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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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How to use
Pick a preset or set the starting tilt. Watch the orange eccentricity curve rise as the blue inclination curve falls: the two are locked together by j_z. Try: (1) Classic 65 deg and read the period in t_K; (2) Hot-Jupiter route and watch the pericentre graph dive towards the star (red line = stellar surface); (3) switch GR on and then to x20 to see the cap; (4) Below critical 35 deg where nothing happens; (5) Libration of omega where the purple point in the e-vector plot rocks back and forth instead of circling the origin. The dashed grey curve in that plot is the predicted path. Drag to orbit, wheel to zoom. Space pauses, R resets.
The Blender 5.2 scene contains the primary star with baked granulation, the companion star, the planet with baked bands, the reference plane disc with tick ring, the axis gizmos and the baked sky; the orbit lines, trails and snapshots are drawn live from the same integrator as the 2D page.
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.