Kepler's World

A simple simulation (in Python) of movements of celestial bodies around the Sun for the purpose of studying Kepler's laws and how they emerge from Newton's law of gravity. Designed for high school students or lower level undergraduate students.

Movements of the eight planets are simulated, as well as movements of some celestial bodes made up for the purpose of experiments. The simulation uses a discrete approximation of the real-world continuous effect of the Newton's law of gravity (F = GMm/(r^2) on the moving body, taking into account only that body and the Sun (the "two-body problem"). The calculations are done for the actual celestial body, and the results are scaled down to be visualized in turtle graphics. The simulation tests if the resulting orbits and bodies' movements obey Kepler's laws.

Concepts of the methodology of science:

• Branches of natural science: physics, chemistry, biology, Earth and atmospheric (planetary) science.
• Natural science vs. mathematics and logic
• Empirical evidence.
• Methods of science: experimental/observational, theoretical/mathematical, computational.
• Laws of science vs theorems of mathematics Newton's laws and Kepler's laws, theorems about ellipses.
• The scientific method.
• Units of measurement and dimensional analysis: units of the gravitational constant G, Vis Viva Equation: v^2 = G(M+m)(2/r - 1/a) Orbital period T = 2 pi sqrt( a^3 / (G(M+m) )
• Discrete simulation of a continuous process.
• Approximation and approximation error.
• Absolute error and relative error.
• Rounding numbers.
• Floating-point arithmetic
• Recurrent/periodic process - for instance planet's movement around the Sun.
• Qualitative vs. quantitative statements ("a planet moves fastest when close to the Sun" vs. Kepler's 2nd law).
• Local effects (of the law of gravity) vs. global properties (of orbits described by Kepler's laws).
• Visualization vs. simulation.
• Emergence (of Kepler's laws from Newton's laws) or reduction of (Kepler's laws to Newton's laws). Note: Newton gave a (mathematical) proof, using calculus.

The visualization shows orbits to scale, but disregards some details not relevant to Kepler's laws. Namely, in this simulation (unlike in reality):

• the orbits of all planets are in the same plane, and
• the foci of elliptical orbits of all planets are on the same straight line.

The period in which the turtle completes the orbit is proportional to the actual orbital period of the planet, but the scaling factor is not the same as for the orbit's size.

A slideshow is also available: Discovering the Mechanics of the Solar System.