The transfer that costs more the gentler it is

The Hohmann transfer is the cheapest way between two circular orbits and the proof is short. Two impulses, both along the velocity, both at the ends of an ellipse tangent to each circle. There is no arrangement of two burns that costs less.

Electric propulsion cannot perform an impulse. A gridded ion engine produces a few hundred millinewtons; on a two-tonne spacecraft that is an acceleration of $10^{-4}\ \mathrm{m/s^2}$, about a hundred-thousandth of the local gravity, and a manoeuvre that a chemical stage completes in four minutes takes it several months. Nothing about the conic the vehicle is on changes suddenly at any point. What it flies instead is a spiral: thrust applied continuously along the velocity, the orbit expanding a little on every revolution.

That spiral costs more, and the amount is not small.

The result with no thrust level in it #

For a thrust small enough that the orbit stays nearly circular at every instant, the spiral’s total velocity change has a closed form of unusual simplicity:

$$\Delta v = |v_1 - v_2| = \left|\sqrt{\frac{\mu}{r_1}} - \sqrt{\frac{\mu}{r_2}}\right|.$$

The difference of the two circular speeds. That is it. No thrust level appears, no transfer time, no mass.

The derivation is a page and the reason is one line. Tangential thrust at rate $a_T$ raises the specific orbital energy at $\dot\varepsilon = a_T v$; the circular-orbit energy is $-\mu/2r$; combining gives $\dot r = 2 r a_T / v$. Integrating $dv$ against $dr$ along the way and using $v = \sqrt{\mu/r}$ throughout gives the difference above, and $a_T$ cancels because it appears once in the rate and once in the time.

So the thrust decides how long the transfer takes and nothing else about it. Halve the acceleration and the manoeuvre takes twice as long, makes twice as many revolutions, and costs exactly the same.

Where the extra twenty per cent goes #

The Hohmann transfer’s advantage has a name and a mechanism.

A burn is worth more when the vehicle is already moving fast, because kinetic energy goes as $v^2$: adding $\Delta v$ to a speed $v$ raises the energy by $v\,\Delta v + \tfrac12\Delta v^2$, and the first term dominates. So the same propellant, expended at periapsis where the vehicle is fastest, buys more orbital energy than the same propellant expended anywhere else.

A Hohmann transfer is arranged to collect that advantage twice. The first burn happens at the periapsis of the transfer ellipse, which is the fastest point of the inner orbit; the second at the apoapsis, which is where circularising is cheapest.

A spiral collects it nowhere. The thrust is on continuously, at every point of every revolution, and the average of $v$ over the transfer is far below the peak the impulsive manoeuvre exploits. The 20 per cent is the difference between spending at the best moment and spending at the average one.

Why the dearer manoeuvre is the cheaper mission #

The rocket equation converts a velocity change into a mass ratio:

$$\frac{m_0}{m_f} = \exp\!\left(\frac{\Delta v}{g_0 I_{sp}}\right).$$

Everything above compared the numerators. The denominators are not the same.

A high-performance chemical stage burning hydrogen and oxygen has a specific impulse of about 450 seconds, so $g_0 I_{sp} \approx 4.4$ km/s. A gridded ion engine reaches 3,000 seconds, so $g_0 I_{sp} \approx 29.4$ km/s — a factor of nearly seven.

Put the two transfers through it, for a low-orbit-to-geostationary manoeuvre where the Hohmann costs 3.92 km/s and the spiral 4.72:

• Chemical, impulsive: $\exp(3.92/4.4) = 2.43$, so the propellant is 59 per cent of the departure mass.
• Electric, spiral: $\exp(4.72/29.4) = 1.17$, so the propellant is 15 per cent.

The manoeuvre that costs 20 per cent more in velocity uses a quarter as much propellant. The exponential that makes a chemical mission hard is what makes the electric one easy, because it is an exponential in $\Delta v$ divided by the exhaust speed, and only one of those two numbers changed.

What was actually measured #

Three flown missions established the numbers, and each measured something the ground could not.

SMART-1, the European lunar orbiter launched in 2003, spiralled from a geostationary transfer orbit to lunar capture over thirteen months, making several thousand revolutions. Its Hall-effect thruster produced 68 millinewtons and consumed 82 kg of xenon for a total velocity change of about 3.9 km/s. A chemical stage of the same performance would have needed roughly 500 kg.

Dawn is the sharpest case, because it did something no chemical mission could. It entered orbit around Vesta, spent fourteen months there, left, and entered orbit around Ceres — a total velocity change of about 11 km/s, delivered over eleven years by three ion thrusters running one at a time at up to 92 millinewtons. It carried 425 kg of xenon on a 1,220 kg spacecraft. At 450 seconds of specific impulse the same 11 km/s would have demanded a propellant fraction of 92 per cent, leaving nothing for the spacecraft.

The commercial measurement is the one that changed the industry. Since about 2015 a substantial fraction of geostationary communications satellites have used electric propulsion for orbit raising. The saving is roughly 40 per cent of the launch mass for the same payload, and it is paid for in time: the spiral takes three to six months, during which the satellite earns nothing and passes repeatedly through the radiation belts. Operators buy the smaller launch vehicle and accept the delay, which is a commercial statement about the exchange rate between the two currencies.

The power plant is part of the payload #

The velocity budget is only half of the trade, and the other half is what makes the gentle transfer worth its extra twenty per cent.

An electric thruster’s thrust and its exhaust velocity are tied together by the power available:

$$F = \frac{2\eta P}{v_e},$$

with $P$ the electrical power and $\eta$ the efficiency. So for a fixed power supply, doubling the exhaust velocity halves the thrust. A chemical engine has no such constraint because its power comes from the propellant itself and arrives with it; an electric engine’s power comes from a solar array or a reactor that has to be carried.

That is what sets the design point. Raising the exhaust velocity saves propellant, by the exponential in the rocket equation; but it lowers the thrust for a given power, which lengthens the trip; and buying the thrust back means a larger power plant, which is dry mass carried the whole way.

The optimisation therefore runs over three quantities that trade against one another — propellant mass, power-plant mass and trip time — and the answer depends on a single figure of merit for the power system: its specific mass, in kilograms per kilowatt. At 20 kg/kW, typical of large solar arrays, the optimum exhaust velocity for an inner-solar-system mission is around 30 km/s; at 5 kg/kW it is considerably higher and the trips are shorter.

So the choice of engine is really a choice about the power supply, and a mission’s exhaust velocity is a conclusion rather than a specification. That is why the same thruster is flown at different operating points on different missions, and why the interesting engineering in electric propulsion has always been in the arrays and the radiators rather than in the thruster.

What the spiral flies through #

The trajectory’s shape has consequences that the velocity budget does not contain, and they have cancelled missions.

A spiral from low orbit to geostationary crosses the radiation belts slowly — months rather than the hours a chemical transfer takes — and the belts are full of trapped electrons and protons that degrade solar cells. A spacecraft that arrives with fifteen per cent less array output than it left with has lost part of the power supply the previous section made the centre of the design, and it has lost it permanently.

There is a second geometric cost. A spacecraft in a low orbit passes through the Earth’s shadow on every revolution, and a solar-electric thruster cannot fire in shadow. Early in the spiral that is a third of each orbit lost; the fraction falls as the orbit rises, but the cumulative effect over hundreds of revolutions is weeks of added transfer time that no continuous-thrust calculation predicts.

And the thrust direction is not free. The efficient direction is along the velocity vector, which for a spiral means the spacecraft must rotate continuously to keep the thruster pointed — while also keeping the arrays facing the Sun and the antenna facing the Earth. Those three requirements are generally incompatible, and the resolution is either articulated arrays, or a thrust direction slightly off-optimal, or periods of no thrust at all.

The idealised spiral has one parameter and the flown one has an operations plan, and the gap between them is why the achieved transfer times run longer than the textbook figures by a substantial margin.

The twenty per cent penalty is not a constant, and the two ends of the radius ratio show how far it moves.

What happens when the thruster stops #

A chemical mission’s engine fires for minutes and coasts for months; a low-thrust mission’s engine fires for years, and that inverts which failures matter.

A thruster that stops for a week on a chemical mission is a scheduling problem. On a spiral it is a week of the trajectory that did not happen, and because the trajectory is the integral of the thrust there is no way to make it up except by extending the mission — the spacecraft is not where the plan said it would be, and every subsequent target date moves with it.

That has two design consequences. Low-thrust missions carry more thrusters than they need and cycle between them, both to spread the wear over the fleet and to survive a failure outright; and their trajectories are re-optimised in flight, repeatedly, from wherever the spacecraft actually is, rather than being flown as a fixed plan laid down before launch.

The re-optimisation is possible precisely because the trajectory is continuous. There is no missed manoeuvre window to recover from and no burn that had to happen at a particular point in an orbit; there is only a slightly different starting state and a new solution computed from it. The property that makes the transfer expensive — that the thrust is spread over everything instead of concentrated where it is worth most — is the same property that makes it forgiving, and for a mission measured in years that trade has generally been worth making.

It is also why these missions publish their trajectories as achieved rather than as planned. The flown path is the integral of everything that actually happened to the thruster over several years, and no two are alike even between identical spacecraft.

The comparison a designer actually makes is therefore not between two trajectories but between two mission profiles, one of which has a fixed cost and a fixed date and the other a cost that depends on how the hardware behaves over several years of continuous operation.

Both profiles have a fixed launch mass, and that is the only quantity the two share, which is why the comparison is made in kilograms delivered rather than in metres per second spent.

Where the model stops #

The near-circular assumption. The closed form $|v_1 - v_2|$ assumes the orbit stays circular throughout, which is true only for very low thrust. At higher accelerations the spiral develops eccentricity and the true cost falls somewhat below the closed form — towards the impulsive answer, continuously, as the thrust rises. The two results above are the two ends of one family and there is no discontinuity between them.

Shadowing. A spacecraft spiralling outward from low Earth orbit passes through the Earth’s shadow on every revolution, and a solar-electric system produces no thrust there. Early in the spiral the shadow is a large fraction of the orbit, so the effective duty cycle is well below one and the transfer time is correspondingly longer. Nothing in the closed form knows about it.

The radiation belts. Between about 1,000 and 20,000 km the spiral spends months inside the trapped-particle belts, and the accumulated dose degrades the solar arrays that are supplying the power. That degradation reduces the thrust, which lengthens the transfer, which increases the dose. It is the practical limit on the technique for large spacecraft, and it appears in no equation on this page.

Plane changes are different. Everything here is a coplanar transfer. Turning the orbital plane costs $2v\sin(\Delta i/2)$ and is proportional to the speed it is performed at, so a low-thrust vehicle can do it far out where the speed is low — and combining the plane change with the spiral is a genuine optimisation problem rather than a formula. Continuous thrust turns out to be better than impulsive for large plane changes, which reverses the whole comparison above.

What the picture cannot show #

The number of revolutions. The spiral in the first figure makes 26 turns because it was drawn at fifty times the real thrust. A drawing at $3\times10^{-5}$ would be 1,300 turns and would be a solid annulus. The one thing about a low-thrust transfer that everyone finds surprising is the thing no drawing of it can carry.

The time. Months, against hours for the impulsive transfer. Every figure here has radius or velocity on its axes and none has a clock.

The power. An ion engine’s thrust is limited by the electrical power available, so the whole technique is a statement about solar arrays and nuclear sources rather than about orbital mechanics. Doubling the array doubles the thrust and halves the transfer time, at no cost in Δv — which means the interesting design variable is on none of these plots.

The engine was tested before the trajectory made sense #

Electrostatic propulsion was proposed by Tsiolkovsky in 1911 and by Goddard in 1906, before any of the trajectory analysis existed, and for the same reason both of them worked on the rocket equation: they had noticed that the exhaust speed is in a denominator.

The first flight test was SERT-1 in 1964, which ran a caesium ion engine for 31 minutes on a suborbital hop. The first operational use for station-keeping was in the 1970s, and the first use for primary propulsion was Deep Space 1 in 1998 — nine decades after the idea and thirty-four years after the first firing.

What took the time was not the engine. It was electrical power. An ion engine’s thrust is proportional to the power it is fed, and a kilowatt in orbit was, for most of the twentieth century, a substantial spacecraft in itself. The technique became practical when solar cells became cheap enough per watt that a few kilowatts stopped being the mission — which is a fact about photovoltaics, arriving in celestial mechanics by way of a denominator. And the other end of the range, where the comparison reverses.

Where the ladder goes next #

Later rungs on this anchor: the optimal-control formulation, in which the thrust direction is a free function and the answer comes from Pontryagin’s principle rather than from a closed form. Edelbaum’s combined plane-change-and-raise solution, which is the one equation covering the case the last figure above is about. Low-energy transfers through the Lagrange points, where the invariant manifolds of the restricted three-body problem provide almost-free routes at the cost of very long flight times. Solar sails, which have no propellant at all and therefore no rocket equation. And the specific-impulse-versus-power trade, which is where an electric propulsion system is actually designed: higher exhaust speed needs more power per newton, so the optimum depends on the mission duration in a way neither the trajectory nor the engine decides alone.

The comparison in this essay is often stated as “electric propulsion is more efficient”. It is not more efficient in the sense the trajectory measures — it uses a fifth more velocity change to do the same job. What it is, is efficient in a different currency, and the entire art is in noticing that the two currencies have an exchange rate and that it is not one.