Dark matter could meet its nemesis on Earth

Introduction. One of the high points of elementary calculus is the derivation of Kepler's empirically deduced laws of planetary motion from Newton's Law of Gravity and his second law of motion. However, the standard treatment of the subject in calculus books is flawed for at least three reasons that I think are important. First, Newton's Laws are used to derive a differential equation for the displacement vector from the Sun to a planet; say the Earth. Then it is shown that the displacement vector lies in a plane, and if the base point is translated to the origin, the endpoint traces out an ellipse. This is said to confirm Kepler's first law, that the planets orbit the sun in an elliptical path, with the sun at one focus. However, the alert student may notice that the identical argument for the displacement vector in the opposite direction would show that the Sun orbits the Earth in an ellipse, which, it turns out, is very close to a circle with the Earth at the center. That would seem to provide aid and comfort to the Church's rejection of Galileo's claim that his heliocentric view had more validity than their geocentric one. Second, by placing the sun at the origin, the impression is given that either the sun is fixed, or else, that one may choose coordinates attached to a moving body, inertial or not. However, Newton's equations in their usual form hold precisely in inertial coordinates. Furthermore, it is an immediate consequence of Newton's Laws that the center of gravity of the two bodies can serve as the origin of an inertial coordinate system. One then finds that both the Earth and the Sun describe elliptical paths with a focus at the center of gravity of the pair. Third, by considering only the displacement vector, the treatment misses an opportunity to present one of the greatest recent triumphs of Newton's derivation of Kepler's Laws: the discovery of extrasolar planets during the past decade, none of which has been observed directly, but whose existence and orbital parameters have been deduced by detecting and analyzing the motion of the stars that they orbit. Two other considerations make the correct way of treating the subject particularly desirable. First, the current treatment for the displacement vector could be left as is, and only a little algebra is needed to show that the same vector referred to the center of gravity is a constant scalar multiple of the one referred to the sun. Second, it connects with an exciting area of current research in astronomy where the methods of elementary calculus have direct application. The purpose of this note is to spell out the slight modifications to the standard treatment of Kepler's Laws that are needed to obtain the additional and accurate information, and to provide a user's guide to the website http://exoplanets.org where a list of all the extrasolar planets discovered so far is maintained and constantly updated. The Basic Equations. Let r(t) be a (3-dimensional) vector function of the parameter t, and suppose that r(t) satisfies the differential equation

Various theories, such as MOND, MOG, Emergent Gravity and theories avoid dark matter by assuming a change in General Relativity and/or in Newton's law. Galactic rotation curves are typically described well. Here the application to galaxy clusters is considered, focussed on the good lensing and X‐ray data for A1689. As a start, the no‐dark‐matter case is confirmed to work badly: the need for dark matter starts near the cluster centre, where Newton's law is still supposed to be valid. This leads to the conundrum discovered by Zwicky, which is likely only solvable in his way, namely by assuming additional (dark) matter. Neutrinos with eV masses serve well without altering the successes in (dwarf) galaxies.

First Page

Intramolecular (13)C kinetic isotope effects were determined for the dimerization of methacrolein. Trajectory studies accurately predict the isotope effects and support an origin in Newton's second law of motion, with no involvement of zero-point energy or transition state recrossing. Atomic motion reaction coordinate diagrams are introduced as a way to qualitatively understand the selectivity.

Bubble size distributions (BSDs) near a reference electrode (RE) in aqueous glycerol solutions of an electrolyte NaCl have been investigated under various gas superficial velocities (Us). BSD and voltage reading of the solution were measured by using a high-speed digital camera and a pH/voltage meter, respectively. The results show that bubble size (b) increases with liquid viscosity (μc) and Us. Self-similarity is seen and can be described by the log-normal form of the continuous number frequency distribution. The result shows that b controls the voltage reading in each solution. As b increases, the voltage increases because of gas bubbles interrupting their electrolyte paths in the solutions. An analysis of bubble rising velocity reveals that Stokes' Law should be used cautiously to describe the system. The fundamental equation for bubble formation was developed via Newton's second law of motion and shown to be the function of three dimensionless groupsWeber number, Bond number, and Capillary number. Af...

Influencing Cyber Laboratory Conceptual Change Through Laboratory Learning

The Rapid Evaluation of Potential Fields in Particle Systems presents a group of algorithms for the computation of the potential and force fields in large-scale systems of particles that are likely to revolutionize a whole class of computer applications in science and engineering. In many areas of scientific computing, from studying the evolution of galaxies, to simulating the behavior of plasmas and fluids, to modelling chemical systems, a numerical scheme is used to follow the trajectories of a collection of particles moving in accordance with Newton's second law of motion in a field generated by the whole ensemble. Extending the earlier work of Rokhlin, Greengard has developed general, numerically stable methods for evaluating all pairwise interactions in linear time, a great improvement over the quadratic time required by the naive approach, and significantly better than any other proposed alternative. The "Rokhlin-Greengard" algorithm promises to make previously prohibitive simulations feasible, with speedups of three to four orders of magnitude in a system of a million particles. Moreover, the algorithm is well-suited for vector and parallel machines, and should make full use of their capabilities. The author presents his work with great clarity, and demonstrates the superiority of his methods both by mathematical analysis and by the results of numerical experiments. The Rapid Evaluation of Potential Fields in Particle Systems is a 1987 ACM Distinguished Dissertation.

Noise and vibration from cars is fairly a complex field occurring in a right frequency range: vibration to noise. In practice, there is a variety of reasons causing noise inside a normal sealed gearbox. According to the J.D.Smith’s (2) analysis, the primary cause of noise is from backlash gaps between teeth when mashing. The friction effects of gear teeth generate vibration, and are then transmitted to a primary rolling bearing (4). In the gearbox internal responses, all of the manufacturing errors, design errors and tooth and gear deflections, etc…are called Transmission Error (TE). TE will be passing the gearbox whole system to the end rolling bearing of a primary shaft. The gear operation of a modeled gearbox showed in Fig.4 vibrates with four free levels. Generalized coordinates j 1 … j 5 are the gear angular velocity from 1 to 5 respectively indicated in Fig.4. The symbols J­­ 1 (J­­ 21 , J­­ 22 ), (J­­ 31 , J­­ 32 ), (J­­ 41 , J­­ 42 ), J­­ 5 are the gear moments from 1 to 5. Responding to the wheel circle radius symbolized r 1 , r 21 , r 22 …r 5 and I 1 , I 2 , I 3 , I 4 , are vibrating forces. The symbols c 1 ,..c 4 ; k 1 ,…k 4 are the viscous fluid damper and the mating gear elasticity (Fig.5). Applying the Hezt’s collision theory, the momentum principle, the Newton's second law of motion, and the author has developed a vibrating equation of the gearbox transmission of four vibrating levels as the following equation (01) To survey vibration in a modeled gearbox, ones need to consider the equation (01) and receive j 1 … j 5 then deal with and according to the formula (12) gives the velocity vector in XYZ of primary bearing housing. With constructing mathematical models, ones can recognize vibration in a certain gearbox when changing the gearbox theoretical design and specifications with convenience, which supports design processes more effectively and quickly.

In recent years, with the advancement in computation power, computer-aided simulation and modeling are at the forefront of food process engineering research. The discrete element method (DEM), a proven numerical method, is commonly used to study particulate or granular systems. In DEM, the motion of distinct granules is defined by Newton's second law of motion. DEM captures the dual nature of granular media that behaves both like solid and liquid. This chapter discusses the basics of DEM, governing equation, contact models, DEM input parameters, and their application in modeling grain post-harvest processes.

Time-dependent raising and lowering operators for the ageing, forced and damped oscillator in quantum mechanics

In the Einstein’s theory of special relativity, the relativistic mechanics is concerned with the motion of bodies whose velocities approach the speed of light. It is understood the velocity of a moving particle with mass is less than the speed of velocity and velocity of a massless particle like photon is equal to the speed of light. This paper presents the Einstein’s mass-energy equivalence and the equations of relativistic mass, momentum and energy from the Newton’s second law of motion.

The “Phantom” Force1

An orifice equation is derived relating the effective aortic valve area, A, the average aortic valve pressure gradient, dP, the stroke volume, SV, and the heart frequency, FH, through considerations of momentum conservation across the aortic valve. This leads to a formula consistent with Newton's second law of motion. The form of the new equation is A = (7.5 X 10(-5)) SV FH2/Pd, where A, VS, FH and Pd are expressed in cm2, ml, s-1 and mmHg, respectively. Aortic valve areas computed with the new orifice equation are found to correlate with those computed by the Gorlin formula in conditions of resting haemodynamic states at a level of r = 0.86, SE = 0.25 cm2, N = 120. The results suggest that the new formula may be considered as an independent orifice equation having a similar domain of validity as the Gorlin formula. The new equation offers the possibility of deriving additional useful haemodynamic relationships through combination with established cardiological formulas and applying it in a noninvasive Doppler ultrasonic or echocardiographic context.

A general theory of the thermoelasticity of stressed materials is presented. The theory is based on the geometry of strain, Newton's second law of motion, the first and second laws of thermodynamics, and the invariance of the internal energy and Helmholtz free energy with respect to an arbitrary finite rigid rotation of the material. Three different sets of physically significant thermoelastic coefficients are discussed. These are (a) the second-order elastic constants, which contain the rotational invariance conditions and always have the Voigt symmetry, (b) the equation-of-motion coefficients, which govern small-displacement wave propagation and have Voigt symmetry only when the stress vanishes, and (c) the coefficients which relate the variation of stress to the variation of strain from the initial (stressed) configuration. Relations between these sets of coefficients are presented for the case of arbitrary initial stress, and also for initial isotropic pressure. In addition, these second-order elastic coefficients for a stressed material are expressed as series in the second-, third-, and fourth-order elastic constants evaluated at zero stress; the expansion parameters in these series are the parameters which measure the strain from the state of zero stress to the stressed state. All of the general relations are illustrated and tabulated for the example of a cubic material under isotropic pressure. A detailed comparison of the present results with previous theories is given. The two types of elastic constants defined by Fuchs and Voigt are generalized to conditions of initial stress, and compared with the three basic sets of elastic coefficients of the present paper. Finally some comments are made regarding the interpretation of thermoelastic measurements on crystals in terms of static and dynamic calculations based on atomic models.

Various approaches have been proposed to model the gas–solid two‐phase flow at different time and length scales, including the so‐called two‐fluid model (TFM), direct numerical simulation (DNS) and combined continuum and discrete model (CCDM). This paper briefly discusses the key features of these models and their relative merit with special reference to modelling gas fluidisation. Focus is then given to CCDM in which the motion of individual particles is obtained by solving Newton's second law of motion and fluid flow by the Navier–Stokes equation based on the concept of local average. The applicability of CCDM is highlighted by its successful simulation of complicated phenomena associated with the transition between fluid‐like and solid‐like behaviour in raceway formation and bed expansion. At the same time, the usefulness of the resulting particle‐scale information is demonstrated in elucidating the fundamentals governing the gas–solid flow. Finally, areas for future development are discussed.© 2003 Society of Chemical Industry