A tumble stopped by the field it tumbles through

A satellite pushed out of a deployer by a spring does not come out still. The spring is never quite centred, the satellite never quite leaves cleanly, and it drifts away turning at a few degrees a second about some axis nobody chose. Nothing aboard can point an antenna, a camera or a solar panel until that tumble is gone.

The order matters more than it might seem. A tumbling satellite’s solar cells face the Sun only part of the time, so it charges its batteries slowly or not at all. Its antenna pattern sweeps past the ground station instead of resting on it, so the first contact is broken into fragments. Any boom, panel or antenna that has to unfold is safer unfolded on a still body than flung out by a spinning one. Until the tumble is removed, the satellite is spending energy it was launched with and cannot easily replace. Tumbles of a few degrees a second are typical, and tens of degrees are not unusual.

A large spacecraft would stop it with thrusters or wheels. A small one often has neither, or has wheels far too weak to absorb the tumble without saturating, and so the first thing it does is magnetic. It carries a magnetometer, which measures the Earth’s field in the satellite’s own frame, and three coils of wire, which turn current into a magnetic dipole. A dipole in a field feels a torque. A torque from outside is the only thing that can change a spacecraft’s total spin, and the Earth’s field is the one outside thing a coil can push against.

The law most such satellites run was described in the 1970s and is almost absurdly short. Measure the field. Work out how fast the measured field is changing. Drive the coils to make a dipole pointing the opposite way to that change. The law is called B-dot, after the notation $\dot{\mathbf B}$ for the rate of change of the field, and it needs no knowledge at all of which way the satellite is facing.

A law that needs no attitude #

The reason it works is that a tumbling satellite sees the field swing round. The Earth’s field changes slowly along the orbit, but in the frame of a body turning at several degrees a second it sweeps round much faster, at a rate set almost entirely by the tumble:

$$\dot{\mathbf B}_{\rm body} \approx -\,\boldsymbol\omega\times\mathbf B .$$

A dipole opposing that change, $\mathbf m = -k\,\dot{\mathbf B}$, is then $k\,\boldsymbol\omega\times\mathbf B$, and the torque it feels is

$$\boldsymbol\tau = \mathbf m\times\mathbf B = -\,k\,B^2\,\boldsymbol\omega_\perp ,$$

where $\boldsymbol\omega_\perp$ is the part of the rotation perpendicular to the field. The torque points against the rotation. It removes energy at the rate $kB^2\omega_\perp^2$ and cannot add any. That is the whole of the law’s stability argument, and the satellite’s orientation appears nowhere in it.

The simulation integrates the full rotation of the satellite, including the way an unevenly shaped body wobbles as it turns, and computes the field at every half-second from the satellite’s position on its orbit. The coils see only what a magnetometer would see.

The torques involved are tiny. A dipole of 0.2 ampere square metres in a field of a few tens of microtesla makes a few millionths of a newton metre, and even a four-kilogram satellite needs a good part of an orbit to be brought to rest with it. But the law is patient, it costs only electrical power, and it cannot run away: a coil that opposes the change it sees can only ever brake.

Two of the three orbits drawn end almost stationary, and the third does not. The difference is not in the law or in the satellite. It is in the geometry of the field the satellite passes through.

The axis a coil cannot reach #

A torque made by a dipole in a field is a cross product, $\mathbf m\times\mathbf B$, and a cross product with $\mathbf B$ is always perpendicular to $\mathbf B$. So at every instant there is one axis about which the coils cannot exert any torque at all: the local field line. Whatever part of the satellite’s spin lies along that line, no current in any coil can change.

With the dipole along the Earth’s axis, the field at every point of the equator points the same way — parallel to that axis — so the direction the coils cannot reach is the same all round the orbit. The law strips away the rest of the spin and leaves this part not slowed but untouched, and the satellite ends up turning steadily about an axis parallel to the Earth’s.

The general statement is a short piece of vector calculus. The spin component along the field can change only if the field’s direction changes, since

$$\frac{d}{dt}\left(\mathbf L\cdot\hat{\mathbf B}\right) = \boldsymbol\tau\cdot\hat{\mathbf B} + \mathbf L\cdot\frac{d\hat{\mathbf B}}{dt} = \mathbf L\cdot\frac{d\hat{\mathbf B}}{dt},$$

and the first term is zero. How much of the unreachable spin a coil can eventually remove is therefore a question about how fast the field’s direction, seen from inertial space, turns along the orbit.

The real dipole is not along the rotation axis. It is tilted by a little over nine degrees, and the Earth turns underneath the orbit once a day, so in an equatorial orbit the field direction wobbles slowly.

Half the along-field spin survives six orbits, and that is with the real tilt helping. An equatorial satellite detumbled this way is left rotating slowly about something close to the field line, and must wait on the Earth’s rotation to finish the job.

The inclination of 97.4 degrees is not arbitrary. At 500 kilometres it is the inclination at which the Earth’s equatorial bulge turns the orbit’s plane once a year, keeping it at a fixed angle to the Sun, and it is where a very large share of small Earth-observation satellites fly. By a happy geometric accident, the orbit chosen for its lighting is also nearly the best orbit in which to stop a tumble with a magnet.

Stillness, as a magnetometer defines it #

The polar orbit does not bring the satellite to rest either. It settles at a small, steady rate, and the rate is not arbitrary.

The law opposes changes in the measured field. A satellite that is still with respect to the stars is not still with respect to the field, because along a near-polar orbit the dipole’s field direction turns through two complete circles for every circuit of the orbit: once because the orbit itself goes round, carrying the local vertical with it, and once more because the field’s angle to the vertical runs from horizontal at the equator through vertical at the pole and back. A satellite that turns with the field sees nothing change, and the law then asks for no dipole at all.

So the stillness the law reaches is stillness as a magnetometer defines it, and in a polar orbit that is a rotation at about twice the orbital rate. It is close to that value rather than equal to it because the field turns unevenly, faster over the poles than over the equator, and a rigid body cannot follow a turn that speeds up and slows down twice an orbit. The residual falls with height only as the orbital rate does — about a quarter from 300 to 2,000 kilometres — and it is small enough to be harmless: a detumbled satellite rotating a tenth of a degree a second can be handed over to a sun sensor, a wheel or a boom that the gravity gradient will hold, and this is exactly the order in which many small satellites bring themselves under control.

The residual also shows what the law is. It does not remove angular momentum relative to the stars; it removes rotation relative to the field. In an orbit where the field hardly turns, those are nearly the same thing and the law’s weakness is the unreachable axis. In an orbit where the field turns fast, the unreachable axis is swept away, and the law’s floor becomes the field’s own rotation.

How long, and how hard #

The pace can be estimated before anything is simulated. The braking torque is $kB^2\omega_\perp$, so the rotation decays on a timescale of roughly $I/(kB^2)$. At 500 kilometres the dipole field is about 25 microtesla over the magnetic equator and twice that over the poles, and along a polar orbit the average of its square is about two and a half times the equatorial value. On average two-thirds of the square of a random rotation lies across the field. With the gain used in these figures and the satellite’s two larger moments of about 0.04 kilogram square metres, that gives one factor of e roughly every 0.18 orbits, and a fall from 8.8 to one degree a second — just over two factors of e — in about four-tenths of an orbit. The two inclined orbits in the first figure took 0.32 and 0.58. The estimate is not the answer, but it says what the answer depends on: in proportion to the moment of inertia, inversely to the gain, and inversely to the square of a field whose strength halves between the pole and the equator.

The detumbling time is set by a torque that grows with the rotation it is opposing, and that has a consequence for fast tumbles.

While the coils have room, the dipole the law asks for is proportional to the rotation, so the torque is too, and the rotation decays exponentially: each halving takes about the same time. Doubling the starting tumble then adds one more halving rather than doubling the wait. The measured additions do not come out equal — they run from 0.10 to 0.46 orbits — and the reason is again the field. A dipole field is twice as strong over the poles as over the equator, the braking torque goes as the square of the field, and a halving that happens to fall over the equator takes several times as long as one over a pole. How long a detumble takes depends on where in the orbit it begins.

At the fastest tumble the law asks for more dipole than the coils can make, and while the coils are pinned at their limit the torque stops growing with the rotation. The time to stop then grows in proportion to the tumble rather than with its logarithm. That is the regime a stronger coil would help with, and only that one.

Above about a tenth of an ampere square metre, bigger coils buy almost nothing for this tumble. The time is set by the law’s gain — how strongly it responds to a given rate of change — and by the field, not by the coils’ ceiling. That is useful to know, because coils cost mass and power, and also because the gain cannot simply be turned up. The magnetometer measures the coils’ own field as well as the Earth’s, so in practice the coils are switched off for a moment every time the field is sampled, and the rate of change is estimated from noisy samples a fraction of a second apart. A high gain amplifies that noise into a jittering dipole. The law is simple; the numbers it is given are not.

There is a striking imbalance in the energy involved. The tumble itself, 8.8 degrees a second about a moment of 0.04 kilogram square metres, holds about half a millijoule of rotational energy. A coil holding a dipole of a fraction of an ampere square metre typically draws a few tenths of a watt, and over the half-orbit a detumble takes it spends hundreds of joules. Nearly all of that becomes heat in the coils’ own resistance; roughly a millionth of it is the tumble being removed. The cost of stopping a satellite magnetically therefore has almost nothing to do with how much spin it has. It is the price of holding a dipole for as long as a field of a few tens of microtesla needs to act, and the quantity that matters in a magnetic coil is how much dipole it makes per watt. What limits the speed of a detumble is torque, not energy: the battery could pay for the braking many times over, and the field will only lend so much leverage per second.

A field that is not quite a dipole #

Everything above uses a tilted dipole, and the Earth’s field is a dipole only to a first approximation. The dipole carries most of the field’s strength at the surface, but the remainder is uneven, and one part of it matters for a detumbling satellite. Over the South Atlantic the surface field falls to about 22 microtesla, where a centred dipole would give more than 35. A satellite crossing that region brakes more slowly while it is there, since the torque goes as the square of the field. The same weakness is why the trapped particles of the inner radiation belt reach lowest over the South Atlantic, and why satellites passing through it collect more upsets in their electronics than anywhere else on the orbit.

The field also changes with time. The dipole has weakened by several per cent over the last century and its axis wanders, so a satellite that uses a stored model of the field to work out where it is pointing must carry an up-to-date one. The B-dot law needs no model. It uses only the change it measures, so it works identically in a field that is stronger, weaker, tilted or lumpier than expected, and it begins working the moment the magnetometer is switched on. That robustness is the main reason it is the law a new satellite runs before anything else: at the moment of release, nothing aboard knows its attitude, its rates or whether its sensors are calibrated, and a law that asks none of those questions is the one that can be trusted.

Where the spin goes #

Every detumble ends the same way in the accounting a spacecraft keeps: the angular momentum has to go somewhere outside the vehicle. When a satellite’s own flexing antennas walk its spin onto a new axis, nothing leaves the satellite, because internal friction cannot change the total. A magnetic coil is different. The torque on the satellite is matched by an equal and opposite torque on the source of the field, and the angular momentum taken from the tumbling satellite is handed, through the geomagnetic field, to the Earth.

A star does something with a family resemblance on a vastly larger scale. A star’s magnetic field forces its wind to turn with it out to large distances, and the wind carries the star’s spin away. In both cases a magnetic field is the lever that moves angular momentum out of a rotating body, and in both cases the field’s geometry decides how effective the lever is.

The geometry is the Earth’s dipole, and its strength falls off as the inverse cube of distance. At 500 kilometres it gives a coil a few tens of microtesla to push against. At geostationary height it is more than two hundred times weaker, and the braking torque, which goes as the square of the field, is some fifty thousand times weaker. Magnetic detumbling is a technique of low orbits for the same reason magnetic momentum dumping is, and above a few thousand kilometres a tumbling satellite has to spend propellant instead.

Still open: how well can coils alone point a satellite? #

Detumbling asks only that rotation be removed, and a law that can only brake is guaranteed to do that. Pointing asks for a chosen orientation, and coils alone cannot supply a torque about the field line at any instant. Over an orbit, though, the field line moves, so in an inclined orbit every axis is reachable some of the time. Laws exist that exploit this, and there are proofs that the averaged system can be brought to a chosen attitude. But the guarantees are slow and conservative, they weaken as the orbit approaches the equator where the field line barely moves, and the real field departs from a dipole in ways that matter. How good a pointing can be promised to a satellite with nothing but coils, and in which orbits, is still being worked out — and for the many small satellites that carry nothing else, it is the difference between an instrument and a passenger.