Normalized L1 Equilibrium
Solve the normalized rotating-frame L1 equilibrium numerically; separation is one and mass fraction refers to the smaller primary.
Description
L1 lies between two primaries in the normalized circular restricted three-body model.
L1 lies between two primaries in the normalized circular restricted three-body model. Barycentric position and distance from the smaller primary are distinct coordinates.1
Solve the normalized rotating-frame L1 equilibrium numerically; separation is one and mass fraction refers to the smaller primary.
When to use Normalized L1 Equilibrium
Use this tool to solve the normalized rotating-frame L1 equilibrium numerically; separation is one and mass fraction refers to the smaller primary.
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 L1 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 smaller-primary mass divided by total mass in [10⁻¹⁵,0.5]. The helper uses the normalized primary locations and 150 bisection iterations in the interior bracket. It returns barycentric x and 1−μ−x as the smaller-primary distance. To obtain physical distance, scale by the intended primary separation independently.
solve rotating-frame collinear force balance between x=−μ and 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": 5.551115123125783e-17,
"distanceFromSmallerPrimary": 0.49999999999999994
}
}Limitations and assumptions
This is an ideal circular restricted equilibrium calculation, without stability analysis or a real ephemeris. L1 and L2 distances need not be equal.
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 L2 Equilibrium: Solve the normalized rotating-frame collinear L2 equilibrium numerically instead of treating it as an exact mirror of L1. 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 L1 Equilibrium — tool.io
Similar or alternative tools
- Normalized L2 Equilibrium
Solve the normalized rotating-frame collinear L2 equilibrium numerically instead of treating it as an exact mirror of L1.
- Cr3Bp Jacobi State
Evaluate the normalized barycentric rotating-frame Jacobi integral and reference drift; separation, total mass and angular speed are normalized to one.