Astronomy & Celestial Mechanics
the sky, computed
Concept

High speed photometry — where it appears

The recording of a single object's brightness many times a second, which resolves events too brief for any image. Its resolution is set by the detector and the geometry rather than by aperture, so a small telescope in the right place beats a large one elsewhere.

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 Below 1.3 km a shadow stops getting smaller. The relative rate at which a star is occulted, against the smallest body a survey can detect, for size distributions with slopes 3.5, 4, 4.5. The cross-section of a body is not its own diameter: diffraction gives every shadow a minimum width of about the Fresnel scale, √(λD/2), which at 40 AU and 550 nm is 1.28 km. Above that the cross-section grows with the body, so lowering the limit gains events as the limit to the power -1.5 for the shallowest distribution drawn; below it the cross-section stops shrinking and only the number of bodies keeps rising, which is a shallower gain by one power. Pushing the limit from 40 km to 0.2 multiplies the rate by 3106196 at the steepest slope and 14131 at the shallowest — so the rate a survey measures is a measurement of the size distribution, which is the quantity a collisional history predicts and nothing else can reach at these sizes.

A population counted by shadows that never repeat

A body a kilometre across at forty astronomical units is a hundred million times too faint to image and casts a shadow just as dark as a large one. Monitoring enough stars fast enough catches those shadows — each one a single unrepeatable event of a fraction of a second, and the measurement is not any event but the rate.

sky · Occultations

Named alongside it

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

The Fresnel scaleAngular diameterCollisional cascadeDetection thresholdDiffractionDiffraction limitEvent rateFalse-positiveFresnel diffractionFringe contrastKnife edgeKuiper belt

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