Illustrated Physics
physics, drawn
Concept

Inverse problem — where it appears

The task of inferring the causes of a set of measurements from the measurements themselves, when each reading is an average over many causes. Such problems usually have many solutions consistent with noisy data, and a unique answer needs extra assumptions about what is plausible.

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

Three bodies of one mass, one field outside, three fields inside. The gravitational field against distance from the centre, both in units of the body's surface values, for 3 spherical bodies of the same mass and radius: a uniform ball; a dense core under a light mantle; a hollow shell. Outside the surface the three curves are one curve — checked at 1.7 radii by adding up the pull of every mass element in each body, which agrees with the pull of a point of the same mass to better than a part in five hundred. Inside they part: the uniform ball's field falls in a straight line, the layered body's rises to 1.24 times the surface value at the top of its core, and the hollow shell's is zero throughout its cavity. Their moment-of-inertia factors are 0.400 (uniform ball), 0.330 (dense core under a light mantle), 0.551 (hollow shell) — a number that no measurement of the field outside can supply.

The field outside that cannot find the core

Gauss's law gives the field outside a body from what it encloses, and read backwards it is a limit. A uniform ball, a planet with an iron core and a hollow shell of the same mass have identical gravity everywhere outside. The field fixes a list of numbers — the mass, the flattening, higher moments — and leaves free everything else, including the moment of inertia; the Earth's core was weighed by its wobble, not by its pull.

electromagnetism · Gauss's law Each wavelength sees from a different height. The weighting functions of 7 infrared channels in a carbon-dioxide band — how much each height contributes to the radiance a satellite measures in that channel — for absorption strengths chosen so that the peaks sit at 3, 8, 14, 20, 27, 35, 43 km. Each is scaled to its own maximum. A channel's peak is where the carbon dioxide above that height amounts to one optical depth: deeper, its emission is absorbed on the way out; higher, there is too little gas to emit. Each function is about 17 km wide at half its peak, the same for every channel, because the absorber's density falls with one scale height, 7 km. The dotted curve is the temperature profile of the standard atmosphere, on its own scale from 180 to 300 K, whose cold tropopause and warm stratosphere the channels between them straddle.

The spectrum that is a thermometer at every height

A satellite looking down at the Earth in the infrared sees, at each wavelength, the glow of whatever layer of air that wavelength escapes from. In the wings of carbon dioxide's great absorption band the air is transparent and the satellite sees the ground; towards the band's centre the gas grows opaque and each wavelength escapes from higher up. A single spectrum is therefore a stack of thermometers at different heights, and weather forecasts are built on reading it. What cannot be read from it is anything much thinner than the ten to twenty kilometres of air each of those thermometers averages over.

quantum · Blackbody

Named alongside it

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

Blackbody radiationBrightness temperatureGauss's lawGravity fieldLapse rateMoment of inertiaMultipole expansionNon uniquenessOptical depthPlanck lawPrecessionRemote sensing

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