Quadrupole Kozai Timescale

Estimate a quadrupole Kozai–Lidov timescale with the 8/(15π) convention in the same units as supplied periods; not a cycle-existence test.

Description

A quadrupole Kozai–Lidov timescale estimates the secular exchange time in a hierarchical triple.

A quadrupole Kozai–Lidov timescale estimates the secular exchange time in a hierarchical triple. Its numerical convention and the supplied mass and period definitions matter when comparing estimates.^[1]

Estimate a quadrupole Kozai–Lidov timescale with the 8/(15π) convention in the same units as supplied periods; not a cycle-existence test.

When to use Quadrupole Kozai Timescale

Use this tool to estimate a quadrupole Kozai–Lidov timescale with the 8/(15π) convention in the same units as supplied periods; not a cycle-existence test.

Prepare the orbital state or model parameters in one reference frame. Confirm compatible distance/time units and the stated angular and gravitational-parameter conventions.

How to use Quadrupole Kozai Timescale and understand its results

Prepare the orbital state or model parameters in one reference frame. Confirm compatible distance/time units and the stated angular and gravitational-parameter conventions.
Enter innerPeriod, outerPeriod, totalTripleMass, perturberMass, outerEccentricity using the field table and calculation rules, then run the tool.
Read cycleTimescale. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
innerPeriod	Inner period	number	Required
outerPeriod	Outer period	number	Required
totalTripleMass	Total triple mass	number	Required
perturberMass	Perturber mass	number	Required
outerEccentricity	Outer eccentricity	number	Required

Enter positive periods and matching-unit masses, with outer period longer than inner, perturber mass below total and outer eccentricity 0≤e<1. The tool evaluates the stated convention and returns a timescale in the same unit as the periods.

timescale = [8/(15π)](outerPeriod²/innerPeriod)(totalMass/perturberMass)(1−outerEccentricity²)^1.5

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
cycleTimescale	Cycle timescale	number

json

{
  "input": {
    "innerPeriod": 1,
    "outerPeriod": 10,
    "totalTripleMass": 3,
    "perturberMass": 1,
    "outerEccentricity": 0
  },
  "output": {
    "cycleTimescale": 50.92958178940651
  }
}

Limitations and assumptions

It does not test whether cycles occur or model octupole effects, relativity, tides or actual cycle amplitudes. The hierarchy and mass definitions must describe the intended system.

Numeric inputs must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Hierarchical Triple Stability Screen: Screen the Mardling–Aarseth hierarchical-triple empirical axis-ratio boundary; not a general N-body stability proof. For a complementary task, use Keplerian Two Body Period Ratio: Compare orbital periods around possibly different gravitating systems using semimajor axes and total gravitational parameters.

References

Quadrupole Kozai Timescale — tool.io

Similar or alternative tools

Hierarchical Triple Stability Screen
Screen the Mardling–Aarseth hierarchical-triple empirical axis-ratio boundary; not a general N-body stability proof.
Keplerian Two Body Period Ratio
Compare orbital periods around possibly different gravitating systems using semimajor axes and total gravitational parameters.