Patent Number: 050842323
Section: summary

The present invention provides: (1) The precise and the unique solution of a previously unsolved P.sub.2 targeting problem; (2) Impacts to the governmental NRC nuclear safety standards, DOD weaponary systems development and many other activities in NASA and in the Department of Energy; (3) Impacts to update the contents of text books of physics and mathematics of all levels for education; (4) Impacts to the designs of scientific intrumentations with applications in high technologies. In conclusion, the invention of TRAJECTORY SOLID ANGLE provides a revolutionary concept in the fundation of physics. It it is confirmed to be true, it will open a new era for researches in physics, mathematics, engineering, advanced instrumentations and scientific measurements. BACKGROUND OF THE INVENTION Back to 1974 and earlier, the U.S. Atomic Energy Commission (AEC) (now the Department of Energy (DOE) and the Nuclear Regulatory Commission (NRC) openly solicited the solution of an unsolved problem which can be equivalently stated as: TO DETERMINE AND DEFINE THE PROBABILITY FUNCTION P.sub.2 FOR A PARTICLE TO HIT A PREDESIGNATED AREA, GIVEN ALL ITS PARAMETERS OF GENERATION AND EJECTION. Responding to the solicitation to solve the problem, a committee of scientists, mathematicians, and engineers from many companies was formed. The companies included: Westinghouse, General Electric, Raytheon, Stanford Research Institute, . . . and many other companies. They produced many reports to solve the problem. Those reports were circulated from one company to the other for the participants to provide mutual reviews and feedbacks in order to obtain the TRUE solution. The inventor, working then in 1974 at Stone & Webster Engineering Co. in Boston, Mass., was also assigned to review and evaluate some of those reports. The inventor was also requested to provide his own solution of the problem in addition to the assignment of reviewing others' reports. As a result, the author invented in October 1974 a new physical term "TRAJECTORY SOLID ANGLE" (TSA) to solve the solicited problem. The TSA was a new name having been called by the inventor in order to identify for its difference from the very well-known Geometric Solid Angle (GSA). Before October 1974, there was no such name as (TSA) in the nomenclature of science and engineering. On the other hand, the Geometric Solid Angle (GSA) has been very well known to all. The original hand-written report in which the TSA was first invented has been kept and saved by the Stone & Webster Engineering Co. since October 1974. Only copies of this original work together with two other topics original work (muti-reservoir transient problems' formulation and solution, and the solution of indeterminate structural systems' problem) were returned to the inventor in early 1975 by Stone & Webster Engineering Co. when the inventor was separated from the company. All these can be seen from the evidences of a copy of the inventor's Mar. 29, 1975 letter to Mr. V. A. Suziedelis, Senior Engineering Manager and Vice President, Stone & Webster Engineering Co. and the inventor's Sept. 8, 1977 letter to Dr. Saul Levine, Director, Office of Nuclear Regulatory Research, Nuclear Regulatory Commission (NRC). These important letters can be found from the cited reference No. [17] which are documented with: (1) reponses to the review and comments about the paper 82-IHTC-86" ON THE INITIATION OF TRAJECTORY SOLID ANGLE AND ITS INFLUENCE TO RADIATIVE HEAT TRANSFER" in 1981; (2) reponses to the review from NRC about the TSA proposal; (3) responses to the U.S. Army Missile Research and Development Command's review about TSA; (4) responses to the U.S. Army Ballistic Research Laboratory of Aberdeen Proving Ground's review about TSA; (5) reponse to the review from Professor Walter Hauser of Northeastern University. The responses to the reviews for DOE proposal No. P7900450 (cited reference [7]) can be seen from cited reference No. [15]. As indicated in cited references [17] and [15], the TSA concept has been rejected from one agency to the other. It was enrouted for review from NRC to BRL to U.S. Army Misslie R&D Command; to National Science Foundation, AFSC-AFAL, AFSC-SAMSO, AFSC-AFOSR, AFSC-RADC, AFSC-AFGL, AFSC-ESD, EPRI, ERDA, JCM-20, SER 211/103 and back to DOE high energy physics division in Jan. 17, 1979 again. It was unfortunate that the reviewers from DOE rejected the proposal again and DOE advised that the inventor should send it to National Science Foundation for support. Again the reviewers of DAR of NSF rejected the proposal and it was transferred to the Physics Division of NSF. It is unfortunate that the reviewers rejected the proposal again. The continuous rejections to accept the concept of TSA by the Federal Agencies; Academic institutions; numerous scientific journals have forced the inventor to take two actions: (1) Formally file the patent of invention of TRAJECTORY SOLID ANGLE; (2) Openly challenge scientists in the 1989 AAAS Annual Meeting, Jan. 14-19, 1989 in San Francisco by sending the cited reference [19] to 11 session organizers; by presenting the TSA papers (cited reference [1] and [2]); by appearing himself in most of the related sessions to discuss what, why, how the TSA concept being important to them and that they have been taking the wrong track to solve their problems. The action of filing the TSA patent was advised by Dr. John Lyons, Director of Engineering, and Dr. George A. Sinnott, Associate Director of Technical Evaluation, both of National Bureau of Standards (NBS) in April 1985. As a result of their advices, the patent for the invention of TRAJECTORY SOLID ANGLE was initially filed in the attachments of an open letter dated Jan. 7, 1986 sent to the Commissioner of Patents and to the Executive Heads both in the Federal and State government of Mass. and to some members of the U.S. Congress. The action of open challenges to the scientists in the AAAS 1989 January Meeting have been done in last month. The TSA proposal is again submitted back to NSF for support. The current proposals pending supports from NSF are cited reference No.: [20], [21], [22]. Thus far, since the invention of the TRAKECTORY SOLID ANGLE by the inventor in October 1974, it has been continuously developed alone solely by the inventor. It has never been funded by any organizations; Federal; state; local and public. All the proposals and technical information, papers and data are strictly proprietary. SUMMARY OF THE INVENTION As indicated in the BACKGROUND OF THE INVENTION, the most relevant work proposed before 1974 in U.S.A. to solve the P.sub.2 targeting problem can be traced from the references of and the reports by Sermanderes [3], Bush [4], Shaffer et al [5], and by the partial notes [6] of a PSAR (Preliminary Safety Analysis Report) of Delmarva Power & Light Company Summit Stattion, April 1974. These four reports concentrated in solving the same example of the solicited targeting problem that is to find the probability function P.sub.2 for a ejected turbine missile to hit a predesignated area assuming it moves in vacuum and under a constant gravitation. A solution of this looks-like-a-simple problem was also proposed by the inventor [7] who was simultaneously assigned to make a detail technical review of those reports. As can be seen from the 20 pages hand-written summary that was in the proposal by the inventor [7], the Geometric Solid Angle (GSA) approach to solve the P.sub.2 targeting problem was discussed in the reports by Sermanderes [3] and by [6]. However, as indicated by the inventor at that time the Geometric Solid Angle (GSA) is only one of the very special case of the invented TRAJECTORY SOLID ANGLE (TSA). The (TSA) was invented without proof at that time making an analogy with the (GSA) that had been used for years for calculation of the scattering & collision cross-section of particles; radar scanning cross-sections; geometric shape factors in radiative heat transfer; and in optics. The (TSA) was invented in a mood of intant reflection at a critical time when the author was told to write a final report on the P.sub.2 targeting problem after he had been working and reviewing the work by others for a continuous period of three months. Even since then the (TSA) has been continuously developed alone by the inventor to examine its validity and truth in solving various kind of problems related to Statistical Mechanics despite of the confrontation and oppositions from all sources. After his separation from Stone & Webster Engineering Company, the inventor was forced to form SYSTEMS RESEARCH COMPANY in order to survive from being unemployed. The SYSTEMS RESEARCH COMPANY's Jan. 17, 1979 unsolicited proposal [7] had provided the various targeting problems having been solved and had indicated the approaches and research plans step by step through all the responses by the inventor to the mutual reviews and discussions with professionals in U.S. governmental, academic, and industrial organizations. It was and still is the current opinion of the inventor that a new statistical method has been found through the definition of (TSA). This new parametric statistics, being characterized and derived from fractional ratio or from the intersection and union of the acceptable laws of physics and the set theory of mathematics, is applicable for a macropic body as well as for a microscopic particle of mathematically defined infinitestimal size under the action of any force and moment fields. This new parametric statistics, being different from the usual consequential statistics or non-parametric statistics, could have been also the essential element that was missed by Einstein in his incomplete work for the unified field theory. With the precise definition of (TSA) and the definite procedures to find the probability function P.sub.2, the probability density function (pdf) for the particle hitting on a surface can be also determined. Thus the physical quantities like dynamic pressure, density, linear momentum, angular momentum and kinetic energy distribution of the particle at any location can also be obtained by integration of the product of the physical quantity with the (pdf) over the predesignated area. This is a standard procedure in statistics to find the expectation values of any functions once the (pdf) is obtained. The important role of Geometric Solid Angle (GSA) in classical and modern physics and its relationship to the inventor's TRAJECTORY SOLID ANGLE (TSA) can further be compared and described in the following: The Geometric Solid Angle (GSA), a mathematical definition from differential geometry well known to scientists and engineers for years, has been applied to study the theory of scattering of particles that was summarized and provided with detail references by Watson [8]. It has been also appied to study the kinetic theory of ideal gas as shown by Lee et al [9] and the radiative heat transfer by Hottel et al [10]. The six-dimensional phase space, used in kinetic theory of gases and originated by Maxwell and Boltzmann, has become the foundation of statistica and quantum mechanics. The main idea of statistical mechanics is to apply the laws of probability and methods of statistics to study the mechanics of particles and bodies. Thus the problem of studying the probability of a particle striking on a predesignated area or on another particle in various force fields has been the central issue in statistical mechanics. When this is applied to the photon that carries a definite amount of energy, it becomes the subject of quantum mechanics. However, as asserted by Park [11], the establishment of a rigorous footing on statistical quantum mechanics from the point of view of quantum mechanicists seems to be difficult. This explained why, aside from all known approaches for the P.sub.2 targeting problem, alternative means were continuously sought for a firm answer before and even after 1974 when the (TSA) was first invented. As one traces through all the references as cited, one will find that the (GSA) has been used in various topics for fundamental analysis, calculation and comparison with experiments in both classical and modern physics. Since it has been widely used for a long period of time and thus it was and still is considered by many that the (GSA) being used in various topics of classical and modern physics is the solution of the P.sub.2 targeting problem. This can be challenged by the invention of the (TSA) which contains (GSA) as one of its special case of millions. In summary, the advantages of the invention over all other methods in the past to solve the P.sub.2 targeting problem are: 1. The invention of TRAJECTORY SOLID ANGLE provides the most precise definition to solve the problem for the first time in October 1974. Comparing the (TSA) method with all other methods at that time, all other methods became approximate. For examples: the Monte Caro Methods; the Geometric Solid Angle (GSA) method are all conditionally acurate in some given ranges of parameters but not precise in all ranges of the given parameters. 2. The definition of (TSA) is explicitly defined with all parameters implicitly contained within the definition while all the other methods do not. 3. Due to the precise definition of (TSA), it is applicable for macroscopic bodies as well as for microscopic particles of mathematically defined infinitestimal size under the actions of any force and moment fields between and among the bodies and particles. Therefore, the (TSA) provides great imacts to the entire range of physics; from the calculation of the collision cross sections of sub-nucleus particles in high energy physics and to that of galaxies in astronomy. The applicabilities of all other methods are relatively limited. 4. The Geometric Solid Angle (GSA) of any targeted area, being finite or infinitesimally small, is unchanged with respect to the location of the source where the particle is ejected. The (GSA) is not related to the parameters of ejection of the particle at all. It is a pure mathematical quantity. The TRAJECTORY SOLID ANGLE (TSA) is a term containing all the parameters of generating the particle and the targeted area to be hit. Thus the (GSA) of any targeted area is always finite and unchanged while that of (TSA) can be zero. This explains why the (TSA) can be and should be used to solve the P.sub.2 targeting problem for particles and bodies under the action of any force and moment fields and that the (GSA) can not and should not be considered as the correct solution for the P.sub.2 problem. There will be errors comparing the use of (TSA) between the use of (GSA) to solve the same problem. The errors will range from 0% to 100%. 5. Since the collision cross sections of many problems in central force fields (which include: the hydrogen model; Alpha scattering; moon-earth model; Comet Halley scatters around the solar system . . . etc.) have been based on the use of (GSA) for calculation and have been published in text books around the world, the furture assertion the truth of (TSA) will provide a great impact to all those results in the past. 6. The (TSA) concept and its definition not only confirms the well known Heisenberg's principle of uncertainty in physics, but also provides the precise definition and procedures to calculate the uncertainty in term of numbers as precise as we want. 7. The most important concept of (TSA) is that the definition can be applied to discover new laws and new particles by comparison and matchings of the unknown results with the already confirmed and proved results. If there are new laws of physics that describe the particle motions other than those of Newton's classical mechanics and Einstein's narrow and general relativity, the present (TSA) concept is still applicable to obtain the precise P.sub.2 function for the problem. 8. Four examples are selected to illustrate how to obtain the probability distribution functions by means of (TSA): These examples are; Alpha scattering; particle in uniform, isotropic linear motion; particle under assumed constant gravitatonal pull on a plane; particle in a medium where the resistance force is linearly proportional to the velocity of the particle and under a uniform gravitational field. These examples are selected on the basis that they are well known and can be found from the open literatures. They were selected with the intention to show that even with such simple well known examples, the correct probability functions and cumulative distribution functions of these problems have never been obtained before. Whether exact solutions can be obtained from the equations of motion that govern other problems will not be the issue because the equation of motions can always be solved by means of numerical analysis together with computer programming. The key issue is that through the definition of (TSA), the P.sub.2 functions can be precisely defined and obtained. The (TSA) can be applied to solve the most fundamental problems in physics that include all the subjects listed as cited references in this application.