Astronomy & Celestial Mechanics
the sky, computed
Concept

Mutual inclination — where it appears

The angle between two orbital planes in a hierarchical system. It is the quantity the secular exchange of eccentricity for inclination depends on, and for a massive inner orbit it is changed by the exchange itself.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

How many planets a star has is the hardest thing a catalogue measures. The multiplicity distribution a transit catalogue would contain, for systems that all truly hold 5 planets, at four mutual inclination dispersions. 40,000 systems are drawn per dispersion with an isotropic viewing direction and Rayleigh-distributed inclinations about a common plane, at semi-major axes of 12, 16, 21, 27, 34 stellar radii; the bars are conditioned on at least one planet transiting, which is what makes a system appear in a catalogue at all. At 0.5° of dispersion 33 per cent of the detected systems show all 5 planets and the mean apparent multiplicity is 3.13; at 10° it is 1.39, with 68 per cent of them showing exactly one. Every one of those systems has 5 planets. The entire difference between a catalogue of singles and a catalogue of compact multiples is one number that nothing in the light curve measures. And the two effects run in opposite directions: the fraction of stars showing any planet RISES with the dispersion — 8%, 9%, 12%, 19% across the four — because scattering the orbits gives more of them a chance to cross the line of sight, while the number seen per detected star falls by a factor of 2.2. A survey that scatters its systems finds more stars with planets and fewer planets per star, and neither number on its own says which has happened. What no figure here can show is the true dispersion, because the observable is the ratio of those two and a system with fewer planets and a tighter plane reproduces it exactly.

How many planets a star has is not a measurement

Draw five thousand identical five-planet systems, scatter their orbital planes by half a degree, and a third of the detections show all five. Scatter them by ten degrees and two thirds show exactly one. Every system has five.

exoplanets · Detection bias A ceiling the outer orbit imposes. The greatest eccentricity a cycle can drive an inner orbit to, against the mutual inclination it started at, for four ratios of the inner orbit's angular momentum to the outer one's. The bottom curve is the test-particle case and it is the familiar closed form, √(1 − (5/3)cos²i₀), reaching one at ninety degrees — recovered here by a solver that knows nothing about it, which is the check. Every other curve is lower. The reason is conservation: the cycle works by moving angular momentum out of the inner orbit, and it has nowhere to put it but the outer one. A test particle can give up all of its angular momentum because the outer orbit is infinite by comparison; an inner binary carrying 1 times as much as its companion cannot, because tilting the outer orbit by the required amount changes the mutual inclination and shuts the cycle down. At ninety degrees the ceiling falls from 1.000 to 0.866. The pericentre a mechanism can deliver is bounded by what the perturber can absorb, and the test-particle formula is an upper bound rather than a prediction.

A ceiling the outer orbit imposes

The closed form for how far a cycle can drive an eccentricity assumes the inner orbit is a test particle, so the angular momentum it gives up goes nowhere. A real inner binary of comparable mass has to be paid for: the outer orbit absorbs what the inner one loses, tilts, and changes the very inclination the cycle depends on.

orbits · Kozai–Lidov The ground an eccentric Jupiter sweeps on its way in. The periastron and apoastron of a Jupiter delivered on an eccentric orbit, against its semi-major axis as tides shrink it at fixed angular momentum from 5 au to 0.060 au, both on logarithmic scales. The periastron stays at 0.03 au throughout; the apoastron falls from 10.0 au toward the final circular orbit, a 5.3-day hot Jupiter. Every radius between the two curves is crossed twice an orbit for the whole of the delivery. The shaded strip below the dashed line is where a small companion would have to orbit to be found transiting with the hot Jupiter — periods under 50 days, inside 0.27 au — and the migrator's orbit covers it until the semi-major axis has fallen to 0.15 au, after which the periastron alone keeps passing through its inner edge. A planet there is scattered out, thrown into the star, or swallowed; none survives a delivery that takes thousands of orbits.

A neighbour an eccentric arrival cannot leave

A hot Jupiter that arrived on an orbit of eccentricity near one swept every distance between its closest and farthest point, thousands of times, while tides shrank its orbit — and anything small in that range was scattered, swallowed or thrown into the star. A hot Jupiter carried in by the disc can keep its neighbours. So a hot Jupiter with a close companion came through the disc, and the share of them that have one puts a floor under how many did — a floor whose height is set less by the count than by three chances nobody has measured.

exoplanets · Planet migration

Named alongside it

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

Occurrence rateTransit probabilityAngular momentumConserved quantityCoplanarityCritical inclinationDetection biasDetection limitDisc migrationEccentricityHierarchical tripleHigh-eccentricity migration

All concepts