Astronomy & Celestial Mechanics
the sky, computed
Concept

Inverse problem — where it appears

The task of working back from observed effects to the causes that produced them, such as recovering a distribution from its blurred or projected image. It is often unstable, so small errors in the data become large errors in the answer unless smoothed.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

One setting Sun drawn through a layer of warm air. The apparent shape of the Sun at five moments of its setting, seen from 20 m above the sea through an atmosphere with an inversion — 0.15 K per metre over 40 m starting 30 m up — computed by tracing rays from the observer out through the layer and mapping each apparent elevation back to the point of the solar disc it came from. The true elevation of the Sun's centre is given under each panel; the dotted outline is the Sun as a standard atmosphere would show it at the same moment, where it is above the sea horizon at all. The layer does not merely flatten the disc: it cuts it into two separate strips at some moments, stacks an inverted image above the upright one, and draws a thin neck between them, because each strip of apparent elevation maps to a strip of the true disc through a curve that folds. The shapes have been recorded for centuries — the Etruscan vase, the omega, the flattened stack — and each is a picture of the transfer curve for the air that night.

A sunset's shape is a sounding of the air

A layer of warm air over a cold sea bends the rays of the setting Sun into shapes — a flattened stack, an inverted copy balanced on the upright one, a vase with a thin neck — and every shape is a drawing of one curve: the map from where light appears to come from to where it really came from. Run backwards, the curve is a temperature profile. It can be read, but only for the lowest air, only from a known height, and only up to a trade between how strong the warm layer was and how thick.

sky · Refraction The kick speeds read back out of the periods: median 100 m/s against the 100 m/s put in. The distribution of kick speeds recovered from the fragments' periods alone, by p(v) = −2v g′(v) applied to the along-track kicks, for 300 and 3000 fragments (points), against the log-normal distribution actually used, median 100 m/s and width 0.3 dex (line), on a logarithmic speed axis with the density per logarithmic interval. With 3000 fragments the recovery follows the shape of the truth and puts the median at 100 m/s, though differentiating a measured distribution twice smooths its peak and roughens its tails; with 300 it is noisy but already in the right place. The inversion uses one assumption, that the kicks were isotropic, and one measurement per fragment, its period — the quantity tracking networks determine first and best.

A kick speed read off the slope of a histogram

Throw fragments at one speed in random directions and the component of each kick along the orbit is spread evenly between plus and minus that speed — Archimedes' theorem about the area of a sphere, applied to debris. Each fragment's period records that component, so the histogram of periods is a stack of flat bands, one per speed, and its slope gives the speeds back: a break-up weighed by its own violence from a catalogue of orbits, if the event was isotropic, and fragment by fragment if it was not.

orbits · Vis-viva A feature read back from the limb. A bump in the source function at optical depth 0.1, a quarter of a decade wide (black dashed), and what an inversion of the limb profile recovers on average when the profile is measured at 24 angles with errors of 1, 0.1, 0.01, 0.001 per cent, the inversion assuming only that the departure from a grey atmosphere is smooth. At one per cent the recovered bump has 56 per cent of its height and is spread over about twice its width; at a thousandth of a per cent, 82 per cent, and still wider than the truth. A thousandfold gain in precision buys a modest gain in resolution — each factor of ten a step, not a factor.

A profile that averages a decade of depth

Each point on a star's disc reports the temperature at a different depth, so a limb-darkening profile looks like a depth sounding of the atmosphere waiting to be read. It is one, but a blurred one. Every angle averages a whole decade of optical depth, the averages overlap, and recovering the temperature structure from them is the inversion of a Laplace transform — a problem whose resolution grows only as the logarithm of the data's precision.

starlight · Limb darkening

Named alongside it

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

Atmospheric refractionAveraging kernelDegeneracyDuctingEccentricityEddington barbierGabbard diagramInclinationLaplace transformLimb darkeningMirageNovaya zemlya effect

All concepts