Astronomy & Celestial Mechanics
the sky, computed
Concept

Free eccentricity — where it appears

The part of a planet's eccentricity that is not imposed by a resonance or a secular forcing — whatever the system was left with by its formation. In transit-timing analysis it is the nuisance parameter that a measured amplitude cannot be separated from.

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

The angle that has nowhere to be measured from. An eccentricity vector carried round a circle of radius 0.034 centred at 0.031 — which is what a secular perturbation does to one, a forced eccentricity with a free one turning about it. Below, the two components e cos ϖ and e sin ϖ, which are smooth, bounded and perfectly ordinary throughout. Above, the longitude of pericentre read off them, which is not: as the eccentricity passes its minimum of 0.0030 the pericentre sweeps through most of a circle, at up to 4080° per unit time against the 12° the free vector itself turns in the same interval. Nothing has happened to the orbit. The pericentre is a place on the orbit, and a nearly circular orbit does not have one — so ω, and Ω with it at zero inclination, are angles measured from a feature that is not there. The equinoctial elements are the pair drawn below, and a propagator written in them steps through this instant without noticing it.

The elements that stop existing

An orbit needs six numbers, and three of the usual six are angles measured from features a perfectly ordinary orbit may not have. At zero eccentricity there is no pericentre to measure from, and the arithmetic knows it.

orbits · Orbital elements Three perturbing masses drawing one curve. Transit-timing residuals for three systems whose perturbing planets are 12, 8, 5 Earth masses — a factor of 2.4 apart — each given the inner-planet eccentricity that the near-resonant theory says will compensate: 0.0000, 0.0166, 0.0363. The three curves have amplitudes of 1.7, 1.7, 1.7 minutes, within 0 per cent of each other, and they are drawn by three separate integrations that were told nothing about the theory used to pick the eccentricities. The eccentricity enters the near-resonant term divided by Δ, the fractional distance from exact resonance — here 0.0267 — so a hundredth of an eccentricity does the work of a factor of two in mass. This is why a transit-timing mass is not a mass until something else fixes the eccentricity, and why the masses that came out of the first years of such fits were systematically lower than the radial-velocity masses of the same planets.

A mass that is only a mass once the eccentricity is known

The near-resonant part of a transit-timing signal carries the perturber's mass and the pair's free eccentricity in the same bracket, divided by the distance from resonance. A hundredth of an eccentricity therefore does the work of a factor of two in mass, and three quite different systems draw one curve.

exoplanets · Transit-timing

Named alongside it

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

Chopping signalConditioningCoordinate singularityElement setEquinoctial elementsForced eccentricityLongitude of pericentreMass eccentricity degeneracyMean motion resonanceN-body integrationNumerical propagationOsculating elements

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