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.

Averaged eccentricity $e=0.503$

Mutual inclination $i=60.8^{\circ}$

Periapsis / initial semimajor axis $q/a_1=0.497$

Averaged axial angular momentum $j_{\parallel}=0.422$

Tilt the distant companion

Initial mutual inclination

65^{\circ}

i_0\;[{}^{\circ}]

Outer / inner semimajor axis

12.0

a_2/a_1

Drag the purple orbit grip to change inclination. Arrow keys change its angle; Shift takes a larger step.

Initial seed and conservative GR precession

Initial eccentricity

0.0500

Averaged relativistic strength

0

\epsilon_{\mathrm{GR}}

This dimensionless strength specifies a point primary’s compactness. It changes the averaged periapsis precession; the direct reference stays Newtonian. The averaged record stops before the gravitational radius reaches one percent of periapsis.

\epsilon_{\mathrm{GR}}=3\left(\frac{a_2}{a_1}\right)^3\frac{GM}{a_1c^2}

r_g=GM/c^2,\qquad r_g/q<10^{-2}

Read the course, actual trajectory tutorial and paper guide

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.

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.

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 misconceptionA small periapsis means the semimajor axis must have shrunk.

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

Run the experiment

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.
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.
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.
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.

Kozai–Lidov Eccentricity and Inclination Cycles

Fast trajectories and their slow average

How does a distant companion trade shape for tilt?

Core mathematical model

One conserved projection

A near-circular limiting maximum

The primary-centered direct force

Common difficulties

A stretched orbit has not necessarily migrated

Run the experiment

Predict the low-tilt branch

Change only the outer plane

Test the slow average

Recognize a partial record

Related experiments and articles

Slow torques and orbital eccentricity