Astronomy & Celestial Mechanics
the sky, computed
Concept

Inverse variance weighting — where it appears

Combining independent estimates of one quantity with weights proportional to one over each estimate's variance. It gives the combination of least variance, and the combined variance is the reciprocal of the sum of the individual reciprocals.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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 A gauge good to 7 per cent with a tenth of the load left, and to 23 per cent with three hundredths. The uncertainty in the propellant remaining in a spacecraft tank, as a percentage of what remains, against the fraction of the 450-kilogram load still in the tank, on a logarithmic uncertainty axis with the tank emptying to the right. Bookkeeping — summing every thruster firing through a flow-rate model — carries an error common to all burns of 2 per cent of the mass used, plus an independent 5 per cent per burn that averages down over 2000 firings; its absolute error grows with the mass used. Gauging by pressure and temperature infers the empty volume of the tank from the gas law applied to a known mass of pressurant, with a combined 0.66 per cent uncertainty in n R T / P and a 0.2 per cent uncertainty in the tank's volume; its absolute error grows as the gas fills the tank. The two methods are independent and are combined by inverse variance. With a tenth of the load left the combined estimate is uncertain by 2.9 kg, 7 per cent of what remains; with three per cent left, by 3.1 kg, 23 per cent. Near empty the absolute error barely changes, so halving what is left doubles the relative error — the gauge is at its worst exactly when the last manoeuvre has to be planned from it.

A fuel gauge that is worst when it is needed

A spacecraft's tank has no float and no dial. The propellant left is estimated by adding up every burn or by reading the pressure and temperature of the gas above the liquid, and both methods' errors grow with the propellant used. Relative to what remains, the error doubles every time what remains halves — so a geostationary satellite has to hold back months of station-keeping as a margin against a gauge that cannot see the last few kilograms.

spaceflight · Station-keeping

Named alongside it

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

Photon noiseSystematic errorAperture photometryAtmospheric extinctionBackground limitedBookkeepingCommon mode noiseComparison starCramer rao boundΔvDifferential photometryEnd of life

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