Astronomy & Celestial Mechanics
the sky, computed
Concept

Mixture model — where it appears

A statistical model in which a population is a weighted sum of several component distributions, each with its own shape. Fitted to data, it estimates the share of each component, such as the fraction of planets brought in by each migration route.

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

A nearly aligned population, seen in projection and in three dimensions. 60,000 transiting systems whose stars are tilted from their planets' orbits by a small random amount: each of the tilt's two components is Gaussian with a width of 5°, and its direction around the orbit normal is random. Orange: the true obliquity ψ, which is never negative and is almost never near zero — its median is 5.9°, because a tilt with two components is small only when both of them are small at once. Blue: the projected angle λ the line profile measures, which is symmetric about zero and peaked there. The projected angles look better aligned than the true ones, and they are not hiding anything: for small tilts λ is one component of a two-component tilt, so the rms true angle, 7.04°, is √2 times the rms projected angle, 5.01°, with no knowledge of any star's inclination.

A spread the projection does not hide

A single projected spin–orbit angle says little about the true one, because the star's tilt is unknown. A population of nearly aligned systems is different: for small tilts the projected angle is one component of a two-component vector, so the rms true obliquity is exactly √2 times the rms projected angle — no star's inclination needed.

exoplanets · Spin–orbit alignment Half and half, before and after the errors. The distribution of hot Jupiters' semi-major axes in units of each planet's own Roche distance, for a population half delivered on eccentric orbits and half halted in the disc. Shaded: each channel's share when every radius and mass is exact — the eccentric planets begin sharply at twice the Roche distance and pile up between 2 and 5, the disc planets spread over a broader hump peaking near 5.1. Line: the same population measured with 10% errors on radius and mass and with radii that have changed by ±15% since the planets arrived. The edge at 2 is gone: 6% of the eccentric planets now appear inside it, where with exact parameters only the 2% delivered on the least eccentric orbits did. The two channels' shapes still differ, but the step that told them apart has become a slope.

A fraction read in each channel's own units

Hot Jupiters arrive by two routes, one that ends at twice the planet's Roche distance and one that ends where the disc stopped. Fitting the mixture in Roche distances alone, as the edge invites, gives a fraction that drifts by almost half if the planets weigh more than assumed. Each route has to be read in the units in which it does not move — and then the answer is limited not by how well the planets are measured but by how their radii have changed since they arrived.

exoplanets · Planet migration A second peak, seen through three precisions. The distribution of measured core fractions for a population in which 90 per cent of planets sit at their stars' prediction (0.32, with an intrinsic scatter of 0.03) and 10 per cent at 0.65 (scatter 0.08), each planet compared with its own star, observed with measurement errors of 0.04, 0.08, 0.15 in core fraction. At 0.04 the dense group is a separate hump; at 0.08 it is a shoulder; at 0.15 it has become a long upper tail on a single peak, indistinguishable by eye from a population that is simply broad. The current measurements sit between the second and third curves.

A second peak the errors have to let through

Whether the densest small planets are a population of their own or the far edge of the ordinary one is not a question about any single planet. It is a question about the shape of a histogram seen through measurement errors as wide as the gap it is looking for, and the number of planets it needs rises fifteen-fold for every factor of three in lost precision — which puts the answer a few dozen planets beyond the samples that first suggested it.

exoplanets · Planet composition

Named alongside it

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

Hot jupiterMeasurement errorCore mass fractionDisc migrationGiant impactsHigh-eccentricity migrationLikelihood ratioPlanet compositionPlanet migrationPopulation inferenceProjected obliquityRadius inflation

All concepts