Patent Number: 050842323
Section: description

FIG. No. 1 is applicable for the general targeting problems in mass spectrometers particle accelerators, super-colliders; actual missiles and rockets targeting problems scattering and collision of astronomical bodies, chaos of classical dynamics and quantum mechanics, fluid dynamics and the weather prediction . . . etc. FIGS. No. 2, 3, 4 are demonstrated in great details how to apply the invention of (TSA) to solve a specific well-known-simple problem but is Probability Density Function (pdf) and Cumulative Distribution Function (cdf) have never been precisely obtained by all other methods before the invention of (TSA). DETAILED DESCRIPTION OF THE INVENTION The detailed description of the invention can be referred to the cited reference No. [1] and [2] which are technical papers contributed for the Poster Sessions in the American Association for the Advancement of Science (AAAS) 1989 Annual Meeting, Jan. 14-19, 1989 in San Francisco, Calif. The formal responses byy the inventor to all reviewers who reviewed and evaluated the inventor's papers and technical proposals about the concept and the application of TRAJECTORY SOLID ANGLE (TSA) can be found from references No. [15] and [17]. The open challenges to confirm the truth of (TSA) in the 1989 AAAS Annual Meeting can be referred to reference No. [19]. References No. [7], [16], [18], [20], [21], and [22] are previous technical proposals produced by the inventor requesting the supports from the U.S. federal governmental organizations. All these documents provide the evidences of impacts and resistances how the professionals react to accept the truth of TRAJECTORY SOLID ANGLE that should be used to solve the P.sub.2 targeting problems in the past 14 years since the invention of TSA in October 1974. The cited references No. [1], [2], [15], [17], and [19] are submitted as the supporting documents with the patent application. As required in CFR S1.71, a description of the preferred embodiments are provided in the followings: As indicated before, the TRAJECTORY SOLID ANGLE (TSA) deviates from but includes the Geometric Solid Angle (GSA) as a special case. (TSA) is defined as the integral of the dot product of the unit tangent of the particle's trajectory to the vectorial area divided by the square of the position vector connecting between the point of ejection and that of the surface area to be hit. The formal derivation and definition of (TSA) can be obtained by means of Fractional Ratio. Define: T, the unit tangent of a trajectory on which the particle travels along at any time. It is a function of its initial time t.sub.o, initial position vector r, initial velocity vector r.sub.o, initial mass m.sub.o, speed of light c, sum of all forces .SIGMA.F.sub.i acting on the particle, sum of all moments .SIGMA.M.sub.i acting on the particle and a function of time t. The unit tangent can be obtained from solving the governing equations for dynamics of rigid bodies and particles including but not limiting to the general relativity as well as the special relativity equations which are shown in the following as an example ##EQU5## The unit vector T can be obtained from solving the above set of equations after satisfying the initial conditions at t=t.sub.o, r=r.sub.o (t.sub.o), r=r.sub.o (t.sub.o), m=m.sub.o (t.sub.o) its magnitude is always 1 at any time t. Thus the unit tangent vector contains all the parameters of generation and ejection of the particle. EQU r=r (t.sub.o, r.sub.o, r.sub.o, m.sub.o, .SIGMA.F.sub.i, .SIGMA.M.sub.i, t) EQU T=r/ir) PA0 A, any vectorial surfaces in the universe, open or closed, its magnitue and direction can be obtained from the position vector R.sub.s which defines the equations of the surface. PA0 R, is the magnitude of the position vector connecting the points on the surface A to the coordinate system's origin from which the particle is released. PA0 dA, is the differential surface area ##EQU6## is defined and called TRAJECTORY SOLID ANGLE (TSA)=.OMEGA. In particular, when the sum of all forces .SIGMA.F.sub.i (r, r, t) and the sum of all moments .SIGMA.M.sub.i (r, r, t) acting on the particle are zero, the solution of the governing equation is obtained for the particle moving uniformly and isotropically from the origin. Then the trajectory coincides with the position vector. Under this condition, the unit tangent T can be related to the position vector R as ##EQU7## With this particular condition, the TRAJECTORY SOLID ANGLE (TSA) becomes Geometric Solid Angle (GSA) which is degenerated from TSA as ##EQU8## This integral is equal to 4.pi. for a surface large enough to enclosed the entire universe including the origin of the coordinate system. In general, the magnitude of this integral is less than 4.pi. for any finte surfaces in the universe. PA0 1. Solve the set of governing equations that govern the trajectory of the particle and obtain the position vector, the velocity vector and the trajectory equation in terms of initial conditions and all other parameters in the governing equations. PA0 2. Find the unit tangent vector from the velocity vector or from the trajectory equation. PA0 3. Find the unit normal vector and the differential surface area from the governing equation of the surface to be struck. PA0 4. Find the intersection of the trajectory on the surface and set the intersection coordinates in terms of the two independent variables that define the surface. PA0 5. The incident angle of the particle on the trajectory striking at the surface and be defined from the inner product of the unit tangent vector to the nuit normal surface vector expressed in terms of the two independent variables at the intersection. PA0 6. The trajectory solid angle for the problem can be obtained from integration over the cosine of the incident angle multiplying the differential surface area divided by the square of the position vector of the surface. PA0 7. The probability distribution function can be defined as the ratio of the trajectory solid angles (TSA). PA0 1. In the (TSA) is confirmed to be the TRUE solution of the P.sub.2 targeting problem, then: the impacts to the nuclear industries are: PA0 2. The (TSA) impacts to the governmental NRC safety standards and DOD weaponry system development are: PA0 3. It provides impacts to many activities in the Department of Energy (DOE) and in NASA: PA0 There are 11 separate tasks proposed to be done in the SYSTEMS RESEARCH COMPANY's 143 pages technical proposal DOE No. P7900450 to the Department of Energy, High Energy Physics Division. PA0 There are also at least 3 technical proposals having been submitted to NASA for funding and support. PA0 4. It provides impacts to update the contents of text books of physics and mathematics of all levels. PA0 [1] Wong, Po Kee, "The Invention of TRAJECTORY SOLID ANGLE and Its Impacts to the Role of Precision Measurement in Physics" American Association for the Advancement of Science (AAAS) 1989 Annual Meeting Contributed Paper Poster Sessions, Physical Sciences Abstract No. 218, page 180, Jan. 14-19, 1989 San Francisco, Calif. PA0 [2] Wong, Po Kee, "Evidences of Impacts and Resistances to Implement a Solved but Controversial Scientific Problem in Curriculum Development" American Association for the Advancement of Science (AAAS) 1989 Annual Meeting Contributed Paper Poster Sessions, Science & Technology Education Abstract No. 804, Jan. 14-19, 1989, San Francisco, Calif. PA0 [3] Sermanderes, S. N., "Methods of Determining the probability of a Turbine Missile Hitting a Particular Plant Region" Westinghouse Topical Report. WCAP-7861, February, 1972. PA0 [4] Bush, S. H., "Probability of Damage to Nuclear Components Due to Turbine Failure" A report of U.S. AEC and Battelle Memorial Institute, November 1972. PA0 [5] Shaffer, D. H., Chay, S. C., McLain, D. K., and Powell, B. A., "Analysis of the Probability of the Generation and Strike of Missiles from a Nuclear Turbine" Westinghouse Research Laboratories Report, March 1974. PA0 [6] "Turbine Missile Probability Analysis" Partial Notes of a PSAR of Delmarva Power & Light Company Summit Power Station, April 1974. PA0 [7] Wong, Po Kee, "Initiation of the Definition of Trajectory Solid Angle and its Influence on Classical, Quantum and Statistical Mechanics" A SYSTEMS RESEARCH COMPANY's Jan. 17, 1979 unsolicited technical proposal to the U.S. Department of Energy, 143 pages, DOE Proposal No.: P7900450. PA0 [8] Watson, K. M., "Collision of Particles" in Encyclopedia of Physics ed. by Besancon, R. M., 2nd, edition, pp. 140-144, Van Nostrand Reinhold Company, New York, Cincinnati, Toronto, London, Melbourne, 1974. PA0 [9] Lee, J. F., Sears, F. W., and Turcotte, D. L., "Statistical Thermodynamics" Addison-Wesley Publishing Company, Inc. 1963. PA0 [10] Hottel & Sarofim, "Radiative Transfer" McGraw-Hill Book Company, Chapter 2, pp. 25-71, 1967. PA0 [11] Park, D., "Introduction to the Quantum Theory" McGraw-Hill, Inc. 2nd. ed., 1974. PA0 [12] Hauser, W., "Introduction to the Principles of Mechanics" Addison-Wesley, Reading, Mass. 2nd. printing, April 1966. PA0 [13] Pierce, Walter D. and Lorber, Michael, "Objectives & Methods For Secondary Teaching" Prentice-Hall, Inc., Englewood Cliffs, N.J., 1977, pp. 104-106, 92-141. PA0 [14] Hass, Glen et al, "Curriculum Planning: A New Approach" Allyn and Bacon, Inc., 470 Atlantic Avenue, Boston, Mass. 1980, third edition, pp. 3-11, 40-50, 92-96, 145-150, 182-187, 156-258. PA0 [15] Wong, Po Kee, "Initiation of the Trajectory Solid Angle and its Influence on Classical, Quantum and Statistical Mechanics" abstract & summary submitted to 15th International Congress of Theoretical and Applied Mechanics for review and evaluation to have been presented on Aug. 17-23, 1980, Toronto, Cannada, 3 pages. plus 14 pages of formal responses by the inventor to answer the questions by the reviewers who reviewed the inventor's DOE Proposal No.: P7900450 (reference No. 7). Total 17 pages. PA0 [16] Wong, Po Kee, "The New Statistical Mechanics and its Impacts in Science Education" a SYSTEMS RESEARCH COMPANY's unsolicited technical proposal submitted to National Science Foundation Proposal No.: R0466, June 3, 1981. PA0 [17] Wong, Po Kee, "Dec. 7, 1981 letter and 43 pages of SYSTEMS RESEARCH COMPANY's documents to Professor R. J. Goldstein, Senor Editor, ASME Contributions to the 7th International Heat Transfer Conference to have been held at the Technische Universitat, Munchen, Fed. Rep. of Germany from Sept. 6, 10, 1982 for presentation and publication of reference No. 1." PA0 [18] Wong, Po Kee, "Nuclear Weapon Effects Simulation by Means of Continuum Mechanics and Paticle Dynamics Approaches and their Comparisons" a SYSTEMS RESEARCH COMPANY's SBIR proposal submitted to Defense Nuclear Agency (DNA) for support, Dec. 30, 1983. PA0 [19] Wong, Po Kee, Oct. 17, 1988 open letter to 11 session chairmen organizers of the 1989 AAAS Annual Meeting in San Francisco, Calif., 7 pages. PA0 [20] Wong, Po Kee, "The Application of the Invention of TRAJECTORY SOLID ANGLE for the Designs of Scientific Instrumentations." SYSTEMS RESEARCH COMPANY's SBIR technical proposal to National Science Foundation (NSF) No. ISI-8860922 June 20, 1988. PA0 [21] Wong, Po Kee, "ADVANCED AUTOMOTIVE RESEARCH-Vehicle Structures and Materials" SYSTEMS RESEARCH COMPANY's proposal to NSF No. ISI-8860538. June 20, 1988. PA0 [22] Wong, Po Kee, "PDE Solvers for Visco-Elasto-Dynamics (VED) and Their Applications in Space Materials and Structures." SYSTEMS RESEARCH COMPANY's SBIR proposal to NSF No. ISI-8860180. June 20, 1988. 4.pi. is the maximum value of both (TSA) and (GSA). GUIDED is equivalent to aiming at a target with a (TSA) from point to point on the trajectory. UNGUIDED is equivalent to release the flying object in any direction at a (TSA) of 4.pi. from a point. GUIDED PROJECTILE is a projectile aiming at the target by control through a (TSA) enclosed by the envelope-surface formed by the projectile's trajectories and tangented to the target surface. UNGUIDED PROJECTILE is a projectile being released and flying in any direction at a (TSA) of 4.pi. from a point. With all these terms, the probability targeting function P.sub.2 can be derived and obtained with the BRIEF DESCRIPTION OF THE DRAWINGS on page 13 from FIG. No. 1. One should notice that the (TSA) has been shown derived from with respect to the ejected particle source from a fixed point for convenience to demonstrate the basic principle. However, there are serveral important points from the present derivation of the (TSA) should be cleared: (a) The (TSA) is not limited to be referred to a fixed point source. In fact, both the source point and the targeted surface can be in motions. When this is the case, the position vector becomes EQU R=R.sub.2 -R.sub.1 and the unit tangent vector becomes ##EQU9## where, r.sub.1 and R.sub.1 are the velocity vector and the position vector of the source point, r.sub.2 and R.sub.2 are the velocity vector and the position vector of the targeted point. (b) The generation and ejection of the particle is not necessary limited to a point source. It can be generated and ejected from a surface source. When this is the case, defining the surface intensity function (a function of any physical quantities/unit surace area of ejection) which can be the irradiance E.sub.e (W/m.sup.2), the pressure p (N/m.sup.2), the heat flux density q (W/m.sup.2) . . . etc. Then the expectation values of these physical quantities from the source surface A.sub.1 to the targeted surface A.sub.2 is ##EQU10## (c) Generalizing from the Largrance description of the motion of a particle to the Eulerian description of the physical quantities of a continuum at a field point, the (TSA) can also be defined for the continuum in fluid dynamics, elastodynamics, visco-elasto-dynamics (VED), Magneto-Viscoelasto-Dynamics (MVD) . . . etc. Maintaining the same definition of (TSA), the unit vector can also be obtained from the solution of the governing equations of the continuum after satisfying the boundary and initial conditions of a physical problem. The unit vector in the Largrance particle description represents a particular particle's trajectory as a function of time while the unit vector in the Eulerian continuum description represents different particles' trajectories as a function of both space and time. In both descriptions, the (TSA) is a dimensionless probability function. The integrand of (TSA) is intrinsically always positive definite for a particle or a continuum under the actions any force and moment fields. Thus, the (TSA) definition will be applicable for the governing equations of fluid dynamics, elastodyanmics, visco-elasto-dynamics (VED), (MVD), the electromagnetic wave equations, quantum electrodynamics which was generalized from the well-known Schrodinger equation in quantum mechanics . . . etc. According to the correspondence principle (which states, roughly, that quantum mechanics makes the same predictions that classical physics does for systems in which classical physics is applicable), the turbine missile problem investigated by Sermanderes [3], Bush [4], Shaffer et al [5], [6], and by the inventor [7] should also be predicted as having the same result as obtained by the QM. However, one will be disappointed if he tries to use the quantum mechanics (QM) to solve the simple problem. The specific turbine missile probability problem had led to the invention of (TSA) and further to question the validity and the applicability of quantum mechanics. This is the most controversial issue opened to be settled by all scienties, mathematiccians and engineers in the world. The procedures to find the distribution function P.sub.2 for a particle striking a predesignated area, given all its parameters of generation and ejection can be systematically summarized in the following steps: As mentioned before, the cited reference No. [1] provides 4 examples of very well-known problems but their (pdf) and (cdf) have never been precisely obtained before. Reference No. [7] also provides the transient solution of a positive ion under electromagnetic field action with consideration of the relativistic effects being considered in the closed form solution. The ion is found in spiral motion and it is accelerated to the speed of light. This particular example is important for the calculation of collision cross section of ions by means of (TSA). There are almost hundreds of very well-known problems whose (pdf) and (cdf) must be recalculated again by means of (TSA). The (TSA)'s impacts in applications can further be summarized in the following: (a) It will force many companies in the nuclear business to re-examine their Preliminary and final Safety Analysis Report (PSAR & FSAR). For each nuclear power plant that was built, is being built, and shall be built. PA1 (b) It will affect the analysis of design which was based on other method of solution for many facilities in the nuclear power plants including the new nuclear core designs and developments. PA1 (a) It provides a direct impact to the missile probability standard that was already established in the NRC Report (WASH-1400) which is cited in reference No. [17] about The Reactor Safety Study. PA1 (b) It provides a direct impact to the guidance and control systems for targeting in our defense missile systems. PA1 (c) It provides a direct impact to the conventional weaponry targeting analysis. PA1 (d) It provides solutions of problems in Target Detection and Localization in: PA1 (e) Ocean physics and engineering research in: PA1 (f) Computer and software engineering in: PA1 (g) Materials researches. Special purpose acoustic transducers and sensors; PA2 Electromagnetics and Broadband Antennas; PA2 Theoretical and experimental tools with which to detect and classify nuclear surface/air burst at sea. PA2 Oceanographic instrumentation; PA2 Remote sensing techniques; PA2 Ocean volume reverberation modeling; PA2 Acoustic response of the ocean bottom PA2 Inexpensive photolithograph techniques for microcircuit fabrications by alternative approaches in the resist materials or for ion beam machinery. The values of the invention depends on whether the solution of the P.sub.2 targeting problem by means of the (TSA) is TRUE. If it is, it will provide all the impacts to practitioners, public and private decision makers, and the general public especially involved in education: It will affect many previous Nobel Laureates' work in scattering and collision crossections of particles; in Statistical Mechanics and Quantum Mechanics. Specific work of interest includes: Rutherford's Alpha Scattering; Hofstadter's electron scattering; Yang's p-p collision and its scattering and the geometric picture; Fermi-Dirac, Bose-Einstein, Maxwell-Boltzmann statistics; quantum mechanics based on Schrodinger's equation; Schwinger and Feynman's quantum electrodynamics and Heisenberg's uncertainty priciple . . . etc. All these topics are in the current text books of physics for graduate and undergraduate levels in all universities in the world. It follows that will influence the selection of materials for the secondary curriculum planning and development according to the impacts. REFERENCES CITED