An orbit lost to a burn that did not fire

A rocket stage with a weak engine cannot leave low orbit in one burn without wasting much of its propellant. A burn that cannot happen at one point spreads along an arc of the orbit, and the farther the arc reaches from perigee, the less of the engine’s work goes into raising the orbit’s energy and the more is lost to gravity. The remedy is to split the departure. Burn for a while at perigee, coast round the larger ellipse that leaves, burn again at the next perigee, and repeat, each burn short enough to stay near the bottom of the orbit, until the last one sends the stage away. Eight burns instead of one can cut the gravity loss from a third of the impulsive cost to about one per cent.

The cost of splitting is time: the stage spends hours to days coasting round its intermediate orbits. The essay on steering a long burn ended on a question that the time raises. A plan that optimises the sequence of burns as one problem may be fragile — sensitive to a burn that underperforms or fails to fire — and a slightly less efficient plan that degrades gracefully might be the better one to fly.

This essay works out what a missed burn actually costs. It costs a revolution, in two currencies: the time the stage spends going round once more, and the angle through which the Earth’s equatorial bulge turns the orbit while it does. The two behave quite differently along the staircase of ellipses, and which one matters depends on where the stage is going.

Why the departure is split at all #

The reason to burn at perigee is the one that makes the same burn worth more when moving fast: a change of speed adds kinetic energy in proportion to the speed already there, so a kilometre a second added where the stage is fastest raises its orbital energy most. A long burn cannot stay at perigee; it smears across an arc on either side, where the stage is slower and the thrust is partly wasted against gravity, and the longer the burn the larger the waste.

Splitting keeps every burn short. Each one is centred on a perigee pass and lasts only as long as the stage stays near the bottom of its orbit, and the orbit between burns carries the stage away and back for free. The penalty is paid in time and in exposure — to the radiation belts, which a staircase of ellipses crosses twice a revolution, and to the chance that one of the burns does not happen. The first is predictable and planned for; the radiation dose is a known cost of the staircase’s geometry; the third is what this essay is about.

The same logic appears on every scale of spaceflight. A transfer that is cheapest between two orbits is also the slowest, and a departure that spends least propellant spends most time. Each saving buys a longer interval in which something can go wrong.

The staircase of ellipses #

A split departure from a circular orbit three hundred kilometres up adds speed at perigee in steps. Each step leaves the perigee where it was and raises the apogee.

The steps are equal in speed but not in effect. Near circular speed, adding a few hundred metres a second at perigee raises the apogee by a few thousand kilometres; near escape speed, the same increment raises it by tens of thousands, because the orbit’s energy approaches zero and its size grows without bound. The eight-burn staircase climbs gently through its first orbits and steeply through its last, from eight thousand kilometres to eighty-six thousand in seven steps, and the eighth burn takes the stage out altogether.

The staircase is the same for any stage that splits its departure this way, whether it is leaving for the Moon, for another planet or for a high Earth orbit. What differs is the target, and whether the target waits.

The time a missed pass costs #

If the burn that should end an intermediate orbit does not happen — the engine fails to ignite, an attitude fault delays it, the ground commands a hold — the stage simply goes round that orbit again and arrives at the next perigee one period later. Nothing is lost but the period.

A missed burn early in the staircase costs an hour and a half or two. A missed burn on the last step costs more than a day. The period grows as the three-halves power of the semi-major axis, and the semi-major axis grows fastest at the top of the staircase, so the time at risk is concentrated in the final orbits — exactly where the plan’s efficiency is gained, since the last burns are the ones that would lose most to gravity if combined.

There is a second-order cost. The gravity loss of the remaining burns does not change when one is postponed, but the plan’s later steps may need to be redistributed if the departure has a deadline. A departure for another planet has a window of weeks, and a day’s delay changes the required departure speed by metres per second. A departure for a high Earth orbit has no deadline at all.

A bulge that turns every ellipse by about the same angle #

While the stage coasts, the Earth’s equatorial bulge turns its orbit. The bulge’s extra mass near the equator — the second zonal harmonic, the largest term in the field a satellite is allowed to feel — pulls the orbit’s plane round the Earth’s axis — the node regresses westward, the effect that lets the Earth’s shape be read off a satellite’s node — and turns the ellipse within its plane, advancing or regressing the line of apsides. Both rates are fastest for low, nearly circular orbits, which pass close to the bulge often.

Measured per day, the turning falls steeply as the orbit grows: a large orbit spends most of its time far from the Earth, where the bulge’s pull is negligible, and completes few revolutions a day. Measured per revolution, it hardly changes. The figure shows why. The turn in one revolution depends on the orbit’s semi-latus rectum — the radius at which it crosses the line perpendicular to its major axis — and for an ellipse whose perigee is fixed that quantity lies between one and two perigee radii, however far the apogee reaches. Nearly all the turning happens during the brief passage near perigee, where the stage sweeps past the bulge, and that passage looks almost the same for every step of the staircase.

So a missed burn costs about the same angle whichever step it is missed on: a few tenths of a degree of the line of apsides and a tenth or two of the node, at the inclinations launch sites give. The angle depends on the inclination as $5\cos^2 i - 1$ for the apsides, which vanishes at 63.4 degrees — the critical inclination at which highly elliptical communication orbits are flown so that their apogees stay over the same hemisphere — and as $\cos i$ for the node, which vanishes for polar orbits.

An inclination at which the apsides stand still #

The inclination dependence offers a way out for the larger of the two turns. A stage launched into an orbit inclined at 63.4 degrees feels no turning of its apsides from the bulge at all; only its node regresses.

That is the inclination of the Molniya orbits, chosen in the 1960s so that a communications satellite’s apogee would stay over the northern hemisphere for years without fuel spent holding it there, and the same property protects a departure staircase. At the critical inclination a missed burn costs only the node’s turn, and the node itself turns more slowly there than at low inclinations, because its rate goes as the cosine of the inclination; the correction at apogee falls to an eighth of its value for a launch from a low-latitude site. A departure that is not constrained in inclination by its launch site, or whose target accepts any plane, could choose it.

Few departures can. A stage launched due east from a site at twenty-eight degrees’ latitude is in a twenty-eight-degree orbit unless it spends propellant to change plane, which costs far more than the corrections it would save. The critical inclination is a curiosity for most departures and a design choice for the few — high-latitude launch sites, missions to orbits that are themselves highly inclined — that start near it.

What undoing the turn costs #

A turned ellipse departs in a turned direction. The final burn’s hyperbolic excess velocity points along a direction fixed relative to the perigee, so a line of apsides rotated by half a degree sends the stage off half a degree from its planned asymptote. That can be corrected, and where it is corrected sets the price.

The cheapest place to turn an ellipse is at apogee, where the stage moves slowest, and the cost is roughly twice the apogee speed times the sine of half the angle. For the first step of the staircase, with an apogee at eight thousand kilometres and an apogee speed of several kilometres a second, undoing one revolution’s turn costs eighty metres a second — a large fraction of a weak stage’s margin. For the last step, with an apogee at eighty-six thousand kilometres and a speed of a few hundred metres a second, it costs under four.

The angle per revolution is nearly the same everywhere; its price falls by a factor of twenty along the staircase. A missed burn high up is cheap to correct in angle and expensive in time. A missed burn low down is cheap in time and expensive in angle, and in practice the early-step angle is usually not corrected at all but absorbed, on the principle that makes the first weeks off a transfer nearly free for corrections — an error caught early is spread over the rest of the trajectory at little cost: the remaining burns are re-planned slightly, and the plan’s later orbits, which turn less per day, are allowed to bring the departure asymptote back towards where it was meant to be.

When the target moves #

For a departure into interplanetary space, a day’s delay hardly matters to the target: the planets move a degree a day or less relative to the Sun, and the departure window is weeks wide. For a stage heading for the Moon, the target is moving fast.

The Moon moves thirteen degrees a day along its orbit. A stage that misses the last burn of a lunar staircase and goes round once more, for a day and a bit, finds the Moon fifteen degrees further on — sixty times more than its own orbit has turned. The planned translunar injection, aimed at where the Moon would be four or five days later, no longer meets it. The stage has to wait for the geometry to come round again — which, for an orbit whose period is not a simple fraction of the month, can take days or weeks of extra revolutions — or spend propellant to re-phase.

That is the reason lunar missions that use perigee-raising staircases build waiting into their plans. The Indian lunar missions of the last decade climbed to the Moon through five or six Earth-bound orbit-raising burns before the translunar injection, and their schedules were set by the geometry of those orbits relative to the Moon’s motion — the constraint a missed burn would disturb, and one that is restored more cheaply by waiting for the geometry to recur than by spending propellant to force it.

Why the plans keep their margins at the top #

The arithmetic gives a simple rule for building a robust staircase. The early steps are cheap to repeat in time and expensive to correct in angle; the late steps are the reverse. A plan can protect itself against early failures by doing nothing — the angle is small and the later burns absorb it — and against late failures by carrying time: an extra revolution allowed for before the final burn, or a departure date chosen so that a day’s slip still lands within the window.

The optimisation that minimises gravity loss pushes the other way. It favours more burns, each shorter, which adds steps at the top where each revolution is long. The trade between efficiency and robustness is therefore not abstract. Each additional burn saves a little propellant and adds a long orbit in which a failure costs a day. For a departure to Mars with a three-week window, that is free; for a stage aiming at the Moon with no margin, it is the dominant risk. The same square root that raises an orbit and turns it for an electric engine applies here in miniature: the cheapest way up is the slowest, and slowness is exposure.

The radiation the staircase crosses #

The time a missed burn costs is not only schedule. Every step of a departure staircase from low orbit passes twice per revolution through the Earth’s inner radiation belt, a few thousand kilometres up, and the higher steps reach the outer belt. A stage that goes round an extra time collects an extra revolution’s dose. For the early steps, which pass quickly through the inner belt near perigee, the extra dose is small; for the middle steps, whose slow climb towards apogee lingers in the belts for hours, it is the largest; for the top steps, which spend most of each revolution far beyond the belts, it falls again.

For a stage with sensitive electronics or a crew, that dose can be the binding constraint rather than the propellant or the schedule, and it peaks on steps where the time cost is moderate and the angle cost already small. A plan built to be robust against failures has to weigh three costs of an extra revolution, not two, and they peak at three different places on the staircase.

What the arithmetic leaves out #

Only the Earth’s flattening turns the orbit. At the top of the staircase the Moon and the Sun pull on the stage’s orbit too, and for apogees of tens of thousands of kilometres their tides turn the orbit at rates comparable with the bulge’s. The figures include only the bulge.

The burns are impulsive. Each burn is drawn as an instantaneous change of speed at perigee. A real weak stage burns for minutes to tens of minutes, and a burn that is cut short rather than missed leaves the stage on an intermediate step the plan did not contain.

Equal steps are one plan among many. Real staircases are optimised, with larger steps where they lose least, and the periods and apogees differ from the figure’s. The pattern — short early orbits, long late ones — is general.

Failures are drawn as clean. A burn that does not fire at all is the simplest failure. More common in practice is a burn that starts late, after a fault is cleared, or ends early, and leaves the stage on an orbit between two steps; the plan’s remaining burns then have to be recomputed for an orbit the staircase did not contain, and the time cost depends on how quickly the ground can do that and uplink it. The figures give the cost of the cleanest failure, which is a lower bound on the others.

The correction is drawn at apogee. Undoing a turn can also be folded into the remaining perigee burns, at a cost that depends on how much those burns’ direction can change without losing efficiency. The apogee figure is a ceiling for a small correction, not an optimum.

Still open: a staircase planned for its failures #

A split departure trades propellant for time, and the time is where its fragility lives. The arithmetic says where: in the long orbits at the top, where a missed burn costs a day, and where a target that moves — the Moon above all — turns that day into a missed rendezvous. The turning of the orbit by the Earth’s bulge, which looks like the obvious hazard of waiting, costs nearly the same angle on every step and is cheapest to undo exactly where the time is most expensive.

What remains open is the planning method that weighs those risks together. Optimising the sequence of burns for minimum propellant, and then adding margins, is what is done now. Optimising it for the expected propellant and time, given the probability that any burn fails, would choose different staircases — fewer, larger burns near the top, perhaps, and more small burns near the bottom where failure is cheap — and nobody has yet shown how much such a plan would save, or cost, on a real mission.