The spin that has to be put somewhere

Pointing a telescope in orbit sounds like a control problem, and control is the easy part of it. The hard part is bookkeeping, and it is hard for the same reason every other angular momentum problem in this collection is hard: the quantity cannot be destroyed, only moved, and a spacecraft in vacuum has very little to move it onto.

A satellite feels four torques that are worth counting. Three of them average nearly to zero around an orbit and one of them does not, and the part that does not is what fills the spacecraft up.

The four torques #

None of these is exotic. Each is a well-understood effect that would be negligible if it did not act in the same direction for hours at a time.

The gradient of gravity across the vehicle. A spacecraft is not a point, and gravity is not uniform across it. The far end is pulled less than the near end, which is exactly the difference that raises tides, and if the body’s principal axes are not aligned with the local vertical the difference exerts a couple. The torque is $\tfrac{3}{2}(\mu/r^3)\,\Delta I \sin 2\theta$, where $\Delta I$ is the difference between two principal moments — so a long thin spacecraft feels far more of it than a compact one, and a spacecraft made deliberately long can use it as a free stabiliser.

The last of the atmosphere. Below a thousand kilometres there is enough gas for drag, and drag acts at the centre of pressure while the vehicle turns about its centre of mass. The offset between them, times the drag force, is a torque. It falls off with altitude at the rate the density does, which is faster than anything else in the problem.

Sunlight. Photons carry momentum, and reflecting or absorbing them exerts a force of a few micropascals times the illuminated area. Same geometry as the drag: force times the offset between the centre of pressure and the centre of mass. It does not depend on altitude at all.

The magnetic field. Every spacecraft has current loops and magnetised materials and therefore a residual dipole moment, and a dipole in a field feels a couple. The field falls as the cube of the distance from the Earth’s centre.

Why the store fills rather than sloshing #

The distinction between the oscillation and the ramp in the opening figure is the whole design problem, and it is easy to state.

A torque that reverses as the vehicle goes round its orbit integrates to zero over a revolution. The wheel it acts on speeds up and slows down and comes back to where it began. It uses capacity — the wheel has to have room for the swing — but it costs nothing.

A torque that does not reverse integrates to a ramp. The gravity gradient on a spacecraft that holds a fixed inertial attitude reverses; on one that holds a fixed attitude relative to the Earth, it does not. Sunlight on a Sun-pointing vehicle does not reverse. Drag on a spacecraft with a permanent centre-of-pressure offset does not reverse. Whatever fraction survives averaging — typically a third or so — accumulates linearly, and the wheel reaches its limit after $h_{\max}/T_{\rm sec}$.

For the representative vehicle above, at five hundred and fifty kilometres, that is a little under five days.

Emptying it #

A reaction wheel does not remove angular momentum from a spacecraft. It relocates it: spinning the wheel one way turns the body the other, and the total is unchanged. When the wheel reaches its maximum speed there is nowhere left to put anything, and the vehicle must push against something outside itself.

There are two options and they are not equivalent.

A magnetic torquer runs current through a coil to make a dipole, and pushes against the Earth’s field. It costs power and no mass. It works only where there is a field to push against, which in practice means low Earth orbit; the field falls as the cube of the distance and by geostationary altitude there is not enough of it.

A thruster fires. That costs propellant, permanently, and the cost has a shape worth knowing.

That null result is the useful part. It is tempting to size a wheel for endurance — a bigger wheel between dumps — and the arithmetic says a bigger wheel buys fewer dumps and nothing else. What actually reduces the propellant is reducing the torque: moving the centre of pressure towards the centre of mass, balancing the vehicle’s moments, demagnetising it, or flying higher.

The shape of the orbit gets a vote #

The environment is not the same everywhere along an orbit, and two orbital properties change the accounting directly. The Earth’s own shape enters too, and not only through the gradient term. The bulge is a thousandth of the field and it dominates every long-term change to an orbit’s orientation, which decides the illumination geometry the solar torque is written in.

A spacecraft with too few wheels #

The most interesting case is the failure case, and it happened.

Three wheels hold three axes. A fourth is usually carried as a spare. A spacecraft that loses two of four has two axes under control and one free, and conventional wisdom says the mission is over — the free axis drifts, the pointing degrades, and a telescope with a drifting roll cannot do photometry.

The alternative is to notice that there is one torque left which is large, steady, and pointed in a known direction: sunlight. If the vehicle is oriented so that the solar pressure torque about the unheld axis is zero, the axis stays put without any wheel at all. It is a balance rather than a control, in the sense that a pencil balanced on its point is: the equilibrium exists, it is found rather than commanded, and it has to be maintained.

The cost is stated in the figure and is the whole design of such a mission: the spacecraft can only observe fields near the ecliptic plane, and it must be repointed every few months as the Sun moves. What it buys is that a photometric survey continues, on a vehicle whose pointing hardware has half failed, using an effect that on a healthy spacecraft is one of the nuisances the wheels exist to absorb.

The store is itself a machine that misbehaves #

A reaction wheel is a flywheel on bearings, and it has two properties that make it a source of the very problem it exists to solve.

It is a gyroscope. A wheel spinning at several thousand revolutions a minute has substantial angular momentum of its own, and a vehicle that rotates about an axis perpendicular to it feels a gyroscopic couple. That couples the three control axes together, so a spacecraft with fast-spinning wheels does not respond to a torque about one axis by rotating about that axis alone — a complication the control law has to carry, and a reason wheels are kept as slow as the momentum budget allows.

It vibrates. No rotor is perfectly balanced, so a wheel exports a small oscillating force and torque at its spin frequency and at harmonics of it. The amplitude is milli-newtons; the consequence is that a telescope’s line of sight jitters by milliarcseconds, which for a long exposure is a smeared image. Observatories therefore isolate the wheels on soft mounts, avoid running them at speeds whose harmonics coincide with a structural resonance, and — for the most demanding instruments — schedule the wheel speeds around the observation.

There is also a zero crossing problem. Bearing friction is largest at very low speed, and a wheel passing through zero does so with a torque discontinuity that the control loop sees as a disturbance. The remedy is to bias the whole set of wheels so that none of them ever needs to stop, which costs stored momentum — capacity spent on avoiding a mechanical defect rather than on absorbing an external torque.

So the store’s capacity is not the number in its specification. Part of it is reserved for the bias, part is unusable because of the speeds that shake the instrument, and what is left is what the secular torque may fill before an unload becomes necessary.

Torque amplification, and the singularity that comes with it #

For a large vehicle the reaction wheel runs out of authority, and the alternative is a device that produces torque a different way.

A control moment gyro holds its rotor at a constant speed and mounts it in a gimbal. Tilting the gimbal changes the direction of the rotor’s angular momentum, and a change of angular momentum is a torque — of magnitude equal to the rotor’s momentum times the gimbal rate, which for a large rotor is enormous compared with what a motor accelerating a wheel can produce.

The gain is real and it is not free. A set of such devices has singular configurations: gimbal arrangements in which all the available momentum changes lie in a plane, so no torque can be produced perpendicular to it. The steering law has to anticipate and avoid those configurations, which is a genuinely hard control problem and the reason the devices are used only where the torque demand justifies it.

The largest installation is on the space station, and its unloading arrangement is the most elegant part of the whole subject. The station cannot fire thrusters routinely — the propellant has to be launched, and the plumes contaminate its experiments — so instead it is flown at a torque equilibrium attitude: an orientation, found by search rather than by formula, at which the gravity-gradient torque and the aerodynamic torque cancel over an orbit.

The vehicle is turned until the disturbance averages to zero, and then the store never fills. That is the only solution in this essay that removes the problem rather than managing it, and it is available precisely because the station is large enough for the gravity gradient to be comparable to everything else acting on it.

The attitude in question is not a fixed orientation. The atmospheric torque depends on the density, which varies with solar activity and with the season, and the gravity-gradient torque depends on the vehicle’s mass distribution, which changes every time a module is added or a visiting vehicle docks. So the equilibrium attitude is recomputed rather than looked up, and the station is flown a few degrees off the orientation an aerodynamicist would choose for minimum drag because those few degrees are what keeps the momentum store from filling.

It is also a solution that scales the wrong way for small spacecraft. The gravity-gradient torque grows with the difference between the vehicle’s principal moments, so a large extended structure has a great deal of it to balance against and a compact satellite has almost none — which leaves the small vehicle with a store to empty and nothing free to empty it into.

The general rule the two cases share is that a spacecraft can only dump momentum into something outside itself, and the candidates are the atmosphere, the magnetic field, the gravity gradient and its own propellant. Which of them is available is decided by the orbit and by the vehicle’s size, and every spacecraft’s attitude system is a choice among those four.

What none of this can be avoided by #

There is no configuration that escapes the accounting. A spacecraft with no wheels at all still feels the torques and still has to spend propellant. A spacecraft with perfect wheels still has to dump. A spinning spacecraft — stabilised gyroscopically, with no wheels — has the same problem in a different form: the environmental torques precess its spin axis, and correcting the precession costs the same impulse.

None of it can be evaded by leaving the vehicle uncontrolled either, since an uncontrolled satellite is a collision hazard priced by area. What can be done is to make the torque small, and the leverage there is entirely geometric. The gravity gradient term is proportional to the difference of two principal moments, so a spacecraft that is nearly symmetric feels almost none of it. The drag and radiation terms are proportional to the offset between the centre of pressure and the centre of mass, so a vehicle balanced to a centimetre feels a tenth of what one balanced to a decimetre does. These are decisions made on the ground, years before launch, and they set the mission’s lifetime as firmly as the propellant tank does.

There is one more escape and it is only partial. Angular momentum can be exchanged between wheels for nothing, so a vehicle with four wheels on skewed axes can redistribute a saturating store across all four and buy itself time. What it cannot do is reduce the total, and the total is a vector: a spacecraft can be at its limit about one axis while two wheels sit nearly idle, and the dump is then required by a direction rather than by an amount. Choosing wheel axes so that the accumulating direction is shared as evenly as possible is a real design lever, and it is one of the few in the problem that costs nothing.

The other partial escape is scheduling. Since the accumulation is a ramp with a known slope, the dumps can be placed where they do least harm — between observations rather than during one, on the night side rather than in daylight, at a point in the orbit where the residual thrust perturbs the trajectory least. That is why a dump is an event in an observing plan rather than a reflex, and why the schedule of a pointed observatory has the sawtooth in the opening figure written into it.

Scheduling has a limit of its own, and it is set by the same ramp. The interval between dumps is the wheel’s capacity divided by the accumulation rate, and both ends of that ratio are fixed years before launch — the capacity by the wheel’s inertia and its maximum speed, the rate by the vehicle’s area, its balance and its orbit. A mission that wants long uninterrupted stares needs a large store, and a large store is a heavy wheel that takes longer to spin up and imposes a slower slew. So the two things a pointed observatory most wants, long integrations and fast repointing, are traded against each other through a component that does neither directly.

That trade is visible in the hardware. A survey instrument that moves constantly carries small, fast wheels and dumps often; a deep-field observatory carries large ones and dumps between targets. Neither is a better choice; they are the two ends of one ratio, and which end a mission sits at is decided by the observing programme rather than by the attitude engineers.

There is one disturbance that cannot be traded against the others, and it is worth separating from the four above. Everything so far is external — atmosphere, gravity gradient, sunlight, magnetism — and a spacecraft also disturbs itself. A scan mirror that steps, an antenna that slews, a tape recorder that spins up, a crew member who pushes off a wall: each exchanges angular momentum with the vehicle internally, so the total is conserved and the attitude is not. Internal torques therefore fill no store and still demand wheel authority, and they are the reason a pointing budget is written against the instrument’s own duty cycle as well as against the environment. On a large station they dominate, which is why the reaction wheels of a crewed vehicle are sized by the crew rather than by the air.

The one lever left, once the geometry and the schedule are fixed, is what the dump is made of.

That is the cleanest trade in the problem and the one most often unavailable. A higher specific impulse means a hotter or an electric thruster, and both bring constraints of their own: a monopropellant system is simple and low-performing, a bipropellant one performs better and needs two tanks and a mixture ratio held for a decade, and an electric thruster performs enormously better and delivers its impulse so slowly that it cannot be used for anything with a deadline. A momentum dump has no deadline, which is exactly why electric propulsion is used for it on the vehicles that carry any.

Where it is not available, the moment arm is the remaining lever, and it is geometric rather than chemical: a thruster mounted further from the centre of mass produces the same torque for less thrust, so the propellant spent scales inversely with the arm. That is why attitude thrusters sit on the corners of a spacecraft rather than beside its engine, and it is one of the few places in this subject where a design choice buys a factor rather than a percentage. The limit on the arm is the vehicle itself — a thruster cannot be mounted beyond the structure — so the leverage available is set by the spacecraft’s own size, and a small satellite is penalised twice: it has a short arm and it usually cannot carry an efficient thruster either. That is the reason momentum management, rather than power or communications, is what most often sets the life of a small pointed spacecraft. The vehicle continues to work in every other respect and simply loses the ability to hold still, which is a failure mode that looks like nothing at all from the ground until the pointing budget stops closing.

Where the ladder goes #

The natural next step is the other budget the same vehicle keeps: the orbit that has to be paid for every year, where the quantity being maintained is position rather than orientation, and the same distinction between what averages out and what accumulates decides the cost.

A second direction runs the other way, into using the environment rather than resisting it. A long spacecraft is stabilised by the gravity gradient for free; a solar sail is a vehicle for which radiation pressure is the propulsion rather than the disturbance; a tether both stabilises and generates power against the same magnetic field a torquer pushes on. In each case a term in the table above has changed sign in the design, and the accounting is unchanged: the angular momentum still has to go somewhere, and something outside the vehicle still has to take it.