Michael Martin Nieto

Exact State and Fugacity Equations for the Ideal Quantum Gases

The fully relativistic expressions for the density, pressure, and energy density of the ideal quantum gases are solved exactly for the fugacity. This allows the exact, fully relativistic equations of state to be obtained for the ideal Fermi and Bose gases. All these expressions are studied in detail, including a discussion of physical implications and various limiting cases.

Quantum phase operators and the Josephson plasma resonance

SummaryUsing the recently developed quantum phase operators, a new derivation is given of the Josephson plasma resonance. The discussion emphasizes the natural and rigorous role that phase operators have in describing coherence phenomena.RiassuntoUsando gli operatori di fase quantistici recentemente sviluppati, si dà una nuova derivazione della risonanza del plasma di Josephson. Si sottolinea nella discussione il ruolo naturale e rigoroso che gli operatori di fase hanno nella descrizione di fenomeni di coerenza.РеэюмеИспольэуя недавно раэвитые квантовые фаэовые операторы, приводится новый вывод плаэменного реэонанса Джоэефсона. Обсуждение укаэывает на естественную и важную роль, которую фаэовые операторы играют при описании когерентных явлений.

Evidence for the Law's Validity

This chapter provides an overview of the evidence for the Titius–Bode Law validity. Recently analyses have been done on the statistical significance of apparent regularities in the solar system, to be explicit, on (1) commensurabilities in mean motion, (2) geometric progressions for planetary and satellite distances, and (3) resonance relations. These regularities are all pertinent to the existence of a generalized Titius–Bode type Law. Molchanov showed that there exist circumstances under which a maximum resonance structure is inevitable in a dissipative harmonic medium if enough time is given. Thus, one is left with the facts that commensurabilities and geometric progressions exist to a statistically significant degree with possibly a tendency toward total commensurability. The closest commensurability in the Blagg-Richardson progression is the bad fifth-order commensurability between bodies that are two orbits apart.

Search for a Geometric Progression Theory

This chapter discusses the search for a geometric progression theory. It describe theories that have been proposed to explain the geometric progression of the Titius–Bode Law or other related laws. In point of fact, in the Titius–Bode context, these theories consider the Law a simple progression and not a progression multiplied by what is called the evolution function. Progression theories can roughly be categorized as (1) electromagnetic, (2) gravitational, or (3) nebular theories, even though many of them have characteristics that could place them in more than one group. It is a fact that any claim that a particular theory of the solar system is correct as it explains the Titius–Bode Law is not valid because, as of yet, no theory has properly explained this progression.

Origin of the Solar System and Hoyle's Theory

This chapter discusses the origin of the solar system and the Hoyles theory. The solar system theories can be divided into two classes: (1) nebular theories and (2) encounter-capture theories. Hoyle gave an explanation of the very striking fact that the Sun has very little angular momentum compared with the planets. The theory also predicts that dwarf stars in general have planetary systems. In contradistinction, it further predicts that massive stars will be rotating rapidly (as is observed) as the lack of a deep convective zone in these stars will prevent the magnetic brake from working effectively. The periodic or evolution function represents a tendency towards commensurability due to a point gravitational or tidal evolution.

CHAPTER 6 – Early Modifications of the Law

This chapter discusses the early modifications of the laws of Titius–Bode Law. In 1785, von Zach pointed out in a sealed letter to Bode that there was a large deviation from the Law for the outer planets to the distances of Mercury and Venus. To rectify this, the Laws distance between Mercury and Venus needed to be smaller. Two years later Wurm accomplished this when he proposed what were the first modifications and further applications of the Titius–Bode Law. More importantly, Wurm was the first to take the point of view that if the Law has a significant dynamical origin, one would expect that a related Law exists for the satellite systems of the major planets. The next modification was made in 1802 by Gilbert who realized that the Law need not be tied to a geometric progression ratio of 2. A similar observation was made in 1828 by James Challis (1803–82), who suffered a bitter fate elsewhere in this story. Soon after the discovery of Neptune, a continuing interest in problems of this kind was manifested by Daniel Kirkwood. Kirkwood was fascinated with regularities in the solar system and ways to observe them. All of these modifications, however, lacked a fine degree of agreement with observation, often missing by quite a few percent, and there was a fair amount of arbitrariness in their parameterizations.

Blagg-Richardson Formulation

This chapter discusses the Blagg-Richardson formulation. The modifications of the Titius–Bode Law were mainly of the form of few equations and none were ever in exceptionally good agreement with all four systems of the Sun. This was changed by two formulations done in this century, which were impressively accurate and quite similar, although formulated independently. The first was done by Miss Mary Adela Blagg in 1913. Miss Blagg also looked at the satellite systems. Three decades later, Richardson did a similar analysis and came to the conclusion that the distance of some form. In any event, however, for the systems where the number of objects is large, the fit is striking. The results of Blagg and Richardson concludes that (1) a general Titius–Bode type Law can be obtained that fits all four systems of the Sun to a much better degree than the original formulations, (2) this Law is a geometric progression in multiplied by a periodic function, (3) with the progression in 1–73, there is no need for the first term of the original Titius–Bode Law and a much better fit is clearly obtained for the planetary system, and (4) the periodic function represents the deviation from a pure geometric progression. The relative deviation is not the same for the four systems but to good accuracy, it can be represented by the same mathematical function.

Periods of the Law's Creation

This chapter describes various periods of the laws creation. It describes the period in which the phenomenon which created the two portions of the Titius–Bode Law occur. The chapter describes the geometric progressions. The geometric progression in the Titius–Bode Law originated during Period I and is a manifestation of some fluid dynamical and/or electromagnetic process, that is, not a purely gravitational or tidal mechanism. In the solar system, commensurabilities between two solid-body satellites are stable but commensurabilities between a satellite and the small bodies of a disk system are not necessarily bound. In fact, some are destructive resonances. The evolution function was produced during Period III by tidal or point gravitational interactions.

Formulation of the Law

The fascinating story of the creation of the Titius Law began in Amsterdam in 1764 when the famous natural philosopher, Charles Bonnet, published his Contemplation de la Nature. This chapter discusses the Laws early successes and failures. When the orbits of Uranus and the asteroid belt were found to fall into place, the Law was highly accepted. But then questions arose concerning the lack of planets between Mercury and Venus and later the lack of agreement between the Law and the orbit of Neptune, let alone Pluto. The chapter also highlights the significance of the Law with respect to the origin of the solar system. It is interesting to note that although Titius called for a new planet in the position between Mars and Jupiter, he could not get himself to call it a chief planet, thinking it better to predict satellites. The proposals that the solar system had a nebular period, that the geometric progression originated in it, and that the progression was caused by some fluid and/or magnetohydrodynamical process were also made.

CHAPTER 12 – Electromagnetic Theories

Publisher Summary This chapter provides an overview of few electromagnetic theories. The first modern astrophysical theory of the Titius–Bode Law was proposed by Olaf Kristian Birkeland (1867–1917) in 1912. For two main reasons, the theory could not stand in its original form. First, it turns out that the value of the Suns field is not nearly strong enough to produce limiting radii within the present solar sys-tem. Also, according to this idea the chemical composition of all the planets should be quite different instead of being in two main groups, the terrestrial and the major planets. The second theory was the Berlages electromagnetic theories. His theories were forerunners of many others but did not receive the attention or credit due to them. The third theory was the Alfvens Theory. Alfven made the assertion that the outer major planets and inner satellites were due to a gas cloud condensing inward, whereas the inner terrestrial planets and the outer satellites were due to a separate encounter with a dust cloud which condensed outward.