Normalized L2 Equilibrium
Solve the normalized rotating-frame collinear L2 equilibrium numerically instead of treating it as an exact mirror of L1.
Description
L2 lies beyond the smaller primary in a normalized circular restricted three-body rotating frame.
L2 lies beyond the smaller primary in a normalized circular restricted three-body rotating frame. Its location must be solved from the force balance and is not an exact reflection of L1.1
Solve the normalized rotating-frame collinear L2 equilibrium numerically instead of treating it as an exact mirror of L1.
When to use Normalized L2 Equilibrium
Use this tool to solve the normalized rotating-frame collinear L2 equilibrium numerically instead of treating it as an exact mirror of L1.
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 Normalized L2 Equilibrium 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 massFraction using the field table and calculation rules, then run the tool.
- Read barycentricX, distanceFromSmallerPrimary. 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 |
|---|---|---|---|
| massFraction | Smaller-primary mass divided by total primary mass; numerical support is [1e-15,0.5]. | number; minimum: 1e-15; maximum: 0.5 | Required |
Enter the smaller primary's total-mass fraction in [10⁻¹⁵,0.5]. With separation one, the primaries lie at −μ and 1−μ. The helper bisects the exterior bracket for 150 iterations and returns barycentric x and its positive distance beyond the smaller primary.
solve x − (1−μ)(x+μ)/|x+μ|³ − μ(x−1+μ)/|x−1+μ|³ = 0 beyond x=1−μ Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| barycentricX | Barycentric x | number |
| distanceFromSmallerPrimary | Distance from smaller primary | number |
{
"input": {
"massFraction": 0.5
},
"output": {
"barycentricX": 1.19840614455492,
"distanceFromSmallerPrimary": 0.6984061445549199
}
}Limitations and assumptions
Distances are normalized, not kilometers. The circular restricted model omits eccentricity and other bodies, and this root does not establish trajectory stability or stationkeeping requirements.
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.
Alternative or Complementary methods
Use Normalized L1 Equilibrium: Solve the normalized rotating-frame L1 equilibrium numerically; separation is one and mass fraction refers to the smaller primary. For a complementary task, use Cr3Bp Jacobi State: Evaluate the normalized barycentric rotating-frame Jacobi integral and reference drift; separation, total mass and angular speed are normalized to one.
References
-
Normalized L2 Equilibrium — tool.io
Similar or alternative tools
- Normalized L1 Equilibrium
Solve the normalized rotating-frame L1 equilibrium numerically; separation is one and mass fraction refers to the smaller primary.
- Cr3Bp Jacobi State
Evaluate the normalized barycentric rotating-frame Jacobi integral and reference drift; separation, total mass and angular speed are normalized to one.