The thrust that rounds an orbit turns once a revolution

An electric thruster pushes a spacecraft with an acceleration of a few tenths of a millimetre a second squared — enough, over months, to change its velocity by kilometres a second, at a tenth of the propellant a chemical engine would use. The price of the gentleness is that the push is spread around the whole orbit instead of being concentrated where it is worth most, and the comparisons of cost that justify flying it at all assume that the thrust is pointed well.

Pointing it well is a problem in its own right, and it has a surprisingly clean structure. The orbit is described by its elements — its size, its shape, its orientation — and the equations Gauss wrote down two centuries ago say exactly how fast each element changes for a push of a given size in a given direction at a given point in the orbit. Those equations are linear in the push, so for any job that can be written as a combination of element changes, the best direction at each instant is a closed form. What comes out is not obvious: for the most common job in commercial spaceflight, the best direction turns through a full circle every orbit.

How a push changes an orbit #

Gauss’s variational equations give the rate of change of each orbital element under a small acceleration, split into a part along the radius from the central body and a part along the direction of motion perpendicular to it. Which direction moves which element reads them one element at a time; what matters here is what happens when they are read for a job that needs two elements moved at once. For the two that matter in the plane of the orbit:

$$\frac{da}{dt} = \frac{2a^2}{h}\left(e\sin\nu\,R + \frac{p}{r}\,S\right), \qquad \frac{de}{dt} = \frac{1}{h}\Big(p\sin\nu\,R + \big((p+r)\cos\nu + re\big)\,S\Big),$$

with $a$ the semi-major axis, $e$ the eccentricity, $\nu$ the position around the orbit measured from perigee, $p$ the semi-latus rectum, $h$ the angular momentum, $r$ the distance, and $R$ and $S$ the radial and along-track accelerations.

Each rate is a weighted sum of $R$ and $S$, and for a push of fixed size pointed at an angle $\alpha$ from the direction of motion, $R$ and $S$ are the sine and cosine of $\alpha$ times the push. Each rate is therefore a sinusoid in $\alpha$, and the angle that maximises it is the angle of the vector of its two weights. The same is true for any weighted combination of the two rates, so whatever the job, the best direction at each point of the orbit is a single arctangent of quantities known from the current orbit.

That is the whole mathematical content of a locally optimal steering law. What makes it interesting is what the arctangents turn out to be, and that they disagree with each other almost everywhere in the orbit.

Two jobs, two directions #

For raising the semi-major axis, the best direction is the one whose radial and along-track parts are in the ratio $e\sin\nu$ to $p/r$ — which is the direction of the velocity itself, tilted from the local horizontal by the flight-path angle. Pushing along the velocity adds energy at the greatest possible rate, because the work done is force times velocity and that product is largest when the two are parallel. The angle never strays more than about seventeen degrees from the along-track direction on an orbit of eccentricity 0.3, and it is exactly along-track at perigee and apogee.

For reducing the eccentricity the answer is completely different. At apogee the best direction is forward: a push there raises the perigee, which rounds the orbit. At perigee it is backward: a push against the motion there lowers the apogee, which also rounds it, at the cost of some energy. In between, the best direction swings through the radial — outward on one side of the orbit, inward on the other — so the thrust vector makes one complete turn relative to the direction of motion every revolution.

No fixed direction serves both jobs. A spacecraft that needs to grow its orbit and round it off at the same time must point its thrust somewhere between the two curves at each moment, with the balance set by how much of each job remains.

A direction that turns in the orbit and stands still in space #

The turning is a statement about the frame the angle is measured in. Measured from the direction of motion, which itself turns once a revolution as the spacecraft goes round, the best thrust for rounding an orbit swings through a full circle. Measured against the stars, it barely moves.

On a nearly circular orbit the arithmetic is short enough to do in full. Put the line from the centre to perigee along one axis and let the spacecraft sit at angle $\theta$ around the orbit. A push of size $f$ fixed in space, pointed at right angles to that line, has a radial part $-f\sin\theta$ and an along-track part $-f\cos\theta$, and the near-circular form of Gauss’s equation for the eccentricity along the perigee line,

$$\frac{de}{dt} \approx \frac{1}{v}\left(R\sin\theta + 2S\cos\theta\right),$$

then gives a rate of $-(f/v)(\sin^2\theta + 2\cos^2\theta)$, which averages to $-\tfrac{3}{2}f/v$ over a revolution and never changes sign. A push that is never re-pointed reduces the eccentricity steadily. Steering perfectly at every instant, along the vector $(\sin\theta, 2\cos\theta)$, does only slightly better: the average of $\sqrt{\sin^2\theta + 4\cos^2\theta}$ is about 1.54, against 1.5 for the fixed direction. A spacecraft that holds one attitude does within three per cent of the best that steering can do.

That is why the title’s turning is not a mechanical burden. A spacecraft does not have to slew its engine through a circle every orbit; it has to hold its engine still against the stars while the orbit carries it round, which is what a spacecraft’s attitude control does most easily. The turn is in the frame of the orbit, and it is the orbit that turns.

Raising the orbit is the opposite case. Its best direction is fixed in the orbit’s own frame, along the velocity, and so it turns once a revolution against the stars. The two jobs are each simple in a different frame, and a law that does both at once is simple in neither.

Where a push is worth most #

The steering law says which way to push at each point. It does not say how much good the push does there, and that varies round the orbit too.

Raising an orbit is most effective at perigee, four times more so than at apogee on this orbit, because the energy a push adds is proportional to the speed it is applied at — the same burn is worth more when moving fast, which is why a chemical stage fires at perigee to raise an apogee. Changing the eccentricity is as effective at apogee as at perigee per unit time, and weakest about 115 degrees from perigee where the best push has turned partly sideways.

Two further facts about time change the picture. The spacecraft spends far longer near apogee than near perigee, so a continuous thrust delivers most of its total effect on the orbit’s shape near apogee whatever its rate per unit time. And a low-thrust spacecraft can choose not to thrust: coasting through the parts of the orbit where the push is worth least saves propellant at the cost of a longer trip. Every steering law makes that trade, explicitly with a rule for when to switch the engine off, or implicitly by thrusting always and accepting the cost.

Two spirals out of a transfer orbit #

The opening figure puts the steering to a test that commercial spacecraft face constantly. A geostationary satellite is launched into a transfer orbit with its perigee a few hundred kilometres up and its apogee at geostationary height, 42,164 kilometres from the Earth’s centre — an elongated orbit of eccentricity 0.73 with a semi-major axis little more than half the target’s. It has to be raised and rounded off at once.

Under a constant acceleration of 0.2 millimetres a second squared pointed along the velocity, the spiral raises the semi-major axis to geostationary size in sixty-four days. But tangential thrust barely changes the eccentricity: it raises the perigee and the apogee together, and the orbit grows like a stretched ellipse. When the semi-major axis arrives, the eccentricity is still 0.59, and the orbit swings between seventeen thousand and sixty-seven thousand kilometres — nowhere near a geostationary orbit, and with no cheap way to fix it from there.

Under a steering law that, at every moment, points the thrust to reduce a weighted measure of how far the orbit is from the target in both size and shape, the spiral takes a different path in the figure: the eccentricity falls steadily as the semi-major axis grows, and the orbit arrives circular at geostationary height after 129 days. The law is greedy — it does what is best for the measure right now, with no look-ahead — and it is a simplified form of the proximity-quotient laws used to design real electric transfers.

Why pushing along the velocity keeps the shape #

The tangential spiral’s failure follows from the eccentricity equation with $R = 0$. Along-track thrust changes the eccentricity at a rate proportional to $(e + \cos\nu)$ divided by the speed: positive through the half of the orbit near perigee, where the push raises the apogee and stretches the orbit, and negative through the half near apogee, where it raises the perigee and rounds it.

On a circular orbit those two halves cancel exactly, and along-track thrust leaves a circle a circle — which is why it is the right law for the spiral out from a low circular orbit. On an elongated orbit they nearly cancel. The spacecraft lingers near apogee, which favours rounding, but it moves slowly there and the rate carries the speed in the denominator, which favours the stretching near perigee. What is left over is a slow drift: from 0.73 to 0.59 across sixty-four days in the figure, while the semi-major axis nearly doubled.

So tangential thrust does exactly what its derivation promised. It adds energy at the fastest possible rate and pays no attention to anything else. The trouble is that from a transfer orbit the energy is the smaller part of the job; the satellite’s apogee is already where it needs to be.

The low point is the whole job #

What distinguishes the two spirals is what happens to the perigee. The transfer orbit’s apogee is already at the target height, so the job is almost entirely to raise the perigee from a few hundred kilometres to forty-two thousand. Along-velocity thrust spends a large part of its effort raising the apogee as well — beyond the target, to sixty-seven thousand kilometres by the time the semi-major axis arrives — and that effort is wasted. The steered law, which knows that the eccentricity has to fall, thrusts mostly forward near apogee, where a push raises the perigee, and its apogee stays near the target throughout.

Both spirals spend propellant at the same rate, since the engine runs continuously at the same thrust. The steered one runs twice as long and arrives at the right orbit; the tangential one arrives sooner at the wrong one. Measured by the velocity change delivered, the steered spiral costs more, and that is the honest comparison: the cheapest way to raise a semi-major axis is not the cheapest way to reach a particular orbit.

Coasting where the push is weak #

A greedy law that thrusts all the time still wastes some of its push. Near the points of the orbit where the best available direction does little for the measure it is reducing, the engine is spending propellant at the full rate for a small return. Every flown steering law therefore carries a second rule, for when to thrust at all: compare the rate at which the best direction reduces the distance to the target here with the best rate available anywhere on the current orbit, and switch the engine off where the ratio falls below a threshold.

Raising the threshold trades time for propellant. A threshold of zero is the continuous spiral drawn here. A threshold near one thrusts only in a narrow arc around the single best point, which approaches the impulsive transfer’s cost and takes correspondingly longer, since the engine runs for a small fraction of each revolution. For the transfer-orbit problem the best arcs are around apogee, where rounding and raising the perigee are both efficient, and a threshold in the middle of the range recovers a useful part of the gap between the spiral and the chemical burn for a modest increase in duration.

The figures leave the coast arcs out deliberately, so that the comparison between the two directions is not confused with a comparison between two duty cycles. In a real design the two choices are made together.

What the gentle route costs #

The steered spiral delivers about 2.24 kilometres a second of velocity change whether the thrust is 0.1, 0.2 or 0.4 millimetres a second squared, taking 258, 129 or 65 days. A chemical stage does the same job with one burn at apogee of about 1.47 kilometres a second, which rounds off the orbit and raises its perigee at the point where doing so is cheapest. The spiral is fifty per cent more expensive in velocity, because it thrusts all the way round the orbit and not only at apogee, and the cost hardly depends on the thrust level once the thrust is small enough that the orbit changes little in one revolution.

It is flown anyway, increasingly, because the engine is chosen by the calendar and by mass. An electric thruster’s exhaust leaves five to ten times faster than a chemical engine’s, so by the rocket equation half again the velocity change costs a fraction of the propellant. A satellite that saves a tonne of propellant on its way to geostationary orbit can be launched on a smaller rocket or carry more payload, and the price is several months of waiting.

From the best direction now to the best path #

The steered law chooses the best direction at each instant for a measure of distance from the target. It does not guarantee the best path, and the difference is the subject of optimal control.

The globally optimal path — the one that reaches the target with the least propellant in a given time, or in the least time with a given thrust — is found by Pontryagin’s principle, which introduces a costate for each element and shows that the optimal thrust direction at each instant is the direction of a combination of the costates weighted exactly as Gauss’s equations weight $R$ and $S$. The costates are themselves governed by differential equations, and finding the optimal path means solving a boundary-value problem: costates chosen at the start so that the spacecraft arrives at the target. For impulsive transfers the same construction is Lawden’s primer vector, whose magnitude says where burns belong.

A locally optimal law is the same construction with the costates replaced by fixed weights on the distance to the target. That replacement is what makes it greedy, and it is why it can be computed in one line at every step instead of by solving a boundary-value problem. For orbit-raising problems like this one the greedy laws come within a few per cent of the true optimum, which is why they are used to design missions and to fly them, and the true optimum is computed afterwards to see how much was left on the table.

What was actually measured #

All-electric satellites. In 2015 two communications satellites built with no chemical engine for orbit raising were launched together and raised themselves from transfer orbits to geostationary orbit with xenon ion thrusters over about six months each. Since then electric orbit raising has become common for large commercial satellites, and the transfer durations of four to six months are set by exactly the arithmetic drawn here: velocity changes of about two kilometres a second at accelerations of a few tenths of a millimetre a second squared.

A spiral to the Moon. A European spacecraft launched in 2003 spiralled out from a transfer orbit to the Moon using a solar-electric thruster, taking over a year and many revolutions, with a steering law that switched between raising the orbit and reshaping it as the mission progressed.

Flight software. The steering laws flown are versions of the locally optimal laws drawn here, with weights tuned by optimisation beforehand and with coast arcs where the efficiency of thrusting falls below a threshold.

Where the model stops #

No shadow. A solar-electric spacecraft loses power in the Earth’s shadow, which it passes through on most revolutions of a transfer orbit, and the thrust stops. Real transfers plan around eclipses, which lengthen them, and the shadow falls on whichever part of the orbit faces away from the Sun — which drifts through the year, so the arcs lost are sometimes the valuable ones near apogee.

No oblateness, no plane change. The Earth’s equatorial bulge perturbs the orbit continuously and turns its node and its perigee at rates that are large while the perigee is low, and most transfers must also change the orbital plane — the part of the problem Edelbaum’s analysis covers. Both are left out; the spirals here are in one plane around a point mass.

Constant acceleration. The thrust is held at a fixed acceleration. A real spacecraft gets lighter as it burns propellant, so its acceleration rises through the transfer.

One weighting. The blended law weights the size and shape errors equally. Other weightings give different paths and slightly different costs, and choosing them is part of mission design.

A rule that is linear in the control #

The general lesson is about a structure that makes control problems tractable. When the rate of change of everything that matters is linear in a control of fixed size, the best control at each moment for any linear measure of progress is a closed form — the direction of a vector of weights — and the whole difficulty moves into choosing the weights. Gauss’s equations have that structure for thrust, and so does every problem in which a small force acts on an orbit, from drag to a third body’s pull, where the elements drift rather than staying constant and the drift is computed in exactly the same way — with nobody choosing the direction.

When an effect is linear in a push, the best direction to push is wherever the weights point, and what changes from one job to another is only the weights. Raising an orbit and rounding it off are the same arithmetic with different weights, and the difference between them is a thrust vector that stays still in one case and turns once a revolution in the other.

Still open: thrust that comes from sunlight alone #

An electric engine still carries propellant, and its total velocity change is limited by the mass it can afford to throw away. A solar sail carries none: its push comes from the momentum of sunlight, with no rocket equation at all. But a sail cannot point its thrust anywhere it likes. Its force is always directed roughly along the normal to the sail and always has a component away from the Sun, so the steering law for a sail is the same problem with a constraint on the directions available — and the constraint changes which orbits can be reached at all.