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 2 essays across one field — 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

Named alongside it

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

OblatenessCritical inclinationGauss's lawGeoidGravitational fieldInverse square lawJ2Legendre polynomialsNodal regressionOrbital elementsPerturbationsSun-synchronous orbit

All concepts