Astronomy & Celestial Mechanics
the sky, computed
Concept

Mean motion resonance — where it appears

A commensurability between two orbital periods, maintained because a particular combination of angles librates rather than circulating. The lock keeps the geometry of successive close approaches from averaging away, so a small perturbation accumulates instead of cancelling.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

The Kirkwood gaps. Asteroid numbers against semi-major axis, with the resonant radii marked. Each gap sits where the orbital period is a simple fraction of Jupiter's, and each of those radii is computed from the harmonic law rather than placed by eye.

Resonance clears a gap in one place and locks a moon in another

When two orbital periods are in a simple ratio, small tugs stop averaging away and start accumulating. Sometimes that empties a region entirely. Sometimes it holds three moons together for the age of the solar system.

gravitation · Resonance An edge where two torques balance, 21 kilometres from the shepherd. Two torques on the edge of a ring, against distance from a shepherding moon, both axes logarithmic and both scaled by the same combination of surface density, radius and orbital rate so that only their shapes are being compared. The flat line is the viscous torque, which comes from collisions between ring particles and does not care how far away anything is; it is drawn for a kinematic viscosity of 12 square centimetres a second, within the range ring seismology gives. The falling line is the moon's, summed over the first-order resonances that crowd together as the gap narrows, which makes it an inverse cube. A flat curve and an inverse cube cross once, and the crossing is where an edge can sit: closer in the moon wins and pushes the material back, further out viscosity wins and the ring spreads. For Daphnis and the Keeler gap the balance lands 20.8 kilometres out against a measured half-width of 21, which is agreement to well inside the uncertainty on the viscosity — and it is the only handle anybody has on that viscosity, since the quantity being inferred is the collision rate among particles a metre across, a billion kilometres away.

An edge is a balance, not a boundary

A ring of particles spreads, because collisions move angular momentum outward. Something has to push back, and the something is a small moon whose torque falls as the inverse cube of the gap. A flat curve and an inverse cube cross once, and the crossing is the sharp edge — which is why an edge exists at all rather than a gradient.

gravitation · Planetary rings A resonance keeps what convergence brings it and releases what divergence takes away. The resonant angle of a body inside a resonance whose strength is changing, for the two signs of that change. The angle obeys a pendulum, and the strength of the pendulum is set by how close the two orbits are; migration changes it slowly compared with the swing, which is the condition under which the area a trajectory encloses is conserved. Convergent migration strengthens the resonance, so the separatrix grows around a trajectory of fixed area and the swing narrows — from 1.05 radians to 0.77 across the figure, the body ending more deeply locked than it began. Divergent migration weakens it, the separatrix shrinks, and it eventually passes inside the trajectory: the angle stops oscillating and begins to run, 37.5π in the last quarter of the drawing alone, and the lock is gone for good. Nothing here is dissipative and nothing is random. The whole asymmetry is the sign of one derivative, which is why a chain of planets in resonance is direct evidence that they migrated toward one another, and why a chain cannot survive a phase in which they moved apart.

Capture is a direction, not a strength

A resonance holds a body that drifts into it from one side and lets go of one that drifts out the other way, and the asymmetry is not about how strong the resonance is. It is the sign of a derivative — whether the trapped region is growing or shrinking — which is why a chain of planets in resonance is direct evidence that they migrated toward each other.

exoplanets · Resonance 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 Jupiter and Saturn meet every 19.86 years, tracing a three-cornered figure that turns 8.5° each round. The heliocentric longitudes at which Jupiter and Saturn are in conjunction — the same longitude seen from the Sun — for 21 successive conjunctions from 1800 to 2200, computed from Keplerian elements and dotted in three colours for the first, middle and last thirds of the span. The mean interval is 19.857 years, the synodic period the two mean motions give. Each conjunction falls 242.8° further round the orbit of Saturn than the one before, so 3 of them come back within 8.5° of where they started: the conjunctions sit near the corners of a 3-sided figure, and the figure itself rotates by 8.5° every 59.6 years. At that rate it returns to its starting orientation — a figure with 3 identical corners only needs to turn by a third of a turn — after about 838 years. The drawn corners are not exactly repeated because the orbits are ellipses: the planets move faster near perihelion, and the conjunction longitudes cluster where both are slow. That near-return is not a coincidence of dates. It is the statement that 3 synodic periods are close to a whole number of each planet's years, which is a near-commensurability of the two mean motions — and near-commensurabilities are where planets perturb one another most. The elements are a fit valid between 1800 and 2050; outside those years they are carried as fixed ellipses turning at their mean rates, which is right for the pattern and not for any individual date.

A triangle of meetings that turns in eight centuries

Jupiter and Saturn meet every twenty years, and each meeting falls about two-thirds of the way round the sky from the last, so the meetings trace a triangle. The triangle turns a third of a turn in 838 years because five of Jupiter's years almost equal two of Saturn's — and that same near-fit is the largest perturbation in the solar system, the one that made Saturn appear to be slowing down.

sky · Apparent motion

Named alongside it

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

Synodic periodChopping signalN-body integrationResonant angleSuper-periodTransit-timing variationLaplace resonanceLibrationMass eccentricity degeneracyPlanet massRadial velocityAdiabatic invariant

All concepts