Astronomy & Celestial Mechanics
the sky, computed
Concept

Shell theorem — where it appears

The result that a spherical shell attracts an external body as though its mass were at its centre and exerts no force at all inside itself. The interior result is what makes a tunnel through a uniform sphere a harmonic oscillator, with a period that does not depend on where the tunnel goes.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Why a shell pulls a point inside it not at all. A double cone from a point inside a uniform shell. In the narrow-cone limit the far patch is 2.80 times further away and 2.80 times wider — the same number, checked to a part in a thousand before this figure is drawn. Its mass is greater by the square of that ratio and its pull weaker by the same square, so the two cancel in every direction.

A sphere pulls exactly like a point, and the proof is a pair of cones

Every orbit ever computed treats the Sun as a dot. That is not an approximation — for a spherical body it is exact, and the reason is a cancellation between two patches of a shell.

gravitation · Shell theorem A thousandth of the field, and all of the precession. Left: the Earth's figure against a sphere of the same equatorial radius, with the flattening drawn 28× its true value. The real difference between the equatorial and polar radii is 21.4 km on 6378 — 1 part in 298 — which at this size would be 0.5 pixels and invisible, so the drawing is a schematic and the number is here instead. Right: the two components of the J₂ perturbation at the surface, each as a fraction of the monopole μ/r², both differentiated from the potential rather than quoted. The radial one strengthens the inward pull by 1.62×10⁻³ over the equator, where the extra mass is, and weakens it by 3.25×10⁻³ over the poles, vanishing at ±35.26° where P₂ does. The transverse one is zero at the equator and at both poles and peaks at ±45°, at 1.62×10⁻³ — and that is the component that does the work. It pulls an inclined orbit back towards the equatorial plane, which is a torque about the line of nodes, and a torque applied to something already turning moves it sideways rather than back. Averaged over an orbit the pair leave a, e and i untouched and turn the whole plane instead, which is why a term a thousandth of the field is the largest single perturbation on almost every satellite ever flown.

The Earth's shape, read off a satellite's node

The Earth is a thousandth of a part from being a sphere, and that thousandth turns every satellite's orbital plane. Vanguard 1 measured it in 1959 — and one retrograde inclination turns the plane at exactly the rate the Sun moves, which is a perturbation used as a design constraint rather than corrected for.

gravitation · Oblateness The ratio that is √2 only for a point. Escape speed divided by circular speed, against distance in units of the Sun's, for a point mass and for flat-rotation-curve haloes truncated at four different radii. The point-mass value is √2 at every distance and it is a floor rather than a rule. Mass outside an orbit contributes nothing to the circular speed — a spherical shell exerts no force inside itself — and everything to the depth of the well, so any extended distribution has a ratio above √2, by an amount that measures how much lies beyond. For a flat rotation curve the escape speed is not even defined without an outer edge: the potential grows as the logarithm of radius and diverges, so the ratio is √(2 ln(R/r)) with R the truncation radius, and the answer depends on a boundary nobody can see. The measured value near the Sun is about 2.27, which on this curve puts the edge at 13 times the Sun's distance — and the strength of that inference is exactly the weakness of the method, because a ratio of 2.50 would put it at 23.

The ratio that is not √2

Escape speed is exactly √2 times circular speed at every distance from a point mass, and the Sun's own escape speed from the Galaxy is 2.3 times its orbital speed. The excess is not an error in either number. It is a measurement of the mass that lies outside the Sun's orbit, which pulls on nothing and holds everything.

spaceflight · Escape The pull of a ring, a shell and a point, in the plane. Radial force in the plane of a thin ring of radius 1, against distance from its centre, in units where the same mass at a point would pull with one over the distance squared; positive is outward. Inside a spherical shell of the same radius and mass the force is exactly zero. Inside the ring it is not: it points outward, toward the nearer part of the ring, at 0.34 at half the radius and 2.75 at nine tenths, growing without limit toward the rim. Outside, the ring pulls inward harder than a point would, by 25 per cent at twice its radius and 9 per cent at three times, because the near side of the ring is closer than the centre. The shell is indistinguishable from the point everywhere outside it.

Inside a ring the pull points outward

A hollow sphere exerts no force on anything inside it, and the proof is a pair of cones whose areas grow exactly as fast as gravity weakens. Cut the sphere down to a ring and the arcs grow only as the distance, the nearer side wins, and a planet set in the middle of a rigid ring falls into it within a handful of orbits — which is why Saturn's rings cannot be solid, and why a flat galaxy spins faster than its own enclosed mass allows.

gravitation · Shell theorem A flattened shell that pulls nothing inside it. The strength of the gravitational field of a thin homoeoid — the shell between two similar concentric ellipsoids, here with axis ratio 0.5 — along its long axis (solid) and its short axis (dashed), in units of GM/a² against distance from the centre in units of the long semi-axis a, from 20,000 equal masses placed on it. The dotted curve is a point mass. Inside the shell the field is zero along both axes to within 0.0001 — the numerical residue of a finite sampling — however flattened the shell is: this is Newton's theorem for a homoeoid, the flattened generalisation of the spherical shell's empty interior. Outside, the shell does not pull like a point: at 1.2 a along the long axis the field is 1.003 against the point's 0.694, and at 0.60 a along the short axis, just outside the shell there, 0.901. The exterior field remembers the shape; the interior does not know there is one.

The shell can be flattened if it thins at the poles

A spherical shell exerts no force anywhere inside it, and a ring exerts a force that points outward. Between them lies a shape Newton found that keeps the sphere's empty interior however far it is flattened: the homoeoid, a shell bounded by two similar ellipsoids, thick at the equator and thin at the poles in exactly the right proportion. It is why the gravity inside any flattened body can be computed at all — and why the planes of stellar streams measure the shapes of dark haloes.

gravitation · Shell theorem The flattening a halo has, and the flattening an orbit feels. Three model haloes with flat rotation curves, each built from thin similar-shell layers whose axis ratio changes with radius (dashed), and the axis ratio a nearly circular orbit at each radius responds to (solid) — the a²-weighted mean over the shells inside the orbit, because a shell exerts no force on anything within it. A halo flattened to 0.85 everywhere is felt as 0.85 everywhere. A halo round inside and flattened to 0.8 beyond about 30 kpc is felt as round by every orbit within 20 kpc — 0.988 at 20 — and at 60 kpc as 0.83, where its own shells are 0.80. A halo flattened inside and round outside is felt as flattened well beyond where its shells have become round. An orbit reports a running average of everything inside it, weighted towards its own radius, and nothing at all about what lies outside.

A shell outside the stream turns nothing

A flattened halo turns the plane of every orbit inside it, and a stream of stars several orbits long records how far. But the turning is set only by the halo inside the orbit — a flattened shell enclosing it exerts no force on it at all — and within that, mostly by the shells just inside. So a stream measures a running average of the halo's shape out to its own radius, a halo round inside and flattened outside looks round to every stream in its core, and reading the shape of each layer back out of the streams costs more precision the further out the layer lies.

gravitation · Shell theorem

Named alongside it

The objects these essays reach for when they reach for this one.

Dark matter haloOblatenessGauss's lawGravitational potentialHomoeoidInverse square lawNodal precessionRotation curveStellar streamCritical inclinationEllipsoidEnclosed mass

All concepts