Sitnikov Problem Simulator

Overview

This program simulates and visualizes the Sitnikov problem - a special case of the three-body problem in celestial mechanics. It consists of two primary bodies moving in elliptical orbits around their center of mass, while a third body moves along a line perpendicular to the orbital plane of the primaries.

Features

• Parallel computation using MPI (Message Passing Interface)
• Configurable orbital parameters via YAML configuration
• Phase portrait generation
• Multiple numerical integration methods support
• Progress tracking for parallel computations

Requirements

• Python 3.x
• NumPy
• SciPy
• Matplotlib
• mpi4py
• PyYAML

All requirements are listed in requirements.txt file for Python environment.

Installation

1. Clone the repository:

git clone https://github.com/StepanRepo/sitnikov-problem.git
cd sitnikov-problem

2. Create and activate a virtual environment (optional but recommended):

python -m venv venv
source venv/bin/activate  # On Unix/macOS
# or
.\venv\Scripts\activate  # On Windows

3. Install dependencies:

pip install -r requirements.txt

Configuration

The simulation parameters can be configured in config.yaml:

• orbit: Primary bodies' orbital parameters

  • a: Semi-major axis
  • e: Eccentricity (0 ≤ e < 1)

• grid: Integration parameters

  • nz: Number of position grid points
  • ndot: Number of velocity grid points
  • z0, z1: Position range
  • dotz0, dotz1: Velocity range
  • t0, tf: Time range

• integrator:

  • method: Integration method. Supported methods from scipy.integrate.solve_ivp:

    • 'RK45' (default): Explicit Runge-Kutta method of order 5(4)
    • 'RK23': Explicit Runge-Kutta method of order 3(2)
    • 'DOP853': Explicit Runge-Kutta method of order 8
    • 'Radau': Implicit Runge-Kutta method of the Radau IIA family of order 5
    • 'BDF': Implicit multi-step variable-order method
    • 'LSODA': Adams/BDF method with automatic stiffness detection

    For detailed information about these methods, see SciPy's solve_ivp documentation

• plotter: Visualization parameters

  • zmin, zmax: Position plot range
  • dotzmin, dotzmax: Velocity plot range
  • dpi: Output image resolution

• parallel:

  • track: Enable/disable progress tracking

Usage

Single-core execution:

python main.py

Parallel execution:

mpirun -n <number_of_processes> python main.py

Plotting results:

python plot.py

Output

The program generates:

• Data files in the result/ directory containing trajectories
• A phase portrait plot showing the system's dynamics

Mathematical Background

The Sitnikov problem describes the motion of a massless particle moving perpendicular to the plane of two equal masses (primaries) in elliptical orbits. The equations of motion are:

$$\ddot{z} = -\frac{z}{(r(t, e)^{2} + z^{2})^{3/2}}$$

where:

• $z$ is the position of the third body
• $r$ is the distance of the primaries from the center of mass
• $t$ is time
• $e$ is orbital eccenrisity