Engineering & Science
Kozai-Lidov Mechanism Formula: Maximum Eccentricity & Critical Inclination
Calculate secular Kozai-Lidov conserved vertical angular momentum L_z and peak quadrupole eccentricity e_max = SQRT(1 - (5/3)*cos^2(i_0)) in hierarchical triple systems.
Above i_c = 39.23 degrees, an outer companion pumps circular orbits to extreme eccentricity!
Yoshihide Kozai (1962) and Mikhail Lidov (1962) independently discovered that a distant tertiary perturber on a mutually inclined orbit conserves the secular vertical Delaunay integral L_z = SQRT(1 - e^2) * COS(i) while trading inclination for eccentricity! Whenever an initially near-circular inner orbit (e_0 -> 0) is tilted beyond the critical Kozai angle i_c = ACOS(SQRT(3/5)) = 39.2315 degrees, quadrupole perturbations drive eccentricity all the way up to e_max = SQRT(1 - (5/3) * COS^2(i_0))!
Lidov famously discovered this instability when realizing Sputnik-era artificial lunar orbiters placed on polar orbits (i_0 -> 90 degrees -> e_max -> 1) crash into the Moon within months! For a hot-Jupiter progenitor with initial e_0 = 0.05 and mutual inclination i_0 = 65 degrees, conserved L_z = 0.4221 and peak eccentricity reaches e_max = 0.8380 (dropping to 0.5580 at i_0 = 50 degrees, and 0 below 39.23 degrees)!
| Initial Mutual Inclination i_0 (e_0 = 0.05) | Conserved L_z = SQRT(1-e_0²)cos(i_0) | Quadrupole Peak Eccentricity e_max | Dynamical Regime |
|---|---|---|---|
| i_0 = 65.0° (High-Obliquity Exoplanet) | 0.4221 | 0.8380 | Strong tidal circularization / Hot Jupiter |
| i_0 = 50.0° (Moderate Triple) | 0.6420 | 0.5580 | Moderate secular e–i cycles |
| i_0 = 30.0° (< 39.23° Critical Angle) | 0.8649 | 0 (No Kozai Pumping) | Apsidal precession without instability |
Quadrupole Kozai-Lidov Maximum Eccentricity e_max (Circular Limit)
Computes SQRT(MAX(0, 1 - (5/3) * COS^2(i_0))) from initial eccentricity B2 (0 <= e_0 < 1) and mutual inclination B3 (0 <= i_0 <= 90 degrees).
=IF(OR(B2 < 0, B2 >= 1, B3 < 0, B3 > 90), "Invalid Orbital Elements", ROUND(SQRT(MAX(0, 1 - (5 / 3) * (COS(B3 * PI() / 180) ^ 2))), 4))Conserved Kozai-Lidov Vertical Angular Momentum Integral L_z
Evaluates SQRT(1 - e_0^2) * COS(i_0 * PI() / 180), invariant throughout secular quadrupole evolution.
=IF(OR(B2 < 0, B2 >= 1, B3 < 0, B3 > 90), "Invalid Orbital Elements", ROUND(SQRT(1 - B2 ^ 2) * COS(B3 * PI() / 180), 4))Try the example
Use the original values or change them to test your understanding.