Astronomy & Celestial Mechanics
the sky, computed
Generator

The shell-field generator

The field of a uniform sphere, inside and out
The field of a uniform sphere, inside and out. Gravitational field strength against distance from the centre of a uniform sphere, in units of the surface value. Inside, only the enclosed mass counts and the field rises linearly; outside, it falls as the inverse square of the distance from the centre.

Gravitational field strength against distance from the centre of a uniform sphere, in units of the surface value. Inside, only the enclosed mass counts and the field rises linearly; outside, it falls as the inverse square of the distance from the centre.

2 essays call shell-field. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

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. Gravitation

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.

 Forty-two minutes, from anywhere to anywhere. Left, the gravitational field inside the Earth: a straight line for a uniform sphere, because the enclosed mass grows as r³ and the field as r, and the PREM curve for the real one, which is nearly flat through the whole mantle at about 9.94 m/s² because the dense core is already all below. Right, what falls through it. A body dropped down a diametric tunnel through a uniform Earth executes simple harmonic motion with ω = √(g/R), reaching the far side in 42.2 minutes; a body dropped down a chord at 0.6 of the radius feels only the component along the tunnel, which is the same ω times a smaller distance, so it arrives in the same time from a shorter trip. The period does not contain the length of the tunnel, its direction, or where the body starts. And 2π/ω is also the period of a circular orbit grazing the surface, 84.3 minutes — the tunnel and the orbit are one ellipse, seen twice. The real Earth is not uniform and gets there in 38.2 minutes instead, which is the honest number and is 9% quicker. Gravitation

The tunnel that takes the same time from anywhere

Inside a uniform sphere the field grows in proportion to the distance from the centre, which is Hooke's law. A body dropped down any straight tunnel arrives in the same time — and that time is the period of an orbit skimming the surface.

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