Astronomy & Celestial Mechanics
the sky, computed
Concept

Kozai–Lidov — where it appears

A secular exchange in which a distant, inclined companion trades a body's inclination for eccentricity while its semi-major axis stays fixed. It can drive a planet's periastron close enough to its star for tides to circularise it, which is one route to a hot Jupiter.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 3 billion years. It falls at 4.5 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.

A planet where one cannot form

A Jupiter at four days orbits inside a region that was too hot to hold ice and too small to hold the material. It did not form there — and the distribution of eccentricities says which of two journeys brought it in.

exoplanets · Planet migration Circularised at twice the closest approach. Eccentricity against semi-major axis, measured in units of the planet's Roche distance 2.16 Rₚ(M⋆/Mₚ)^⅓, for planets delivered on orbits of eccentricity 0.97 with periastra of 1.2, 1.6, 2.2, 3 Roche distances. Tides raised on the planet dissipate orbital energy at periastron and exert almost no torque over the rest of the orbit, so the orbit's angular momentum is nearly conserved while its energy is removed: a(1 − e²) stays fixed and each track runs down and to the left along it. Each ends on a circle of radius q(1 + e₀), which for e₀ near one is twice the periastron — 1.2 → 2.36, 1.6 → 3.15, 2.2 → 4.33, 3 → 5.91. The shaded region above the curve is where the periastron is inside the Roche distance and the planet is stripped on its first close passage. So the channel cannot deliver a circular orbit inside twice the Roche distance: the inner edge of the hot Jupiters is a prediction, stated in units that depend on each planet's own radius and mass rather than on its period.

A planet circularised at twice the distance that would destroy it

A giant planet thrown onto an orbit of eccentricity 0.97 by a distant companion loses its orbital energy to tides at every close passage and keeps almost all of its angular momentum. The orbit therefore shrinks to a circle at exactly twice its closest approach — and since a closest approach inside the Roche distance destroys the planet, the hot Jupiters delivered this way should begin at twice the Roche distance, measured in units of each planet's own size.

exoplanets · Planet migration An inner orbit's inclination, with and without the octupole. The inclination of an inner orbit to the plane of a distant perturber against time, in units of the quadrupole timescale, starting at 65 degrees with a nearly circular orbit, for octupole strengths ε = 0 and 0.03. At ε = 0 — a circular outer orbit — the inclination oscillates in the regular Kozai–Lidov cycle and never crosses ninety degrees, because the angular momentum along the outer orbit's axis is conserved and cannot change sign. With the outer orbit eccentric, the inclination drifts from cycle to cycle and crosses ninety degrees, the orbit turning from prograde to retrograde, first at t ≈ 120, 2 times in all within the time shown. The integration conserves the secular energy to 7·10⁻⁶.

An orbit that turns over

A distant third body makes an inner orbit trade inclination for eccentricity in slow, regular cycles, and one quantity — the inner orbit's angular momentum along the outer orbit's axis — keeps the trade bounded and forbids the orbit from ever tipping past ninety degrees. That conservation holds only if the outer orbit is a circle. Make it eccentric and the quantity drifts, the cycles stop repeating, and at a moment of extreme eccentricity the inner orbit turns over, from prograde to retrograde, with its pericentre pressed a thousand times closer than the regular cycle allowed.

orbits · Kozai–Lidov How much of a population is driven how close. The fraction of an isotropic population of inner orbits whose Kozai–Lidov cycling brings 1 − e below a given value within 150 secular timescales, against that value, both logarithmic, for octupole strengths ε = 0, 0.01, 0.03; each curve is 128 integrated orbits with random pericentres and nodes. The thin line is the quadrupole closed form, √((3/5)(1 − (1 − q)²)), which the ε = 0 population follows. At 1 − e = 0.01 the fractions are 9.4, 20.3, 40.6 per cent; at 0.001, 3.1, 18.0, 38.3. The octupole matters most where the quadrupole is weakest — at the extreme closeness a planet needs to be captured by tides, or a pair of compact objects needs to merge — because it adds orbits whose quadrupole cycle stopped far short of it.

Eleven in a hundred by geometry, and the rest by waiting

A distant companion drives an inner orbit through cycles of eccentricity, and whether a given system reaches a dangerously close approach depends on its inclination. Counted over a population with random orientations, the answer at quadrupole order is pure geometry — about eleven per cent reach one per cent of their semi-major axis, all at the same inclination. An eccentric outer orbit multiplies that share, spreads the inclinations, and makes the count depend on how long the systems have been waiting.

orbits · Kozai–Lidov

Named alongside it

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

Hot jupiterEccentricityHierarchical tripleOctupoleOrbital flipRoche limitSecular perturbationTidal circularisationAngular momentumBinding energyChaosConserved quantity

All concepts