Statement of Problem Solar eclipses happen when the Moon passes between the Earth and the Sun, which then casts a shadow on the Earth's surface. The prediction of these events needs

a precise understanding and simulation of the complex gravitational interactions between the Earth, Moon, and Sun, governed by Newton's law of gravitation. The primary challenge lies in developing a computational model that accurately simulates these celestial bodies' orbital mechanics. The model must account for various factors such as the gravitational forces between the bodies, their initial positions and velocities, and the relative alignments necessary for an eclipse to occur. The goal is to predict not only the occurrence of a solar eclipse, but also its specific characteristics: the type of eclipse (total, partial, or annular), the date and time of occurrence, and the geographical regions on Earth where it will be visible. The problem is to create a reliable and user-friendly computational simulation based on Newtonian mechanics that can accurately predict solar eclipses./nScientific Merit Application of Classical Mechanics: Demonstrates the practical application of Newton's laws and universal gravitation in a real-world context. Understanding Celestial Mechanics: The project offers a deeper understanding of the dynamics of celestial bodies and provides insights into how gravitational interactions influence the orbits of planets and moons. Precision and Accuracy: The project highlights the importance of precision and accuracy in astronomical predictions. It can serve as a case study in how small changes in initial conditions or model parameters can significantly impact outcomes in dynamical systems. Verification and Validation of Models: By comparing the simulation results with real-world data, the project teaches the importance of model verification and validation. Public Engagement and Educational Value: Solar eclipses are events of significant public interest. The project can therefore be used as a tool for public education and outreach, making complex scientific concepts accessible and engaging to a broader audience. Foundation for Advanced Study: For those continuing in the field, the project is the simple model that can be advanced for more complex studies in orbital dynamics, space mission design, and even exoplanet discovery and study./nMethod 1. Theoretical Framework: - Use Newton's law of universal gravitation to calculate the gravitational forces between the Earth, Moon, and Sun. - Implement Newton's second law of motion to update the velocity and position of each body. 2. Numerical Simulation: - Employ numerical integration methods (Euler or Runge-Kutta) to solve the equations of motion. - Initialize the system with current positional data of the Earth, Moon, and Sun. 3. Eclipse Prediction: - Define geometric conditions under which a solar eclipse occurs (alignment of the Sun, Moon, and Earth). - Detect instances when these conditions are met during the simulation. - Calculate the path of totality on Earth's surface to determine where the eclipse will be visible. 4. Data Analysis: - Analyze the simulation data to find the next date and time when a solar eclipse occurs. - Determine the type of eclipse (total, partial, or annular) based on the relative positions and distances of the Moon and Sun from the Earth. Sub Ch Co/nEenpse Prediction. - Define geometric conditions under which a solar eclipse occurs (alignment of the Sun, Moon, and Earth). - Detect instances when these conditions are met during the simulation. - Calculate the path of totality on Earth's surface to determine where the eclipse will be visible. 4. Data Analysis: - Analyze the simulation data to find the next date and time when a solar eclipse occurs. - Determine the type of eclipse (total, partial, or annular) based on the relative positions and distances of the Moon and Sun from the Earth. 5. Visualization: - Create visualizations of the Earth, Moon, and Sun's orbits. - Develop a graphical representation of the solar eclipse's path of totality on the Earth's surface. 6. Verification: - Compare the predicted eclipse date with known solar eclipse data for accuracy verification. Proof of Concept Proof_of_Concept.m Sur Ch Co