Astronomy & Celestial Mechanics
the sky, computed
Concept

Photon noise — where it appears

The irreducible uncertainty from the discreteness of light: N detected photons carry a standard deviation of the square root of N. It sets the best precision achievable on any brightness measurement, and for a faint source the sky contributes more of it than the source.

Named by 5 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 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 faint star is measured against a brighter sky

For anything at the edge of detection the dominant source of noise is not the object. It is the sky in the same aperture, which is brighter than the star and is subtracted rather than measured — and once that is true, every rule of thumb about apertures, exposure times and image quality changes.

starlight · Photon noise What comes back is a ramp, not a threshold. Detection efficiency against signal-to-noise: the fraction of synthetic transits injected into real photometry that the pipeline afterwards finds. The measured curve is a gamma cumulative distribution of shape 4.65 and scale 0.98 beginning at 4.1, which is the form a survey's own injection tests are fitted with; the dashed line is the step at 7.1 that a threshold calculation assumes instead. Half the injections are recovered at 8.33, 1.2 units above the nominal threshold — the ramp is a property of the search and the cut is a separate decision, so the two need not meet anywhere in particular. The rest of the disagreement is the area between the curves. The pipeline does not reach 99 per cent efficiency until 14.9, four units above the threshold, and it recovers 45 per cent one unit above it. Over a population whose signal-to-noise falls as s^-2 — which is what a planet population looks like, because there are far more small planets than large ones — the step function counts 1.21 times as many detections as the ramp does. That factor is not an error bar. It multiplies every occurrence rate computed without it, and it is larger for the small planets than for the large ones, because the small ones live where the ramp is.

The threshold that is not a threshold

A survey's detection limit is quoted as a number — seven point one — and a pipeline does not behave that way. Half the injected signals come back at the threshold, and full efficiency arrives four units above it.

exoplanets · Detection bias A bright star wants a wide aperture and a faint one wants 0.68 of the seeing. Signal-to-noise of simple aperture photometry against the aperture radius, in units of the seeing's full width at half maximum (1″), each divided by what optimal pixel weighting achieves for the same star, for stars of V = 12, 17, 20, 23 observed for 60 s through a 1 m telescope under a sky of 21 mag/arcsec². A small aperture loses starlight; a large one admits sky, and the balance depends on which dominates. For a bright star its own photons are most of the noise, so a wider aperture keeps gaining light almost for free and the best radius is large — 1.63 FWHM at V = 12, reaching 100.0 per cent of the optimum. For a star fainter than its sky the best radius shrinks to 0.680 FWHM and the best aperture reaches only 90.5 per cent of what weighting each pixel by its share of starlight divided by its variance achieves. That residual is exact in the background-limited limit: the best aperture captures 71.5 per cent of the light and 0.902 of the optimal signal-to-noise, so optimal weighting is worth 11 per cent in signal-to-noise, or 23 per cent in exposure time, and no more. The image is taken to be Gaussian; a real point-spread function has broader wings, which makes a fixed aperture a little worse and the optimal weights harder to know.

The best aperture throws away a tenth

Aperture photometry counts every pixel inside a circle equally and every pixel outside it not at all. For a faint star against its sky the best circle is two-thirds of the seeing wide, catches 71.5 per cent of the light, and reaches 90.2 per cent of the signal-to-noise that weighting each pixel by what it is worth achieves — a loss of 23 per cent in exposure time that no algorithm can beat by more.

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

Named alongside it

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

Aperture photometryRead noiseSignal-to-noiseSky backgroundBackground limitedDifferential photometryInverse variance weightingThe point-spread functionScintillationSystematic floorAirglowAtmospheric extinction

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