, because the burn’s arc shrinks in proportion to it and the loss grows as the square of the arc. The constant-rate turn sits a little below thrust along the velocity, and the fixed attitude well above both.

The saving is up to a tenth of the loss, and it comes from turning more slowly than the velocity does. Following the velocity rotates the thrust at the vehicle’s own angular rate about the planet, which rises as the burn adds speed while the vehicle is still low. The best constant rate is about two-thirds of the starting orbital rate, so the thrust lags behind the velocity’s turning — staying nearer the direction it had where the vehicle was lowest and fastest, and so where each kilogram of propellant bought most.

For the weakest stages the comparison reverses. At a thrust-to-weight of a twentieth the burn lasts a hundred minutes, longer than the orbit itself, and the vehicle’s angular rate changes so much through the burn that no single rate approximates it; the velocity, which turns at whatever rate the trajectory dictates, is the better guide. Between those regimes, near a thrust-to-weight of a tenth, the two are equal.

Where the engine points #

The difference between the rules is easiest to see in the direction itself.

Both of the alternative rules begin with the thrust pitched above the velocity — pointed outward of the direction of motion — while the vehicle is lowest and fastest. That seems wasteful, since thrust off the velocity changes direction rather than speed, but the turning is what the burn’s later part needs: by pitching outward early, the vehicle raises its path, flattens the climb it would otherwise make later, and arrives at the end of the burn moving nearly along the thrust. The constant-rate turn does this gently and closes onto the velocity as the burn ends. The fixed attitude, which cannot turn, has to start far above the velocity to be correct on average and finishes far below it, spending the first and last minutes of the burn pushing well off the direction of motion.

What a pointing error costs #

The modesty of the savings has a reason, and it is visible in what happens when a good rule is flown badly.

The cost of an error grows as the square of the error, because the optimum is a minimum and its neighbourhood is flat. A two-degree error costs a metre and a half per second out of a departure of three and a half kilometres per second, a fraction too small to measure against the propellant’s own uncertainties. Ten degrees costs a few tens of metres per second, comparable to the saving the better rule bought.

The two facts are the same fact. Steering laws that differ by a few degrees in where they point differ by a few metres per second in what they spend, and the better law’s advantage is real but second order. That is why a simple rule, flown accurately, is close to the best, and why a guidance system that holds its attitude to a degree gains more than one that follows a cleverer law loosely.

Faster departures, longer burns #

The departure speed sets how long the burn is, and so how much is at stake in steering it.

Most of a departure’s cost is the escape itself, the speed needed to reach zero energy, so the burn’s length grows only slowly with the excess speed left over beyond escape: twenty minutes for half a kilometre per second, twenty-six for six. The loss grows faster than the burn’s length, because the later part of a long burn is applied further from the bottom of the well, and the room for better steering grows with it in proportion. For the fastest departures — the ones that reach the outer planets directly — the difference between following the velocity and the best steady turn is a few tens of metres per second, and the difference between either and an unsteered burn is several hundred.

Guidance that aims as it burns #

Real stages do not fly a precomputed law open-loop. Their guidance measures the acceleration the engine actually delivers, predicts where the current trajectory will end, and re-aims the thrust repeatedly through the burn to arrive at the target — a cut-off energy, an orientation of the departure asymptote, a time. The steering laws used by launch vehicles and upper stages for half a century are of exactly the constant-rate family drawn here, in a form whose tangent of the thrust angle varies linearly with time, re-solved every few seconds as the burn proceeds. Their success is the practical evidence for the small gap between a two-parameter law and the true optimum over burns of a few minutes.

For the Moon-bound departures of the late 1960s, the third stage’s escape burn lasted about six minutes at roughly two-thirds of a gravity, a short burn in the sense of the figures, with a gravity loss of a few tens of metres per second; guidance of this kind brought it within a fraction of a per cent of the target. For the weak stages that small spacecraft now carry, burns of tens of minutes are the norm, and the same guidance, applied to a long arc, sits in the regime where the figures say a two-parameter law starts to fall behind.

The spin-stabilised stage #

The fixed attitude deserves its own accounting, because it is not a strawman. For decades, small solid upper stages have been spun up before ignition and fired while spinning, so that their axis is held by gyroscopic stiffness and any thrust misalignment is averaged away — provided the spin stays about the axis it was given, which a flexible or fuel-sloshing body need not. Such a stage cannot steer at all.

The penalty is larger than the whole gravity loss of a well-steered burn of the same thrust. A fixed attitude is a good approximation only when the burn sweeps a small arc, which is why spinning stages were built with high thrust, burning out in a minute or two from a low parking orbit, where the arc is a few degrees and the penalty a few metres per second. At the thrust-to-weight of the weak, efficient liquid stages now favoured for upper stages and for small spacecraft, holding the attitude still would waste a hundred and fifty metres per second or more, which is why such stages carry guidance that turns them.

A departure is a direction as well as a speed #

Everything above has asked only that the burn reach a given energy. A real departure must also leave in a given direction: the hyperbola’s outgoing asymptote has to point along the path to the target planet, fixed by the launch window and the transfer, and the orientation of that asymptote is set by where in the orbit the energy is added. An impulsive burn at the lowest point fixes it directly. A long burn smears it out, and the asymptote of the final hyperbola points in a direction that depends on the steering as well as on the timing.

That couples the two choices. A steering law that minimises the propellant for a given energy leaves the asymptote somewhere; if it is not where the transfer needs it, the burn’s start must be shifted round the orbit to compensate, and the shifted burn may no longer be centred where the propellant does most good. For short burns the coupling is weak and the two can be optimised separately. For long ones the best departure is a compromise between where the energy is cheapest to add and where the asymptote has to end, and the steering that reaches it is no longer the one that wins the energy-only comparison drawn here.

The parking orbit’s own orientation adds a third constraint. A departure for a planet must leave from an orbit whose plane contains the outgoing asymptote, which ties the launch time to the departure date, and every day of delay rotates the required plane relative to the orbit’s. Steering cannot fix a plane mismatch cheaply; it is paid for in launch timing or in a plane change, and the cost of either dwarfs the few metres per second of the steering’s advantage.

Splitting the burn instead #

There is another way to reduce the loss that does not depend on steering at all: split the departure into several burns, each centred on a perigee pass, coasting round the increasingly eccentric orbit between them. Each burn sweeps a shorter arc, and the loss, which grows as the square of the arc, falls roughly as one over the number of burns. The arithmetic of splitting the departure is the larger lever for weak stages: four perigee burns cut the loss of a stage at a tenth of a gravity by roughly three-quarters, where the best steering of a single burn recovers a few per cent.

Steering and splitting interact. Each of the shorter burns in a split departure is in the regime where a constant-rate turn helps, so the two savings stack; but the shorter each burn, the smaller the steering’s share. The choice between more perigee passes, which cost time and expose the spacecraft repeatedly to the radiation belts, and better steering of fewer burns, which costs guidance complexity, is made mission by mission.

What the calculation leaves out #

The calculation is planar, from a circular orbit, at constant thrust and exhaust speed, with each steering law’s parameters optimised for the single departure drawn. Real departures must also arrive in the right direction, not only with the right energy, which constrains the burn’s centre relative to the orbit and couples steering to timing; they start from orbits that are often already eccentric; and they are flown by guidance that re-optimises its targets during the burn as it measures what the engine has delivered. The search over each rule is a grid followed by golden sections, which finds the best member of each family to within a few metres per second. The true optimum, computed by the primer-vector theory with the burn’s start time free, lies below all three families, and its advantage over the best constant-rate turn is expected to be small for short burns and appreciable for long ones.

A loss with its own geometry #

The general point is about why gravity losses exist and why they are hard to steer away. A rocket is efficient where it is fast, and a planet makes it fast only at the bottom of an orbit. A burn that takes time cannot stay at the bottom, so some of its thrust is applied where the vehicle is slower and climbing; steering can choose which direction that thrust is applied in, but not where. The fraction steering recovers is bounded by how much of the loss is a matter of direction rather than of position, and for short burns that is a tenth. The cheapest transfers between orbits were derived for impulses; every real one flown with an engine pays the loss, and pays a little less for being steered well.

The same geometry governs the launch from the ground, where the losses are larger still, and the very weak, very long burns of electric propulsion, where the engine never stops and the steering problem becomes a spiral whose thrust direction turns once a revolution. The departure burn sits between them: long enough for steering to matter, short enough for a two-parameter rule to come close.

Still open: steering through several perigee passes at once #

The best departure for a weak stage is a sequence of perigee burns, each steered, with the coast arcs between them chosen together. Optimising the whole sequence as one problem — how many burns, how long each, how each is steered, and how the orbit’s orientation is allowed to drift between them — is a problem in optimal control whose solutions are computed numerically and whose general structure is understood. What is not settled is how robust those solutions are to a missed burn or an engine that underperforms on one pass, which for small spacecraft with tight propellant budgets matters more than the last few metres per second of the optimum. A steering law that is slightly worse but degrades gracefully may be the better one to fly.