The pull that grows on the way down

The shape decides the falloff ends its section on gravity with the interior of a uniform sphere. Gauss’s law counts only the mass enclosed, which grows as the cube of the radius, while the area it is spread over grows as the square, so inside a uniform ball the field falls in a straight line to zero at the centre. A body dropped down a shaft through such a planet would oscillate with the period of a low orbit, and the result is one of the most pleasing in the subject.

It is also the answer usually given for the Earth, and for the Earth it is wrong. Not slightly wrong: wrong in sign, over the outer 2,891 kilometres, which is nearly half of the Earth’s radius and five-sixths of its volume.

Counting only what is inside #

The law that does the work is the one counting what comes out introduces for charge, with mass in place of charge and a sign changed. For any closed surface, the flux of the gravitational field through it is $-4\pi G$ times the mass inside. For a spherical surface in a spherically symmetric body, the field is the same all over the surface and points inward, so

$$g(r) = \frac{G\,M(r)}{r^2},$$

with $M(r)$ the mass inside radius $r$. Nothing outside the sphere contributes — the shells outside pull equally in every direction and cancel — so the field at any depth is decided by how much mass lies deeper.

For a uniform body $M(r)$ grows as $r^3$ and the field falls in proportion to $r$. For the Earth, what $M(r)$ does depends on how its density varies with depth, and that is not something anybody can measure by going there. The deepest hole ever drilled reaches twelve kilometres.

Where the density comes from #

The density inside the Earth is inferred from how the Earth vibrates. Earthquakes send seismic waves through the interior, and the times they take to arrive at stations around the world fix the speeds of compressional and shear waves at every depth. After a very large earthquake the whole planet rings like a bell for days, and the frequencies of those free oscillations depend on the density as well as on the wave speeds. Fitting both gives a density profile.

The figures here use the Preliminary Reference Earth Model, published by Adam Dziewonski and Don Anderson in 1981, which describes the density as a cubic polynomial in radius within each layer. The profile was fitted to seismic data. It was not fitted to gravity — and that makes three numbers it has to reproduce into genuine checks, which the figure performs before drawing anything.

Integrated over the Earth, the model’s density gives a mass within 0.02 per cent of the Earth’s measured mass. It gives a surface gravity of 9.82 metres per second squared. And it gives a moment of inertia of 0.3308 times the Earth’s mass times its radius squared, against 0.3307 measured from the slow precession of the Earth’s axis. That last number is the sharpest of the three. A uniform ball has a moment-of-inertia factor of 0.4; the Earth’s is lower because its mass is concentrated towards the centre, and the precession of the equinoxes had shown that the core must be dense long before any seismologist mapped it.

The density jumps by almost a factor of two at the core–mantle boundary, from the iron alloy of the outer core to the silicate rock of the mantle. The mantle above it is less dense than the average of everything it covers, and in the lower mantle the difference is large: rock at 5.6 grams per cubic centimetre sitting on a core whose average is 11.

How the core was weighed before it was seen #

The rise of gravity through the mantle is a consequence of a dense core, and the core was known to be dense long before anybody could draw a profile of it. In the 1890s Emil Wiechert pointed out that the Earth’s mean density, five and a half grams per cubic centimetre from the measurement of its mass, is twice that of the rocks at its surface, so something much heavier had to be inside — an iron core, he proposed, by analogy with iron meteorites. The precession of the equinoxes said the same thing in a different way, through the moment of inertia: a planet whose mass is concentrated towards its centre is tipped by the pull of the Sun and Moon in a way a uniform one would not be.

Seismology then located it. Richard Oldham noticed in 1906 that compressional waves crossing the deep interior arrive late, as though slowed by something at depth; Beno Gutenberg put the boundary at 2,900 kilometres in 1913, almost exactly where the reference model has it; and Inge Lehmann found in 1936, from faint arrivals inside the shadow the core casts, that a solid inner core sits within the liquid outer one. The waves travel at speeds set by the medium they cross, so their arrival times map where the medium changes, and the density profile drawn here is that map turned into masses.

Why gravity rises through the mantle #

Going down through the mantle does two things to $g = GM(r)/r^2$. It removes the shell of mantle just crossed from the enclosed mass, which weakens the pull. And it brings the remaining mass closer, which strengthens it. Which effect wins depends on how much mass the shell held compared with what lies beneath.

The shell just crossed contains mass $4\pi r^2\rho\,dr$, a fraction $3\rho\,dr/(\bar\rho\, r)$ of the enclosed mass, with $\bar\rho$ the mean density inside. Moving inward by $dr$ increases $1/r^2$ by a fraction $2\,dr/r$. So gravity increases going down whenever $3\rho/\bar\rho < 2$ — whenever the local density is less than two thirds of the mean density beneath it.

That is the statement the density figure draws as its third curve. Through the entire mantle the local density lies below two thirds of the mean enclosed, so gravity rises all the way down; at the core–mantle boundary the local density jumps above it, and through the core gravity falls. A uniform Earth has its local density equal to the mean everywhere, which is above two thirds of it, and so gravity falls — the familiar straight line is the special case of a body that is not stratified at all.

Gauss’s law at a point #

The same condition comes out directly from the local form of Gauss’s law, which counting what comes out obtains by shrinking the closed surface to a point: the divergence of the field at any point is $-4\pi G$ times the density there. For a spherically symmetric field that becomes

$$\frac{dg}{dr} = 4\pi G\rho - \frac{2g}{r} = 4\pi G\left(\rho - \tfrac23\,\bar\rho\right),$$

and the sign of the bracket is the sign of the slope.

The figure computes the slope both ways and compares them, and the agreement is the local law being verified on a real body. It is also worth reading the equation for what it says about locality. The slope of gravity at a given depth responds to the rock at that depth — through $\rho$ — and to everything beneath it through a single number, $\bar\rho$. Nothing above the point enters at all. That is Gauss’s law’s statement that a field is set by its sources and by nothing else, written for a particular depth, and it is the same statement that makes the potential satisfy Poisson’s equation in the rock and Laplace’s in the air above it.

The criterion is not special to gravity. The electric field inside an atom’s electron cloud obeys the same equation with charge density in place of mass density, and there the local charge density — the negative cloud — is always less than two thirds of the mean enclosed charge, which is dominated by the positive nucleus. So the electric field grows steadily as the nucleus is approached through the cloud where the electron probably is, less and less screened, which is why the innermost electrons of a heavy atom are so tightly bound.

Falling through the Earth #

The oscillation in a shaft through a uniform Earth is the result the shape decides the falloff singles out, because a force proportional to displacement is the restoring force of a pendulum at small angles and the motion is simple harmonic. The real Earth’s gravity is not proportional to distance, so its shaft is not a harmonic oscillator.

The real crossing is four minutes faster. The body feels more than surface gravity for its first 2,891 kilometres, so it is travelling faster than in a uniform Earth when it reaches the core, and through the core the pull falls away and it coasts. It passes the centre at nearly ten kilometres a second — faster than orbital speed at the surface, and not far short of escape speed from it, which is a statement that most of the Earth’s gravitational binding energy is concentrated in its interior rather than spread evenly through it.

The uniform Earth has one further property the real one loses. In a uniform ball every straight frictionless chord, not only the one through the centre, takes the same 42.2 minutes, because the component of gravity along any chord is proportional to the distance from the chord’s midpoint. A chord through the real Earth that stays in the mantle and one that dips into the core take different times, and there is no longer a single answer to how long a gravity train between two cities would take.

The pressure at the centre #

The pressure that only knows depth is the rule inside a planet too, with the two changes that the density varies and the gravity does as well. Integrating the weight of the overlying rock down from the surface, with the density of the model and the gravity from Gauss’s law, gives 136 gigapascals at the top of the core and 364 at the centre — the value the reference model’s own tables give.

A uniform Earth with the same mass gets half of that. It puts too little mass near the centre, so the deep interior has too little gravity pulling the overlying layers down; and the layers it does put there are too light. The dense core works twice: it raises the gravity everywhere above it, which makes every layer of the mantle heavier than it would otherwise be, and then adds its own great weight on top of that at the bottom.

Pressures like these can now be reached in a laboratory, for a moment and in a speck of material squeezed between the tips of two diamonds and heated by lasers, which is how the melting temperature of iron at core pressures has been measured. The pressure figure is the target those experiments aim at, and it comes out of the same two lines — Gauss’s law for the gravity, hydrostatic balance for the pressure — as the rest of this page.

Pressure on that scale also closes the loop with what makes a body round. The size at which a body becomes round compares the pressure at a body’s centre with the strength of its rock, and for the Earth the comparison is not close: 364 gigapascals is a thousand times what any rock can hold up as a mountain.

One number that says where the criterion flips #

Every planet has a moment-of-inertia factor, and that single number says a good deal about where its gravity peaks. A uniform ball has 0.4, and inside it gravity falls straight to the centre. The Moon’s is 0.393, barely below uniform, because its iron core is small, and its interior gravity falls almost linearly from the surface. Mars’s is 0.364, and the Earth’s is 0.331 — low enough that gravity climbs through nearly half the radius before it turns.

A star is the extreme case. The Sun’s factor is about 0.07: its mass is packed towards the centre, where the density is more than a hundred times the mean, so gravity inside the Sun rises inward through most of its volume and reaches its maximum deep inside, where the enclosed mass is already most of the star. The mass no cold matter can hold up meets the same concentration in a white dwarf, where it decides whether the star can support itself at all. In each case the two-thirds rule is the local reading of what the moment of inertia summarises in one integral: the more centrally a body’s mass is gathered, the deeper its gravity peaks.

Where the model stops #

The Earth is taken as spherically symmetric. It is flattened by its rotation, by about a third of a per cent, and its surface gravity varies from 9.78 at the equator to 9.83 at the poles, partly from the flattening and partly from the rotation itself, which subtracts a centrifugal term this treatment leaves out. Inside, the flattening changes the field by amounts of the same order, and Gauss’s law no longer gives the field from a single enclosed mass once the symmetry is gone.

The density is an average over each depth. The reference model is one-dimensional, and the real mantle has hot rising regions and cold sinking slabs with density differences of a per cent or so, and two continent-sized regions near the base of the mantle whose nature is still argued. Those change local gravity by small amounts and change the picture of the mantle a great deal.

The density is inferred, not measured. Seismic data constrain wave speeds far better than densities, and the free oscillations that constrain density are compatible with a range of profiles. Different reference models differ by a few tenths of a gram per cubic centimetre in places; they all reproduce the rise of gravity through the mantle, because the mass and the moment of inertia force the core to be dense.

And the shaft is imaginary. No material could line a hole through temperatures of five thousand kelvin and pressures of hundreds of gigapascals, and the Earth’s rotation would press a falling body against the wall. The fall is a way of reading the gravity profile as a motion, not a proposal.

What the pictures cannot show #

The gravity curve is drawn as a smooth function of radius and cannot show how it is known. Nothing has ever measured gravity more than a few kilometres below the surface; every point below that is a consequence of a density that was itself inferred from waves. The curve is a chain of inferences drawn with the confidence of a measurement, and its most striking feature — the rise through the mantle — is trustworthy precisely because the Earth’s mass and its precession would each be violated by any profile without it.

Nor does any figure show what the core is. The density jump at 2,891 kilometres is drawn as a step, and the step is iron meeting rock: a liquid metal ocean whose convection generates the Earth’s magnetic field, sitting under a solid mantle that flows at centimetres a year. The gravity profile knows only that the material below the step is heavy.

Still open: how much a field outside a body can say about the body inside #

Gauss’s law determines the field outside a spherically symmetric body completely from its total mass, and that completeness is also a limit: every spherically symmetric arrangement of the same mass gives exactly the same external field. Gravity measured on the surface cannot, by itself, tell a dense core from a uniform ball. The interior on this page had to be supplied by seismology before Gauss’s law could say anything about it.

Departures from symmetry make the question richer and harder. The Earth’s external field is not exactly that of a sphere, and satellites measure its irregularities with great precision; the irregularities constrain where mass is, but a given pattern outside is produced by infinitely many arrangements inside. Which features of an interior a field can determine, which it cannot, and what combination of fields and waves removes the ambiguity, is the question Gauss’s law leaves once it is turned round and asked about sources it cannot see.

The habit worth carrying away is the one the two-thirds rule makes concrete. Before trusting a result derived for a uniform body, ask which way the real body departs from uniform, and whether the result depends on the local value or on an average. Gravity inside a planet depends on both, the uniform ball hides the difference between them, and in the Earth the difference is large enough to reverse the answer.