A turn spread where the climb leaves room

A satellite launched from Cape Canaveral into a low parking orbit starts with an inclination of 28.5°, the latitude of the launch site, and must end in geostationary orbit over the equator with none. A chemical upper stage does the turn where it is cheapest, at the top of a transfer ellipse, in one burn. An electric engine cannot: it thrusts for months, gently and continuously, and it must remove the inclination while it raises the orbit. Edelbaum’s closed form gives the cost of doing both at once, about 5.95 kilometres a second, by tilting the thrust out of the orbital plane by an angle that grows through the transfer.

The tilt in Edelbaum’s law has one size through each revolution. It switches from one side of the plane to the other at the two antinodes — the points a quarter of a turn from the nodes — because an out-of-plane push turns the plane in one sense before the antinode and the other sense after it. But its size does not change round the orbit, and that is a simplification made to keep the problem solvable in closed form, not a property of the optimum.

The hero figure draws what the optimum actually does. Round one revolution, at the start, half-way through and near the end of the transfer, the optimal tilt (solid) follows the argument of latitude: largest at the nodes, zero at the antinodes, smoothly between. Edelbaum’s (dashed) is a square wave of the same average sense. Half-way through, the optimal thrust leans 44° out of the plane at the nodes, where Edelbaum’s leans 34° all round.

Why the tilt should follow the nodes #

An out-of-plane force turns an orbit’s plane by rotating its angular momentum, and how much a push rotates it depends on where in the orbit it is applied. At the nodes — where the orbit crosses the reference plane — a sideways push tilts the plane directly. At the antinodes, a quarter of a turn away, the same push swings the line of nodes round instead and changes the inclination not at all. In between the efficiency falls as $|\cos u|$, with $u$ the argument of latitude measured from the ascending node. A pure plane change exploits this by thrusting only in arcs about the nodes and coasting elsewhere, which recovers most of the $\pi/2$ that a continuous turn wastes, at the cost of time.

The in-plane thrust has no such geometry. On a nearly circular orbit, a push along the velocity raises the orbit by the same amount wherever it is applied. So a transfer that both raises and turns has two jobs with different maps over the orbit: one indifferent to where it is done, one done well only near the nodes. The best division of the thrust between them is the one that does the turning near the nodes and the raising everywhere else.

Written as an optimisation, each instant’s thrust direction maximises a weighted sum of the two rates, the weights being fixed by how far the transfer has to go in each. The in-plane rate is proportional to the cosine of the tilt and the out-of-plane rate to its sine times $|\cos u|$, so the optimum satisfies

$$\tan\beta = k\,|\cos u|,$$

with $k$ set by the ratio of the two weights and the orbital speed. Near the antinodes $\cos u$ vanishes, so the thrust lies in the plane and raises; near the nodes it leans out as far as $k$ allows and turns. Edelbaum’s law replaces $|\cos u|$ by its average, $2/\pi$, and so spends out-of-plane thrust at the antinodes, where it does nothing, and too little at the nodes.

The same transfer, both ways #

The two laws are flown here in the orbit-averaged problem: circular orbits throughout, the thrust a small acceleration, the transfer computed in terms of the velocity change spent, so that the answer is independent of how strong the engine is. For each law the free constant is found by shooting — adjusting the initial ratio of the weights until the transfer arrives at geostationary speed with exactly the inclination removed. For the constant-yaw law the result reproduces Edelbaum’s closed form to four figures, which checks the method.

Both laws defer most of the turning to the end of the transfer, where the orbit is high and slow and a degree of plane change costs least — the same reason a chemical stage turns at apogee. What differs is the efficiency of every revolution along the way. The varying law extracts more turning per metre per second at every stage, because it never spends out-of-plane thrust where the geometry wastes it, and it pays nothing for that on the climb, because the thrust it keeps in the plane near the antinodes raises the orbit exactly as well as it would anywhere.

Four ways to pay #

The comparison is clearest against the alternatives a mission planner would actually weigh.

Raising first and turning at the end is the obvious sequence and the worst. It is the low-thrust cousin of the chemical mistake of turning at the worst speed there is, inverted: a chemical stage should turn high and slow, and does; a low-thrust spacecraft also turns high and slow if it raises first, but it then has to turn continuously, at every point of every revolution, and most of those points are bad ones. It does the turning where it is cheapest per degree, at the slow top of the transfer, but a continuous turn wastes the factor $\pi/2$ that thrusting at every point of the orbit costs. Turning in arcs about the nodes at the top recovers most of that, at the cost of coasting for two-thirds of each orbit, and brings the total to 6.25 kilometres a second. Blending the turn into the climb does better than either, because it begins turning as soon as the out-of-plane thrust is worth anything and uses the climb to fill the rest of each orbit; Edelbaum’s constant yaw gets 5.95 that way. The varying yaw gets 5.80.

The structure of that last improvement is the point of the essay. The arcs solution for a pure plane change saves its factor by switching the engine off where turning is poor, and pays for it in time. The blended solution has nowhere it needs to switch off, because there is always something useful for the thrust to do: where it cannot turn, it raises. The coasts of the pure plane change become climbing in the combined one. Edelbaum’s constant yaw half-exploits this — it blends, but uniformly — and the varying yaw exploits it fully.

The difference in the budget comes entirely from the turning. Both blended laws raise the orbit by the same 4.65 kilometres a second of speed change, the difference between the circular speeds at the two ends, because on a circular spiral the gentle transfer pays the full difference in speed rather than the smaller sum a two-burn Hohmann transfer pays. What separates them is how much additional velocity change the 28.5° costs: about 1.30 kilometres a second with constant yaw and 1.15 with varying yaw, on top of the climb. Measured that way the variation saves more than a tenth of the cost of the turn.

The same comparison at a smaller turn #

At ten degrees the ordering is the same and the gaps shrink. Raising first is still worst by more than half a kilometre a second, because the $\pi/2$ penalty of turning continuously at the top applies to whatever turn there is; the arcs still recover most of it; blending still wins. The varying yaw’s margin over Edelbaum falls to thirty-three metres a second, because with so little turning to do the out-of-plane thrust is a small part of every revolution and its misplacement at the antinodes costs little.

How much the variation is worth #

The gain depends on how much of the transfer’s work is turning.

For a small turn the out-of-plane thrust is a small fraction of the total, and wasting a third of it at the antinodes costs little. As the turn grows the waste grows with it, until near thirty degrees the saving levels off at about two and a half per cent. Larger turns than about forty degrees change the character of the transfer — the optimal thrust leans so far out of the plane early on that the orbit is barely raised at first — and the circular-orbit model used here is no longer the right one to compare.

For a launch from a site nearer the equator the turn is smaller and the gain correspondingly less.

Ten degrees of inclination — a launch from a site near the equator, or a parking orbit chosen to leave part of the turn for the upper stage — brings the yaw down to under twenty degrees at the nodes and the saving to a few tens of metres a second. The shape of the law is unchanged: follow the nodes. Only the amplitude depends on how much turning there is to do.

What 150 metres a second buys #

An electric engine with an exhaust speed of twenty kilometres a second spends about three-quarters of a per cent of the spacecraft’s mass to deliver 150 metres a second. For a five-tonne satellite that is about forty kilograms of xenon — not decisive for a mission’s design, but not nothing either, and it is propellant that would otherwise have to be launched, stored, and accounted for in fifteen years of station-keeping afterward. The time saved is proportional: about two and a half per cent of a transfer that takes several months, roughly a week.

Against those savings sit the costs of executing the law. The thrust vector has to yaw continuously round each orbit, following a smooth curve rather than flipping between two fixed angles twice a revolution. A spacecraft whose solar arrays must stay pointed at the Sun while its thruster swings has to coordinate the two through the whole transfer, which Edelbaum’s law already demands in simpler form. The smooth law is gentler on that coordination than a square wave, which asks for an abrupt flip twice per orbit, so the practical cost of the variation may well be negative.

The real limits on such transfers lie elsewhere. A spiral from low orbit to geostationary passes slowly through the radiation belts, and the damage the solar arrays accumulate there grows with the time spent; the engine’s exhaust speed is chosen by the calendar as much as by the propellant. Shadow passes, during which a solar-electric engine cannot thrust, cut out parts of each early orbit and break the symmetry the steering law assumes. A steering law that saves two and a half per cent of the velocity change while those effects move the timeline by tens of per cent is a refinement, but it is a free one.

What Edelbaum gave up, and why #

Edelbaum’s analysis, published in 1961, was a piece of engineering mathematics designed to give a closed form a mission planner could evaluate by hand, and the constant-yaw assumption is what made it possible. With the tilt fixed within each revolution, the averaged equations separate and integrate to one square root — the law of cosines with the inclination change multiplied by $\pi/2$. With the tilt varying as $|\cos u|$, the averages become integrals of the inverse tangent and the problem has to be solved numerically, as here.

The trade was a good one. The closed form is within two and a half per cent of the circular-orbit optimum for any realistic turn, it exposes the structure of the problem — why the turning is deferred to the end, why the cost is a triangle — and it gives an initial guess that a numerical optimiser converges from in a few iterations. Applying optimal control to the same averaged problem, without the constant-yaw restriction, gives the varying law directly; the figures compute it rather than quote it. That a sixty-year-old approximation is that close to the optimum it simplified is a measure of how well its author chose what to give up.

There is a family resemblance to the thrust that rounds an orbit, the steering law that removes eccentricity by tilting the in-plane thrust once a revolution. There too the efficient law varies round the orbit because the effect of a push depends on where it is applied, and there too the constant-direction law gives a usable approximation. Every low-thrust steering problem has this shape: a quantity the thrust changes at a rate that depends on the orbital phase, and an optimal law that concentrates the effort where the rate is highest.

Checking the law against its own claim #

A steering law derived from an optimisation makes a claim that can be tested without trusting the derivation: no small change to it should reduce the cost. The test applied here is the simplest one. Take the varying law’s shape, $\tan\beta = k|\cos u|$, and replace the exponent on $|\cos u|$ — one in the derived law — by other values, flattening the shape toward Edelbaum’s square wave at zero or sharpening it toward pure node-thrusting at large exponents, and re-solve the transfer each time.

The cost is lowest at an exponent of one and rises on both sides: toward Edelbaum’s 5.95 as the shape flattens, and toward 5.90 as it sharpens. A flatter shape wastes out-of-plane thrust near the antinodes; a sharper one turns efficiently at the nodes but takes the thrust there away from the climb, which then has to be made up elsewhere at a worse ratio of turning to raising.

That the minimum falls where the derivation put it is a check on the arithmetic of the optimal-control argument rather than on its assumptions; the assumptions — circular orbits, continuous thrust, a fixed node — are tested only by flying something more realistic.

Where the model stops #

The orbits stay circular. Every result here is for transfers through a sequence of circular orbits, the assumption that makes the problem orbit-averageable. The true optimum allows the orbit to become eccentric and concentrates the turning near apoapsis, which can save a few per cent more for large plane changes; the varying yaw drawn here is the best circular-orbit law, not the best law.

The thrust is continuous. Real transfers lose thrust in the Earth’s shadow, which for low orbits is a third of each revolution. Where the shadow falls relative to the nodes changes which parts of each orbit can be used, and an optimal law with shadows has to be computed for the actual geometry.

The node is held fixed. The Earth’s equatorial bulge turns the line of nodes of an inclined orbit by several degrees a day in low orbit, which moves the points where turning is efficient. The law follows the nodes wherever they are, but the transfer’s total cost changes slightly when they move.

The steering is orbit-averaged. Within one revolution the orbit changes slightly, and the law is computed as though it did not. For a transfer of several hundred revolutions the error is far smaller than the gain.

Still open: shadows that fall on the nodes #

The varying law has a vulnerability the constant law shares and magnifies. It concentrates out-of-plane thrust at the nodes; if the Earth’s shadow falls on a node, the turning that should have been done there cannot be, and the law must reallocate it to parts of the orbit where it is less efficient. Because the orbit’s plane and the direction of the Sun change through a months-long transfer — the line of nodes regressing under the bulge, the Sun moving round the sky — the shadow drifts across the nodes and away again, and the cost of the transfer depends on the date of launch.

Computing that dependence means solving the steering problem with shadows for every launch date, and the answer would say how much of the varying law’s saving survives a real calendar: whether there are seasons in which launching to a turn-and-raise transfer is distinctly cheaper because the shadows avoid the nodes through the critical last month, when most of the turning is done.