Astronomy & Celestial Mechanics
the sky, computed
Concept

Scintillation — where it appears

Rapid flickering of a star's brightness caused by high-altitude turbulence focusing and defocusing its light. It is a fixed fraction of the flux, so it dominates the noise budget of bright stars on small telescopes and weakens as the aperture grows.

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

What the error bar is made of, on a 1 m in 60 s. The four contributions to a photometric error, against the brightness of the star, for a 1-metre aperture, a 60-second exposure and a sky of 21 magnitudes per square arcsecond. The star's own photons give a line of slope exactly 0.2 — σ ∝ N^−1/2 and N ∝ 10^−0.4m, so a magnitude of extra faintness costs a fifth of a magnitude of precision, and no instrument changes that. The sky and the read noise are fixed counts, so their lines have slope 0.4, twice as steep, and they overtake the star at V = 18.25 — that crossing is the faint limit of the night, and it moves when the Moon rises rather than when the telescope changes. Scintillation is flat, because the atmosphere modulates a bright star and a faint one by the same fraction: at 4.09e-4 relative it is 0.44 millimagnitudes here and it is what caps the bright end, up to about V = 10.8. Below all of them is the systematic floor at 0.3 millimagnitudes, which is flat-fielding and colour terms and does not integrate down at all.

The error bar that comes from counting

A brightness is a number of photons, so the precision of the measurement is fixed before any instrument is chosen. What follows is a slope of exactly 0.2 magnitudes of error per magnitude of star, a slope of 0.4 once the sky wins, and a floor that neither of them explains.

starlight · Photon noise Ten comparison stars as bright as the target cost 5 per cent in precision; ten 2 magnitudes fainter cost 23. The precision of a V = 12 target measured relative to an ensemble of comparison stars on the same 60-second frames, against the number of comparison stars, on a logarithmic precision axis, for comparisons 1 mag brighter, as bright as the target, 1 mag fainter, 2 mag fainter. A change in the atmosphere's transparency of 2.0 per cent — which would put the target's raw brightness out by 20.0 mmag — multiplies every star by the same factor and vanishes from the ratio. What is left is the target's own noise, 0.89 mmag, plus the ensemble's, which falls as the inverse square root of the number of stars in it. With comparisons as bright as the target the result is σ√(1 + 1/N): one comparison costs 41 per cent, ten cost 5. Fainter comparisons are noisier and need many more to reach the same point; brighter ones help, but the target's own noise is a floor the ensemble can only approach. Scintillation is treated as independent from star to star, which is right for stars more than a few arcseconds apart on a large telescope and makes it part of the noise that does not cancel. The figure also cannot show the defining weakness: every comparison star is assumed constant, and a variable among them injects its variability into every measurement made against the ensemble.

The comparison stars are part of the measurement

Measuring a star against others on the same frame cancels everything the atmosphere and the instrument do to all of them at once — a two per cent change in transparency vanishes completely. What does not vanish is the comparison stars' own noise, which the target inherits, and the variability of any comparison that is not constant, which the target reports as its own.

starlight · Photon noise Lucky frames chosen by the twinkle instead of the sharpness. The mean Strehl ratio of a target against its separation from the reference star, drawn as the high layer's shift in units of the aperture (32 arcseconds at the right for a 2.5-metre telescope and a layer at 8 km), for the best 2 per cent of 1600 frames chosen three ways: sharpest on the reference (orange), least scintillation of the reference's light (green), and best on both ranks together (blue); half the turbulence at the ground, both layers varying in strength frame to frame. Chosen by sharpness, the frames give 0.52 on axis and 0.34 at the edge. Chosen by scintillation, 0.22 and 0.22: the same everywhere, because the high layer — the only one whose blur differs between directions — has been selected directly, and lower everywhere, because the ground layer, which blurs every direction alike, has not been selected at all. Both ranks together give 0.44 and 0.34, a little more at the edge than sharpness alone at some cost at the centre. All frames average 0.10.

A twinkle that knows the high air and not the low

Lucky imaging keeps the frames in which a reference star is sharp, and that choice happens to favour moments when the high turbulence was quiet, which is why it reaches further across the field than it should. A star's twinkling reports the high layer directly. Choosing frames by the twinkle alone gives a field that is equally sharp everywhere and sharp nowhere, because the twinkle cannot see the air at the ground. Choosing by both buys a little more field for a little less centre, and only where the high air comes and goes.

sky · Seeing How fast to switch. The rms false polarisation left by a 1 per cent rms brightness fluctuation, against the modulation frequency, both on logarithmic scales, for fluctuations whose spectrum turns down at 5, 20, 100 Hz. Below the turnover the false signal is the whole fluctuation, about 0.7 per cent; above it, it falls steeply as the pairs are taken faster than the air changes. To reach a part in ten thousand (dashed) — the precision at which stellar and planetary polarisation becomes interesting — the switching must run at 3,311 Hz, 13,183 Hz, 66,069 Hz respectively: switching at hertz is useless, and a single beam needs kilohertz or more.

A polarisation measured faster than the air changes

Polarisation is a difference between two brightnesses, and the brightness of a star seen through the air changes by a per cent from one millisecond to the next. Measure the two one after the other and the air's flicker becomes a false polarisation a hundred times larger than the signals worth having. The cure is to switch between them faster than the air can change, or to measure both at once and then swap them — turning a calibration problem into a differential one.

starlight · Polarimetry

Named alongside it

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

Photon noiseDifferential photometrySystematic errorAnisoplanatismAperture photometryAtmospheric extinctionCommon mode noiseComparison starDifferential measurementDual beam polarimetryFaint limitFried parameter

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