A spin that left the axis it was given

A spacecraft keeps its orientation by putting angular momentum somewhere: into wheels, into the Earth’s magnetic field, into propellant thrown overboard. The oldest and cheapest store is the vehicle itself. Spin it, and its own angular momentum holds its axis fixed in space the way a gyroscope’s does, with no wheel, no sensor and no electricity. The first American satellite was stabilised exactly that way, and it is the reason the method comes with a rule.

Explorer 1 went up at the end of January 1958. It was a slender cylinder about two metres long and fifteen centimetres across, still attached to the casing of its final rocket stage, and it was spun about its long axis so that it would fly like a rifle bullet. The spin was set before launch and nothing aboard was meant to change it. The tracking stations soon found that it had changed anyway. The satellite was no longer turning about its length; it was tumbling end over end, turning about an axis across its body.

No torque from outside had done it. The explanation, worked out by Ronald Bracewell and Owen Garriott that year, is short enough to state in a sentence and deep enough that every spinning spacecraft since has been designed around it: a body that cannot change its angular momentum but can lose energy will end up spinning about the axis of its greatest moment of inertia. Explorer 1 had four flexible wire antennas sticking out from its sides. As the body wobbled, they flexed, and the flexing turned a little of the rotation into heat. That was enough.

Two conserved quantities, and one of them leaks #

A satellite in orbit is, to a very good approximation, free of torques on the timescale of a spin. The gradient of gravity, sunlight and the thin atmosphere all act, but they take hours or days to change a spin that turns several times a second. So over the minutes that matter here the satellite’s angular momentum is conserved — its size and its direction in space are both fixed.

Its kinetic energy of rotation is not fixed in the same way, and the difference is the whole story. For a spin about one of the body’s principal axes the two quantities are related by

$$T = \frac{L^2}{2I},$$

where $I$ is the moment of inertia about that axis. Hold $L$ fixed and the energy depends on which axis carries it. About the axis of least inertia — the long axis of a rod — the energy is as large as that angular momentum can make it. About the axis of greatest inertia — any axis across the rod — it is as small as it can be. The ratio of the two is the ratio of the moments.

Anything inside the body that flexes, sloshes or rubs takes energy out of the rotation and leaves the angular momentum alone, because internal forces come in equal and opposite pairs and cannot change the total. The energy can only go down. The only way for it to go down while the angular momentum stays put is for the angular momentum to move, relative to the body, onto an axis with a larger moment of inertia. Relative to the stars the angular momentum never moves at all; it is the body that turns underneath it until a different axis lines up with it.

The figure is a numerical integration of Euler’s equations for a rigid body, with one term added to stand for the antennas. The term removes energy at a rate proportional to how far the body’s rotation is from being aligned with its angular momentum, and it is written so that it cannot change the angular momentum — which is the only property of a real flexing antenna the argument needs. Its strength is a free choice, and it has been chosen here so that the whole migration fits into a few hundred turns. A real satellite with a real antenna takes much longer, and the path it follows is the same.

Three things happen, in order. For roughly the first two hundred turns the wobble grows slowly and nothing appears to change: the long axis still carries nearly all the angular momentum, and the curves hardly move. Then, within a few tens of turns, the shares swap. After that the body settles onto a spin about the transverse axis, with the wobble dying out.

What the energy does #

The same run can be read as a single number falling.

With moments of 1 and 8.6, the body has to give up 88 per cent of its rotational energy to get from the start to the finish, and it does so without losing any angular momentum. Something has to have absorbed that energy, and for Explorer 1 it was the antennas and the joints that held them, warming by an amount far too small to measure. The large number is the point: the spin a designer chose was the most energetic state available, and every dissipative process aboard was quietly working to leave it.

The curve is not a steady decline. The loss rate is small at first, because a small wobble means only slightly misaligned rotation and a sink proportional to misalignment has little to act on. It grows as the wobble grows, peaks around the crossing of the dashed line, and falls away again as the body approaches a clean spin about its new axis. That is the shape of a runaway followed by a settling, and it is why a spinning satellite can look perfectly behaved for a long time before it fails.

The end state also turns more slowly. With the angular momentum unchanged and the moment of inertia 8.6 times larger, the rotation rate falls by the same factor. A satellite spun at hundreds of revolutions a minute about its length finishes turning end over end at a small fraction of that rate — a change that shows up immediately in the fading pattern of its radio signal, which is how the tracking stations saw it.

A sphere, and the curves drawn on it #

The migration is easiest to see in the body’s own frame, where the angular momentum is the thing that moves.

Two facts fix the picture. The angular momentum has a constant size, so seen from the body its tip always lies on a sphere. The kinetic energy is a sum of three squares, $T = L_1^2/2I_1 + L_2^2/2I_2 + L_3^2/2I_3$, so at a fixed energy the tip also lies on an ellipsoid. A rigid body with no dissipation has both quantities fixed, and its angular momentum must travel along the curve where the sphere and the ellipsoid cut each other. Those curves are called polhodes, and drawing all of them on the sphere draws every possible torque-free motion of the body at once.

Most polhodes are closed loops around one of two axes. Low-energy loops circle the axis of greatest inertia; high-energy loops circle the axis of least inertia. The two families are divided by a single curve through the intermediate axis, the separatrix, which belongs to neither and is the subject of the most surprising motion a rigid body makes.

With a small energy sink the body follows a polhode almost exactly for one circuit, loses a little energy, and moves to a slightly lower one. The path is a slow spiral across the family of loops, and the direction of the spiral is set by the sign of the energy change and nothing else. The loops around the long axis widen until the path reaches the separatrix. On the other side the loops around the transverse axis narrow, and the path winds down to a point.

The crossing hides a small coin toss. There are two ends to the transverse axis, one on each side of the separatrix loop, and which of them the body ends up spinning about depends on exactly where along the separatrix it happens to cross. The angular momentum in space does not care, since the two outcomes are the same spin seen from opposite ends of the body; but a satellite with a sensor on one side will find it pointing one way or the other, and no measurement of the starting state precise enough to predict which is ever available.

The same dissipation, the other way round #

Start the body about its axis of greatest inertia instead and give it the same wobble.

Now the energy sink has almost nothing to take. The spin is already the lowest-energy rotation its angular momentum allows, and the small wobble is the only energy above that floor. Dissipation removes the wobble and stops. The spin that was fragile when it began about the long axis is protected when it begins about the heaviest one, by exactly the same physical process.

This is why a spin-stabilised spacecraft is built the other way from Explorer 1. The drum-shaped communications satellites of the 1960s were squat cylinders spun about their axis of symmetry, and a squat cylinder has its greatest moment of inertia about that axis. Many carried nutation dampers — a ring of viscous fluid or a ball rolling in a curved tube — whose only purpose is to dissipate energy whenever the body wobbles. On a spin about the heaviest axis that is a stabiliser; on a spin about the lightest axis the same damper would have been an accelerant.

A rule with no ratio in it #

The rule is an inequality, not a threshold with a margin. That can be checked directly by changing the shape of the body a little at a time.

A body whose spin axis is only ten per cent lighter than its transverse axes still loses its spin, slowly. A body whose spin axis is ten per cent heavier keeps it. There is no shape between the two that is safe and no degree of slenderness at which the rule begins to apply: it applies from the first per cent. That is what makes it a rule rather than a design guideline, and why it is easy to get wrong in a vehicle that looks roughly symmetric. A spacecraft that grows a long boom or a solar array after launch, or burns off the propellant that made it squat, can move from one side of the line to the other without anybody deciding it should.

The rate is the part that is not universal. The integrations use one simple model of dissipation, and the growth rates in the figure are rates for that model. A flexing wire, a sloshing tank and a viscous damper each take energy from a wobble in their own way, at their own frequency and with their own dependence on spin rate, and real rates can differ from the drawn ones by orders of magnitude. The analysis that decides whether a spinning upper stage survives its burn is almost entirely an analysis of that rate, because the sign was settled in 1958.

There is a way around the rule, found in the decade after it, and it does not contradict it. A spacecraft can be built in two parts, one spinning and one held still, with its dampers mounted on the still part. The energy argument then applies to the two parts separately, and a long vehicle can be kept spinning about its long axis if the dissipation on the despun part is made to dominate. The rule has not been broken; the body it is applied to has been redefined.

The same inequality in bodies nobody built #

Nothing in the argument is specific to spacecraft. Any rotating body that is not perfectly rigid is an energy sink, and over a long enough time it ends up rotating about its axis of greatest inertia.

The Earth does. Its moment of inertia about the polar axis is larger than about any equatorial axis, because of the equatorial bulge, and the planet spins about that axis to within a few metres at the pole. The residual is the free wobble of the rotation axis around the figure axis, which should damp out in the same way the transverse component damped out in the figures above, and the fact that it has not is itself a measurement: something keeps feeding it.

Asteroids mostly do. The great majority of measured asteroid light curves repeat with a single period, which means the body is turning about one principal axis. The exceptions are overwhelmingly slow rotators. A slowly turning body stores little energy in its wobble and flexes little under it, so the internal friction that should remove the tumble works on a timescale that can exceed the time since the last collision set it tumbling. The asteroid Toutatis turns in a complex, non-principal-axis way with periods of several days, and the interstellar object 'Oumuamua appears to have been tumbling as it passed the Sun. Both are what a body looks like when the clock on its energy sink is longer than the clock on whatever disturbed it.

Moons and planets that are spun down by tides are a different case with a family resemblance. There the torque is external and the angular momentum does change, but the final state is still chosen by an energy argument, and the body is still found turning about its axis of greatest inertia with its long axis pointed where the tide can grip it. The dissipation that heats a tidally worked moon spends energy the orbit cannot replace; the dissipation in Explorer 1’s antennas spent energy the spin could not replace, and the accounting was the same.

What a designer can and cannot choose #

The practical content of the rule is a short list. A body spun about its heaviest axis is stable in the presence of dissipation and can be helped by adding more. A body spun about its lightest axis is unstable in the presence of dissipation and will eventually turn over, on a timescale set by how much it flexes. A body spun about its middle axis is unstable even with none, which is a separate result with its own figure. And the moments that decide which case applies are those of the whole vehicle as it is at that moment, including propellant, deployed hardware and anything still attached from launch — Explorer 1’s moments included its spent rocket casing.

What a designer cannot choose is the conservation law underneath. The total angular momentum of a satellite is set when the last external torque stops acting, and in a solar system where most of the spin sits in a few places, as in a single spacecraft, redistributing it inside the system is free and removing it is not. The antennas on Explorer 1 did not slow its angular momentum by any amount at all. They only moved it — and in doing so they chose, from every orientation the satellite could have held, the one it would hold for the rest of its life.

Still open: how fast does a tumbling asteroid forget its tumble? #

The sign of the effect is certain, and the rate is not. For a small body the damping time depends on its rigidity, its internal friction and the square of its spin rate in ways that have to be modelled rather than measured, because no one has watched an asteroid relax. Classic estimates put the time to damp a tumble at many millions of years for a kilometre-sized body turning once a day; later analyses argued that the strain in a wobbling body is distributed in a way that makes dissipation considerably faster. Which is right decides whether the tumbling asteroids now known were disturbed recently, or have simply never had time to settle — and so whether a light curve with two periods in it is evidence of a recent collision or only of a slow spin.