Kepler’s law

Historical prelude Kepler s laws Historical prelude Maxwell Theory Axiomatic teaching of Quantum Mechanics Problem lack of reference points Problem imprecise boundaries Problem inaccurate formulation Solution reference points from a journey Solution precise boundaries Solution accurate formulation Intuitive teaching of Quantum Mechanics Conclusion... 

In any course of NM, one of the first applications is the derivation of Kepler s laws of planetary motion. Historically this is one of the great triumphs of NM. Kepler s laws state that the orbits of the planets around the sun are ellipses with the sun in one of the focal points and that the speed of the planets is such that equal areas inside of the ellipse are swept in equal times. 

Now consider the hypothetical problem of trying to teach the physics of space flight during the period in time between the formulation of Kepler s laws and the publication of Newton s laws. Such a course would introduce Kepler s laws to explain why all spacecraft proceed on elliptical orbits around a nearby heavenly body with the center of mass of that heavenly body in one of the focal points. It would further introduce a second principle to describe course corrections, and define the orbital jump to go from one ellipse to another. It would present a table for each type of known spacecraft with the bum time for its rockets to go from one tabulated course to another reachable tabulated course. Students completing this course could run mission control, but they would be confused about what is going on during the orbital jump and how it follows from Kepler s laws. 

The principles of Kepler s laws and orbital jumps in isolation would leave students confused. Alternatively suppose students were taught that only free-falling space flight can be understood from Kepler s laws, and that the tables for course corrections had been constructed from careful experimentations and observations. In this case, students would not be confused either. The confusion comes from stating that everything will be explained theoretically and then only explaining half. 

However, in Maxwell s days everyone assumed that there had to be a mechanical underpinning for the theory of EM. Many researchers worked on very detailed hidden variable theories for the EM field, in an attempt to prove that the laws of EM were in fact a theorem in NM, just like Kepler s laws are a theorem in NM. No one noticed that it was impossible to do this, since Maxwell s equations are not Galilei invariant and Newton s laws are. That includes Lorentz who discovered around 1900 that the Maxwell equations are invariant under another transformation that now bears his name. 

This separation will allow the students to properly assess the measurement process, which plays a special and complex role in QM that is different from its role in any classical theory. Just as Kepler s laws only cover the free-falling part of the trajectories and the course corrections, essential as they may be, require tabulated data, so too in QM, it should be made clear that the Schrbdinger equation governs the dynamics of QM systems only and measurements, for now, must be treated by separate mles. Thus the problem of inaccurate boundaries of applicability can be addressed by clearly separating the two incompatible principles governing the change of the wave function the Schrbdinger equation for smooth evolution as one, and the measurement process with the collapse of the wave function as the other. 

The coefficient k is one of the fundamental constants of physics and astronomy. From Kepler s laws it is possible to express the masses of all planets in terms of the mass of the sun. However, in order to find the mass of the sun and. 

In 1687, Newton summarized his discoveries in terrestrial and celestial mechanics in his Philosophiae naturalis principia mathematica (Mathematical Principles of Natural Philosophy), one of the greatest milestones in the history of science. In this work he showed how his (45) principle of universal gravitation provided an explanation both of falling bodies on the earth and of the motions of planets, comets, and other bodies in the heavens. The first part of the Principia, devoted to dynamics, includes Newton s three laws of motion the second part to fluid motion and other topics and the third part to the system of the (50) world, in which, among other things, he provides an explanation of Kepler s laws of planetary motion. 

To treat this example by dimensional analysis, the acceleration due to gravity (g = 9.81 m s-2) had to be known, this being calculated from the gravitational law. Sir Isaak Newton derived it from Kepler s laws of planet movements. Bridgman s ([1], p.12) comment on this situation is particularly appropriate ... 

See also Celestial mechanics Kepler s laws Space probe. 

See also Celestial mechanics Geocentric theory Heliocentric theory Kepler s laws. 

As an example of the above rules we shall now give a discussion of the hydrogen atom, the complete quantisation of whicli was carried out by Sommerfeld. By Kepler s laws, the orbit of the electron round the nucleus is an ellipse it is therefore simply periodic. Since the... 

Our treatment of acid-base equilibria so far has been based on the mass action law, i.e., on the constancy of the equilibrium constants. Comparison with experiment shows that this relatively simple model is by and large correct, just as it would be essentially correct to say that the earth rotates around the sun according to Kepler s laws. If one looks much closer, one will find that it is not quite so, but that the influence of the moon must be taken into account as a small correction if a more precise description is required. In fact, there is a hierarchy of corrections here, starting with the influence of the moon, then that of the planets, and eventually that of all other heavenly bodies. Although it might appear to be a hopeless task to include an almost endless number of stars and galaxies, in practice the list of effects we need to include is restricted by the limitations on the experimental precision of our measurements, and a simple hierarchy of corrections suffices for all practical purposes. A similar situation applies to acid-base equilibria. 

The mass action law formalism, through its equilibrium constants, takes into account the interactions of the solvent with the various acids, bases, and salts these certainly are the dominant effects, comparable to Kepler s law in the above analogy. However, the formalism of the mass action law does not explicitly consider the mutual interaction of the solute particles, nor the effect of these solutes on the concentration of the solvent. Activity coefficients /have therefore been introduced in order to incorporate such secondary effects they are individual correction factors that multiply... 

Kepler s laws - The three laws of planetary motion, which established the elliptical shape of planetary orbits and the relation between orbital dimensions and the period of rotation. 

With the law of gravity, Newton was able to explain why Kepler s laws described the planetary motion. The law of gravity is an example of a central force. The force is directed along the center of mass of two bodies. The mathematical formulation is given as follows ... 

Galileo formulated some mechanical principles that enabled Newton to establish the physical basis of Kepler s laws. By noticing that a ball which rolls on a plane that slopes downward is accelerated and one on a plane that slopes upward is retarted, he argued that motion along horizontal planes should be uniform and perpetual. This view represents an important advance... 

The development in celestial mechanics after Newton was largely in the hands of the French mathematician Pierre-Simon Laplace (1749-1827). The stability of the solar system was the major unsolved problem. Neither Kepler s laws nor Newton s mechanics could be applied successfully to more than a single orbit at a time. The imiversal law of gravitation must clearly apply to any pair of celestial bodies and with several planets and moons circling the sun it is inevitable that mutual perturbations of the predicted perfect elliptical orbits should occur. Newton himself could never precisely model not even the lunar motion and concluded that divine intervention was periodically necessary to maintain the equilibrium of the solar system. 

Heavenly bodies are observed to move Apparent heavenly movement appears to be related to seasonal changes Kepler s laws describe planetary motion... 

Kepler s laws Three laws of planetary motion formulated by Johannes Kepler on the basis of observations made by Tycho Brahe. Kepler pubUshed the first and second laws in 1609 and the third in 1619. The laws state that (1) the orbits of the planets are elliptical with the sun at one focus of the ellipse (2) each planet revolves around the sun so that an imaginary Une (the radius vector) coimecting the planet to the sun sweeps out equal areas in equal time periods (3) the ratio of the square of each planet s sidereal period to the cube of its distance from the sun is a constant for aR the planets. 

These are problem relative - there is a model for the blood system of a mammal and another for the nervous system, both abstracted from the body of the animal in question. To create an analytical model the anatomist must be able to observe the body of the animal in question as a concrete source of the abstract lay out of the various anatomical systems that can be represented in diagrams. Newton s model of the solar system as a system of perfect material spheres obeying Kepler s laws of motion and the inverse square law of gravitational attraction was derived by abstraction and ideahzation of observable fears of the actual solar system (Frigg 2010 251-268). Analytical models can be wholly pictorial as in anatomy or they can be abstract and partially mathematical as in the Newtonian cosmology. 

For science the enlightened approach was to reject the mystical in favor of clear, Cartesian logic. Newton and Descartes had made it appear that all systems in the natural world could eventually be described by mathematical formulas, and for such disciplines as physics, the approach brought gratifying results Newton s gravity and his laws of motion described the behavior of objects on Earth, and Kepler s law described the behavior of the planets. But then physicists were dealing with the macroscopic world—a world they could see or manipulate. Chemists have never had this luxury. The scale of their world is atomic, and the activity of atoms has to be inferred rather than observed. So in the early 1700s chemists did not yet have the tools to reduce their systems to exact mathematical models. But they had the desire. 

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