AC09
Kozai–Lidov cycles from torque to planetary orbits
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.
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.
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.
L2
Derive the quadrupole eccentricity cycle
Average the companion’s tidal potential, derive the vector equations and recover the critical inclination and maximum eccentricity.
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.
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.
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.
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
- Tremaine & Yavetz, Why do Earth satellites stay up?, arXiv: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.
- Liu, Muñoz & Lai, Short-range forces in triple systems, arXiv: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.
- REBOUND 5.2.2 official Kozai example and IAS15 integrator
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.
- Gupta et al., A hot-Jupiter progenitor on a super-eccentric retrograde orbit, Nature 632 (2024)
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.
- Klein & Hamilton, Short-range forces beyond double averaging, arXiv: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.