Astronomy & Celestial Mechanics
the sky, computed
Concept

Fresnel diffraction — where it appears

Diffraction observed at a finite distance from the obstacle, where the curvature of the incoming wavefronts still matters. It sets how sharp a shadow's edge can be and governs the light that leaks around an occulting disc into the shadow behind it.

Named by 2 essays across one field — 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 A shadow 77 mm across at 144.3 m: an occulter 1.42 m wide and a 50 mm aperture inside its umbra. Side view of an eclipse made by two spacecraft, distances along the axis in metres and widths across it in metres, the vertical scale magnified about fifty times. Sunlight arrives from the left. The occulting disc, 1.42 m across, casts an umbra — the cone from which none of the Sun's disc is visible — which narrows because the Sun is not a point: its edges converge at a rate set by the Sun's angular radius, 959.6 arcseconds, and would meet 153 m behind the disc. At 144.3 m, where the coronagraph sits, the umbra is 77 mm across and the aperture 50 mm; the penumbra, from which part of the Sun is visible, extends to 2.76 m. From the aperture the occulter's edge hides the Sun out to 1.058 solar radii, so the corona is visible from just beyond the limb.

An eclipse held in place to a millimetre

A total eclipse of the Sun works because the Moon's disc happens to match the Sun's almost exactly, and it lasts a few minutes. Two spacecraft flying 144 metres apart can make one that lasts six hours: a disc 1.4 metres across, with a coronagraph sitting in its shadow. The geometry that sets the shadow's size also sets the price. At that separation the umbra is 77 millimetres across around a 50-millimetre aperture, so the formation has to hold itself in line with the Sun to a centimetre — and to two millimetres in January, when the Sun is at its largest.

sky · Phases and eclipses

Named alongside it

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

Angular diameterCoronagraphDiffraction limitFormation flyingThe Fresnel scaleFringe contrastHigh speed photometryKnife edgeLimb darkeningLunar occultationOccultationPenumbra

All concepts