The compass that finds the axis the Earth turns about

The same push, further out introduced torque, and the arguments after it followed rigid bodies through their free motions — the axis that will not hold, the push that comes out sideways, the top that nods before it settles, the axis a leak of energy chooses and the wobble that is slow because the Earth gives. Two of those ended by naming the same next step: the gyrocompass, where a constrained gyroscope on a rotating Earth finds north on its own.

The gyrocompass was invented in the first decade of the twentieth century, by Hermann Anschütz-Kaempfe in Germany and Elmer Sperry in the United States, because steel warships had made the magnetic compass unreliable and submarines, sealed in a steel hull, had made it useless. It is the most practical application of the physics of spinning bodies there is, and it contains two surprises. One is that a device built from a wheel and a weight finds true north — the axis of the Earth’s rotation — with no reference to the sky. The other is that making it work on a moving ship forces its behaviour to have a particular period, fixed not by the instrument but by the size of the Earth.

A gyroscope that is not allowed to tilt #

A free gyroscope, spinning fast in frictionless gimbals, keeps its axis fixed in space: nothing exerts a torque on it, so its angular momentum does not change. Fixed in space means fixed relative to the stars, and the Earth turns under it. An observer on the ground sees a free gyroscope’s axis turn slowly, once a day; the forces that are not there described the same apparent motion as the work of fictitious forces in the Earth’s rotating frame. That makes a free gyroscope a detector of the Earth’s rotation, which is what Léon Foucault built one for in 1852 and named. It does not make it a compass.

A gyrocompass adds a constraint: the axis must stay nearly horizontal. Hang a weight below the gyroscope’s housing, so that tilting the axis raises the weight, and gravity resists any tilt. Now consider what the Earth’s rotation does. At latitude $\lambda$ the Earth turns about its axis at $\Omega$, and the part of that rotation about the local north–south horizontal line, $\Omega\cos\lambda$, tilts the local horizon: the eastern horizon sinks and the western one rises, continuously. A spinning axis pointing somewhat east of north keeps its direction in space, so as the eastern horizon sinks under it, its east end is left higher above the horizon. The pendulous weight resists that tilt with a torque, and a torque on a gyroscope does not tip it — as the push that comes out sideways found, it turns the axis at right angles to the torque. The geometry works out so that the axis turns in azimuth, towards the meridian.

Swinging into the meridian #

The result is an oscillation about north.

Released twenty degrees east of north, the axis rises at its east end, the weight’s torque turns it westward, it overshoots north, its west end now rises, and it is turned back. Undamped, it swings about north for ever, as the dashed curve does in the figure. The period depends on how strongly the weight resists tilting compared with the gyroscope’s angular momentum, and on the Earth’s rate of rotation at that latitude; this compass has been built so that at latitude fifty degrees the period is 84.4 minutes, for a reason that comes later. With damping — in real compasses a small viscous coupling, or a second, liquid-filled reservoir that sloshes behind the tilt — the swing dies away and the axis settles on the meridian.

The only direction in which the axis stays put is along the meridian, because only there does the tilting of the horizon, which is a rotation about the north–south line, leave the axis level. An axis pointing north is lying along the very line the horizon tilts about. Everywhere else the tilting lifts one end, and the constraint turns that into a swing. The instrument does not detect north; it detects the direction in which the Earth’s rotation fails to disturb it, and that is north by definition.

The path of the axis #

The swing is clearer when both of the axis’s small angles are drawn at once.

The figure plots the direction of the axis against its tilt. Undamped, the end of the axis traces an ellipse round north, forty degrees wide and only a degree and a half high: the pendulous weight resists tilt stiffly, so a small tilt is enough to drive a large swing in azimuth. Tilt and azimuth are a quarter of a cycle apart, like position and velocity in any oscillator — the tilt is greatest as the axis passes through north, which is where the swing in azimuth is fastest. Damping makes the ellipse a spiral.

It is worth noticing that the compass could not work without tilting. A gyroscope held rigidly level would have no torque on it about any horizontal axis and so no precession; it would simply be dragged round by the Earth’s rotation with the platform. The small tilt is the signal. The designer’s task is to make the constraint strong enough that the tilt stays small and weak enough that the Earth’s rotation can drive it at all.

Why the settled axis is not quite level #

The damped compass settles on north, but not quite level, and the reason is the other component of the Earth’s rotation. At latitude $\lambda$ the Earth turns about the local vertical at $\Omega\sin\lambda$ — the rate at which a Foucault pendulum’s plane appears to turn — and that means the meridian itself, the north–south line on the ground, turns relative to the fixed stars at that rate. A gyroscope pointing along the meridian would drift off it, relative to the ground, by fifteen degrees an hour times $\sin\lambda$, if nothing kept it there.

What keeps it there is the pendulous torque, which can only act if there is a tilt. So the settled compass stands with its north end slightly raised, just enough that the torque precesses the axis in azimuth at exactly $\Omega\sin\lambda$ and keeps it on the turning meridian. For a compass tuned as this one is, the settled tilt at fifty degrees comes to about six minutes of arc — invisible, and essential; the figures leave the vertical component out, which is why they show the axis settling exactly level. It also explains a design detail of real compasses: the damping must be applied so that it does not shift where the axis settles, or the compass would come to rest a little east or west of north by an amount depending on latitude, and some early designs did.

The history of that detail runs through an unexpected name. In 1915 Hermann Anschütz-Kaempfe sued Elmer Sperry’s company for infringing his gyrocompass patent, and the Berlin court appointed as its expert a patent examiner turned physicist, Albert Einstein, who studied the instruments, found for Anschütz, and became interested enough to work on the design himself. The compass Anschütz’s firm built in the 1920s, with its gyroscopes floating in a liquid-filled sphere and centred electromagnetically, carried improvements Einstein had suggested, and the royalties from it supported him for years. A man who had made acceleration indistinguishable from gravity spent part of the decade on an instrument whose whole difficulty was that it could not tell them apart.

Weaker towards the poles #

The north-seeking torque comes from $\Omega\cos\lambda$, the horizontal component of the Earth’s rotation, and so it weakens with latitude.

At the equator the whole of the Earth’s rotation is about the north–south horizontal line, and the torque is at its largest. At seventy-five degrees it is four-tenths of its value at fifty, and the swing takes half as long again. Any small disturbing torque — friction in the bearings, an imbalance in the wheel — shifts the settled direction by an angle inversely proportional to the north-seeking torque, so an error that is negligible at mid-latitudes grows steadily towards the poles. At the pole itself the Earth’s axis is vertical, the horizon does not tilt at all, and there is no north for the instrument to find. Ships and aircraft in the high Arctic rely on other methods.

The settled direction also carries a small systematic error that depends on the ship’s own motion. A ship steaming north is itself moving round the Earth’s axis, and the compass responds to the combined rotation, which points slightly off the true axis. The correction depends on speed, course and latitude and is applied from a table; for a ship steaming north at twenty knots at fifty degrees it is about two degrees, which is large enough to matter and small enough to correct.

The period a ship forces on the compass #

Why 84.4 minutes? The reason is not in the gyroscope at all. It is in the problem of telling which way is down aboard something that accelerates.

A compass’s pendulous weight hangs along the apparent vertical, the direction of gravity combined with the ship’s acceleration. The floor that cannot be told from gravity established that no local measurement can separate the two. When the ship speeds up, the apparent vertical leans backward, the weight swings, the compass feels a spurious tilt, and its heading is disturbed. For a gyrocompass whose swing period is short, the disturbance at every change of speed or course throws it off north by several degrees. Max Schuler, a cousin of Anschütz-Kaempfe, found the cure in 1923, and it applies to anything that is meant to indicate the true vertical on a moving vehicle.

When a vehicle moves a distance $x$ over the Earth’s curved surface, the true vertical at its position turns by $x/R$. A horizontal acceleration $a$ turns it at an angular acceleration $a/R$. The same acceleration pushes on an indicating pendulum and swings it. Schuler showed that the two effects exactly cancel — the pendulum turns precisely as the true vertical turns, and stays aligned with it — if the pendulum’s equivalent length is the Earth’s radius. Its period is then $2\pi\sqrt{R/g}$, 84.4 minutes.

The figure shows three indicating references aboard a ship that gains ten knots in a minute. A one-metre pendulum leans to the apparent vertical for as long as the acceleration lasts: thirty minutes of arc, which at one nautical mile per minute of arc is an error of thirty miles in any position worked out from it. A reference a hundred kilometres long is barely moved at first and then rings, with an error of up to seventeen minutes of arc swinging for hours. The one tuned to the Earth’s radius does not move off the true vertical at all. No pendulum can be six thousand kilometres long, but an equivalent length is a ratio of moment of inertia to pendulous moment, and a gyroscope’s angular momentum makes a small device behave like an enormously long pendulum. Tuning the compass’s period to 84.4 minutes is the same thing as giving it that equivalent length.

A period that belongs to the planet #

The number turns up elsewhere, and not by accident.

A satellite in a circular orbit just above the ground, if the air allowed it, would go round in $2\pi\sqrt{R^3/GM}$, which is $2\pi\sqrt{R/g}$ with $g = GM/R^2$: 84.4 minutes. A stone dropped into a straight tunnel through a uniform Earth oscillates to the far side and back in the same time, because inside a uniform sphere gravity grows in proportion to distance from the centre — the pull that grows on the way down found that the real Earth departs from uniformity, which shortens the fall somewhat — and a force proportional to displacement gives simple harmonic motion with $\omega^2 = g/R$. That is the same force law the orbit that does not come back to itself found to be one of only two that close orbits, and the satellite is the tunnel’s motion seen from the side.

A Schuler-tuned reference is, in effect, a pendulum whose bob sits at the centre of the Earth. Any residual error it has does not grow; it oscillates with the planet’s own period. The same is true of the errors in an inertial navigation system, which integrates accelerations twice to find position and uses a Schuler-tuned platform — physically or in software — to keep its sense of the vertical. Left alone, its position error does not drift away but swings back and forth every 84.4 minutes, which is how such systems stay usable over hours.

The period is the planet’s, not the instrument’s, and it changes from world to world. On the Moon, with a radius of 1,737 kilometres and a surface gravity of 1.62 metres per second squared, it is 108 minutes; on Mars, 100 minutes. It depends only on the planet’s mean density, since $R/g$ is proportional to $1/G\rho$, so every body of the Earth’s density, whatever its size, has the Earth’s Schuler period — and a navigation system built for one world has to be retuned for another.

What the drawing leaves out #

The figures treat the compass as a two-variable oscillator, azimuth and tilt, linearised for small angles, with an idealised damping term. Real compasses have more degrees of freedom: the gimbals, the liquid ballistic that supplies the pendulous torque in many designs, and the ship’s rolling, which can couple into the compass through its own period if the design is careless. Rolling at sea is fast compared with 84.4 minutes, and the design keeps the compass’s response to it small, but in some sea states and headings the rolling produces a steady error, the quadrantal error, which is cancelled by a second gyroscope or by arranging the masses symmetrically. The ship’s speed error mentioned above is not in the figures, and neither is the error from a turn, which the Schuler tuning removes only when the compass is at its design latitude; away from it, the period differs from 84.4 minutes and a residual error appears after every manoeuvre.

Modern ships increasingly use compasses with no spinning wheel at all — ring-laser and fibre-optic gyroscopes, which measure rotation by the Sagnac effect, the difference in travel time for light going both ways round a loop that the ring where the two beams disagree described. They find north by measuring the Earth’s rotation directly and computing where its horizontal component points. The physics of the constraint is replaced by computation, and the Schuler period survives in the software.

Still open: how small a north-finder can be #

A gyrocompass finds north by detecting a rotation of fifteen degrees an hour, and the precision with which it does so depends on how well the instrument can measure a slow rotation against its own drifts. Mechanical gyrocompasses reach a fraction of a degree; fibre-optic ones reach a few hundredths. Making north-finders small enough for a handheld device, with gyroscopes etched in silicon a millimetre across, runs into the drift of such tiny sensors, which is typically larger than the Earth’s rotation itself, and whether micromechanical gyroscopes can be brought below that level — by new resonator designs, by averaging, or by rotating the sensor to cancel its bias — is an active engineering question. Atom interferometers, in which the Sagnac phase is picked up by matter waves rather than light, offer far greater sensitivity per unit area and are being developed as navigation-grade rotation sensors, but are not yet practical at sea.

The habit worth carrying away is to find the direction in which a disturbance vanishes. A level gyroscope on a turning Earth is tilted by the rotation in every direction but one, and a constraint that converts tilt into turning drives it to that one — true north — while the need to ignore a ship’s accelerations forces its swing to the period of a pendulum as long as the Earth’s radius, 84.4 minutes, which is also the period of a grazing satellite and of a fall through the planet. A compass that knows nothing of geography finds the Earth’s axis by being the one instrument aboard the ship that the planet’s own turning leaves alone.