Astronomy & Celestial Mechanics
the sky, computed
Concept

Temperature inversion — where it appears

A layer in which temperature rises with height instead of falling. In the air one bends light downward more strongly than normal and can form a duct that carries light round the Earth's curvature; in a stellar atmosphere one turns absorption lines into emission.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The sign of one gradient decides which way the disc is shaded. The emergent intensity across a stellar disc, against the fractional radius, for four gradients of the source function with depth. Every curve is the Eddington–Barbier relation evaluated at the appropriate μ and checked against a numerical integration of the transfer equation. The contrast between centre and limb is the gradient, and its sign is the gradient's sign. A source function rising inward — the ordinary case, since temperature rises inward and the source function follows it — darkens the limb: a sight line at the edge leaves from higher up, where the gas is cooler. A flat source function produces a uniformly bright disc whatever the geometry. And a source function that falls inward brightens the limb, because now the shallower ray is reading a hotter layer. That last case is not hypothetical: above a star's photosphere the temperature stops falling and begins to rise, so at wavelengths that see those layers — millimetre continuum, the cores of strong lines, the ultraviolet — the Sun's limb is brighter than its centre. Limb darkening is therefore not a property of stars but of a gradient, and the same atmosphere shows both signs at different wavelengths.

A limb brighter than the middle

Limb darkening is not a property of stars. It is a property of a gradient, and the same relation that produces it produces the opposite whenever the source function falls inward — which it does above every stellar photosphere. The Sun's limb is darker in the visible and brighter at a millimetre, and both are the same equation.

starlight · Limb darkening Rays that follow the curve of the Earth. Rays of light leaving an observer five metres above a cold surface at elevations between −0.2° and +0.45°, traced through an atmosphere whose lowest 150 metres warm upwards by 0.15 K per metre, drawn as height above the ground against distance along it — so the Earth's curvature has been removed and a ray that follows the curve runs flat. Rays that start downwards or only slightly upwards are bent back by the inversion before they can leave it, glance off the surface, and are guided along the layer in shallow hops: 5 of the 8 traced remain inside it for the whole 300 km. Only the steeper rays escape upwards. Traced backwards, the trapped rays are lines of sight: an observer inside the duct sees objects hundreds of kilometres away that lie far below the geometric horizon, lifted into view.

An atmosphere that bends light round the Earth

A horizontal ray of light curves towards denser air, and in an ordinary atmosphere it curves at a fifth of the rate the ground falls away. Put eleven degrees of warmer air above every hundred metres of cold air over sea ice and the ray curves as fast as the Earth does. It follows the curve, and a Sun five degrees below the horizon rises two weeks early.

sky · Refraction 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 Four atmospheres, one refraction, until the last few degrees. Refraction against zenith angle, traced through the four atmospheres of the first figure. The curves lie on top of each other from the zenith to beyond 80 degrees, and separate only near the horizon, where they reach 37.3 arcminutes (isothermal), 34.0 arcminutes (standard lapse), 50.8 arcminutes (inversion), 28.7 arcminutes (hot ground). Everything above the first few hundred metres of air is different between them, and almost none of it shows.

A bend fixed by the air at the telescope

Starlight crosses fifty kilometres of air whose temperature nobody at the telescope knows, and yet the refraction it suffers can be computed to a hundredth of an arcsecond from a thermometer and a barometer on the dome. The structure of the air overhead cancels out of the first term exactly, and out of the second because of hydrostatics. It returns only within a few degrees of the horizon, and there it takes over completely.

sky · Refraction

Named alongside it

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

Atmospheric refractionNovaya zemlya effectRefractive indexSuperior mirageAtmospheric ductingBrightness temperatureChromosphereDegeneracyDuctingEddington barbierEmission measureFata morgana

All concepts