Astronomy & Celestial Mechanics
the sky, computed
Concept

Visibility — where it appears

The complex quantity a pair of telescopes measures, equal to one Fourier component of the sky brightness at the frequency their separation sets. Rotating the Earth sweeps each baseline through an arc of the transform plane, which is how an array of a few dishes samples enough of it to make an image.

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

A shadow edge 10.3 metres wide, and a stellar diameter read off how blurred it is. A star disappearing behind the Moon, drawn as intensity against position across the shadow. The horizontal axis is in Fresnel scales of √(λD/2) = 10.3 metres at 550 nm and 3.844e+5 km, which is the only length the problem has; at a limb speed of 0.62 km s⁻¹ one of them takes 16.6 milliseconds to pass, so the whole event is over in a tenth of a second and needs photometry at a kilohertz. The Moon has no atmosphere and its limb is a knife edge, and a knife edge does not cast a shadow with an edge: the intensity at the geometric boundary is 0.250, a quarter rather than a half, and outside it the light overshoots to 1.37 before ringing down. Every one of those numbers is a property of the wave and of nothing else. What the star contributes is the blurring. Each point of the stellar disc casts its own copy of the pattern, displaced by its own position, so the observed trace is the pattern convolved with the star's projected disc — 22.4 metres wide for the 12 milliarcsecond curve, against a 10.3-metre fringe. The contrast falls from 0.28 to 0.04 across the four curves drawn, and inverting that fall is how several hundred stellar diameters were measured with a single telescope, no interferometer, and no resolution at all. The picture cannot show the limitation that ended the technique's dominance: the Moon goes where it goes, so only stars within a few degrees of the ecliptic are ever occulted, and each is occulted at whatever position angle the geometry happens to offer.

The edge of a shadow is a wave

An asteroid's shadow has an edge because the asteroid is large. The Moon's does not — at visible wavelengths and lunar distance the edge of a shadow is ten metres wide, so a lunar occultation is a diffraction pattern sweeping past at half a kilometre a second, and how blurred its fringes are is the star's own diameter.

sky · Occultations The limb is 40% as bright as the centre, and that is a temperature gradient. Left: a stellar disc shaded by the grey-atmosphere law I(μ)/I(1) = (2 + 3μ)/5, in 26 steps, with μ = cos θ read from the centre outwards. Right: that law against μ, with linear laws at the measured solar coefficients from 400 to 1600 nm. The grey law's coefficient is exactly 3/5 and its very limb is exactly 2/5 of the central brightness — both read off the drawn curve rather than quoted — because the Eddington–Barbier relation makes the emergent intensity at angle μ the source function at optical depth τ = μ, and in radiative equilibrium that source function is linear in τ. The limb is not a cooler part of the star. A sight line entering at the edge reaches unit optical depth higher up, where the gas is cooler, so what the darkening measures is the run of temperature with depth; a star with an isothermal atmosphere would show a uniform disc, and one with a steeper gradient a darker limb. The measured coefficients fall from 0.9 at 400 nm to 0.35 at 1600 — the same gradient seen through a less steep Planck function — which is why a radius measured from a transit or a fringe null has to say which colour it was measured in.

The light that is missing from the edge

The Sun's limb is forty per cent as bright as its centre, and the reason is not that the edge is cooler. A sight line entering at the edge stops higher up, so what the darkening measures is the temperature gradient — and it is worth seventeen per cent on Betelgeuse's radius.

starlight · Limb darkening The transform plane an array of 9 actually samples. Every point in this plane is a spatial frequency the array has measured, in thousands of wavelengths. The 9 antennas make 36 pairs, each pair measures one point at any instant, and turning the Earth sweeps each of them along an ellipse — so eight hours of tracking turns 36 measurements into the arcs drawn here. Two properties are structural rather than chosen. The plane is Hermitian: a real sky forces V(−u,−v) = V(u,v), so half the points are free and the coverage is symmetric through the origin. And every ellipse has axis ratio exactly sin δ = 0.707* at this declination, measured off the longest track as 0.707 — an array is squashed in one direction by where the source is in the sky, and at the equator the tracks collapse to lines whatever the array. What the figure cannot show is the hole in the middle: no baseline is shorter than an antenna is wide, so the largest structures on the sky are simply not measured, and no processing recovers them.

Resolution without a mirror

Two telescopes a kilometre apart do not make a kilometre-wide telescope. They measure one number — the Fourier component of the sky at the spatial frequency their separation sets — and an image is what you get by collecting enough of those.

starlight · Interferometry The visibility precision a one-per-cent diameter needs. The precision in fringe visibility needed to measure a uniform disc's angular diameter to 1%, on the best baseline up to 330 m, against the diameter, at three wavelengths. Below the resolution limit the required precision falls as the square of the diameter. Even at its best, a visibility measures a diameter only about as precisely as it is itself measured, so a one-per-cent diameter needs visibilities better than a per cent. The shaded band, one to three per cent, is what a single calibrated visibility typically achieves; averaging ten or more measurements brings the effective precision to a few tenths of a per cent (dotted). At 0.3% the limit is about 0.13 mas in the visible, 0.19 mas in the red and 0.38 mas in the near-infrared H band, where most stellar diameters have been measured.

A diameter smaller than the fringes can see

The surface-brightness method gives distances good to a per cent for binaries of cool giants, and loses that precision for the hot stars that dominate other galaxies' bright binaries. The repair is an empirical calibration from the angular diameters of nearby hot stars — and those diameters are a few tenths of a milliarcsecond, below what the longest optical baselines comfortably resolve. A diameter measured at the edge of resolution is a small departure from perfect fringe contrast, and it costs visibility precision as the square of how small the star is.

starlight · Angular diameter

Named alongside it

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

Angular diameterLimb darkeningAngular resolutionAperture synthesisB starsBaselineCalibrationDeconvolutionDiffraction limitEclipsing binaryEnergy transportFresnel diffraction

All concepts