Triangular Lagrange Linear Stability

Evaluate the Routh criterion μ(1−μ)<1/27 for linear L4/L5 stability in the planar circular restricted model.

Description

The triangular L4 and L5 points of the planar circular restricted three-body model have a linear stability condition depending on the smaller primary's share of the total mass.

The triangular L4 and L5 points of the planar circular restricted three-body model have a linear stability condition depending on the smaller primary's share of the total mass. The condition describes perturbations within that idealized model.^[1]

Evaluate the Routh criterion μ(1−μ)<1/27 for linear L4/L5 stability in the planar circular restricted model.

When to use Triangular Lagrange Linear Stability

Use this tool to evaluate the Routh criterion μ(1−μ)<1/27 for linear L4/L5 stability in the planar circular restricted model.

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

How to use Triangular Lagrange Linear Stability and understand its results

Prepare the orbital state or model parameters in one reference frame. Use compatible distance/time units and the stated angular and gravitational-parameter conventions.
Enter smallerMassFraction using the field table and calculation rules, then run the tool.
Read routhProduct, criticalSmallerMassFraction, linearlyStable. 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
smallerMassFraction	Smaller mass fraction	number	Required

Enter smallerMassFraction as the smaller primary's mass divided by both primary masses combined, strictly above 0 and at most 0.5. The tool calculates the product, the critical fraction and the strict comparison. Equality at the product threshold is not flagged linearlyStable; the output is dimensionless.

routhProduct = μ(1−μ); linearlyStable = routhProduct < 1/27; critical μ = (1−√(23/27))/2

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

Output field	Meaning	Value type
routhProduct	Routh product	number
criticalSmallerMassFraction	Critical smaller mass fraction	number
linearlyStable	Linearly stable	boolean

json

{
  "input": {
    "smallerMassFraction": 0.01
  },
  "output": {
    "routhProduct": 0.0099,
    "criticalSmallerMassFraction": 0.03852089650455137,
    "linearlyStable": true
  }
}

Limitations and assumptions

This does not solve a trajectory, assess nonlinear stability or include eccentric primary orbits and other perturbations. A stable flag is conditional on the planar circular restricted assumptions, not a guarantee of a real object's long-term retention.

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 Tisserand Orbital Parameter: Compute a circular restricted-three-body Tisserand parameter in the supplied planet frame; no universal comet taxonomy cutoff is embedded. For a complementary task, use Soi Handoff Hysteresis: Apply caller-defined sphere-of-influence handoff hysteresis to a sampled trajectory; boundaries are not universal force-dominance limits.

References

Triangular Lagrange Linear Stability — tool.io

Similar or alternative tools

Tisserand Orbital Parameter
Compute a circular restricted-three-body Tisserand parameter in the supplied planet frame; no universal comet taxonomy cutoff is embedded.
Soi Handoff Hysteresis
Apply caller-defined sphere-of-influence handoff hysteresis to a sampled trajectory; boundaries are not universal force-dominance limits.