The one axis the field will not take

A satellite held steady by reaction wheels is storing angular momentum it cannot get rid of. Each small torque from the thin air, from sunlight, or from the gradient of gravity across its body adds to the spin of the wheels. When a wheel reaches its top speed it can no longer absorb anything, and the satellite starts to turn. A satellite with thrusters fires them to unload the wheels and pays in propellant. A satellite in low orbit has a cheaper outlet. Its coils make a magnetic dipole that pushes against the Earth’s field, so the momentum goes into the planet’s magnetic field and the price is electrical power.

The coils have one blind spot, which detumbling has already met. A dipole $\mathbf m$ in a field $\mathbf B$ feels the torque $\mathbf m \times \mathbf B$, and that is always at right angles to the field. At any instant there is one direction in which the coils can do nothing at all, and it is the direction of the local field line. For detumbling this is a nuisance that the moving field slowly cures. For dumping it decides whether the method works at all. A satellite pointed at the Earth has exactly one axis about which momentum keeps accumulating. In the orbits close to the equator, that axis is where the field lies.

A torque fixed to the body is handed back #

A satellite that keeps one face towards the Earth is not still. It turns once every orbit about the orbit normal, its pitch axis, so that the same side always faces down. Its wheels hold momentum in a frame that rotates with it, and in that frame the stored momentum $\mathbf h$ changes as

$$\frac{d\mathbf h}{dt} = \boldsymbol\tau - n\,\hat{\mathbf n} \times \mathbf h,$$

where $\boldsymbol\tau$ is the torque on the body, $n$ is the orbital rate and $\hat{\mathbf n}$ is the orbit normal. The last term is the frame’s own rotation. Its consequences decide everything that follows.

Suppose the torque is fixed to the body and lies about roll, the axis pointing along the direction of flight. Seen from the stars, the satellite turns once an orbit, so that torque points in every direction in the orbit plane in turn. What it adds while it points one way it takes back half an orbit later. The equation shows the same thing: $\mathbf h$ is carried round the pitch axis while the torque adds to it, and its size traces the arches in the figure. The largest the store gets is $2\tau/n$, which for 0.2 micronewton metres at 500 kilometres is 0.36 millinewton-metre-seconds. That is a small fraction of what even a small wheel holds, and it never grows.

A torque about pitch is different, because pitch is the axis the satellite turns about. That direction never changes as seen from the stars, and the store along it grows in a straight line, to nearly nine millinewton-metre-seconds in the twelve hours drawn, and further for as long as the torque lasts. For an Earth-pointing satellite the store that has to be emptied is the pitch store. Steady torques about the other two axes are handed back every orbit at no cost.

This matters because the steady torques an Earth-pointing satellite feels are mostly pitch torques. Drag acts along the direction of flight. If the centre of pressure sits above or below the centre of mass, the drag force times that vertical offset is a torque about the orbit normal, which is pitch. A satellite whose principal axes are slightly misaligned in pitch feels the gravity gradient pulling it back, also about pitch. Torques fixed in inertial space are another matter, such as sunlight on an array that tracks the Sun. They can accumulate in any direction, and they are discussed at the end. The body-fixed ones collect about one axis.

The field over the equator points along the orbit normal #

The Earth’s field near its surface is close to that of a dipole. Over the magnetic equator it runs horizontally, from south to north, parallel to the dipole’s axis. A satellite in an equatorial orbit therefore spends its whole orbit in a field running parallel to the Earth’s axis, and the Earth’s axis is the normal to that orbit, the satellite’s pitch axis. The field points along the one axis whose momentum must be removed, and a coil cannot push about the field direction.

The figure measures this as a single number: the fraction of the field that lies across the pitch axis, $|\hat{\mathbf b} \times \hat{\mathbf n}|$. A coil set can produce pitch torque in proportion to that share, so it is also the share of the coils’ strength that can reach the pitch store. In the equatorial orbit it averages 0.24, and it is not zero only because the real dipole is tilted a little over nine degrees from the Earth’s axis. Where the orbit runs north or south of the tilted magnetic equator, the field picks up a vertical part and leans off the pitch axis by up to about eighteen degrees, twice the tilt. All the dumping an equatorial satellite can do comes from that lean, and the lean is different at every longitude.

Raising the inclination tips the orbit normal away from the Earth’s axis, and the field across it grows. At 28.5 degrees, the inclination reached by launching due east from Florida, the mean share is 0.68 and it varies twice each orbit. At 51.6 degrees, the space station’s inclination, it is 0.89. At 97.4 degrees, the sun-synchronous inclination for 500 kilometres, the orbit passes nearly over the poles and the field lies almost in the plane of the orbit. The pitch axis is then nearly perpendicular to the field everywhere, the share stays above 0.98, and the coils can apply their full strength about pitch for the whole orbit.

How much pitch torque the coils can make #

The share converts into a torque budget. A coil set with largest dipole $m_{\max}$ can, at best, make a pitch torque of $m_{\max}\,|\mathbf B|\,|\hat{\mathbf b} \times \hat{\mathbf n}|$ at each instant. Averaged over a day of orbits, so that the Earth’s rotation carries the tilted dipole all the way round, that is the largest steady pitch torque the coils could cancel indefinitely. Above it the pitch wheel fills whatever the control law does.

The figure gives that budget for coils typical of a small satellite, 0.2 ampere square metres, at 500 kilometres where the field is about 25 microtesla over the magnetic equator. In a near-polar orbit the coils can sustain 7.6 micronewton metres about pitch. The budget falls steadily with inclination, roughly as its sine. A dipole aligned with the Earth’s axis would leave the equatorial orbit with nothing at all, since the field would be exactly along pitch everywhere.

The real dipole’s tilt changes that at the bottom of the curve. With the axis 9.2 degrees off, the equatorial orbit keeps 1.2 micronewton metres, about a sixth of the polar value. That authority comes from the geometry of the planet’s interior and changes as the dipole wanders. A satellite designed for the equator and counting on it is counting on the tilt of a field whose axis has swung by about two degrees in the last century.

Above about 20 degrees the two curves merge, and inclination does the work. A launch from a low-latitude site that settles for a low inclination is also accepting a smaller magnetic dumping budget. Such orbits are chosen because they stay clear of the South Atlantic Anomaly, where the inner radiation belt dips low over the ocean, and so they are favoured by detectors that cannot tolerate its particles. The anomaly is a feature of the same field. An orbit low enough in inclination to avoid it is also an orbit where the field gives little grip on pitch.

The store settles where the gain meets the share #

The capacity is a ceiling. What a satellite actually stores depends on the control law. The usual law is the cross-product law, which sets the dipole to

$$\mathbf m = \frac{k}{|\mathbf B|^2}\,\mathbf h \times \mathbf B, \qquad \mathbf m \times \mathbf B = -k\,\mathbf h_\perp,$$

so that the coils remove the part of the stored momentum lying across the field, at a rate set by the gain $k$, and leave the part along the field alone. With the store pointed along pitch, the part across the field is $h\,|\hat{\mathbf b} \times \hat{\mathbf n}|$, and the component of the coils’ torque that acts back on pitch is $k\,h\,|\hat{\mathbf b} \times \hat{\mathbf n}|^2$. The store stops growing where that equals the disturbance:

$$h_{\text{steady}} \approx \frac{\tau}{k\,\langle|\hat{\mathbf b} \times \hat{\mathbf n}|^2\rangle}.$$

The share enters squared. It decides once how much of the store the coils can see, and again how much of the resulting torque points back along pitch.

The figure at the head of the essay runs this law for three days, with $k = 5\times10^{-4}$ per second and the 0.2 micronewton metre pitch torque. Without coils the store reaches 52 millinewton-metre-seconds. The near-polar satellite holds it at 0.40, the space-station inclination at 0.54, and the Florida-launch inclination at about 1.2, varying through the day as the tilted dipole turns underneath. The equatorial orbit levels off at 7.5. All four are close to the formula: a mean squared share of 0.06 over the equator multiplies the polar store by sixteen and more.

Raising the gain is the obvious response, and in the inclined orbits it works. A tenfold gain lowers their stores almost tenfold, as the formula says it should. Over the equator it barely helps. To produce a torque on a store that lies almost along the field, the law has to command a dipole much larger than the torque it gets back. That is $1/|\hat{\mathbf b} \times \hat{\mathbf n}|$ times larger, so the coils hit their 0.2 ampere-square-metre limit long before the store is small. From then on the gain no longer matters: the coils are working at their limit and the pitch torque they deliver is the capacity in the previous figure. The equatorial satellite has run out of authority, not of gain.

Where no law can keep up #

The last figure raises the disturbance to 2 micronewton metres, above the equatorial orbit’s capacity of 1.2 and below every inclined orbit’s. The outcome splits accordingly. The three inclined satellites hold their stores at a few millinewton-metre-seconds. The equatorial satellite’s store climbs without limit, at a little over two-fifths of the rate it would with no coils at all. No control law can change that, because the shortfall is in the size of $\mathbf m \times \mathbf B$ about pitch, averaged over a day, and a control law chooses only the direction of $\mathbf m$ within its limit. Bigger coils raise the limit in proportion and cost power in proportion. The other choices are to reduce the torque by moving the centre of pressure, to carry thrusters for a periodic dump, or to choose a different orbit.

The capacity also falls with height. The dipole field weakens as the inverse cube of distance, so the same coils at 1,500 kilometres have about two-thirds of the authority they have at 500. At geostationary height the field is more than two hundred times weaker than at 500 kilometres, and dumping into it is no longer practical. There the problem returns to thrusters and to the propellant budget it started from. In low orbit, though, the same descent that strengthens the field also strengthens the drag. Aerodynamic pitch torque rises much faster than the field as altitude falls, so the margin between them depends on the satellite’s shape and on solar activity, not on its height alone.

The coils cannot listen while they push #

The cross-product law needs the field vector at every instant, and a small satellite measures it with a magnetometer mounted a few centimetres from the coils. A coil carrying its full current makes a field at the magnetometer that can be comparable to the Earth’s, so the measurement and the actuation cannot run together. The usual arrangement takes turns: the coils switch off, the magnetometer reads the field, and the coils switch on again with the dipole computed from that reading. The cycle is a second or so, and the coils are on for part of it.

That duty cycle multiplies every capacity in the figures. Coils that push for 70 per cent of the time deliver 70 per cent of the average torque, at every inclination alike, so the ratio between the polar and the equatorial budgets is unchanged while both fall. In a polar orbit the cut leaves a wide margin. Over the equator, where the budget is already a sixth of the polar one, it can be the difference between a store that settles and one that grows.

The same arrangement limits the gain. A law that commands large dipoles to chase a store across a field it can barely see is the law that keeps the coils saturated, and saturated coils need longer silent intervals for the magnetometer to settle. The equatorial satellite is therefore pushed towards the worst version of the trade on each count: the highest gain, the largest commanded dipoles, the least time to deliver them, and the smallest share of the field to push against. The figures here assume coils that never rest. They are an upper bound on what a real small satellite in that orbit can dump.

A telescope that holds still has the same blind axis #

An Earth-pointing satellite is the case worked through here because its own rotation sorts the torques into one axis that accumulates and two that do not. A telescope that holds its attitude fixed among the stars has no such sorting. Every steady torque on it accumulates, on all three axes, and the coils must remove momentum in every direction. The question then is not where the field sits relative to pitch but where it sits in inertial space, and over the equator the answer is the same as before in a different frame.

In an equatorial orbit the field runs parallel to the Earth’s axis everywhere, apart from the lean the dipole’s tilt allows. So for a telescope there, whatever it is pointed at, the direction the coils cannot reach is close to the direction of the celestial pole. A store along that direction can be emptied only as fast as the tilt permits, and a store across it is emptied freely. The blind axis has not gone away; it has moved from the body into the sky. In the body it lies wherever the celestial pole happens to be from the current target, so it changes each time the telescope is repointed.

A low-inclination orbit is often not chosen at all but inherited. A spacecraft launched due east reaches an inclination equal to its launch site’s latitude at no cost, and anything lower costs a plane change. A launch from close to the equator is what makes an equatorial orbit cheap in the first place. The same choice that saves propellant at launch and keeps the detectors out of the radiation belts also gives the attitude system the weakest grip on its momentum it can have in low orbit. Those are three consequences of one decision, and they are usually weighed by three different groups of engineers.

What the arithmetic leaves out #

The figures use the field of a tilted dipole. The real field adds higher-order terms that are a tenth or more of the dipole at the surface and less in orbit, and the most prominent of them is the weak region over the South Atlantic. Over the equator those terms shift the share around the orbit, but they cannot change the fact that the main field there lies along pitch. Nodal regression is also left out. In three days it moves the plane by a few degrees, and over weeks it carries the orbit’s node round relative to the dipole’s longitude, so the equatorial authority averages over all longitudes, as the daily rotation already makes it do.

The stores in the hero figure are for one combination of torque, coils, height and gain. They scale simply. The steady store goes as torque over gain until the coils saturate, and the capacity goes as coil strength times field strength. A satellite with ten times the torque, ten times the coils and ten times the gain has the same figures with every vertical axis multiplied by ten.

Torques fixed in inertial space are the other omission. Sunlight on an array that turns to face the Sun produces a torque fixed relative to the Sun, not to the body, and seen from the rotating frame such a torque turns once an orbit along with everything else. It accumulates in inertial space in whatever direction it points, including directions in the orbit plane. For those the question is not the field’s direction relative to pitch but how widely the field’s direction in inertial space varies over an orbit and a day, which is the question the detumbling essay answered. Near-polar orbits are again the best case, since the field there turns through two full circles every orbit.

Still open: whether an equatorial satellite can rely on its coils alone #

For an Earth-pointing satellite in a high-inclination orbit, the arithmetic leaves little doubt: coils can hold the pitch store indefinitely against torques well above the ones it will meet, and the satellite needs no propellant for its attitude. In an orbit within a few degrees of the equator the outcome hangs on a margin set by the dipole’s tilt. It gives about a sixth of the polar authority, with a daily rhythm as the Earth turns, and it is drifting as the geomagnetic pole moves.

Some instruments want such orbits because the radiation is lowest there, so the question is not idle. Can a mission planned for a few degrees of inclination, with the torques it will really meet over a solar cycle when the atmosphere expands and contracts, keep its pitch wheel empty on coils alone, or will it need a propellant reserve held back for the years when the Sun is active? The figures here bound it. Settling it requires the satellite’s own aerodynamic shape, its own coils, and the field over the years it flies.