---
title: "Kozai–Lidov Cycles Interactive Lab | Sandbox Physics"
description: "How can a distant companion stretch a planet’s orbit? An inclined companion slowly torques a planet’s orbit."
canonical_url: "https://sandboxphysics.com/labs/kozai-lidov-cycles/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/kozai-lidov-cycles/"
last_modified: "2026-10-28"
---

AC09 · A distant companion reshapes an orbit

# Kozai–Lidov Eccentricity and Inclination Cycles

**How can a distant companion stretch a planet’s orbit?** An inclined companion slowly torques a planet’s orbit. For a negligible-mass planet and a distant circular companion, the simplest orbit average trades eccentricity for inclination while preserving angular momentum along the outer axis. Extra precession and fast perturbations change the exchange.

Tilt the outer orbit and watch the inner ellipse stretch and turn. Compare its slow average with saved three-dimensional Newtonian trajectories.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The planet is a massless test particle. A primary and equal-mass companion follow an exact circular Kepler orbit; the direct primary-centered equation includes the companion’s attraction and the primary’s acceleration. Units set the gravitational constant, primary mass and initial inner semimajor axis to one. The displayed geometry is the doubly averaged quadrupole orbit; fast orbital phases have been removed. Inner and outer paths use explicitly stated different display scales. A representative companion marker controls its plane, and body sizes are schematic. Default direct records are frozen REBOUND IAS15 integrations; changed parameters trigger an adaptive Newtonian integration. Optional conservative GR precession is added only to the averaged prediction, so the direct reference remains Newtonian. Close approaches, an unbound inner osculating orbit and numerical limits stop the relevant record. The averaged model truncates at quadrupole order; the direct Newtonian reference retains the full companion force. A conservative weak-field cutoff also bounds the averaged GR term. No dissipative tides, migration, finite planet mass or unique exoplanet formation history is modeled.

## Fast trajectories and their slow average

Amber is the saved direct Newtonian integration; cyan is the separately integrated averaged prediction. Both begin from the same stated inner orbit and companion geometry.

Replay the saved secular history $t/t_K=1.80$

Eccentricity through time

Mutual inclination in degrees

The exchange in eccentricity and inclination

Direct minus averaged eccentricity

Units, comparison and stopping rules

The direct model retains the full companion force; the averaged model also truncates at quadrupole order. The tutorial adds an instantaneous-quadrupole control to separate these approximations. The gravitational constant, primary mass, companion mass and initial inner semimajor axis are one; the planet is massless. The inner Kepler period is two pi. The secular time scale is the cube of the outer-to-inner semimajor-axis ratio. Plotted differences linearly resample saved direct eccentricity values onto the averaged time grid; raw state files remain available in the export.

$$
j_{\parallel}=\sqrt{1-e^2}\cos i,\qquad t_K=\frac{1}{n_1}\frac{M}{m_2}\left(\frac{a_2}{a_1}\right)^3
$$

The averaged axial momentum is conserved in this quadrupole model. Direct circular restricted motion instead conserves a rotating Hamiltonian; the planet’s two-body energy varies. A near-primary distance of 0.03 or a companion distance of 0.2 ends the direct record. The averaged record stops before its periapsis reaches 0.03, or before its gravitational radius reaches one percent of periapsis. No stopped trace is continued.

Physics tutorial

## How does a distant companion trade shape for tilt?

**Background** The companion attracts both a planet and its primary. The difference in those pulls produces a slow torque on the planet’s orbit.

**Why it matters** A planet can make close stellar passages while its averaged semimajor axis stays fixed. Comparing fast trajectories with their slow average reveals when that prediction loses accuracy.

Start with the essentials

- **Focus question**: Why does the inner ellipse stretch while its orbit plane turns?
- **One-sentence intuition**: The circular-outer quadrupole average preserves axial angular momentum. A smaller angular-momentum length therefore changes the mutual inclination.

### Core mathematical model

#### One conserved projection

$$
j_\parallel=\sqrt{1-e^2}\cos i
$$

Eccentricity and mutual inclination come from the same averaged orbit. This projection is conserved in the circular-outer doubly averaged quadrupole model.

#### A near-circular limiting maximum

$$
e_{\max}=\sqrt{1-\frac53\cos^2i_0}
$$

The test-particle, near-circular quadrupole limit grows above its critical angle. The finite initial eccentricity shown by the controls changes the exact maximum.

#### The primary-centered direct force

$$
\ddot{\boldsymbol r}=-\frac{\boldsymbol r}{r^3}+\frac{\boldsymbol R-\boldsymbol r}{|\boldsymbol R-\boldsymbol r|^3}-\frac{\boldsymbol R}{R^3}
$$

The last term removes the primary’s acceleration. The direct reference is Newtonian even when conservative GR precession is active in the averaged prediction.

### Common difficulties

#### A stretched orbit has not necessarily migrated

**Typical misconception** A small periapsis means the semimajor axis must have shrunk.

**Better mental model** At fixed semimajor axis, rising eccentricity moves periapsis inward and apoapsis outward. Dissipative energy loss is needed for the migration scenario.

### Run the experiment

1. 01

   #### Predict the low-tilt branch

   Before choosing the low-inclination preset, predict whether the ellipse will stretch strongly. Compare it with the exchange preset.

   **What to observe:** The small initial inclination leaves a large axial momentum and only modest eccentricity changes.

2. 02

   #### Change only the outer plane

   Drag the purple grip and observe eccentricity and inclination together. Pause near a maximum and check the axial projection.

   **What to observe:** The inner plane turns as its angular-momentum size changes. The averaged projection stays fixed.

3. 03

   #### Test the slow average

   Compare the wide and compact presets at the same initial inclination, then use the tutorial’s instantaneous-quadrupole control.

   **What to observe:** The full-versus-average difference combines averaging and multipole truncation. The third force model separates their roles.

4. 04

   #### Recognize a partial record

   Choose the near-polar preset and watch it stop at the inner boundary. Read the completed time before measuring a period.

   **What to observe:** The stopped excursion has no complete recurrence; no full period is assigned.

Physical conclusion

Strong initial tilt allows large eccentricity excursions. Fast outer motion, higher force orders and additional precession each require their own comparison.

## Related experiments and articles

Slow torques and orbital eccentricity · Reading path

### Slow torques and orbital eccentricity

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why a distant companion stretches an orbit**](https://sandboxphysics.com/stories/secular-orbits/why-a-distant-companion-stretches-an-orbit/)
2. [Article **Angular momentum connects orbital shape and tilt**](https://sandboxphysics.com/stories/secular-orbits/angular-momentum-and-slow-torques/)
3. [Article **Derive the quadrupole eccentricity cycle**](https://sandboxphysics.com/stories/secular-orbits/derive-the-quadrupole-cycle/)
4. [Article **Compare trajectories without confusing three different errors**](https://sandboxphysics.com/stories/secular-orbits/compare-trajectories-and-averaging/)
5. [Article **Measure a secular period from frozen Cartesian trajectories**](https://sandboxphysics.com/stories/secular-orbits/measure-periods-from-frozen-trajectories/)
6. [Article **Read octupole dynamics, relativity and a proposed planet history**](https://sandboxphysics.com/stories/secular-orbits/read-octupole-gr-and-planet-migration/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)