A turn that costs more when the shadow sits on the node

Turning an orbit’s plane is the most expensive thing a spacecraft does with its engine. A turn of a few degrees costs a fixed fraction of the orbital speed, and an impulsive burn does it most cheaply where the spacecraft is slowest and at a node, the two points where the orbit crosses the plane it is turning toward. An electric engine cannot make an impulsive burn. It pushes gently for months, and a push out of the orbital plane does different things at different points of the orbit: at the nodes it tilts the plane; halfway between them it mainly swings the line of nodes around and tilts nothing.

That is why low-thrust steering laws reverse the push twice an orbit, and why the cheapest of them push only near the nodes and coast elsewhere. Every such law assumes the engine can push wherever the law says. A solar-electric engine runs on sunlight. Once an orbit the spacecraft passes behind the Earth, the arrays go dark, and the engine stops. Where in the orbit that happens decides how much of the turning is lost.

Turning is done at the nodes #

A push of acceleration $a$ perpendicular to the plane of a circular orbit changes its inclination at a rate

$$\frac{di}{dt} = \frac{a}{v}\cos u,$$

where $v$ is the orbital speed and $u$ the argument of latitude, the angle from the ascending node along the orbit. At a node, $\cos u = \pm 1$ and the push tilts the plane at its maximum rate; at ninety degrees from a node it tilts nothing. Reversing the push at the antinodes keeps the two halves of the orbit tilting the same way, and averaged over a whole orbit the rate is $2/\pi$ of the maximum.

So the orbit is not equally valuable all the way round. The shaded arcs in the opening figure show what that means for a shadow. On an orbit a little over six thousand kilometres up, the shadow covers a sixth of the circle when the Sun lies in the orbit’s plane. If that sixth is centred on a node, the engine is off for the most valuable stretch of the orbit and a quarter of each orbit’s turning is lost. If it is centred on an antinode, the engine is off where it was doing little, and the loss is seven per cent. The same sixty degrees of darkness costs nearly four times as much in one place as in the other. It is the shape of the cosine that does it: the cosine is flat-topped near a node, so a dark arc there removes nearly its full width of turning, and steep near an antinode, so a dark arc there removes only the thin tails of the turning on either side.

Why the engine stops #

An electric engine accelerates ions through a voltage, and the power for that comes from solar arrays. A thruster producing a few tenths of a newton draws several kilowatts, and there is no practical way to store enough energy to run it through a long shadow. On an orbit a few thousand kilometres up the shadow lasts most of an hour; five kilowatts for forty-five minutes is nearly four kilowatt-hours, which in lithium-ion cells is around twenty-five kilograms of battery before allowing for the margins that keep the cells healthy over thousands of cycles. A spacecraft carrying that much battery could keep thrusting through the dark, and the turning it saved would have to be weighed against twenty-five kilograms it carries for the rest of its life.

For most missions the battery loses. The spacecraft’s batteries are sized to keep its computers and heaters alive through the shadow, not to run its engine, and the engine simply stops when the arrays go dark and restarts when they come back. Every steering law then has a gap in it once an orbit, and the gap is in a place the law did not choose.

The same arithmetic makes the shadow an old problem rather than a new one. The first electric spacecraft to leave Earth orbit, a small European probe that spiralled to the Moon in 2003 and 2004, spent more than a year raising its orbit and planned its thrusting arcs around each orbit’s eclipse. The first all-electric communications satellites, launched in 2015, spent about six months each spiralling from a low transfer orbit to geostationary, and their thrust profiles were built around the shadows of every one of the hundreds of orbits along the way. The shadow has been in every low-thrust trajectory ever flown; what changes with the steering law is how much it costs.

A shadow whose length depends on the Sun #

Where the shadow falls is set by the direction of the Sun. Its length is set by the angle between the Sun and the orbit’s plane, the beta angle.

The Earth’s shadow is, to a good approximation, a cylinder pointing away from the Sun. An orbit whose plane contains the Sun passes through the middle of the cylinder and crosses its full width. An orbit whose plane is tilted to the Sun’s direction crosses the cylinder off-centre, along a shorter chord, and an orbit tilted far enough misses it altogether: above a beta angle of thirty degrees, for an orbit at twice the Earth’s radius, the spacecraft is in sunlight all the way round. The loss curves are therefore a family, each lower and narrower than the last as the Sun rises above the plane, and at twenty-five degrees the worst shadow costs about a sixth of the turning.

Three things fix where on the family a spacecraft sits on any given day: the orbit’s size, the direction of its node, and the direction of the Sun. All three change during a transfer.

The shadow slides round the orbit #

The direction of the node changes because the Earth is not round. Its equatorial bulge turns the plane of every inclined orbit backward around the polar axis — the regression that Vanguard 1 first measured — at a rate that for an orbit at twice the Earth’s radius and an inclination of thirty degrees is about three-quarters of a degree a day. Meanwhile the Sun moves forward along the ecliptic at about a degree a day. The anti-Sun direction, where the shadow is, therefore moves round the orbit relative to the node at nearly two degrees a day, and passes over one node or the other every three months or so.

The figure follows one transfer, removing the twenty-eight and a half degrees of inclination a launch from Florida leaves, at an out-of-plane acceleration of a millimetre a second squared. The inclination falls almost linearly over two months, and the shadow’s cost swings beneath it from a twentieth of the turning to a quarter. Near the start the shadow is far from the nodes and the loss is small; a month in, the shadow has slid onto a node and the loss peaks; toward the end it has moved off again. The changing inclination also changes the beta angle, because the orbit’s plane tilts as the transfer proceeds, so the shadow’s length changes along with its position.

A transfer that depends on its launch date #

Because the shadow’s position at the start depends on the date, and its slide is the same for every start, a transfer that begins on one date meets the shadow on the nodes at a different stage from one that begins a month later.

The cost ranges from an eighth of the transfer’s shadow-free duration to a quarter, depending on when it starts. A launch that begins the transfer with the shadow drifting onto a node pays from the first orbit and pays again as the inclination changes; one that begins with the shadow between the nodes and the Sun high above the plane pays least. The difference between the best and the worst is a week of a seven-week transfer — and in propellant, since an electric engine’s thrust is fixed and its propellant consumption with it, a week of extra thrusting is a sixth more propellant.

The week is not spread evenly. Because the shadow’s position relative to the nodes repeats every three months or so while the transfer lasts seven weeks, a transfer meets the shadow on a node at most once, and the date decides whether that meeting comes early, when the inclination left is large and the beta angle small, or late, when the plane has nearly finished turning and the Sun sits higher above it. The figure’s ripple is the trace of that one meeting sliding through the transfer as the start date moves.

That is the effect that a spiral to geostationary orbit and every low-thrust plane change has to budget for, and it is why mission planners compute the transfer for each candidate launch date rather than once. The arithmetic here assumes a single circular orbit; a real transfer that also raises the orbit changes the shadow’s length and the node’s regression rate together, and the dependence on the date is correspondingly more intricate.

Concentrating the thrust concentrates the risk #

The cheapest steering laws are the most exposed to this, for a reason built into what makes them cheap.

A law that pushes all the way round the orbit spends a good part of its propellant near the antinodes, where it buys little turning, so it gets about two-thirds of the turning per unit of propellant that an impulsive burn at the node would. An arc law that pushes only within twenty-five degrees of each node gets nearly all of it, because every push it makes is near a node. But that is the stretch of orbit a shadow on a node removes. With the shadow on a node, the arc law loses all of that node’s thrust — half its turning for the orbit — while the all-round law loses a quarter. With the shadow between the nodes, the arc law loses nothing, because it was coasting there anyway.

The arc law therefore trades efficiency for exposure. On a transfer whose dates keep the shadow away from the nodes, it is the better law by a wide margin. On one that drags the shadow across the nodes during the transfer, it can lose more time than its efficiency saves — and a law that adapts, widening its arcs while a node is in shadow and narrowing them again afterwards, is better than either. The adaptive version has to know where the shadow will be, which means carrying the Sun’s and the node’s motion in the guidance.

Turning when the shadow is elsewhere #

A spacecraft that knows where the shadow will be has a choice an impulsive spacecraft never has: it can decide when to do its turning. A transfer that both raises the orbit and turns its plane can push in the plane while the shadow sits on a node — raising the orbit, which loses no more to a shadow on a node than to one anywhere else — and do the turning in the weeks when the shadow has slid between the nodes. Neither job is made cheaper by the shadow, but the shadow’s cost is moved from the job that is sensitive to its position to the one that is not.

That reshuffling is a version of an old trade in orbital mechanics. An impulsive mission that needs to turn its plane flies further away in order to turn, spending velocity to raise its apoapsis because the turn is cheaper there. A low-thrust mission spends time instead: it reorders its thrusting so that each kind of push happens when the geometry favours it. The trade costs nothing in propellant if the transfer has slack in its schedule, and a transfer lasting months nearly always does. What it costs is planning, because the right order depends on the launch date and on the node’s regression rate, which depends on the inclination the transfer is changing.

Higher orbits escape it #

The shadow is a fixed cylinder the width of the Earth, and a larger orbit crosses it along a smaller fraction of its circumference.

On a low orbit, five hundred kilometres up, the shadow covers well over a third of the circle, and on a node it takes nearly half the turning. At geostationary distance it covers a few per cent of the circle for a few weeks around each equinox and nothing for the rest of the year. Plane changes are also cheaper in velocity at high altitude, where the orbital speed is lower, which is why steering laws for spirals from low orbit to geostationary defer most of the turning to the outer part of the spiral. The shadow gives a second reason for the same choice: the turning done low down is not only dearer per degree but also more often interrupted, and in the worst place.

The equinox seasons at the top of the spiral #

At the end of a spiral to geostationary orbit the shadow takes its simplest form. A geostationary satellite is in the Earth’s shadow only when the Sun is close to the equatorial plane, which is to say for a few weeks around each equinox; for the rest of the year its orbit passes above or below the shadow and it is in sunlight all the time. During an eclipse season the daily shadow grows from nothing to a maximum of about seventy minutes at the equinox and shrinks again over about forty-four days.

A spacecraft finishing its plane change near geostationary distance in an eclipse season meets the shadow every day at the same local time, which for a nearly equatorial orbit means at a fixed angle from the node; one finishing outside it meets no shadow at all. The final, cheapest degrees of the turn are therefore sensitive to the date in a stepwise way that the low-orbit analysis smooths over. The plane change paid at the worst speed there is is the one done low and in a hurry; the one done high, outside an eclipse season, is the one the shadow cannot touch.

What the geometry leaves out #

The shadow here is a sharp cylinder. The real one has a penumbra in which the arrays deliver partial power, and the atmosphere refracts sunlight into the edge of the umbra, so the transition is gradual over a few degrees of arc; the engine’s controller usually treats the penumbra as shadow for safety, which makes the effective shadow slightly wider than the umbra drawn. A spacecraft carrying batteries can keep thrusting through part of the shadow, at the cost of the batteries’ mass and their cycle life, and some missions do exactly that for the shadows that fall on nodes.

The transfer modelled is also the simplest kind: a single circular orbit, a constant acceleration, and a pure inclination change. Real low-thrust transfers raise the orbit at the same time, and the inclination change is blended with the raising — the steering that turns where the climb leaves room — so the shadow interrupts both. The loss in raising is not concentrated at the nodes and is less sensitive to where the shadow falls.

Still open: steering that knows where the dark will be #

The shadow turns a steering law’s efficiency into a function of the calendar. The law that is cheapest in sunlight is the most exposed to a shadow on a node, the law that is most robust is the least efficient, and the cost of a transfer can differ by a sixth between launch dates a few weeks apart. All of this follows from three motions — the spacecraft round its orbit, the node round the Earth, the Sun round the sky — and could be computed for any transfer in advance.

What is not settled is how much of it a spacecraft’s guidance should do on board. A law that adapts its thrust arcs to the shadow’s predicted position has to know the node’s regression and the Sun’s motion, which it can, and to plan weeks ahead, which most electric-propulsion guidance does not. The alternative is to accept a fixed law and choose the launch date to suit it — which converts an orbital-mechanics problem back into a scheduling one, of the kind that a satellite’s own shadow geometry already imposes on everything in low orbit.