---
title: "Kozai–Lidov cycles from torque to planetary orbits"
description: "Learn how a companion trades orbital shape for tilt through three chapters, a frozen-trajectory tutorial and a guide to extreme planetary orbits."
canonical_url: "https://sandboxphysics.com/stories/secular-orbits/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/secular-orbits/"
last_modified: "2026-10-28"
---

AC09

# Kozai–Lidov cycles from torque to planetary orbits

Sandbox Physics · 2026-10-28

A distant companion can twist a planet’s orbit until its ellipse stretches toward the star. Follow the angular momentum, derive the slow equations, and compare them with actual saved three-dimensional integrations. Then read how extra forces and uncertain observations enter a proposed planet history.

[Tilt the companion and reshape the orbit →](https://sandboxphysics.com/labs/kozai-lidov-cycles/)

[Concept Why a distant companion stretches an orbit An inclined companion can trade the shape of a planet’s orbit for its tilt. Follow the slow torque from a round orbit toward close stellar passages.](https://sandboxphysics.com/stories/secular-orbits/why-a-distant-companion-stretches-an-orbit/) [L1 Angular momentum connects orbital shape and tilt Derive the link between eccentricity, inclination and a conserved angular-momentum component, then estimate the slow torque timescale.](https://sandboxphysics.com/stories/secular-orbits/angular-momentum-and-slow-torques/) [L2 Derive the quadrupole eccentricity cycle Average the companion’s tidal potential, derive the vector equations and recover the critical inclination and maximum eccentricity.](https://sandboxphysics.com/stories/secular-orbits/derive-the-quadrupole-cycle/) [L3 Compare trajectories without confusing three different errors Use saved three-dimensional integrations to separate numerical error, orbital averaging and quadrupole truncation, including explicit stopping boundaries.](https://sandboxphysics.com/stories/secular-orbits/compare-trajectories-and-averaging/) [Tutorial Measure a secular period from frozen Cartesian trajectories Download the fixed orbit records and a standard-library Python script, verify hashes and reproduce the three-model period comparison.](https://sandboxphysics.com/stories/secular-orbits/measure-periods-from-frozen-trajectories/) [Paper guide Read octupole dynamics, relativity and a proposed planet history Follow fixed paper equations and figures from quadrupole limits to octupole forces, observed eccentric planets and the failure of double averaging.](https://sandboxphysics.com/stories/secular-orbits/read-octupole-gr-and-planet-migration/)

The browser evolves a doubly averaged orbit and compares it with a matched Newtonian reference. The tutorial retains full and instantaneous-quadrupole IAS15 records so that force truncation and averaging can be tested separately. The final guide reads fixed papers on octupole dynamics, relativistic effects and a highly eccentric planet.

## Sources and calculation scope

1. [Tremaine & Yavetz, Why do Earth satellites stay up?, arXiv:1309.5244v1](https://arxiv.org/abs/1309.5244v1)

   Fixed 20 September 2013 version, DOI 10.1119/1.4874853. Potential 18, vector equations 23, constraints 24 and section IV give the quadrupole model and near-circular critical-angle limit.

2. [Liu, Muñoz & Lai, Short-range forces in triple systems, arXiv:1409.6717v2](https://arxiv.org/abs/1409.6717v2)

   Fixed 7 January 2015 version, DOI 10.1093/mnras/stu2396. Sections 2–4, quadrupole potential 7, octupole 8, GR 32–33/42 and Figures 3–4/7. The browser retains the circular-outer, test-particle quadrupole; the guide reads the wider model.

3. [REBOUND 5.2.2 official Kozai example and IAS15 integrator](https://rebound.hanno-rein.de/c_examples/kozai/)

   The official example motivates an independent direct reference. Frozen trajectories use REBOUND 5.2.2, source commit c34134716cd484ed9cdb9f7d54ef8d859bdf547d, IAS15 accuracy parameter one hundred-billionth, with separately stated initial conditions and stopping rules. REBOUND software is GPL-3.0-only; no publisher code is copied into the Lab.

4. [Gupta et al., A hot-Jupiter progenitor on a super-eccentric retrograde orbit, Nature 632 (2024)](https://www.nature.com/articles/s41586-024-07688-3)

   Version of record 17 July 2024, DOI 10.1038/s41586-024-07688-3, CC BY 4.0. Observed transits/RVs are Figures 1–2; Extended Data 5–6 give model-dependent initial conditions and an illustrative tidal-migration history. Full publisher-deposited XML, PMC11291287, was read for these distinctions.

5. [Klein & Hamilton, Short-range forces beyond double averaging, arXiv:2606.19456v1](https://arxiv.org/html/2606.19456v1)

   Fixed 17 June 2026 preprint version; an August v2 also exists. Figure 1 separates averaging/force choices, Figures 2–3 show cycle changes, and Appendix B compares approximations with direct N-body examples. This is a reading guide; the browser does not reproduce its single-averaged GR/tidal calculations.