Collinear Lagrange Point Solutions in the Circular Restricted Three-Body Problem with Radiation Pressure using Fortran

A Fortran collinear Lagrange points calculator.

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Table of Contents

• Background
• Key Files
• Software Requirements
• Quality Assurance
• Getting Started
• Acknowledgements
• References
• Citation
• License

Background

ForCLaP (Fortran Collinear Lagrange Points) is a Fortran command line application demonstrating Newton-Raphson convergence on collinear solutions to the circular restricted three-body problem. Optional displacement of the collinear Lagrange points due to radiation pressure, relevant to small bodies such as asteroidal dust particles, is incorporated.

ForCLaP assumes a traditional three-body system with a large, central primary (m₁), a smaller secondary (m₂) and an infinitesimal tertiary mass (m₃). ForCLaP accepts a user-input value for the m₁/m₂ mass ratio of the system. Use of a mass ratio within the Routh value limit, is enforced however.

ForCLaP also accepts a user-input value for a ratio of solar radiation pressure to gravitational force experienced by m₃, β. This allows simulation of a luminous primary such as the Sun, but ForCLaP can just as easily be used for simulation of a non-luminous primary, such as planet, with an input ratio, β, of zero.

Collinear Lagrange point solution calculation is done with a Newton-Raphson algorithm. The program therefore inherits limitations of that algorithm. The algorithm can suffer runaway solution divergence, for example, but ForCLaP explicitly flags this and provides graceful algorithm termination. For algorithm convergence, iteration continues until the solution and the Newton derivative are both stable to four significant figures.

Locations of the Collinear Lagrange Points in the Circular Restricted Three-Body Problem with Radiation Pressure

Formulas for the locations of the collinear Lagrange points L₁, L₂ and L₃ are given below. Symbol definitions and explanatory theory are given in [1].

L₁

$$ \begin{eqnarray} f(r_2) &=& \frac{r_2^5 - 3r_2^4 + 3r_2^3 - \beta r_2^2}{-r_2^5 + 2r_2^4 - r_2^3 + r_2^2 - 2r_2 + 1} - \frac{m_2}{m_1}\\ f'(r_2) &=& \frac{-r_2^8 + 4r_2^7 - 3\beta r_2^6 - (7 - 2\beta)2r_2^5 + (26 - \beta)r_2^4 - 24r_2^3 + (9 - 2\beta)r_2^2 - 2\beta r_2}{r_2^{10} - 4r_2^9 + 6r_2^8 - 6r_2^7 + 9r_2^6 - 12r_2^5 + 9r_2^4 - 6r_2^3 + 6r_2^2 - 4r_2 + 1}\nonumber \\ \end{eqnarray} $$

L₂

$$ \begin{eqnarray} f(r_2) &=& \frac{r_2^5 + 3r_2^4 + 3r_2^3 + \beta r_2^2}{-r_2^5 - 2r_2^4 - r_2^3 + r_2^2 + 2r_2 + 1} - \frac{m_2}{m_1}\\ f'(r_2) &=& \frac{r_2^8 + 4r_2^7 + (6 + 3\beta)r_2^6 + (14 + 4\beta)r_2^5 + (26 + \beta)r_2^4 + 24r_2^3 + (9 + 2\beta)r_2^2 + 2\beta r_2}{r_2^{10} + 4r_2^9 + 6r_2^8 + 2r_2^7 - 7r_2^6 - 12r_2^5 - 7r_2^4 + 2r_2^3 + 6r_2^2 + 4r_2 + 1}\nonumber \\ \end{eqnarray} $$

L₃

$$ \begin{eqnarray} f(r_1) &=& \frac{-r_1^5 - 2r_1^4 - r_1^3 + (1 - \beta)r_1^2 + (1 - \beta)2r_1 - \beta + 1}{r_1^5 + 3r_1^4 - 3r_1^3} - \frac{m_2}{m_1}\\ f'(r_1) &=& \frac{-r_1^6 - 4r_1^5 - (2 - \beta)3r_1^4 - (1 + \beta)14r_1^3 - (1 - \beta)(26r_1^2 + 24r_1 + 9)}{r_1^8 + 6r_1^7 + 15r_1^6 + 18r_1^5 + 9r_1^4}\nonumber \\ \end{eqnarray} $$

Key Files

File	Notes
src/forclap.f90	Fortran program.

Software Requirements

Software	Notes
Fortran	Details here. Free and proprietary compilers available.

Tip

Is Fortran already installed? It's sometimes bundled with other software. Strawberry Perl, for example, includes GNU Fortran (GFortran). Installation can be checked from the command line. E.g., for GFortran on Linux, macOS or Windows:

gfortran --version

Quality Assurance

ForCLaP has been tested in the following environments.

Windows 11, GFortran 14.3.0

Type	Component	Version
Platform	Operating system	Windows 11, 26H2 (OS Build 26300.9550)
Software	GNU Fortran	14.3.0

Windows 11, GFortran 16.2.0

Type	Component	Version
Platform	Operating system	Windows 11, 26H2 (OS Build 26300.9550)
Software	GNU Fortran	16.2.0, run from MSYS2 version 20260611, UCRT64 shell

Getting Started

Compilation

Program source code requires compilation on the target system with a Fortran compiler. E.g., for GFortran:

gfortran src/forclap.f90 -o forclap -fimplicit-none

GFortran Compiler Flags

Compiler Flag	Notes
-o	Name of the output file.
-fimplicit-none	Forces all variables to be declared.

Execution

The compiled Fortran program should be executed from a command line. Execution syntax for some example command lines is given below.

Bash, Windows PowerShell

./forclap

Windows Command Prompt

forclap

Acknowledgements

Microsoft Copilot [2] was used as a licensing management aid.

References

1. T. N. Stenborg, "Collinear Lagrange Point Solutions in the Circular Restricted Three-Body Problem with Radiation Pressure using Fortran", in Astron. Data Anal. Softw. Syst. XVII, in Astronomical Society of the Pacific Conference Series, vol. 394, R. W. Argyle, P. S. Bunclark and J. R. Lewis, Eds., 2008, pp. 734–737.
   View PDF   View at publisher   SciX

2. Microsoft Copilot. (2026). Microsoft. Accessed: 24 September 2026. [AI model output, Microsoft Copilot application (cloud‑based service)] Available: Download the Microsoft Copilot app.

Citation

Citation details are available by clicking Cite this repository in the GitHub sidebar.

License

This repository is licensed under the BSD-3-Clause license. Details are available in the LICENSE file.