A thruster that must arrive slow

A spacecraft arriving at a planet has a speed left over from its journey, and to stay it must get rid of it. The arithmetic of doing so with a rocket was the subject of the cheapest circle to stop in: fall deep into the planet’s gravity, burn at the lowest point, where the craft is moving fastest, and every metre a second of the burn removes the most energy it can. That essay ended by listing the ways of arriving without a large burn — aerocapture, ballistic capture, and the spirals of electric propulsion — and noting that a low-thrust spiral makes the advantage of burning fast an integral rather than a factor.

This essay works out what that means. An electric thruster has an exhaust ten times faster than a chemical rocket’s and a thrust ten thousand times weaker for its mass: it accelerates a spacecraft by a tenth of a millimetre a second squared, which over a day adds about nine metres a second. It cannot burn at periapsis, because it spends a whole orbit there and nothing at all happens quickly. The questions are what it can capture at all, what it pays to get down, and what it gets in return.

Why an efficient engine is a weak one #

The weakness is not an engineering shortfall; it is a consequence of the efficiency. An electric thruster accelerates its propellant with electrical power, and the power needed to throw propellant out at exhaust speed $v_e$ with thrust $T$ is $T v_e/2$ divided by the thruster’s efficiency. At fixed power, a faster exhaust means proportionally less thrust. An ion engine fed five kilowatts of electrical power — a large solar array’s worth at the Earth’s distance — and ejecting xenon at thirty kilometres a second produces about a fifth of a newton: the weight of a sheet of paper. On a spacecraft of two tonnes that is a tenth of a millimetre a second squared.

So the trade is fixed by physics before any design choices are made. A chemical rocket’s exhaust leaves at three or four kilometres a second because that is what the chemical energy in its propellant allows, and its thrust is limited only by how fast it can burn; an electric thruster can push its exhaust to almost any speed, but only as fast as its power supply allows, and every gain in exhaust speed is paid for in thrust. Ten times the exhaust speed means a tenth of the propellant for the same velocity change — and, at the same power, a tenth of the thrust and ten times the duration.

A weak thruster has one pass to do it #

The hero figure is two flights through the same sphere of influence, with the same thruster pointed backwards the whole time. The slower arrival is caught; the faster passes through, bent by the planet and slowed a little, and leaves. What separates them is not the planet’s gravity, which treats both the same, but the time and distance the thruster has to work while the craft is close enough for the planet to keep it.

That can be turned into an estimate. A craft that enters the sphere of influence with excess speed $v_\infty$ has kinetic energy $v_\infty^2/2$ per unit mass to lose before it is bound. Thrusting against its velocity at acceleration $a$, it removes energy at $a$ times the distance it travels, and the distance it can travel while inside the sphere is of the order of the sphere’s radius $R$. So it can be captured only if

$$v_\infty \lesssim \sqrt{2aR}.$$

The estimate is crude — the craft also gains speed falling towards the planet, which makes each metre a second of braking near periapsis worth more — and it is conservative.

The integrated flights confirm the shape and sharpen the number. At a tenth of a millimetre a second squared, a craft can be captured from an excess of about three hundred metres a second; at a millimetre a second squared, from one and a half kilometres a second. A minimum-energy transfer from the Earth reaches Mars with two and a half kilometres a second to spare, and capturing that would take about three millimetres a second squared — thirty times what an electric thruster gives a spacecraft of a few tonnes.

So a low-thrust spacecraft cannot remove an arrival speed at the planet. It has to arrive without one. That is not a small adjustment to the mission: it means that the long heliocentric spiral out from the Earth must end with the craft on almost exactly the planet’s own orbit, moving at almost exactly the planet’s speed, so that it drifts into the sphere of influence rather than falling through it. The electric missions that have orbited other bodies — Dawn at Vesta and Ceres, SMART-1 at the Moon — did exactly this: they matched the target’s orbit far away and let themselves be captured at a walking pace.

Paying the whole circular speed #

The second difference is in the cost of getting down once captured, and it is larger.

A craft that arrives with no excess and spirals down to a low orbit pays almost exactly the circular speed of that orbit — 3.4 kilometres a second at Mars. That follows from a fact about slow spirals between circular orbits: the velocity change is the difference of the two circular speeds, and from far away the outer orbit’s circular speed is nearly zero, so the difference is the whole of the inner one’s.

A single burn from the same arrival costs much less. Falling from far away to the low orbit’s height, the craft arrives at periapsis at escape speed, $\sqrt{2}$ times the circular speed, and a burn there that brings it down to circular speed costs $(\sqrt{2} - 1)$ of the circular speed: 1.4 kilometres a second at Mars. The spiral costs two and a half times as much.

The comparison is fair only as a comparison of velocity changes, and it is not the whole comparison: the electric thruster’s exhaust is ten times faster, so a velocity change two and a half times larger costs it a quarter of the propellant mass. That is why electric propulsion is used anyway. But the factor of two and a half is real and has a definite cause.

Arriving without an excess is not free either #

Requiring the spacecraft to arrive with almost no excess moves cost rather than removing it. A craft leaving the Earth’s neighbourhood and arriving at Mars’s on matching orbits has to change its heliocentric speed from the Earth’s, thirty kilometres a second, to Mars’s, twenty-four; by a slow spiral between the two circular orbits that costs the difference, 5.7 kilometres a second. A chemical transfer from low Earth orbit to low Mars orbit spends about the same total, 5.7, in two burns — and that total includes climbing out of the Earth’s well and settling into Mars’s, because each burn is made deep in a well where the planet’s own escape speed comes almost free with it.

So an electric spacecraft launched onto an escape trajectory by an ordinary rocket spends about 5.7 kilometres a second matching orbits and 3.4 spiralling down at Mars — about nine in all — and one that started its spiral in low Earth orbit would spend another 7.7 climbing out. With an exhaust eight or ten times faster the electric mission still uses less propellant. It uses years rather than months, and it never uses either planet’s gravity to multiply anything. That is the general shape: efficient engines make their savings in propellant and give back part of them in the velocity changes they cannot avoid, because they cannot spend them where the Oberth effect is strongest.

The Oberth effect, averaged #

A thrust applied to a moving craft changes its energy at a rate equal to the thrust times the craft’s speed. So the energy removed per metre a second of velocity change is the speed at which the braking is done, and the slope of each curve in the figure is that speed. The single burn is made at periapsis, at 4.8 kilometres a second, the fastest the craft ever moves; it removes the 5.8 square kilometres per square second that separate the arrival from the low orbit in 1.4 kilometres a second of velocity change.

The spiral begins its braking far out, where the craft is crawling and each metre a second of thrust removes almost no energy. As it descends it moves faster, its thrust becomes more effective, and the curve steepens; but by the time the craft is moving fast it has already spent most of its velocity change at low speed. Averaged over the spiral, the thrust is applied at 1.7 kilometres a second, and the same 5.8 square kilometres per square second cost 3.4 kilometres a second.

That is the precise sense in which a spiral integrates the Oberth effect rather than exploiting it. The effect is not lost — the spiral’s thrust is worth more near the planet than far from it, exactly as a burn’s is — but a spiral cannot choose where its thrust is applied, and it pays at the average speed of its whole descent rather than at the maximum. The same arithmetic, applied to a chemical rocket’s finite burns, is why a burn that cannot happen at one point costs more than an impulse: a burn spread over an arc spends some of its velocity change away from the fastest point. A spiral is the extreme case, spread over the whole descent.

There is a partial remedy, and it is the same one chemical stages use when their burns are too long. A weak thruster can choose when to thrust even if it cannot thrust hard: on an eccentric captured orbit it can switch off for most of each revolution and thrust only on the arcs around periapsis, where the craft moves fastest. Each orbit then removes less energy, so the descent takes more revolutions and more time, but the energy removed per metre a second rises towards the periapsis speed. In the limit of thrusting only at periapsis, an electric craft reproduces the impulsive cost — at the price of a descent that takes many times longer, because it is using a few per cent of each orbit. Real low-thrust trajectories at Mars and the Moon use something between the two: thrust arcs that start wide, when the orbit is large and time matters most, and narrow as the orbit shrinks and the periapsis advantage grows. The figures here draw the simplest case, continuous thrust, which is the slowest in velocity change and the fastest in time.

Deep wells punish spirals most #

Across the planets, the spiral’s cost is the low orbit’s circular speed and the burn’s from rest is $(\sqrt 2 - 1)$ of it, so the spiral always costs 2.4 times the burn when both start from no excess. What changes from planet to planet is how large those numbers are, and how fast the minimum-energy arrival is.

At Jupiter the circular speed just above the cloud tops is forty kilometres a second. No electric spacecraft carries forty kilometres a second of velocity change, and spiralling down to a low orbit at Jupiter is out of the question; the orbiters there arrive fast, burn low and accept highly elliptical orbits, as the arithmetic of the cheapest circle explained. At Mars the numbers are comfortable for both methods. At Mercury the order reverses: a minimum-energy arrival there is so fast — nearly ten kilometres a second, because Mercury moves fast deep in the Sun’s well — that a burn from it costs seven and a half kilometres a second, more than a spiral from rest. The mission sent there in 2018 has spent eight years being slowed by an electric engine and nine planetary flybys so that it can arrive gently; it is a case where arriving without an excess is worth the years it takes.

Paying in time #

What electric propulsion buys with its fast exhaust it pays for in time. At a tenth of a millimetre a second squared the descent from capture to a low Mars orbit takes about a year and winds round the planet some two and a half thousand times, most of them in the last few hundred kilometres where the orbits are short. A chemical capture takes minutes.

The time matters for more than patience. A spiral through a planet’s radiation belts — the Earth’s especially, for a spacecraft spiralling out — accumulates dose on every pass; a spiral at a planet with an atmosphere spends weeks at heights where drag is not negligible; and the electric thruster’s power comes from solar panels whose output falls as the square of the distance from the Sun, so the accelerations available at Jupiter are a twenty-fifth of those at the Earth. The engine is chosen by the calendar: where the mission can wait, it spirals; where it cannot, it burns.

Where spirals have worked #

The electric missions that have gone into orbit around other bodies show the argument at work, because each arrived exactly as the arithmetic requires. SMART-1, a small European spacecraft, spiralled out from a transfer orbit around the Earth over more than a year, and was captured by the Moon’s gravity when its own orbit had grown large enough that the Moon could take it at low relative speed; it then spiralled down to its working orbit over several more months. Dawn left the Earth with an ordinary rocket, spent three years spiralling outward around the Sun on its ion engines, and drifted into Vesta’s neighbourhood with almost no relative speed; it was captured, mapped the asteroid for a year, spiralled out again, and did the same at Ceres. Vesta’s and Ceres’s gravity is so weak — the circular speed near their surfaces is a few hundred metres a second — that the descent spirals cost little, and for small bodies electric propulsion is the obvious choice.

At the planets the balance is closer, and depends on the planet. Mars is within reach of an electric orbiter, at a price in time; Mercury, as the figure shows, is a case where slow arrival is the only reasonable way; the giant planets are out of reach for low orbits altogether. The electric engine changes which destinations are cheap, rather than making all of them cheaper.

What the flights leave out #

The flights are integrated in a planet’s field alone, with the Sun’s influence represented only by the boundary of the sphere of influence and a capture criterion requiring the orbit’s farthest point to lie inside it. Near that boundary the Sun’s pull is comparable to the planet’s, and a real low-thrust capture takes place in the three-body problem, through the regions near the Lagrange points where the manifolds that lead in and out of the planet’s neighbourhood channel slow arrivals. A trajectory designed with those in mind can be captured at lower thrust than the sphere-of-influence estimate suggests, by entering along a tube rather than straight in.

The thrust is always directed against the velocity. That is close to optimal for removing energy but not for every purpose: a spiral that also has to change the orbit’s plane, or to end on an elliptical orbit, steers differently, and its costs follow Edelbaum’s analysis of steered thrust rather than the plain difference of circular speeds.

And the comparison of velocity changes ignores the masses. A spiral costing 3.4 kilometres a second with an exhaust of thirty kilometres a second uses eleven per cent of the spacecraft’s mass in propellant; a burn of 1.4 kilometres a second with an exhaust of three uses thirty-seven. The electric craft’s heavier power system eats part of that difference, and how much depends on the mission.

Still open: capture without the spiral #

The two ways of stopping at a planet without a large burn that remain on the list — aerocapture, which uses the atmosphere and has never flown once, and ballistic capture, which uses the Sun’s own pull to hand a slow arrival to the planet — are both, in different ways, answers to the problem this essay describes. An electric spacecraft arriving with no excess at all can be captured weakly by the Sun–planet three-body dynamics with no thrust, into a loose, temporary orbit far from the planet, and then begin its spiral from there; the spiral’s cost is unchanged, but the arrival’s constraints relax.

What has not been worked out in general is the joint design: a heliocentric spiral that ends exactly where the three-body dynamics will capture it, followed by a planetocentric spiral that uses the remaining thrust as efficiently as the descent allows. For missions to small bodies, whose gravity is so weak that every arrival is a slow one, that design is already routine. For the planets it is the frontier between the electric engine’s efficiency and the gravity well’s indifference to how patiently it is entered.