Interior Resonance Location

Locate an interior Keplerian p:q resonance from the supplied planet orbit, rather than embedding Kirkwood-gap reference positions.

Description

An interior period-ratio resonance places a faster small-body orbit inside a planet's orbit under Keplerian scaling.

An interior period-ratio resonance places a faster small-body orbit inside a planet's orbit under Keplerian scaling. Supplied revolution counts define the ratio, while a radial region can bound the search.^[1]

Locate an interior Keplerian p:q resonance from the supplied planet orbit, rather than embedding Kirkwood-gap reference positions.

When to use Interior Resonance Location

Use this tool to locate an interior Keplerian p:q resonance from the supplied planet orbit, rather than embedding Kirkwood-gap reference positions.

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 Interior Resonance Location 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 planetSemimajorAxis, smallBodyRevolutions, planetRevolutions, regionInnerRadius, regionOuterRadius using the field table and calculation rules, then run the tool.
Read resonanceSemimajorAxis, withinRegion. 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
planetSemimajorAxis	Planet semimajor axis	number	Required
smallBodyRevolutions	Small body revolutions	integer	Required
planetRevolutions	Planet revolutions	integer	Required
regionInnerRadius	Region inner radius	number	Required
regionOuterRadius	Region outer radius	number	Required

Enter a positive planet axis, positive integer counts with small-body count greater than planet count, and ordered nonnegative inner/outer region radii. The tool computes the nominal axis and tests whether it falls inclusively within that region. Use matching distance units.

resonanceAxis = planetAxis × (planetRevolutions/smallBodyRevolutions)^(2/3)

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

Output field	Meaning	Value type
resonanceSemimajorAxis	Resonance semimajor axis	number
withinRegion	Within region	boolean

json

{
  "input": {
    "planetSemimajorAxis": 4,
    "smallBodyRevolutions": 8,
    "planetRevolutions": 1,
    "regionInnerRadius": 0.5,
    "regionOuterRadius": 2
  },
  "output": {
    "resonanceSemimajorAxis": 1,
    "withinRegion": true
  }
}

Limitations and assumptions

This does not embed Kirkwood-gap locations, model resonance widths or demonstrate actual resonant motion. The assumed common central gravity underlies the period-to-axis scaling.

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. A successfully returned false flag is an evaluated outcome; its meaning is explained above.

Alternative or Complementary methods

Use Reduced Resonance Detuning: Reduce a candidate period-ratio resonance to coprime integers and report relative detuning; proximity alone does not show resonant angle libration. 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

Interior Resonance Location — tool.io

Similar or alternative tools

Reduced Resonance Detuning
Reduce a candidate period-ratio resonance to coprime integers and report relative detuning; proximity alone does not show resonant angle libration.
Keplerian Two Body Period Ratio
Compare orbital periods around possibly different gravitating systems using semimajor axes and total gravitational parameters.