Astronomy & Celestial Mechanics
the sky, computed
Concept

Upper limit — where it appears

A statement that a quantity is below some value, quoted when the measurement is consistent with zero. It is a real result rather than a failure, and its value depends on the confidence level and on the prior as much as on the data.

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

The best-fitting eccentricity of a circular orbit. What a fitted eccentricity comes out at when the orbit's true eccentricity is 0 and each component of the eccentricity vector carries an error of 0.03. The distribution is not centred on the truth and cannot be: an eccentricity is the length of the vector (e cos ϖ, e sin ϖ), lengths are not negative, and a quantity bounded below by zero whose components scatter symmetrically has a distribution pushed away from the bound. For a circular orbit the most likely fitted value is exactly one error bar, 0.0300 here, and the mean is 0.0376 — 1.2533 error bars, which is √(π/2) and comes from geometry rather than from any property of the data. The practical consequence is a catalogue of small eccentricities that are all measurements of their own error bars, and the fix is not a better fit but a different question: an upper limit rather than a value.

An eccentricity that cannot be zero

Fit an orbit to noisy data and the eccentricity that comes back is never zero, not even when the orbit is a perfect circle. The reason has nothing to do with the data and everything to do with the fact that a length cannot be negative.

orbits · Orbit determination A 5 per cent continuum error, and an optical depth wrong by 1.1. The mean transmitted flux of the Lyman-alpha forest against redshift, with a 5 per cent uncertainty in the quasar continuum drawn as a band. The continuum is not observed: at these redshifts every part of the spectrum blueward of the emission line is absorbed, so the level has to be extrapolated from the red side across a region where the quasar's own spectrum has structure. A fractional error in that level is a fractional error in the flux, and since the optical depth is minus the logarithm of the flux, the resulting error in the optical depth is the fractional error divided by the flux — which grows without bound as the forest goes black. At z = 2 it is 0.06; at z = 6.2 it is 1.1. That is why measurements of when reionisation ended are quoted as limits rather than values above about redshift six.

A forest with no continuum left

Measuring how much neutral hydrogen sits between here and a distant quasar means measuring the fraction of its light that survives, which means knowing how much light there was. At high redshift nothing survives at the wavelengths that would show it, so the level is extrapolated across the region being measured — and the optical depth is the logarithm of a number divided by a guess.

cosmology · Reionisation

Named alongside it

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

Continuum placementCovarianceEccentricityEccentricity vectorError propagationThe Gunn–Peterson troughJacobianThe Lyman-α forestMarginal distributionMaximum likelihoodMean transmitted fluxNeutral fraction

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