Kepler model

The Kepler model was ceased upon by Sommerfeld to account for the quantized orbits and energies of the Bohr atomic model. By replacing the continuous range of classical action variables, restricting them to discrete values of... 

The capacity to solve novel problems by constructing analogies to already-used visualisations. (Gilbert, 2008). For example, using Kepler s model of the Solar System to explain the electronic structure of an atom, in the manner of Bohr, and hence being able to predict, very approximately, the absorption spectram that it will produce. 

Here Hd is the number of atoms in a unit cell, the volume of which is V, and is the shortest interatomic distance in the arrangement. The definition contains a division by /2 so that the parameter D becomes unity for close-packing structures. Kepler s conjecture ensures that the parameter D is always less than or equal to unity. The fraction of space occupied (fi in the rigid-sphere model, which is often used in the discussion of metallic structures, is proportional to the parameter D and the relation is as follows. 

The supernova models utilized below have all been generated by S. Woosley using the Kepler [7] supernova code. The evolution is followed with Kepler until the expansion becomes essentially homologous. At this point the Kepler information is transferred to the spectrum code. It is important to emphasize that once the supernova model has been selected the predicted spectra are generated without adjustable parameters. The adjustable parameters of the model are on the whole those with direct astrophysical relevance, for example the initial stellar mass and amount of mass loss. It is,.of course, feasible and useful to use the comparison of the observed and predicted spectra to refine the supernova model. This, however, has not yet been done except in a rough way. At present, the nebular spectra have been used only to choose broad classes of feasible models and mle out others. For example, the He detonation models for SNIa have been ruled out on the basis of the spectrum[8]. The results presented, therefore, must be viewed accordingly as generic to a whole class of models whose details are still unresolved. 

For theoretical chemistry to succeed it must develop the power to elucidate the behaviour of chemical substances to the satisfaction of experimental chemists, known to operate at many different levels. Understanding is not promoted by the generation of numbers, however accurate or numerous, without a simple picture that tells the story. It is inevitable that the chain of reasoning must reduce the problem of understanding the behaviour of substances, to the understanding of molecules, atoms, electrons, and eventually the aether. Again, this ladder of understanding should not be obscured by complicated mathematical relationships that cannot be projected into a simple picture. Small wonder that the planetary model of the atom, inspired by Kepler, and discredited almost a hundred years ago, is still the preferred icon to represent nuclear installations and activity in the commercial world. Theoretical chemistry should also communicate with the predominantly nonscientist population of the world, but in order to tell a story it is first of all necessary to know the story. 

The first quantitative atomic model appeared early in the previous century, based on the pioneering work of Lord Rutherford and the Danish physicist Niels Bohr. It was devised in simple analogy with Kepler s model of the solar system and, despite a number of known fatal defects, it has such intuitive appeal that, even today, scientists and non-scientists alike accept it as the most reasonable working model for understanding the distribution of electrons in atoms. Formulation of the model was guided by three important experimental observations which had no obvious explanation in terms of 19th century physics. 

An obvious possible improvement of the Bohr model was to bring it better into line with Kepler s model of the solar sxstem, which placed the planets in elliptical, rather than circular, orbits. Sommerfeld managed to solve this problem by the introduction of two extra quantum numbers in addition to the principal quantum number (n) of the Bohr model, and the formulation of general quantization rules for periodic systems, which contained the Bohr conjecture as a special case. 

Figure 2-54 shows Kepler and his planetary model based on the regular solids [84], According to this model the greatest distance of one planet from the sun stands in a fixed ratio to the least distance of the next outer planet from the sun. There are five ratios describing the distances of the six planets which were known to Kepler. A regular solid can be interposed between two adjacent planets so that the inner planet, when at its greatest distance from the sun, lays on the inscribed sphere of the solid, while the outer planet, when at its least distance, lays on the circumscribed sphere. 

However, the planetary model which is also a densest packing model probably symbolizes Kepler s best attempt at attaining a unified... 

Figure 2-54. Johannes Kepler on Hungarian memorial stamp and his Planetary Model based on the regular solids [87],...

The nuclear model of the atom, as envisioned by Rutherford and Bohr, had much in common with the solar system. In each there is a massive core that exerts a controlling influence over less massive satellites orbiting around the central core. In both the solar system and the atom, the force between the central core and the orbiting satellites decreases as the square of their separation. In the case of the solar system, it was Johannes Kepler, early in the seventeenth century, who first allowed hard data—data he knew to be accurate—to sit in judgment on his speculations about the orbits of the Sun s planets. 

Like Kepler, Bohr assumed circular orbits for the electron s motion around the nucleus unlike Kepler, however, Bohr was not guided by orthodoxy, but by reasons of simplicity. But what about those spectral lines which, upon close scrutiny, are not one line, but two. .. or more This is where Sonunerfeld enters the story. Sonunerfeld generalized the Bohr model by considering the more general orbit— the ellipse. Actually, the ellipse is the more likely orbit for an electron moving under the influence of the force exerted on it by the nucleus. The same is true for the orbital motion of the planets. A planet can have a circular orbit, but the condition for circularity is much more specialized and hence more unlikely. So Sonunerfeld relaxed the specific condition required for a circular orbit and considered an electron moving along an elliptical path. 

Newton tested his own ideas by rederiving the laws of Kepler, while Kepler had deduced his three laws from Tycho s observational data. So in fact, at the very foundation of modem Science we find a this very fruitful relationship between observation and theory. It is all too easy to forget that in the, not so distant past, the computers were humans [6]. To trace the pre-history behind the modem computers is yet another story [7]. In the case of Tycho Brahe, Johannes Kepler and Isaac Newton, using a modem vocabulary, it was Kepler who did the work of a computer , while Tycho Brahe provided the experimental evidence and Newton supplied the theoretical and mathematical models. Thanks to these pioneering scientists we perform our Molecular Dynamics simulations today [8-10]. 

Fig. 14.15. Nano- and macrocosmos/universe similarities schematic representation of the M0132 type cluster on the cover-page of Angew. Chem., highlighting its similarity to Kepler s early model of the universe and...

Regions of high flux predicted by the rat and monkey CFD models correlated with the distribution of formaldehyde-induced squamous metaplasia in nasal passages of rats exposed to 10 or 15 ppm formaldehyde for 6 months (Kimbell et al. 1993, 1997a) and of Rhesus monkeys exposed to 6 ppm formaldehyde for 6 weeks (Kepler et al. 1998). Results from these studies support the hypothesis that airflow patterns are key determinants of the amount of formaldehyde reaching the site of formaldehyde-induced nasal lesions. 

These results have led to the ongoing development of, for each of these species, anatomical models of nasal airflow and uptake, pharmacokinetic models for nasal tissue metabolism, and pharmacodynamic models of development of tumors and preneoplastic tissue changes to be applied to the rodent data to better estimate air levels that will present minimal risks for upper respiratory tract damage in humans (CUT 1998 Cohen Hubal et al. 1997 Conolly et al. 1992 Conolly and Andersen 1993 Kepler et al. 1998 Kimbell et al. 1993, 1997a, 1997b Morgan 1997 Morgan et al. 1991 Subramaniam et al. 1998). 

Kepler GM, Joyner DR, Fleishman A, et al. 1995. Method for obtaining accurate geometrical coordinates of nasal airways for computer dosimetry modeling and lesion mapping. Inhal Toxicol 7 1207-1224. 

Today, we recognize the main problem with Kepler s model Kepler did not understand that neither the number of planets nor their spacings are fundamental quantities that need to have an explanation from first principles. Rather, both are the result of historical accidents in the solar protoplanetary disk. Still, it is perfectly legitimate to give an anthropic explanation for earth s orbital radius. If that orbit were not in the continuously habitable zone around the sun (Kasting et al., 1993), we would not be here to ask the question. 

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. 

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