Reduced Resonance Detuning
Reduce a candidate period-ratio resonance to coprime integers and report relative detuning; proximity alone does not show resonant angle libration.
Description
A candidate orbital resonance is an integer period ratio; equivalent ratios should be reduced before reporting their order.
A candidate orbital resonance is an integer period ratio; equivalent ratios should be reduced before reporting their order. Close periods alone do not demonstrate a librating resonant angle.1
Reduce a candidate period-ratio resonance to coprime integers and report relative detuning; proximity alone does not show resonant angle libration.
When to use Reduced Resonance Detuning
Use this tool to reduce a candidate period-ratio resonance to coprime integers and report relative detuning; proximity alone does not show resonant angle libration.
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 Reduced Resonance Detuning 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 firstPeriod, secondPeriod, p, q, maximumRelativeDetuning using the field table and calculation rules, then run the tool.
- Read reducedP, reducedQ, resonanceOrder, relativeDetuning, nearResonance. 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 |
|---|---|---|---|
| firstPeriod | First period | number | Required |
| secondPeriod | Second period | number | Required |
| p | P | integer | Required |
| q | Q | integer | Required |
| maximumRelativeDetuning | Maximum relative detuning | number | Required |
Enter positive periods in matching units, positive integer p and q, and a nonnegative relative-detuning tolerance. Euclid's algorithm removes their greatest common divisor. The tool reports reduced integers, order, signed fractional detuning and nearResonance using an inclusive absolute-tolerance comparison.
detuning = (secondPeriod/firstPeriod)/(reducedP/reducedQ) − 1; order = |reducedP−reducedQ| Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| reducedP | Reduced p | number |
| reducedQ | Reduced q | number |
| resonanceOrder | Resonance order | number |
| relativeDetuning | Relative detuning | number |
| nearResonance | Near resonance | boolean |
{
"input": {
"firstPeriod": 2,
"secondPeriod": 3,
"p": 6,
"q": 4,
"maximumRelativeDetuning": 0.01
},
"output": {
"reducedP": 3,
"reducedQ": 2,
"resonanceOrder": 1,
"relativeDetuning": 0,
"nearResonance": true
}
}Limitations and assumptions
No resonant dynamics, time evolution or angle analysis is performed. A zero detuning is an exact supplied ratio within numeric arithmetic, not proof of resonance.
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 Interior Resonance Location: Locate an interior Keplerian p:q resonance from the supplied planet orbit, rather than embedding Kirkwood-gap reference positions. For a complementary task, use Resonant Angle Libration Span: Unwrap successive resonant angles using the shortest step and measure their sampled half-range; cadence must resolve steps below 180 degrees and a short span does not prove libration.
References
-
Reduced Resonance Detuning — tool.io
Similar or alternative tools
- Interior Resonance Location
Locate an interior Keplerian p:q resonance from the supplied planet orbit, rather than embedding Kirkwood-gap reference positions.
- Resonant Angle Libration Span
Unwrap successive resonant angles using the shortest step and measure their sampled half-range; cadence must resolve steps below 180 degrees and a short span does not prove libration.