A plane turned only where turning is cheap

Turning the plane of an orbit is the most expensive manoeuvre in spaceflight per degree, because the whole orbital velocity has to be rotated and the cost is the chord of the rotation: twice the speed times the sine of half the angle. An impulsive turn pays that at a node, where the orbit crosses the reference plane. A turn with a weak engine, spread over hundreds of revolutions, pays more. Edelbaum’s analysis of continuous low thrust put the factor at π/2 for small turns — a continuous turn spends fifty-seven per cent more velocity than an impulsive one.

That penalty is usually presented as the price of low thrust, and accepted, because an electric engine’s exhaust is so fast that the extra velocity costs little propellant. But the penalty has a specific origin, and the origin suggests a cheaper way to fly. A continuous turn pushes for the whole of every revolution, including the parts of the orbit where a sideways push does almost nothing to the inclination. Thrusting only where the push is effective costs less velocity and more time, and the trade between the two turns out to be one of the more useful pieces of arithmetic in practical orbit control.

Where a sideways push tilts the plane #

A push perpendicular to an orbit’s plane does two different things depending on where in the orbit it is applied, and the distinction is the whole argument.

At a node, the spacecraft is crossing the reference plane, and a push perpendicular to the orbit rotates the orbit about the line joining the two nodes: the inclination changes and the line of nodes stays where it is. A quarter of an orbit away, at the highest point above the reference plane, the same push rotates the orbit about a different axis — the line through the spacecraft’s current position — and the effect is to swing the line of nodes round the pole while leaving the inclination unchanged. In between, a push does a mixture, in proportion to the cosine and the sine of the angle from the node.

So a continuous turn, to keep changing the inclination in one sense, must reverse its push at every node crossing: push north on one half of the orbit, south on the other. Averaged over a revolution, the useful part of a constant push is the average of the absolute value of the cosine, which is two over pi. That is Edelbaum’s factor, seen from the other end: the engine runs all the time, but on average only two-thirds of its push goes into the plane, and the rest swings the node back and forth, the swings on opposite halves of the orbit cancelling.

Thrusting in arcs #

If the engine runs only within an arc of half-width α about each node and coasts the rest of the way round, the average effectiveness of the push while it runs is the average of the cosine over the arc, sin α divided by α. The velocity spent per unit of inclination is then the orbital speed times α over sin α: one, the impulsive cost, as the arcs shrink to points, and π/2 as they widen to cover the whole orbit.

The function is extraordinarily flat near zero. Arcs of ±30 degrees — the engine running for a third of every orbit — cost less than five per cent more than an impulse. Arcs of ±10 degrees cost half a per cent more. Most of Edelbaum’s fifty-seven per cent is spent in the parts of the orbit within forty-five degrees of the points a quarter-orbit from the nodes, where the push is mostly swinging the node.

The price is time, and it rises steeply as the arcs narrow. A turn made in arcs of ±30 degrees takes twice as long as a continuous one, because the engine is running a third of the time but each second of thrust is more effective; arcs of ±10 degrees take nearly six times as long. Between continuous thrust and very narrow arcs there is a smooth trade, and where on it a mission sits depends on what it values.

The formula, checked against an orbit #

The averaging behind the formula assumes that the push is so weak that the orbit barely changes in one revolution, which is exactly the low-thrust case, and it can be checked directly.

The integration follows the spacecraft through the full equations of motion, switching the push on and off by its position relative to the node and reversing its sign at each crossing, and measures the velocity spent and the inclination gained. The agreement is exact to the precision drawn, because the push changes the orbital speed by only a few millimetres per second over six orbits. The arithmetic of arcs is not an approximation of the averaged theory; it is the averaged theory, and the averaged theory is right.

Propellant against time #

For an electric spacecraft the trade is between propellant and months.

A ten-degree turn in low Earth orbit is a large manoeuvre: two kilometres per second if made continuously, a velocity a chemical stage would spend most of a spacecraft’s mass to deliver. With an electric thruster it is feasible, at a cost of about an eighth of the spacecraft’s mass in propellant and seven months of thrusting. Narrowing the thrust to arcs of ±30 degrees saves almost forty kilograms — a third of the propellant — and doubles the time to fifteen months. Narrowing them further saves almost nothing more, and triples the time again.

The knee of the curve is where most missions would sit. The first halving of the arcs buys most of the saving; after that, each kilogram costs weeks. Whether those weeks are worth it depends on whether the spacecraft has anywhere better to be, whether its thruster’s lifetime is measured in hours of firing, which arcs conserve, and whether the radiation dose of a slow climb through the belts matters, which the time does not.

The satellites that already do this #

The cleanest application is one that happens every day. A geostationary satellite’s orbit is tilted out of the equator by the Sun and Moon at about 0.85 degrees a year, and holding it in the equatorial plane — north–south station-keeping — is by far the largest item in its propellant budget.

Satellites with chemical thrusters make the correction in short burns at a node every few weeks, close to the impulsive budget of about forty-six metres per second a year. Satellites with electric thrusters cannot make short burns; their thrusters are weak. They fire for an hour or two near each node every day and coast the rest of the day. On the figure that is the region of arcs of ±10 to ±15 degrees, within a per cent of the impulsive budget, and the thrusters, running a few hours a day, accumulate more than ten thousand hours of firing over a fifteen-year life — which is the order of what such thrusters are qualified for. A continuously firing scheme would spend half as much again and wear the thrusters out sooner.

The pattern also manages something else. The Sun and Moon tilt a geostationary orbit about a direction that drifts through the year, so the correction has to be aimed in the right direction as well as made in the right amount, and the choice of which node to fire at and for how long sets that aim. The arcs that minimise the budget are also the arcs that give the operators the most control over which way the plane is pushed.

Moving the node instead #

The same arithmetic applies, with sine and cosine exchanged, to the other thing a sideways push can do. Some missions need the orbit’s node moved rather than its inclination changed — a satellite joining a different plane of a constellation, or a spacecraft whose orbit must line up with a target’s. A push perpendicular to the plane swings the node most effectively at the apexes, a quarter-orbit from the nodes, and not at all at the nodes themselves. Thrusting in arcs about the apexes costs α over sin α times the impulsive node change, exactly as for the inclination, and continuous thrust costs π/2 times.

Node changes have a free alternative that inclination changes lack. The Earth’s equatorial bulge swings the node of every inclined orbit at a rate that depends on its altitude and inclination, so a satellite that lowers or raises its orbit for a while drifts in node relative to its neighbours at no cost beyond the altitude change, and then returns. Constellations distribute their planes this way: a spacecraft is released into a parking orbit whose node drifts relative to its operational plane, waits for the drift to carry it to the right node, and then climbs. That is a plane change paid for in time and in altitude rather than in propellant, the same trade as the arcs at a larger scale.

The east–west half of station-keeping is cheaper and works differently. The Earth’s slightly elliptical equator pulls a geostationary satellite towards one of two stable longitudes, and holding it at its assigned slot costs a few metres per second a year, applied along the orbit rather than across it. Electric thrusters handle that too, usually by tilting the same thrusters that make the north–south correction so that each firing does a little of both. The geometry of the arcs makes this nearly free: a thruster firing near a node mostly tilts the plane, and a small along-track component of the same firing adjusts the longitude drift without any separate burn. The daily rhythm of two firings near the nodes is therefore the whole of a modern satellite’s orbit control, and the arithmetic of α over sin α is why the rhythm is two short firings rather than one long one.

Why the node and not the apex #

There is a geometric reason the nodes are special, and it is worth stating directly. The inclination is the angle between the orbit’s plane and the equator, and the plane is rotated about whatever axis the push’s torque on the orbit’s angular momentum points along. At a node, the spacecraft’s position lies along the line of nodes, and a perpendicular push turns the angular momentum about that line, which tilts the plane relative to the equator. At the apex, a quarter-orbit away, the position is perpendicular to the line of nodes, and the push turns the angular momentum about the apex direction, which swings the plane round the pole without changing its tilt.

The same push at the same speed costs the same propellant in both places, and buys a different thing. That is the general shape of every manoeuvre problem in orbit: the cost of a push depends only on how much velocity it delivers, while its effect depends on where it is delivered — which is why the cheapest place to turn an orbit is where it is moving slowest, why an orbit is raised most efficiently at its lowest point, and why the plane is tilted most efficiently at its nodes.

Large turns are chords #

For small turns, narrow arcs approach the impulsive cost. For large turns they cannot, and the reason is geometric rather than a matter of thrust.

An impulse rotates the velocity vector in one step, and its cost is the straight line between the old velocity and the new: the chord of the angle. A gradual turn, however efficiently its pushes are placed, rotates the velocity through a sequence of small steps that follow the arc of a circle, and its cost is at least the length of the arc. For a small angle the arc and the chord are equal. For a turn of ninety degrees the arc is eleven per cent longer than the chord, and no arrangement of low-thrust arcs at a fixed radius can do better than the arc.

Edelbaum’s law, which lets the orbit’s radius change during the turn, does something cleverer for large angles: it raises the orbit partway through, where the velocity is lower and the plane cheaper to rotate, and lowers it again. That is the low-thrust version of flying further away in order to turn, and for large turns it beats continuous thrust at a fixed radius by a wide margin. Combining it with arcs about the nodes would beat both, at a further cost in time. The impulsive chord remains the floor, and for turns of more than a few tens of degrees the cheapest real manoeuvres are those that combine a chemical burn at a high apogee with whatever low thrust can do on the way there and back.

One manoeuvre, several ways to pay #

The arcs complete a family of ways to buy an inclination change, each paying in a different currency. An impulsive burn at the node pays in velocity, at full price. Splitting the rotation between two burns, one at each end of a transfer, pays less because each part is made where it is cheapest. Raising the apogee before turning pays in the velocity of the raise and saves on the turn, because the orbit is slower where the turn is made. Borrowing the turn from the Moon pays in time and in the geometry of the encounter, and spends almost no propellant at all. Low thrust spread over a spiral pays in time and in the π/2 penalty, which the propellant efficiency of an electric engine more than covers.

Thrust arcs sit in that family as the way to recover most of the low-thrust penalty by spending more time. They do not change the chord-and-arc arithmetic for large turns, and they do nothing for a mission that has no time to spare, but for the many missions whose plane changes are small and whose time is cheap — station-keeping above all — they are close to free. For a geostationary satellite whose remaining propellant is hardest to measure exactly when it matters most, a few per cent of the annual budget saved is months of life.

What the arcs cost in other ways #

Thrusting in arcs has costs the velocity budget does not show. A spacecraft that fires for an hour twice a day must slew its thrusters between the two directions at each node, or carry thrusters pointing both ways, and the slews themselves consume time and attitude-control effort. Its power system is sized for the thrusters’ full draw but used for a fraction of each orbit, so the power that could have been thrusting is idle — or used for something else, which for a communications satellite is the payload. And the coast arcs leave the orbit’s other elements to drift under whatever perturbations act on them, which a continuous scheme could have corrected along the way.

Those costs are why the choice is a trade rather than a rule. For a geostationary satellite the trade favours short arcs decisively, because time is free — the drift is slow and the correction is daily for fifteen years — and propellant is the satellite’s lifetime. For a spacecraft that has to reach a new orbit to start its mission, time is the mission, and continuous thrust may be the better choice despite the penalty.

What the calculation leaves out #

The figures treat circular orbits, thrust perpendicular to the plane only, and a push weak enough that the orbit changes little in one revolution. A real low-thrust spacecraft usually raises its orbit while it turns it, and the optimal steering then mixes in-plane and out-of-plane thrust with a yaw angle that varies round the orbit — Edelbaum’s law in full. Thrust arcs in that setting are placed by a combined optimisation: out-of-plane thrust near the nodes, in-plane thrust near the points where it raises the orbit most efficiently, which for a circular orbit is everywhere. The Earth’s oblateness swings the node of any inclined orbit steadily, by degrees a day in low orbit, so a turn lasting months must track a moving node; the arcs follow it, but a mission planned without it would fire in the wrong places within a week. And the eclipses that a spacecraft passes through each orbit switch off a solar-powered thruster for a third of the time in low orbit, imposing arcs of their own that may or may not fall near the nodes.

Still open: arcs for the combined turn and raise #

The cleanest open problem is the combined one. A spacecraft raising its orbit from low Earth orbit to geostationary altitude while removing the launch site’s inclination — the standard electric orbit-raising mission — can thrust in-plane all the time to raise, and add out-of-plane thrust where it is effective, and Edelbaum’s law tells it how to blend the two continuously. Whether thrust arcs, coasting where neither raising nor turning is efficient, can improve on that blend depends on the relative values of time and propellant for the particular mission, and the answer varies from satellite to satellite. The arithmetic of α over sin α says that the inclination part of the bill could be cut by a third; whether a schedule exists that cuts it without lengthening the raise by months is computed mission by mission.