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Eccentricity (orange) and inclination / 90 deg (blue) versus secular time; green line = predicted e_max
Eccentricity-vector plane (e cos w, e sin w): live path (purple) over the predicted closed curve (grey dots)
Pericentre distance q = a(1 - e) in AU (log); orange = 0.05 AU hot-Jupiter zone, red = stellar surface
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The science
The set-up. A planet orbits one star (1 solar mass) of a wide binary; the second star, hundreds of AU away, tugs on the planet's orbit. Averaged over both the planet's period and the companion's period, that tug cannot change the planet's semi-major axis a, but it exchanges angular momentum between the orbit's shape and its tilt. This is the Kozai-Lidov effect (Lidov 1962, Kozai 1962).
Equations used (test particle, quadrupole order, circular outer orbit). With the dimensionless angular-momentum vector j (|j| = sqrt(1 - e^2), along the orbit normal), the eccentricity vector e (towards pericentre) and n = unit normal of the companion's orbit, the doubly averaged motion in units of the Kozai time t_K is
dj/dtau = 3/4 [ (j.n)(j x n) - 5 (e.n)(e x n) ]
de/dtau = 3/4 [ (j.n)(e x n) + 2 (j x e) - 5 (e.n)(j x n) ]
t_K = (1/n_in) (m_tot / m_out) (a_out / a)^3 (1 - e_out^2)^(3/2), tau = t / t_K, n_in = 2 pi / P
The page integrates these six equations with a fourth-order Runge-Kutta scheme whose step shrinks as the angular momentum |j| falls (the rates grow like 1/|j| at high eccentricity).
Two conserved quantities. The z component of j is constant because the averaged potential is axisymmetric about n, and so is the energy:
j_z = sqrt(1 - e^2) cos i = const
Q = (2 + 3e^2)(3 cos^2 i - 1) + 15 e^2 sin^2 i cos 2w = const
As e rises, cos i must rise with it, so the orbit swings towards the binary plane while it stretches. Starting from e0 close to 0 these give the maximum eccentricity in closed form:
e_max = sqrt(1 - (5/3) cos^2 i0), only if |cos i0| < sqrt(3/5), i.e. 39.23 deg < i0 < 140.77 deg
At 87 degrees e_max = 0.9977, so a 5 AU planet is pulled to a pericentre q = a (1 - e) of only 0.011 AU, about 2.5 solar radii. If tides then dissipate energy at pericentre while the angular momentum stays fixed, the orbit freezes at a_f = a (1 - e_max^2), here 0.05 AU: a 4-day orbit, which is how Kozai cycling is thought to make some hot Jupiters (Wu & Murray 2003; Fabrycky & Tremaine 2007). The readout shows this tidal-capture endpoint.
General relativity. The orbit's pericentre also advances by the relativistic rate
d(omega)/dt = 3 (GM)^(3/2) / ( c^2 a^(5/2) (1 - e^2) )
which is small at 5 AU but grows as 1/(1 - e^2), so at extreme eccentricity it out-runs the companion's forcing and caps e_max. The slider multiplies the real value (x1 is the physical number) so you can see where the cap starts; planet-planet precession and stellar oblateness act the same way.
Checks on this page. (1) j_z and Q drift over 120 time units; (2) the numerically measured peak eccentricity against the closed form at 70 degrees; (3) the same at 80 degrees with a non-zero starting e, against the root of the two conservation laws solved independently; (4) no growth below the critical angle.
Honest simplifications. Quadrupole order only: the octupole term (eccentric companion, comparable masses) is what flips an orbit from prograde to retrograde and is not modelled. The planet is a test particle and the companion's orbit is circular and fixed. The inner star is a point mass, tidal bulges and spin are not followed, and the "tidal capture" line is the angular-momentum-conserving endpoint, not a simulation of tidal friction. The orbit drawn is the osculating ellipse with the mean anomaly advanced at an arbitrary display rate; the real planet completes millions of orbits per secular cycle.