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
• References

Background

ForCLaP (Fortran Collinear Lagrange Points Calculator) 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 Stenborg [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
forclap.f90	Fortran program.

Software Requirements

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

Quality Assurance

ForCLaP has been tested with the GNU Fortran (gfortran) compiler, version 15.1.0.

Getting Started

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

gfortran forclap.f90 -o forclap

References

1. Stenborg, TN 2008, "Collinear Lagrange Point Solutions in the Circular Restricted Three-Body Problem with Radiation Pressure using Fortran", in RW Argyle, PS Bunclark and JR Lewis (eds), Astronomical Data Analysis Software and Systems XVII, vol. 394, Astronomical Society of the Pacific, Provo, UT, pp. 734–737.
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