The meaning of the laws of Kepler is well described in their statements, which you can find in any textbook:
- First Law, or law of the elliptical orbits:
The orbit of a planet is an ellipse with the Sun at one of the two foci.
- Second Law, or law of the areas:
A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
- Third Laws, or laws of the periods:
The square of a planet’s orbital period is proportional to the cube of the length of the semi-major axis of its orbit.
It that all? Well, yes… just as Leonardo’s Mona Lisa might be described as a small painting depicting a girl with folded arms and a half smile.
In fact, behind both products of the human genius, there is much more than that. As for Kepler’s laws, over and above their astronomical value, their importance extends to several fields, such as:
- classical Mathematics (Geometry and Trigonometry)
- Physics (they inspired Newton’s Law of universal gravitation, and constitute clear examples of the preservation of physical quantities, such as energy and the angular moment)
- differential or infinitesimal calculus (also invented by Newton, and Leibniz, just to solve those laws of motion),
- philosophy, as well as the history of human progress.
It goes without saying that, without Kepler’s Laws and the scientific progress they inspired, humankind would not have been able to successfully send missions to explore our Solar System, and more broadly, would have a much narrower and incorrect view of the Universe.
The activity we are introducing here does not pretend to be a “course”, but rather a “path” which, with the idea of learning the meaning of these three laws better, will touch many mathematical and physical concepts. At the same time, there will be the chance to face programming challenges for increasingly complex problems. To this aim, we will use the Scratch language, which certainly is not the quickest and most efficient one for numerical calculus, but offers big advantages, such as the intuitive approach to coding and the easy management of interactive interfaces.
To the various stages of this path will be associated programs, not only to be run passively, but whose code should be read and especially understood (that’s why we shall try to make them more understandable, with some explanatory comment) and, in our intentions, these programs should also be used as a starting point for experiments of coding.
Here are the stages, one after the other, with the codes in order to carry them out:
- Draw an ellipse, centered with respect to the intersection of its axes. Identify some significant elements of an ellipse (major axis, minor axis, eccentricity, focal distance).
- Draw an ellipse, centered with respect to one of its foci. Identify other elements of the orbit (right half-side), describe the relationships among them, with rispect to certain dynamical quantities, in the case of an orbit.
- Calculate the orbital motion of a planet around a star, while calculating dynamical equations, first by using a simplified algorithm, then a more sophisticated one (Runge-Kutta method). By calculating several dynamic and geometric quantities, we can carry out a series of tests, such as for example on the total energy constancy and angular momentum, and on the effetcive dependence of orbit parameters on physical constants, in accordance with the formulas introduced in the previus step.
Future Steps (work in progress)
- Dynamical calculus of the orbital motions of two bodies with comparable masses. Introducing the concept of mass centre, and system’s reduced mass.
- Comparison between the law of gravitational attraction, which scales with radial distance 1/R2, and laws scaling as 1/Rn, with n other than 2. Analysis of the precession of the perihelion, as well as of the orbital stability (orbits reaching the central star).
- Reducing the problem of the orbit to a one-dimensional problem. Introducing the concept of Effective Potential, and extending it to cases with n other than 2.
- Open orbits: parabolic and hyperbolic.
- Impact of meteorites upon the Earth. Dependence of the Earth impact section according to relative velocity.
- Motion under the gravitational effect of two major bodies: planets’ orbits around a binary stellar system; the orbits of a satellite subject to the attraction of a star and a planet (Lagrangian points; Greek and Trojan asteroids).
- Motions of 2 planets around a star, and mutual disturbance of their orbits.
- Shepherd moons and Saturn’s rings.
- “Non”-gravitational motions. Double tail of comets.
- Rocket travel in a star-planet system.
- Gravitational slingshot method.
