Planetary motion

Isaac Newton modestly claimed to have stood on the shoulders of giants to explain how he was able to achieve his outstanding discoveries in the fields of gravitational attraction and planetary motion our Giants in the field of electromagnetic induction would probably have said the same. 

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

Hydrodynamic marked the beginning of fluid dynamics—the study of the way fluids and gases behave. Each particle in a gas obeys Isaac Newton s laws of motion, but instead of simple planetary motion, a much richer variety of behavior can be observed. In the third century B.C.E., Archimedes of Syracuse studied fluids at rest, hydrostatics, but it was nearly 2,000 years before Daniel Bernoulli took the next step. Using calculus, he combined Archimedes idea of pressure with Newton s laws of motion. Fluid dynamics is a vast area of study that can be used to describe many phenomena, from the study of simple fluids such as water, to the behavior of the plasma in the interior of stars, and even interstellar gases. 

Some historical data, meaningful in the experience of early scientists, might not be meaningful or accessible to modem students. For example, observational data about planetary motion (or even the nature of planetary motion itself) is not something which easily eonneets with the everyday experienee of students today. 

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. 

Pan mixers Vertical, rotating paddles, often with planetary motion Mixing, whipping and kneading of materials ranging from low viscosity pastes to stiff doughs Food, pharmaceuticals and chemicals, printing inks and ceramics... 

As an example of the insufficiency of present usefulness and self-consistency as grounds for belief in a scientific construct, it may be in order to recall some scientific history. In our own field we have the familiar example of phlogiston and in astronomy the example of epicycles. By the use of epicycle superimposed on epicycle, the geocentric theory was able to give a self-consistent, popular, and accurate description of apparent planetary motions. The epicycle treatment is analogous to a Fourier analysis of the motions and its accuracy did not guarantee the physical reality of epicycles. 

Within the solar system the observable changes are of a different kind, best described as chemical change. The most striking common feature of those chemical reactions driven by solar energy is their cyclic nature, linked to planetary motion. All phenomena, collectively known as life, or growth, are of this type. Their essential characteristic is a state far from equilibrium. For a life process, equilibrium is synonymous with death and chemical change after death is a rapid slide towards equilibrium. The most advanced chemical theories deal with these posthumous effects and related reactions only, albeit rather superficially. A fundamental theory to predict conditions for the onset of elementary chemical change is not available. 

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. 

Metabolic transformations are characterized by high speed and yield, as well as high regio-, diastereo-and enantio-specificity. Errors in the stereochemistry of the molecules that serve to construct the genetic material are smaller than for the planetary motions. With secondary metabolites, however, enantiomerically inq)ure con unds are also encoimtered, typkally with monoterpenes and alkaloids from terrestrial plants even ant dal pathways in the same organism have been found, albeit as rare events (Guella 1998). 

In hydrodynamic columns, there are spools of coiled tube that rotate on themselves and around a central axis. These combined rotations create a planetary motion with a highly variable centrifugal field that produces mixing zones followed by decanfafion zones (Figure 7.3). The stationary phase is partly retained inside the coils if fhe mobile phase is flown the right way. The coil rotation produces an Archimedean force that pushes the liquid phases toward one end of the coil called the head (higher pressure). 

There is an interesting parallel between the introduction of Newtonian mechanics and the introduction of Lavoisierian chemistry. As long as the planetary motions were only described, it was possible to evaluate all systems only on the basis of their accuracy of prediction. By this standard, the Ptolemaic and Copernican systems were not far apart and one might prefer... 

Tn the Rohr model of the hydrogen atom, the proton is a massive positive point charge about which the electron moves. By placing quantum mechanical conditions upon an otherwise classical planetary motion of the electron, Bohr explained the lines observed in optical spectra as transitions between discrete quantum mechanical energy states. Except for hvperfine splitting, which is a minute decomposition of spectrum lines into a group of closely spaced lines, the proton plays a passive role in the mechanics of the hydrogen atom, It simply provides the attractive central force field for the electron,... 

The planetary model of the atom was proposed by Rutherford in 1912 following the a particle scattering experiments of Geiger and Marsden, which showed most the mass of an atom to be concentrated in a tiny positive nucleus. The orbiting of light electrons resembles the problem of planetary motion first solved by Newton. 

In the context of planetary motion, this relationship between the square of the orbital period (inversely proportional to ft)) and the cube of the orbital radius is known as Kepler s Third Law. 

The latter half of the nineteenth century was a time of intellectual triumph in the physical sciences. Most of the material contained in the first year of modem college physics courses was completely understood by then. Newton s laws had been rephrased in different mathematical forms which simplified even complicated many-body problems such as planetary motion. In addition, the description of electric and magnetic fields by Maxwell s equations was an essentially complete success—so much so that these equations and their consequences are the central focus of some graduate physics courses even today. 

The most important problem that has been solved in detail by using action-angle variables is the Kepler problem of planetary motion. The details of the analysis are not important in the present context, but the form of the Hamiltonian for rotation in a central potential V r) = —k/r, obtained as... 

The first known sketches of snowflakes from Europe in the sixteenth century did not reflect their hexagonal shape. Johannes Kepler was the first in Europe, who recognized the hexagonal symmetry of the snowflakes as he described it in his Latin tractate entitled The Six-cornered Snowflake published in 1611 [24], By this time Kepler had already discovered the first two laws of planetary motion and thus found the true celestial geometry when he turned his... 

Intimate mixing is accomplished in change-can mixers in two ways. One method is to have the mixing-unit assembly revolve in a planetary motion so that the rotating blades sweep the entire circumference of the can (Fig. 18-39). The other is to mount the can on a rotating turntable so that all parts of the can wall pass fixed scraper blades or the agitator blades at a point of minimum clearance. 

While these results agreed with planetary motion known at the time there was now an explanation for differences in motions. These solutions were equally valid for applying to any systems body Earth s moon, Jupiter s moons, galaxies, truly universal. 

Planetary mixer instrumentation for direct torque measurement does not substantially differ from that of a high shear mixer. Engineering design should only take into account the planetary motion in addition to shaft rotation. ... 

The cross-axis CPC produces a unique mode of planetary motion, such that the column holder rotates about its horizontal axis while revolving around the vertical axis of the centrifuge.This motion provides satisfactory retention of the stationary phase for viscous, low-interfacial tension, two-phase solvent systems, such as aqueous-aqueous polymer phase systems. Our previous studies demonstrated that the cross-axis CPC equipped with a pair of multiplayer coils or eccentric coil assemblies in the off-center position was very useful for the separation of proteins with polyethylene glycol-potassium phosphate solvent systems.The apparatus is also useful for the separation of highly polar compounds such as sugars,hippuric acid, and related compounds, which require the use of polar two-phase solvent systems. 

The cross-axis CPC produces a synchronous planetary motion of the column holder, which rotates about its horizontal axis and simultaneously revolves around the vertical axis of the apparatus at the same angular velocity. In the X—1.5L type of the apparatus, the column holder... 

The type-J synchronous planetary motion of the coil holder is shown in Fig. 5a (see also instrumentation of... 

The acceleration produced by the planetary motion is then obtained from the second derivatives of Eqs. 8 and 9,... 

Fig. 5 Analysis of centrifugal force field for type-J planetary motion, (a) Planetary motion (b) planetary motion in an x-y coordinate system for the analysis of centrifugal force (c) distribution of the eentrifugal vectors on the column holder. (From Ref. [12].)...

Fig. 6 Mixing and settling zones in the spiral column undergoing type-J planetary motion. (From Ref. [12].)...

The design of the cross-axis CPC is based on the hybrid between type-L and type-X planetary motions, which results in an extremely complex centrifugal force field with a three-dimensional fluctuation of force vectors during each revolution cycle of the holder. The pattern of this centrifugal force field produced by the cross-axis CPC somewhat resembles that produced by the type-J planetary motion (Fig. 5c), but it is superimposed by a force component acting in parallel to the axis of the coil holder. This additional force component acts to improve the retention of the stationary phase. This beneficial effect is greatest in type-L planetary motion and becomes smallest in the type-X planetary motion. A detailed mathematical analysis on this planetary motion is described elsewhere. 

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