Hierarchical Triple Stability Screen
Screen the Mardling–Aarseth hierarchical-triple empirical axis-ratio boundary; not a general N-body stability proof.
Description
A hierarchical triple can be screened using an empirical ratio between outer and inner semimajor axes.
A hierarchical triple can be screened using an empirical ratio between outer and inner semimajor axes. Outer eccentricity, mass ratio and mutual inclination affect the Mardling–Aarseth boundary used here.1
Screen the Mardling–Aarseth hierarchical-triple empirical axis-ratio boundary; not a general N-body stability proof.
When to use Hierarchical Triple Stability Screen
Use this tool to screen the Mardling–Aarseth hierarchical-triple empirical axis-ratio boundary; not a general N-body stability proof.
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 Hierarchical Triple Stability Screen 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 innerSemimajorAxis, outerSemimajorAxis, outerEccentricity, outerInnerMassRatio, mutualInclinationDegrees using the field table and calculation rules, then run the tool.
- Read requiredAxisRatio, actualAxisRatio, aboveEmpiricalBoundary. 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 |
|---|---|---|---|
| innerSemimajorAxis | Inner semimajor axis | number | Required |
| outerSemimajorAxis | Outer semimajor axis | number | Required |
| outerEccentricity | Outer eccentricity | number | Required |
| outerInnerMassRatio | Outer inner mass ratio | number | Required |
| mutualInclinationDegrees | Mutual inclination degrees | number | Required |
Enter positive inner/outer axes, outer eccentricity in [0,1), nonnegative outer/inner mass ratio and inclination 0–180 degrees. The tool compares the actual axis ratio strictly with the empirical required ratio and returns both plus the flag; equality is not above the boundary.
requiredRatio = 2.8(1+q)^0.4(1+e)^0.4/(1−e)^1.2 × (1−0.3 i/180) Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| requiredAxisRatio | Required axis ratio | number |
| actualAxisRatio | Actual axis ratio | number |
| aboveEmpiricalBoundary | Above empirical boundary | boolean |
{
"input": {
"innerSemimajorAxis": 1,
"outerSemimajorAxis": 5,
"outerEccentricity": 0,
"outerInnerMassRatio": 1,
"mutualInclinationDegrees": 0
},
"output": {
"requiredAxisRatio": 3.6946221501641037,
"actualAxisRatio": 5,
"aboveEmpiricalBoundary": true
}
}Limitations and assumptions
This empirical screen is not a general N-body stability proof. System hierarchy, model applicability and neglected perturbations require independent assessment.
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 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. For a complementary task, use Mutual Hill Separation: Compute mutual-Hill separation and screen a supplied boundary; the conventional 2√3 criterion assumes small, circular, coplanar two-planet orbits.
References
-
Hierarchical Triple Stability Screen — tool.io
Similar or alternative tools
- 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.
- Mutual Hill Separation
Compute mutual-Hill separation and screen a supplied boundary; the conventional 2√3 criterion assumes small, circular, coplanar two-planet orbits.