Astronomy & Celestial Mechanics
the sky, computed
Concept

Chopping signal — where it appears

The short-period part of a transit-timing variation, produced by the tug each time two planets pass one another at conjunction. It repeats at the pair's synodic period and depends on the perturber's mass but hardly on its eccentricity, so it breaks degeneracies the slower signal leaves.

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

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 An ephemeris fitted to 120 days, 43 minutes wrong within a year against a band of ±5.1. Transit times of a 6 Earth-mass planet on a 10-day orbit, perturbed by a 14 Earth-mass planet at 15.24 days, integrated for 1460 days and compared with a straight-line ephemeris fitted only to the transits in the first 120 days — the shaded window. Inside the window the line fits to 2.1 minutes. Outside it the pair's 317-day super-period carries the transits away from the line, and within a year of the window closing the prediction is 42.8 minutes early of the observed transit, 347 days after the last fitted one. The narrow band is the formal three-sigma uncertainty of the same line for a timing precision of 0.5 minutes per transit, which at that date is ±5.1 minutes: the error is 8.4 times the band. A statistical uncertainty assumes the residuals are noise, and these are a signal, so the band describes a planet that does not exist.

A forecast that fails on a schedule

A transiting planet perturbed near a resonance keeps a clock that wanders, and a straight-line ephemeris fitted to part of the wander predicts the next transit with a confidence the wander does not deserve. The error is not noise and does not average down; it grows on the pair's super-period, it is many times the formal uncertainty within a year, and how soon it appears depends on which stretch of the wander happened to be observed. When a model that includes the known perturber still fails, the failure has a period, and the period is a planet.

exoplanets · Transit-timing 5 perturbers that draw one 58-day timing signal. Every perturbing planet that gives a 3-day transiting planet the same timing signal — a sinusoid with a 57.8-day super-period and an amplitude of 1.27 minutes — placed wide of the nearest first-order commensurabilities inside and outside its orbit, with its mass found by integrating until the amplitude matched. outside, near 4:3 at 4.070 days needs 7.6 Earth masses and would move the star by K = 3.1 m/s; outside, near 3:2 at 4.620 days needs 12.0 Earth masses and would move the star by K = 4.8 m/s; outside, near 2:1 at 6.329 days needs 25.6 Earth masses and would move the star by K = 9.2 m/s; inside, near 3:2 at 1.966 days needs 6.9 Earth masses and would move the star by K = 3.7 m/s; inside, near 2:1 at 1.462 days needs 45.8 Earth masses and would move the star by K = 26.7 m/s. The period ratio is along the bottom on a logarithmic axis and the required mass up the side. A super-period fixes the distance from some resonance and not which resonance it is, and the amplitude then fixes a mass for each guess — so the timing alone returns a list rather than a planet. The velocity semi-amplitudes differ by a factor of 8.5 across the list, which is one of the two ways the list is shortened.

One timing curve and five planets that could draw it

A transiting planet whose times wander at a 58-day period, by just over a minute, is being pulled by something — but the period says only how far from some resonance the pull comes, not from which. Perturbers inside and outside the orbit, near four different commensurabilities, each with its own mass, reproduce the same curve to a fraction of a per cent. Timing alone returns a list, and even the detail that shortens it hides a coincidence of its own.

exoplanets · Transit-timing

Named alongside it

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

Mean motion resonanceN-body integrationSuper-periodSynodic periodTransit-timing variationMass eccentricity degeneracyPlanet massRadial velocityDegeneracyEphemerisFree eccentricityOsculating elements

All concepts