Mutual Hill Separation

Compute mutual-Hill separation and screen a supplied boundary; the conventional 2√3 criterion assumes small, circular, coplanar two-planet orbits.

Description

Mutual Hill separation expresses neighboring planetary spacing in a mass-dependent scale.

Mutual Hill separation expresses neighboring planetary spacing in a mass-dependent scale. A caller-selected boundary is a screen within assumed orbital conditions rather than a universal stability test.^[1]

Compute mutual-Hill separation and screen a supplied boundary; the conventional 2√3 criterion assumes small, circular, coplanar two-planet orbits.

When to use Mutual Hill Separation

Use this tool to compute mutual-Hill separation and screen a supplied boundary; the conventional 2√3 criterion assumes small, circular, coplanar two-planet orbits.

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 Mutual Hill Separation 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, firstPlanetMass, secondPlanetMass, starMass, minimumHillSeparation using the field table and calculation rules, then run the tool.
Read mutualHillRadius, hillSeparation, aboveSuppliedBoundary. 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
firstPlanetMass	First planet mass	number	Required
secondPlanetMass	Second planet mass	number	Required
starMass	Star mass	number	Required
minimumHillSeparation	Minimum hill separation	number	Required

Enter ordered positive semimajor axes, nonnegative planet masses with positive sum, positive stellar mass and positive minimum separation. Use common mass and distance units. The tool returns Hill radius, dimensionless spacing and a strict above-boundary flag; equality fails the flag.

RH = mean(a1,a2)[(m1+m2)/(3Mstar)]^(1/3); separation = (a2−a1)/RH

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

Output field	Meaning	Value type
mutualHillRadius	Mutual hill radius	number
hillSeparation	Hill separation	number
aboveSuppliedBoundary	Above supplied boundary	boolean

json

{
  "input": {
    "innerSemimajorAxis": 1,
    "outerSemimajorAxis": 2,
    "firstPlanetMass": 1,
    "secondPlanetMass": 1,
    "starMass": 1000,
    "minimumHillSeparation": 3.4641016151377544
  },
  "output": {
    "mutualHillRadius": 0.13103706971044485,
    "hillSeparation": 7.631428283688878,
    "aboveSuppliedBoundary": true
  }
}

Limitations and assumptions

The conventional 2√3 boundary assumes small circular coplanar two-planet orbits. This does not assess eccentric systems, additional planets or long-term N-body stability.

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 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 Laplace Sphere Of Influence: Compute the Laplace sphere-of-influence scale a(m/M)^(2/5), not a Hill-stability radius.

References

Mutual Hill Separation — 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.
Laplace Sphere Of Influence
Compute the Laplace sphere-of-influence scale a(m/M)^(2/5), not a Hill-stability radius.