A wingnut that turns over on its own

In 1985 the cosmonaut Vladimir Dzhanibekov was aboard the Salyut 7 space station, unscrewing a wingnut from a bolt. A wingnut spun off the end of a thread keeps turning about the bolt’s axis, and in weightlessness it drifts away still turning. This one did something else. After a short while it flipped over end for end, kept spinning in the new orientation, and a little later flipped back. It carried on doing so, at regular intervals, for as long as it could be watched.

Nothing touched it. The cabin air exerts almost no torque on a small spinning object over a few seconds, so its angular momentum could not change, and a wingnut has nothing inside it to flex or slosh. A spacecraft that migrates from one axis to another does so because it loses energy, and this was not losing any: it flipped over and came back with the same spin it left with, which is what a body with no dissipation must do.

The explanation had been sitting in Leonhard Euler’s equations for the rotation of a rigid body since 1765, and Louis Poinsot had drawn it as a picture in 1834. It is easy to reproduce on Earth by tossing a book, a phone or a tennis racket in the air with a spin about the right axis. But on the ground the flip lasts one throw. In orbit it continues indefinitely, and it is the reason that one of the three ways of spinning a spacecraft is never used.

Three steady spins, and only two of them steady #

Any rigid body has three principal axes, at right angles, about which it can turn steadily without wobbling. They are the axes of greatest, least and intermediate moment of inertia. A book has them: across its face through the spine and the opposite edge, across its face through the top and bottom, and straight through its covers. For a body with three different moments, each is an equilibrium: spin exactly about any of them and nothing changes.

Equilibria differ in what happens when they are disturbed, and the three are not alike.

The top and bottom lanes are flat because the disturbance stays the size it was given: the body wobbles slightly about its axis and the wobble neither grows nor decays. The middle lane is a square wave. The component of angular momentum along the spun axis holds near one, falls through zero to minus one — at which point the body is spinning the same way in space but upside down in itself — holds there, and comes back.

The figure is a direct integration of Euler’s equations, with energy and angular momentum both checked to be conserved at every step. The instability is not a numerical artefact and is not produced by any loss; it is what the equations of an ideal rigid body say.

Which of the three axes is unstable depends only on the order of the moments. It does not depend on their sizes, or on how carefully the spin is started, or on whether the body is a wingnut, a spacecraft or a planet. That is why the result is sometimes called a theorem: every rigid body with three distinct moments has exactly one unstable steady spin, and it is always the one in the middle.

A pencil on its point #

Linearising Euler’s equations about a spin at rate $\Omega$ about the intermediate axis shows what the disturbance does. The two small components across the spin axis, $\omega_1$ and $\omega_3$, obey

$$\ddot\omega_1 = \Omega^2\,\frac{(I_2 - I_1)(I_3 - I_2)}{I_1 I_3}\,\omega_1 ,$$

and the same for $\omega_3$. When the spin is about the middle axis, both brackets are positive and the coefficient is positive, so the solutions are growing and decaying exponentials. About either end axis one bracket is negative, the coefficient is negative, and the solutions are oscillations. That sign is the whole of the stability result, and the growth rate of the disturbance is

$$\sigma = \Omega\sqrt{\frac{(I_2 - I_1)(I_3 - I_2)}{I_1 I_3}}.$$

For the 1 : 2 : 3 body the disturbance grows by a factor of e every 0.28 turns — by a factor of about thirty-eight in each turn. A one-per-cent knock needs only four and a half e-foldings to become comparable to the spin itself, and so the first flip comes after about a turn and a half. The disturbance does not have to be deliberately applied, either. A real wingnut spun off a real thread cannot be released with its spin exactly on a principal axis, and the error it inherits is enough.

The pencil comparison is exact in one respect and misleading in another. Like a pencil on its point, the intermediate-axis spin is an equilibrium from which any departure grows exponentially at first. Unlike a pencil, the body does not fall over and stay fallen. It has nowhere lower to fall to, because nothing is losing energy, and so it comes back.

Why a tossed phone flips in one throw #

On the ground the effect is usually shown by throwing something flat. A book, a phone or a tennis racket tossed upward with a spin about the right axis comes down having turned over, and it does so within the one or two turns a throw allows. That is not luck in the choice of object. Flat objects sit close to the worst case.

For a thin flat plate, the moment of inertia about the axis through its face is the sum of the moments about the two axes lying in its plane. A phone twice as long as it is wide therefore has moments in the ratio 1 : 4 : 5, about its long axis, its short axis and the axis through its screen. The intermediate axis is the short one, across the screen. Putting those numbers into the growth rate gives a disturbance multiplied by e nearly five times per turn, or more than a hundredfold each turn. A throw that puts the spin five degrees off the short axis has used up its margin in about half a turn, and the phone flips before it is caught.

A tennis racket is the same calculation with a handle. Its least moment is about the handle, its greatest is through the face, and the axis lying in the face at right angles to the handle is the unstable one. Tossed about the handle, or about the axis through the face, it spins cleanly. Tossed about the third axis, it comes back to the hand with its face reversed, which is where the result’s informal name, the tennis racket theorem, comes from.

The plate is close to the worst case for a reason that can be stated exactly. No moment of inertia can exceed the sum of the other two, and a flat plate reaches that limit. For a given pair of smaller moments, the growth rate increases as the largest moment grows, so among all bodies sharing those two smaller moments a flat plate flips fastest. Nothing needs to be thrown carelessly to demonstrate the effect; a careful throw of a flat object shows it just as well.

Why it comes back #

The reason the flip repeats is clearest in the picture Poinsot drew.

Seen from inside the body, the angular momentum has a fixed length, so its tip lies on a sphere. The kinetic energy is fixed too, and that confines the tip to an ellipsoid. The tip must therefore move along a curve where the two surfaces meet — a polhode. The polhodes on this sphere come in two families of closed loops, one circling the axis of least inertia and one circling the axis of greatest, and the two families are divided by a separatrix: a pair of great circles that cross at the two ends of the intermediate axis.

A spin exactly about the intermediate axis sits where the two separatrix circles cross. Any disturbance puts the angular momentum on a neighbouring polhode, and every polhode near that crossing is a long thin loop that follows the separatrix all the way round the sphere to the opposite end of the intermediate axis before closing. So a small disturbance does not produce a small wobble. It produces a trip to the far side of the body and back. Near the crossings the motion is slow, because the flow there is almost stationary, and that is the time the body spends apparently spinning steadily. Away from them the motion is fast, and that is the flip.

Spins about the end axes sit in the middle of the loop families. A disturbance moves the angular momentum onto a small loop around the axis, and a small loop stays small, which is the flatness of the top and bottom lanes in the first figure.

Two consequences come straight out of the picture. The flip goes whichever way the polhode goes, which is set by the sign of the disturbance; a disturbance of the opposite sign sends the body round the other side of the sphere. And the motion is strictly periodic. Conservation of energy and angular momentum pins it to one closed curve, and it retraces that curve forever. A wingnut in a perfectly empty room would flip at perfectly regular intervals until something touched it.

A logarithm, not a threshold #

The steady-looking stretch between flips is the time the disturbance takes to grow from its starting size to the size of the spin. Exponential growth makes that time a logarithm.

The slope carries a factor of two that is worth understanding. Near either end of the intermediate axis the disturbance has a growing part and a shrinking part, and the conserved energy and angular momentum fix the product of the two at roughly the square of the starting disturbance ε. The body approaches the end of the axis along the shrinking direction and leaves along the growing one, and the time it lingers there is the time for one part to fall from order one to the size the product allows: $\tfrac{1}{\sigma}\ln(1/\varepsilon^2)$, which is $\tfrac{2}{\sigma}\ln(1/\varepsilon)$. Each flip is one such lingering and one quick transit whose length does not depend on ε at all, and the measured slope is exactly that two over σ.

The consequence is the whole engineering content of the result. Setting a spin up ten times more carefully buys a fixed, small delay. Setting it up a million times more carefully, which is far beyond what any real release mechanism can achieve, buys about eight turns instead of two. No level of precision makes the wait infinite, because the logarithm of a finite number is finite. There is no tolerance to which an intermediate-axis spin can be built.

That is quite different from the migration driven by energy loss, which depends on how strongly the body dissipates and can be made arbitrarily slow by making the body stiffer. The flip does not depend on anything inside the body except its three moments. A spacecraft designer faced with dissipation can buy time by building a stiffer vehicle; faced with the intermediate axis, time cannot be bought at all.

This is why no one holds a spacecraft’s attitude by spinning it about its middle axis. Spin stabilisation uses the axis of greatest inertia, where both the rigid-body dynamics and dissipation help. The axis of least inertia is used only briefly, when a long rocket stage has to be spun for a burn and the burn is over before the energy loss catches up. The intermediate axis is not used at all.

How far from symmetric is safe #

The growth rate vanishes when the middle moment equals either of the others, because the body then has an axis of symmetry and no distinct middle axis. It is natural to ask whether a body close to symmetric is safe for practical purposes.

The answer is no, and the shape of the curve says why. The rate goes as the square root of the distance to symmetry, so it falls steeply only at the very end. A body whose middle moment is within four per cent of its largest still multiplies a disturbance by nearly five every turn. The flip is slower, not absent.

For a spacecraft the practical hazard is not a vehicle designed around the middle axis but one that crosses into it. A vehicle spun about its axis of greatest inertia that then deploys a boom, extends a solar array or burns off propellant can change the order of its moments without changing its spin. If the axis it is turning about becomes the intermediate one, the spin that was protected an hour earlier is now a pencil on its point. The moments that matter are the moments at each instant, and the checks that catch this are done for every configuration a vehicle passes through, not only for the one it launches in.

The same motion with a little friction #

A real body is never perfectly rigid, and the two results combine. A body with some internal dissipation spun near its intermediate axis flips as above, but each trip round the sphere costs it a little energy. Its polhode drifts towards the axis of greatest inertia, and after some number of flips it drops out of the flipping motion onto a loop around that axis and settles there. The crossing of the separatrix that marked the migration of an energy-losing body from one family of loops to the other is this motion, passed through once.

That is also why Dzhanibekov’s wingnut is the most celebrated but not the most informative example. On a larger scale, the planets are all spinning about their axes of greatest inertia, where dissipation over billions of years has left them, and the only free wobble left in the Earth’s rotation is the small, slowly damped motion about that stable axis that geodesists have followed for more than a century — a wobble that by rights should have stopped. The intermediate-axis flip is a property of bodies that have not had time to settle: tools, spacecraft, fragments of a collision, and asteroids spun up or knocked recently enough that their internal friction has not caught up.

Nor does the Earth risk a sudden flip of the kind the wingnut makes, a claim that resurfaces from time to time. It is spinning about its heaviest axis, the stable one, and the difference between its polar and equatorial moments, set by the bulge its spin raises, is one of the best-measured quantities in geophysics. What is true is subtler: the whole solid planet can drift relative to its spin axis over geological time if mass is redistributed inside it, because the axis of greatest inertia moves with the mass and the spin follows it. That is a slow migration, driven and dissipative, and it belongs to the physics of energy loss rather than to the physics of the flip.

Dead satellites are the case that matters most in practice. A spacecraft that loses power stops controlling its attitude, and whatever spin it is left with evolves under exactly these rules, slowly modified by the environmental torques. A vehicle sent to capture or deorbit a derelict has to match that rotation, and the rotation may be flipping, precessing or migrating from one axis to another. The growing debris population that makes such missions worth flying is mostly tumbling, and the spins of the largest pieces are measured from the ground by timing the glints of sunlight off their surfaces and the returns of laser pulses from any retroreflectors they carry. Reading those measurements means deciding, first, which of the three kinds of motion is in front of the telescope.

Still open: which side of the separatrix do tumbling asteroids choose? #

Asteroids that tumble rather than spin about a single axis can be found on either side of the separatrix, turning in loops round their long axis or in loops round their short axis, and both kinds have been identified from light curves with two periods in them. How the population divides between the two, and why, is not settled. The split carries information about how tumbles are excited — by collisions, by close planetary encounters or by the slow push of re-radiated sunlight — and about how internal friction removes them, which pushes a tumbling body across the separatrix exactly once. But the census is small, each classification rests on a long and difficult campaign of photometry, and whether the ratio found so far describes the asteroids or only which ones are easiest to measure is still a question.