Simulation Explanation:

This simulation visually demonstrates the principles of orbital mechanics governed by Newtonian gravity. At its core, it models the two-body problem, specifically the interaction between a massive central star (the Sun) and a much smaller orbiting planet (the Earth). By adjusting parameters like the masses of the Sun and Earth, their initial distance, and the Earth's initial tangential velocity, you can observe fundamental concepts such as the conservation of angular momentum and specific orbital energy. The simulation calculates key orbital characteristics like eccentricity (e), semi-major axis (a), and semi-minor axis (b). Eccentricity, a measure of how much an orbit deviates from a perfect circle, is crucial: an 'e' value less than 1 results in an elliptical orbit (like planets in our solar system), 'e' equal to 1 signifies a parabolic escape trajectory, and 'e' greater than 1 represents a hyperbolic, unbound path. The simulation dynamically draws the resulting orbital path, tracks the planet's position and velocity, and even includes a polar graph illustrating its angular displacement (theta) and radial distance. A collision detection system highlights when the planet's path would intersect the Sun, demonstrating the dramatic consequences of insufficient orbital velocity, pausing the simulation and resetting parameters for a new exploration.

Physics Principles & Formulas:

• Newton's Law of Universal Gravitation: F = G * M * m / r^2
  This fundamental force dictates the gravitational attraction between the Sun and Earth, underlying all orbital motion.
• Specific Angular Momentum (h): h = r * v_tangential
  A conserved quantity in central force motion, explaining why the planet speeds up closer to the Sun and slows down farther away.
• Specific Orbital Energy (ε): ε = (1/2) * v^2 - G * M / r
  The total energy per unit mass, also conserved, which combines kinetic and potential energy to define the orbit's overall energy state.
• Orbital Eccentricity (e): e = sqrt(1 + (2 * ε * h^2) / (G^2 * M^2))
  Determines the shape of the orbit: e < 1 for an ellipse, e = 1 for a parabola, and e > 1 for a hyperbola.
• Semi-major Axis (a): a = -G * M / (2 * ε) (for ellipses)
  Defines the overall size of an elliptical orbit. For parabolic/hyperbolic, it's a parameter related to curvature.
• Semi-minor Axis (b): b = a * sqrt(1 - e^2) (for ellipses)
  Defines the 'width' of an elliptical orbit. For hyperbolic, it's the semi-conjugate axis.
• Kepler's Third Law (Orbital Period T): T = 2 * π * sqrt(a^3 / (G * M))
  Relates the orbital period to the semi-major axis and the central mass for elliptical orbits.