The tunnel that takes the same time from anywhere

The outside of the shell theorem is the famous half: a spherical body pulls on anything beyond it exactly as though its whole mass sat at its centre, which is why the Moon can be treated as a point and why an orbit is an ellipse rather than something more complicated. The inside is the other half, and it produces a result that sounds like a puzzle and is a theorem.

Drop a stone down a straight tunnel bored through a uniform sphere. It arrives at the far end in a time that does not depend on how long the tunnel is, which direction it runs, or where it starts. For a body of the Earth’s mass and radius that time is forty-two minutes, and it is forty-two minutes for a tunnel through the centre and for a tunnel from London to Paris.

Why the period contains nothing #

Inside a uniform sphere of density $\rho$, the mass enclosed within radius $r$ is $\tfrac43\pi r^3\rho$, so the field is

$$g(r) = \frac{G\,\tfrac43\pi r^3\rho}{r^2} = \frac{4\pi G\rho}{3}\,r.$$

A restoring force proportional to displacement is Hooke’s law, and Hooke’s law gives simple harmonic motion of angular frequency $\omega = \sqrt{4\pi G\rho/3}$ — which contains the density and nothing else. Not the mass, not the radius, not the amplitude. A stone released just below the surface and a stone released a metre from the centre have the same period. The chord is the part that looks like a trick and is not. A tunnel that misses the centre by a distance $d$ feels the full field $-\omega^2 \mathbf{r}$ at each point, and the tunnel wall takes the component perpendicular to itself. What is left along the tunnel is $-\omega^2 x$, where $x$ is the distance from the tunnel’s own midpoint — the same spring constant, a smaller amplitude, and therefore the same period. The stone travels less far and is pulled less hard, and the two effects cancel to the last digit.

The number that appears twice #

Here is the connection worth the essay. The full period of the tunnel oscillation is

$$T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{3}{4\pi G\rho}} = 2\pi\sqrt{\frac{R^3}{GM}},$$

and the second form is instantly recognisable: it is the period of a circular orbit at radius $R$, which is to say an orbit skimming the surface. The stone falling down the tunnel and the satellite going round outside take exactly the same time.

That is not a numerical coincidence, and the reason is prettier than the fact. A straight-line oscillation through the centre is an ellipse of eccentricity one — a degenerate ellipse, squashed until its minor axis vanishes — with the same semi-major axis $R/2$… except that the tunnel case has the mass distributed rather than concentrated, so the two are the same for a different reason: both are governed by $\sqrt{R^3/GM}$ because that is the only combination of the available quantities with the dimensions of time.

The dynamical time #

Read the other way round, $\sqrt{3\pi/G\rho}$ — half the tunnel period — is the single most useful timescale in astrophysics, and it is worth naming.

Any self-gravitating body has one, and it depends on its mean density and nothing else. It is the time a body takes to rearrange itself if its pressure support is removed; it is the free-fall time of a cloud collapsing to form a star; it is the timescale on which a star responds to being poked. For the Sun’s mean density of 1.41 g/cm³ it is about half an hour. For a molecular cloud at $10^4$ hydrogen molecules per cubic centimetre it is a few hundred thousand years. For the Universe at the critical density it is the Hubble time — which is the same statement as saying that the expansion rate and the mean density are two ways of writing one number.

What was actually measured #

The forty-two minutes is a calculation, and the ingredients in it are measurements of two very different kinds.

The mean density of the Earth is the harder one, and it is the whole content of the Cavendish experiment. Weighing the Earth is not something anybody has done; what Cavendish did in 1798 was measure the attraction between two lead spheres in a laboratory, which fixes $G\rho$ against $g$ — and nothing in the sky is ever weighed in kilograms except through that one torsion balance and its descendants. The modern value, 5.514 g/cm³, is known to about five figures; $G$ itself is known to four, which is the worst-determined of all the fundamental constants.

The profile is seismological. The PREM model drawn in the opening figure is fitted to the travel times of earthquake waves, to the Earth’s free oscillation frequencies and to its moment of inertia, and it says the density runs from 2.6 g/cm³ at the surface to 13.1 at the centre. That is a factor of five, and it is why the flat mantle field in the figure is not a small correction: a body falling through the real Earth reaches the centre in 38.2 minutes rather than 42.2, because for most of the descent the field is 12% stronger than the uniform model allows.

The history is the origin of the Principia #

The tunnel is not a modern classroom puzzle. It is the subject of the exchange of letters between Hooke and Newton in the winter of 1679–80, and that exchange is where the Principia comes from.

Hooke wrote asking what path a body would follow if dropped from a great height on a rotating Earth, and — as a second question — what it would do if the Earth were opened to let it through. Newton replied with a spiral into the centre, which is wrong, and Hooke corrected him: the path would be a closed oval, returning to its starting height. The correction stung, Newton went away and worked out what an inverse-square force actually produces, and the ellipse with the primary at a focus came out of it. The question that started modern dynamics was a question about a hole through the Earth.

Hooke’s oval, incidentally, is right for the uniform case and is the one drawn below.

One number, five densities #

Because the dynamical time depends on the mean density alone, a single expression covers objects that have nothing else in common — and reading it across them is the fastest way to acquire a sense of scale in this subject.

The last two are the ones that repay attention. A white dwarf’s dynamical time being under a second is why a white dwarf that exceeds its limiting mass does not sag — it collapses, in less time than it takes to say so. A neutron star’s being tens of microseconds is why the fastest pulsars, spinning seven hundred times a second, are not torn apart: their rotation period is still long compared with the time their own gravity takes to act.

The Earth is not uniform, and the answer moves #

The forty-two minutes is computed for a sphere of uniform density, and the Earth is nothing of the kind: it has an iron core at some thirteen tonnes per cubic metre against a crust at under three, so more than half the mass sits inside half the radius.

That changes the force law along the tunnel. In a uniform sphere the gravitational acceleration falls linearly to zero at the centre, which is what makes the motion simple harmonic and the period independent of everything. In the real Earth the acceleration rises slightly with depth through the whole mantle — from 9.8 at the surface to about 10.7 at the core–mantle boundary, halfway down — and only then falls.

A stronger pull for most of the journey means a shorter journey. Integrating the equation of motion through a measured density profile gives about 38 minutes rather than 42 for a fall through the centre, and the difference is a direct consequence of the concentration.

That is worth noticing because it inverts the usual relationship between a toy model and a measurement. The forty-two-minute result is exact for an object nobody has, and the four-minute discrepancy is a measurement of how centrally condensed the Earth is — the same quantity a moment-of-inertia factor reports, arrived at through a fall time instead of through a spin.

The chord makes no difference to the time and a great deal of difference to everything else, which is worth seeing at the other extreme.

Where the same number governs a cloud #

The expression behind the tunnel is $\sqrt{3\pi/32G\rho}$, and it appears wherever gravity is the only thing acting.

The free-fall time of a gas cloud is exactly that: the time for a pressureless sphere of density $\rho$ to collapse to a point. For a molecular cloud at a few thousand hydrogen molecules per cubic centimetre it works out at a few million years, and for a dense core about to form a star, at a hundred thousand times that density, a few tens of thousands.

Those numbers set up one of the standing problems of star formation. The Milky Way contains roughly a billion solar masses of molecular gas; if all of it collapsed in a free-fall time, the Galaxy would produce stars at something like a hundred solar masses a year. The observed rate is about one.

So something is holding the clouds up, or destroying them, or both — turbulence, magnetic fields, and the feedback of the stars that do form, which heat and disperse the gas around them. The free-fall time is the denominator in every version of that argument, and the ratio of the observed rate to the free-fall rate is the efficiency the whole subject is trying to explain.

A fall time computed for an idealised tunnel is therefore the standard of comparison for the rate at which a galaxy makes stars, which is a long way from where the calculation started and is the same equation throughout.

What such a tunnel would actually be like #

The idealisation is worth puncturing once, because the ways it fails are each a piece of physics.

Air. A tunnel open at both ends fills with atmosphere, and at the centre of the Earth the pressure of a column of air that deep would be enormous. With air in it, drag dominates entirely: a falling body reaches a terminal velocity of a few hundred kilometres an hour rather than the eight kilometres a second the vacuum calculation gives, and the journey takes days rather than minutes. The tunnel has to be evacuated for the result to mean anything.

Rotation. The Earth turns, so a body dropped down a straight shaft does not fall along it: in the rotating frame the Coriolis force pushes it sideways, and for a pole-to-pole tunnel that is zero but for any other the body strikes the wall within the first few kilometres. A tunnel that a body could traverse freely would have to be curved, and the curve depends on the endpoints.

Materials. At the core–mantle boundary the pressure is 136 gigapascals and the temperature about 4,000 kelvin, and there is no substance that holds a cavity open under those conditions.

So the thing being computed is a property of the Earth’s mass distribution, dressed as a journey. That is the honest description of the whole exercise: the tunnel is a device for making a density audible as a time, and the forty-two minutes is a statement about $G\rho$ with a story attached.

The same fall, without the tunnel #

There is a version of the problem that needs no hole at all, and it is the one that turns the result into an instrument.

Consider a satellite in a circular orbit skimming the surface. Its period is $2\pi\sqrt{R^3/GM}$, which is the same combination of quantities as the tunnel’s — the two differ by a factor that is exactly one. A body falling through a hole in a planet and a body circling just above its surface keep the same time, and both depend on the planet’s mean density and nothing else.

That equivalence is the useful form, because low orbits are observable and tunnels are not. Timing a satellite skimming a body gives its mean density directly, with no need to know its size or its mass separately — which is how a spacecraft passing an asteroid establishes what the object is made of, from an orbit rather than from a sample.

The numbers span the solar system’s range. A satellite grazing the Earth takes 84.5 minutes; one grazing the Moon takes 108, because the Moon is less dense; one grazing a comet nucleus at half the density of water would take about six hours; and one grazing a neutron star, at $10^{17}$ kilograms per cubic metre, takes a tenth of a millisecond.

Five orders of magnitude in density, five in time, and one expression. That is the whole content of the tunnel result, stated for objects that exist.

The equivalence has a limit worth stating, and it is the same one throughout: both results assume the mass is spherically distributed and that the orbiting or falling body is at the surface of it. A satellite skimming an irregular asteroid is not on a circle at all, and reading a density off its period requires modelling the shape first — which is why the technique is quoted for round bodies and treated as an estimate for everything else.

That estimate is nevertheless the first thing computed about any newly encountered small body, because it needs only a period and no assumption about composition — and a density near one tells a different story about what the object is than a density near three.

A period, in this subject, is very often the cheapest measurement available and the one that constrains the most.

Where the model stops #

Four things are wrong with the picture, and it is worth being specific about which of them matter.

The rotation of the Earth matters and is the largest. In the rotating frame a falling body is deflected by the Coriolis force, so a straight tunnel is not a path any free body follows; a real gravity train would have to be a curved tube, and would grind against its wall. Hooke’s original question was about precisely this deflection and Newton got that part right.

Air matters enormously. A tunnel full of air at atmospheric pressure would stop a stone in a few kilometres; the standard treatment assumes it evacuated, which is a considerable assumption for a hole through a planet.

The non-uniform density matters at the 10% level, which the figure computes rather than assumes.

And the absence of any such tunnel matters in the way that makes this a thought experiment: the deepest hole ever drilled reached 12.3 km, a fifth of a per cent of the way, and the rock at the bottom was at 180 °C and flowing.

And two readings of a case the uniform sphere does not cover at all, in which the interior field is not linear in the radius and the tunnel would therefore have no period to speak of.

Where this ladder goes next #

The theorem’s two halves have now both been used: the outside one to justify treating bodies as points, the inside one to produce a period out of a density. The rungs above ask what happens when the sphericity fails.

The nearest is the Earth’s own oblateness, where the departure from a sphere is one part in three hundred and is nevertheless the dominant perturbation on every low satellite orbit. Beyond that lies the general problem of a field with a shape — the multipole expansion, of which the shell theorem is the statement that a sphere has only a monopole — and the measurement problem it creates, since a satellite’s orbit responds to the whole tower of moments and an observer has to disentangle them from one drifting node.

The other direction is the timescale. $\sqrt{3\pi/G\rho}$ turns up again as the free-fall time of a collapsing cloud and as the response time of a star that has lost its pressure support, and it is the clock against which a self-gravitating system’s strange thermodynamics is measured.