Astronomy & Celestial Mechanics
the sky, computed
Concept

Marginal distribution — where it appears

The probability distribution of one parameter after integrating over all the others. When parameters trade against each other the marginal is much wider than the joint constraint, and its width reports the degeneracy rather than the measurement's precision.

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 One amplitude, and every point on this curve fits it. The set of distances and inclinations that produce the same measured amplitude in one detector. A binary seen face-on radiates most strongly along its spin axis, so it can be twice as far away as an edge-on binary and still arrive with the same strain — the curve is (1 + cos²ι)/2 and the factor between its ends is exactly two. Nothing in a single detector's data distinguishes the two ends. What makes this worse rather than merely awkward is that the prior pulls the other way: an isotropically oriented population has half its members beyond 60 degrees, so most binaries really are closer to edge-on, and a posterior that combines a flat likelihood along this curve with that prior returns a distance biased low with an error bar that understates the range. The whole business of standard-siren cosmology is the business of cutting across this curve.

A distance tangled with an angle

A gravitational wave carries its own distance, with no ladder underneath it and nothing to calibrate. What it also carries, inseparably, is the orientation of the orbit that made it — and one detector cannot tell a nearby binary seen edge-on from one twice as far away seen face-on.

gravitation · Gravitational waves

Named alongside it

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

Antenna patternCovarianceDegeneracyEccentricityEccentricity vectorElectromagnetic counterpartError propagationHubble constantInclinationJacobianLuminosity distanceMaximum likelihood

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