Mass Ratio and Eccentricity Effects
Madeline Renee Boss
Samhitha Devi Kunadharaju
University of Texas at Austin
Course: CS 323E – Elements of Scientific Computing
This project presents a numerical study of the classical Newtonian two-body problem, focusing on how mass ratio and orbital eccentricity influence the resulting orbital dynamics. The two-body system is reduced to an equivalent one-body problem in the center-of-mass frame and integrated numerically using an adaptive Runge–Kutta (RK45) method via SciPy’s solve_ivp. Orbits are simulated for multiple mass ratios and eccentricities, and the individual trajectories of both bodies are reconstructed and visualized.
| File | Description |
|---|---|
two_body_code.ipynb |
Jupyter/Colab notebook containing the numerical integration, trajectory reconstruction, and orbit visualizations |
Two_Body_Paper.pdf |
Full paper describing the physical background, mathematical formulation, numerical methods, and results |
image.png |
Representative visualization of center-of-mass orbital trajectories |
requirements.txt |
List of dependencies (NumPy, Matplotlib, SciPy) |
- The two-body problem can be reduced to a single effective particle moving in a central gravitational potential.
- Eccentricity primarily controls the shape of the orbit (circular → elliptical).
- Mass ratio determines how orbital motion is distributed between the two bodies.
- In high mass-ratio systems, the heavier body remains nearly stationary while the lighter body traces a wide orbit.
- Numerical integration provides a flexible framework for studying gravitational dynamics beyond analytic solutions.
Mass ratios: 1:1, 1:2, 1:4, 1:16
Orbital eccentricities: 0, 0.25, 0.50, 0.75
Reference frame: Center-of-mass frame
Initial conditions: Orbits are initialized at pericenter with purely tangential velocity.
Numerical method: Adaptive Runge–Kutta integration (RK45)
Solver: scipy.integrate.solve_ivp
Integration interval: One full orbital period for each configuration
Output: Reconstructed trajectories of both bodies and orbit visualizations
This results in 16 total simulated orbital configurations.
- Increasing eccentricity elongates orbits and increases velocity variation between pericenter and apocenter.
- Increasing mass imbalance causes the heavier body’s motion to localize near the center of mass.
- The equal-mass case exhibits symmetric orbital motion, while extreme mass ratios resemble star–planet systems.
- All simulated orbits close cleanly after one period, indicating numerical stability and conservation behavior.
- Install the required dependencies
pip install -r requirements.txt- Launch the Jupyter notebook:
jupyter notebook two_body_code.ipynb- Run all cells to generate the numerical solutions and orbit visualizations.