Astronomy & Celestial Mechanics
the sky, computed
Series

Refraction — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Refraction against apparent altitude. How far the air lifts an object, in arcminutes, against where the object appears to be. The lift is 34.5′ at the horizon — larger than the Sun's own diameter — and falls below one arcminute above 45°. At the horizon it is also the least reliable number in positional astronomy, because it depends on the temperature profile of the air the sight line crosses.

    The Sun sets before it sets

    At the moment the Sun's lower edge appears to touch the horizon, the whole of it is already below. Refraction lifts it by more than its own diameter, squashes it while it is there, and makes every sunrise and sunset time in every almanac a statement about the air rather than about the sky.

    part 1 · sky
  2. A star drawn out into 3.0 arcseconds of spectrum. Atmospheric refraction relative to its value at 550 nanometres, against wavelength, at four zenith angles. The air's refractive index rises towards the blue, so the blue image of a star sits above the red one and the object is smeared into a short vertical spectrum. At 60 degrees from the zenith the separation across an optical band is 2.96 arcseconds — several times the size of the image at a good site, and comparable to the width of a spectrograph slit. Every curve here is the same curve multiplied by the tangent of the zenith angle, which is why one corrector with an adjustable strength works at every airmass. The practical consequences are three: a slit aligned other than vertically loses blue light or red light depending on where it was centred, a photometric aperture contains a different fraction of the light in each band, and an astrometric position depends on the colour of the star it is measured from.

    The atmosphere is a prism as well as a lens

    Refraction lifts a star towards the zenith, and everybody corrects for that. It lifts blue light further than red, and the difference is a short vertical spectrum a few arcseconds long — larger than the image, larger than a spectrograph's slit, and quietly present in every ground-based measurement not taken straight overhead.

    part 2 · sky
  3. A square field imaged as a squashed one. A square grid of stars 1° across, centred 70° from the zenith, and how refraction moves each one relative to the field's centre after the displacement common to the whole field has been removed. The arrows are magnified 200 times. Refraction lifts every star towards the zenith, but it lifts the lower ones more because refraction grows towards the horizon, so the field is compressed vertically by 2377 parts per million; and every star moves along its own vertical circle, which converge at the zenith, so the field is compressed horizontally by 286 parts per million. The corners move by up to 4.4 arcseconds against the centre — a hundred times the precision of a modern relative astrometric measurement, and a distortion that changes as the field crosses the sky.

    A square field imaged as a squashed one

    Relative astrometry measures positions against neighbouring stars so that the atmosphere's refraction, common to them all, cancels. It does not cancel. Refraction changes across the field, so a square patch of sky seventy degrees from the zenith is imaged squashed by a few parts in a thousand — and stars of different colour are moved by different amounts, by more than the parallaxes being measured.

    part 3 · sky
  4. 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.

    part 4 · sky
  5. 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.

    part 5 · sky

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