Inverse Square Law Of Force Research Articles

Superscalability of the random batch Ewald method.

Coulomb interaction, following an inverse-square force-law, quantifies the amount of force between two stationary and electrically charged particles. The long-range nature of Coulomb interactions poses a major challenge to molecular dynamics simulations, which are major tools for problems at the nano-/micro-scale. Various algorithms are developed to calculate the pairwise Coulomb interactions to a linear scale, but poor scalability limits the size of simulated systems. Here, we use an efficient molecular dynamics algorithm with the random batch Ewald method on all-atom systems where the complete Fourier components in the Coulomb interaction are replaced by randomly selected mini-batches. By simulating the N-body systems up to 108 particles using 10 000 central processing unit cores, we show that this algorithm furnishes O(N) complexity, almost perfect scalability, and an order of magnitude faster computational speed when compared to the existing state-of-the-art algorithms. Further examinations of our algorithm on distinct systems, including pure water, a micro-phase separated electrolyte, and a protein solution, demonstrate that the spatiotemporal information on all time and length scales investigated and thermodynamic quantities derived from our algorithm are in perfect agreement with those obtained from the existing algorithms. Therefore, our algorithm provides a promising solution on scalability of computing the Coulomb interaction. It is particularly useful and cost-effective to simulate ultra-large systems, which is either impossible or very costly to conduct using existing algorithms, and thus will be beneficial to a broad range of problems at nano-/micro-scales.

Linearized Boltzmann collision integral with the correct cutoff

In the calculation of the linearized Boltzmann collision operator for an inverse-square force law interaction (Coulomb interaction) F(r)=κ/r2, we found the widely used scattering angle cutoff θ≥θmin is a wrong practise since the divergence still exists after the cutoff has been made. When the correct velocity change cutoff |v′−v|≥δmin is employed, the scattering angle can be integrated. A unified linearized Boltzmann collision operator for both inverse-square force law and rigid-sphere interactions is obtained. Like many other unified quantities such as transition moments, Fokker-Planck expansion coefficients and energy exchange rates obtained recently [Y. B. Chang and L. A. Viehland, AIP Adv. 1, 032128 (2011)], the difference between the two kinds of interactions is characterized by a parameter, γ, which is 1 for rigid-sphere interactions and −3 for inverse-square force law interactions. When the cutoff is removed by setting δmin=0, Hilbert's well known kernel for rigid-sphere interactions is recovered for γ = 1.

MOONLIGHT: A NEW LUNAR LASER RANGING RETROREFLECTOR AND THE LUNAR GEODETIC PRECESSION

Since the 1970s Lunar Laser Ranging (LLR) to the Apollo Cube Corner Retroreflector (CCR) arrays (developed by the University of Maryland, UMD) supplied almost all significant tests of General Relativity (Alley et al., 1970; Chang et al., 1971; Bender et al.,1973): possible changes in the gravitational constant, gravitational self-energy, weak equivalence principle, geodetic precession, inverse-square force-law. The LNF group, in fact, has just completed a new measurement of the lunar geodetic precession with Apollo array, with accuracy of 9 × 10−3, comparable to the best measurement to date. LLR has also provided significant information on the composition and origin of the moon. This is the only Apollo experiment still in operation. In the 1970s Apollo LLR arrays contributed a negligible fraction of the ranging error budget. Since the ranging capabilities of ground stations improved by more than two orders of magnitude, now, because of the lunar librations, Apollo CCR arrays dominate the error budget. With the project MoonLIGHT (Moon Laser Instrumentation for General relativity High-accuracy Tests), in 2006 INFN-LNF joined UMD in the development and test of a new-generation LLR payload made by a single, large CCR (100mm diameter) unaffected by the effect of librations. With MoonLIGHT CCRs the accuracy of the measurement of the lunar geodetic precession can be improved up to a factor 100 compared to Apollo arrays. From a technological point of view, INFN-LNF built and is operating a new experimental apparatus (Satellite/lunar laser ranging Characterization Facility, SCF) and created a new industry-standard test procedure (SCF-Test) to characterize and model the detailed thermal behavior and the optical performance of CCRs in accurately laboratory-simulated space conditions, for industrial and scientific applications. Our key experimental innovation is the concurrent measurement and modeling of the optical Far Field Diffraction Pattern (FFDP) and the temperature distribution of retroreflector payloads under thermal conditions produced with a close-match solar simulator. The apparatus includes infrared cameras for non-invasive thermometry, thermal control and real-time payload movement to simulate satellite orientation on orbit with respect to solar illumination and laser interrogation beams. These capabilities provide: unique pre-launch performance validation of the space segment of LLR/SLR (Satellite Laser Ranging); retroreflector design optimization to maximize ranging efficiency and signal-to-noise conditions in daylight. Results of the SCF-Test of our CCR payload will be presented. Negotiations are underway to propose our payload and SCF-Test services for precision gravity and lunar science measurements with next robotic lunar landing missions. In particular, a scientific collaboration agreement was signed on Jan. 30, 2012, by D. Currie, S. Dell’Agnello and the Japanese PI team of the LLR instrument of the proposed SELENE-2 mission by JAXA (Registered with INFN Protocol n. 0000242-03/Feb/2012). The agreement foresees that, under no exchange of funds, the Japanese single, large, hollow LLR reflector will be SCF-Tested and that MoonLIGHT will be considered as backup instrument.

MOONLIGHT: A NEW LUNAR LASER RANGING RETROREFLECTOR AND THE LUNAR GEODETIC PRECESSION

Since the 1970s Lunar Laser Ranging (LLR) to the Apollo Cube Corner Retroreflector (CCR) arrays (developed by the University of Maryland, UMD) supplied almost all significant tests of General Relativity (Alley et al., 1970; Chang et al., 1971; Bender et al.,1973): possible changes in the gravitational constant, gravitational self-energy, weak equivalence principle, geodetic precession, inverse-square force-law. The LNF group, in fact, has just completed a new measurement of the lunar geodetic precession with Apollo array, with accuracy of 9 × 10−3, comparable to the best measurement to date. LLR has also provided significant information on the composition and origin of the moon. This is the only Apollo experiment still in operation. In the 1970s Apollo LLR arrays contributed a negligible fraction of the ranging error budget. Since the ranging capabilities of ground stations improved by more than two orders of magnitude, now, because of the lunar librations, Apollo CCR arrays dominate the error budget. With the project MoonLIGHT (Moon Laser Instrumentation for General relativity High-accuracy Tests), in 2006 INFN-LNF joined UMD in the development and test of a new-generation LLR payload made by a single, large CCR (100mm diameter) unaffected by the effect of librations. With MoonLIGHT CCRs the accuracy of the measurement of the lunar geodetic precession can be improved up to a factor 100 compared to Apollo arrays. From a technological point of view, INFN-LNF built and is operating a new experimental apparatus (Satellite/lunar laser ranging Characterization Facility, SCF) and created a new industry-standard test procedure (SCF-Test) to characterize and model the detailed thermal behavior and the optical performance of CCRs in accurately laboratory-simulated space conditions, for industrial and scientific applications. Our key experimental innovation is the concurrent measurement and modeling of the optical Far Field Diffraction Pattern (FFDP) and the temperature distribution of retroreflector payloads under thermal conditions produced with a close-match solar simulator. The apparatus includes infrared cameras for non-invasive thermometry, thermal control and real-time payload movement to simulate satellite orientation on orbit with respect to solar illumination and laser interrogation beams. These capabilities provide: unique pre-launch performance validation of the space segment of LLR/SLR (Satellite Laser Ranging); retroreflector design optimization to maximize ranging efficiency and signal-to-noise conditions in daylight. Results of the SCF-Test of our CCR payload will be presented. Negotiations are underway to propose our payload and SCF-Test services for precision gravity and lunar science measurements with next robotic lunar landing missions. In particular, a scientific collaboration agreement was signed on Jan. 30, 2012, by D. Currie, S. Dell’Agnello and the Japanese PI team of the LLR instrument of the proposed SELENE-2 mission by JAXA (Registered with INFN Protocol n. 0000242-03/Feb/2012). The agreement foresees that, under no exchange of funds, the Japanese single, large, hollow LLR reflector will be SCF-Tested and that MoonLIGHT will be considered as backup instrument.

MoonLIGHT: A USA–Italy lunar laser ranging retroreflector array for the 21st century

Since the 1970s Lunar Laser Ranging (LLR) to the Apollo Cube Corner Retroreflector (CCR) arrays (developed by the University of Maryland, UMD) have supplied significant tests of General Relativity: possible changes in the gravitational constant, gravitational self-energy, weak equivalence principle, geodetic precession, inverse-square force-law. LLR has also provided significant information on the composition and origin of the Moon. This is the only Apollo experiment still in operation. In the 1970s Apollo LLR arrays contributed a negligible fraction of the ranging error budget. Since the ranging capabilities of ground stations improved by more than two orders of magnitude, now, because of the lunar librations, Apollo CCR arrays dominate the error budget. With the project MoonLIGHT (Moon Laser Instrumentation for General relativity High-accuracy Tests), in 2006 INFN-LNF joined UMD in the development and test of a new-generation LLR payload made by a single, large CCR (100mm diameter) unaffected by librations. In particular, INFN-LNF built and is operating a new experimental apparatus (Satellite/lunar laser ranging Characterization Facility, SCF) and created a new industry-standard test procedure (SCF-Test) to characterize and model the detailed thermal behavior and the optical performance of CCRs in laboratory-simulated space conditions, for industrial and scientific applications. Our key experimental innovation is the concurrent measurement and modeling of the optical Far Field Diffraction Pattern (FFDP) and the temperature distribution of retroreflector payloads under thermal conditions produced with a solar simulator. The apparatus includes infrared cameras for non-invasive thermometry, thermal control and real-time payload movement to simulate satellite orientation on orbit with respect to solar illumination and laser interrogation beams. These capabilities provide: unique pre-launch performance validation of the space segment of LLR/SLR (Satellite Laser Ranging); retroreflector design optimization to maximize ranging efficiency and signal-to-noise conditions in daylight. Results of the SCF-Test of our CCR payload will be presented. Negotiations are underway to propose our payload and SCF-Test services for precision gravity and lunar science measurements with next robotic lunar landing missions. We will describe the addition of the CCR optical Wavefront Fizeau Interferogram (WFI) concurrently to FFDP/temperature measurements in the framework of an ASI-INFN project, ETRUSCO-2. The main goals of the latter are: development of a standard GNSS (Global Navigation Satellite System) laser Retroreflector Array; a second SCF; SCF-Test of Galileo, GPS and other ‘as-built’ GNSS retroreflector payloads. Results on analysis of Apollo LLR data and search of new gravitational physics with LLR, Mercury Radar Ranging will be presented.

The cumulative measure of a force: A unified kinetic theory for rigid-sphere and inverse-square force law interactions

By introducing a cutoff on the cumulative measure of a force, a unified kinetic theory is developed for both rigid-sphere and inverse-square force laws. The difference between the two kinds of interactions is characterized by a parameter, γ, which is 1 for rigid-sphere interactions and -3 for inverse-square force law interactions. The quantities governed by γ include the specific reaction rates, kernels, collision frequencies, arbitrarily high orders of transition moments, arbitrarily high orders of Fokker-Planck expansion (also called Kramers-Moyal expansion) coefficients, and arbitrarily high orders of energy exchange rates. The cutoff constants are shown to be incomplete gamma functions of different orders. The widely used cutoff constant in plasma physics (usually known as Coulomb logarithm) is found to be exactly the zeroth order of the incomplete gamma function. The well known Arrhenius reaction rate formula comes from the first order of the incomplete gamma functions, while the negative first order can be used for fitting the fusion reaction rate between deuterium and tritium.

Precise computation of acceleration due to uniform ring or disk

We describe a precise numerical method to evaluate the acceleration vector of an inverse-square force law caused by a uniform ring or disk. The method consists of two parts. One is rewriting of the analytic expressions of the vector to reduce the loss of information in its computation. The other is a procedure to compute a special linear combination of the complete elliptic integrals of the first and the second kind appeared in the rewritten expressions.

Energy and the Elliptical Orbit

In the January 2007 issue of The Physics Teacher, Prentis, Fulton, Hesse, and Mazzino1 describe a laboratory exercise in which students use a geometrical analysis inspired by Newton to show that an elliptical orbit and an inverse-square law force go hand in hand. The historical, geometrical, and teamwork aspects of the exercise are useful and important. This paper presents an exercise which uses an energy/angular momentum conservation model for elliptical orbits. This exercise can be done easily by an individual student and on regular notebook-sized paper.

Lower Limit to the Scale of an Effective Quantum Theory of Gravitation

An effective quantum theory of gravitation in which gravity weakens at energies higher than approximately 10(-3) eV is one way to accommodate the apparent smallness of the cosmological constant. Such a theory predicts departures from the Newtonian inverse-square force law on distances below approximately 0.05 mm. However, it is shown that this modification also leads to changes in the long-range behavior of gravity and is inconsistent with observed gravitational lenses.

A microscopic definition of mass

The story in the April 2006 issue of Physics Today April 2006, (page 32) on the redefinition of the kilogram leads me to draw attention to an alternative way of defining mass, which I think deserves further consideration.First of all, the measurement of relative masses can be done much more precisely at the microscopic level than at the macroscopic; the combination of measurements via precision mass spectrometers, magnetic traps, and nuclear–reaction Q values already yields mass ratios for several elementary particles and atoms with a precision of 1 part per billion or better. The actual masses in kilogram units of various microscopic systems—for example, electron, proton, or a monoisotopic atom—that might be chosen for a mass standard are also accurately known, but with a somewhat larger uncertainty, around the 50-ppb mark. The present aim of mass metrology is to reduce that uncertainty to a few parts per billion. But, since atomic-scale measurements are so accurate, why not define the kilogram here and now as a prescribed multiple of the mass of the chosen microscopic standard? We would then have a mass standard with the desired properties of permanence, stability, universal availability, and the embodiment of the concept of mass with a precision that is in principle indefinitely high. Moreover, the standard would be located in the experimental domain where accurate masses are most directly accessible and most important. The standard kilogram artifact and its various copies would then revert to calibrated comparison objects for carrying out macroscopic mass measurements, in which a precision of 50 ppb is far better than needed by commerce, industry, physics experiments, or everyday life.I have another comment, which concerns the choice of microscopic standard and indeed whether one can define mass without having to choose such a standard. Analysis of the systematic change in spacetime orientation of the equal-phase hypersurfaces in an accelerated charged-particle wave-packet shows that a particle's inertia is proportional to the value of its de Broglie angular frequency when the particle is free and in its rest frame. This fact strongly suggests that this frequency be used as an absolute definition 1 1. J. W.G. Wignall, Meas. Sci. Technol. 16, 682 (2005). https://doi.org/10.1088/0957-0233/16/3/009 of inertial mass m, in contrast to the usual mass M, which is relative to the kilogram or some agreed standard atomic object. Such absolute masses can be, and already are being, measured with very high precision. Since m = Mc 2/ħ and c is a defined number, values of absolute mass can be calculated immediately from the measurements of ħ/M obtained by several recent methods. Those methods combine interference or diffraction with determination of velocities of particles or of atomic recoils associated with photon emission or absorption, with uncertainties now approaching the few-ppb level.Following the microscopic-first approach mentioned earlier, the kilogram could then be defined once and for all as the mass of an object consisting of atoms whose summed absolute masses (subtracting 1/c2 times interatomic binding energy) amount to exactly 8.522 467 2 × 1050 s−1. This awkwardly large number would rarely enter calculations. In the microscopic world, it is measured frequencies that are important; macroscopic measurements would use comparison objects calibrated to the new kilogram via direct or indirect determinations of the numbers of atoms they contain, calibrations that would employ a number of modern mass-metrology techniques.An added bonus of the microscopic mass definition is that it eliminates the need for a separate unit—and dimension—not only for mass, but also for electric charge. For an inverse-square force law, charge turns out to be dimensionless; classical and quantum physics can then be expressed entirely in terms of just two kinds of basic units—those of time and length.REFERENCESection:ChooseTop of pageREFERENCE <<1. J. W.G. Wignall, Meas. Sci. Technol. 16, 682 (2005). https://doi.org/10.1088/0957-0233/16/3/009 , Google ScholarCrossref, CAS© 2007 American Institute of Physics.

Cosmological perturbations on local systems

We study the effect of cosmological expansion on orbits---galactic, planetary, or atomic---subject to an inverse-square force law. We obtain the laws of motion for gravitational or electrical interactions from general relativity---in particular, we find the gravitational field of a mass distribution in an expanding universe by applying perturbation theory to the Robertson-Walker metric. Cosmological expansion induces an ($\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{a}/a)\stackrel{\ensuremath{\rightarrow}}{r}$ force where $a(t)$ is the cosmological scale factor. In a locally Newtonian framework, we show that the $(\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{a}/a)\stackrel{\ensuremath{\rightarrow}}{r}$ term represents the effect of a continuous distribution of cosmological material in Hubble flow, and that the total force on an object, due to the cosmological material plus the matter perturbation, can be represented as the negative gradient of a gravitational potential whose source is the material actually present. We also consider the effect on local dynamics of the cosmological constant. We calculate the perihelion precession of elliptical orbits due to the cosmological constant induced force, and work out generalizations to the rotation curve and virial relation applicable to clusters of galaxies.

Spontaneous Lorentz breaking at high energies

Theories that spontaneously break Lorentz invariance also violate diffeomorphism symmetries, implying the existence of extra degrees of freedom and modifications of gravity. In the minimal model (``ghost condensation'') with only a single extra degree of freedom at low energies, the scale of Lorentz violation cannot be larger than about M ~ 100GeV due to an infrared instability in the gravity sector. We show that Lorentz symmetry can be broken at much higher scales in a non-minimal theory with additional degrees of freedom, in particular if Lorentz symmetry is broken by the vacuum expectation value of a vector field. This theory can be constructed by gauging ghost condensation, giving a systematic effective field theory description that allows us to estimate the size of all physical effects. We show that nonlinear effects become important for gravitational fields with strength \sqrt{\Phi} > g, where g is the gauge coupling, and we argue that the nonlinear dynamics is free from singularities. We then analyze the phenomenology of the model, including nonlinear dynamics and velocity-dependent effects. The strongest bounds on the gravitational sector come from either black hole accretion or direction-dependent gravitational forces, and imply that the scale of spontaneous Lorentz breaking is M < Min(10^{12}GeV, g^2 10^{15}GeV). If the Lorentz breaking sector couples directly to matter, there is a spin-dependent inverse-square law force, which has a different angular dependence from the force mediated by the ghost condensate, providing a distinctive signature for this class of models.

Universal dynamics of spontaneous Lorentz violation and a new spin-dependent inverse-square law force

We study the universal low-energy dynamics associated with the spontaneous breaking of Lorentz invariance down to spatial rotations. The effective Lagrangian for the associated Goldstone field can be uniquely determined by the non-linear realization of a broken time diffeomorphism symmetry, up to some overall mass scales. It has previously been shown that this symmetry breaking pattern gives rise to a Higgs phase of gravity, in which gravity is modified in the infrared. In this paper, we study the effects of direct couplings between the Goldstone boson and standard model fermions, which necessarily accompany Lorentz-violating terms in the theory. The leading interaction is the coupling to the axial vector current, which reduces to spin in the non-relativistic limit. A spin moving relative to the "ether" rest frame will emit Goldstone Cerenkov radiation. The Goldstone also induces a long-range inverse-square law force between spin sources with a striking angular dependence, reflecting the underlying Goldstone shockwaves and providing a smoking gun for this theory. We discuss the regime of validity of the effective theory describing these phenomena, and the possibility of probing Lorentz violations through Goldstone boson signals in a way that is complementary to direct tests in some regions of parameter space.

Null test of the inverse-square law of gravity

A differential equivalent of the inverse-square force law is Gauss's law for the field, , where . Thus, by summing the outputs of an in-line gravity gradiometer rotated in three orthogonal directions, one can perform a near-null test of the inverse-square law. The primary advantage of this type of experiment is its reduced sensitivity to the density and metrology errors of the source. We have developed a three-axis superconducting gravity gradiometer and carried out such a test using a lead (Pb) pendulum to produce a time-varying field. This experiment places a new () limit of at , where and are parameters of the generalized potential . This result represents an improvement of an order of magnitude over the best existing limit at . To achieve further improvement in the resolution of with the present instrument, a near-null source is needed. We plan to carry out an improved test of the inverse-square law using a long cylindrical shell as the source. Looking farther into the future, we hope to be able to conduct a geological-scale null experiment by moving the gradiometer up and down a tower above a plain.

Testing the inverse-square law of gravity: a new class of torsion pendulum null experiments

We discuss two null experiments designed to test the inverse-square law of gravity in the critical range between and with higher precision than earlier work. To do so, we have devised a torsion pendulum detector whose mass distribution is specifically configured to provide high-sensitivity detection of a uniquely non-Newtonian derivative of the potential (the horizontal derivative of the Laplacian), rather than looking for a small deviation from the expected power-law dependence on distance of a Newtonian field derivative. This method provides a stronger null test of the gravitational inverse-square law force because it is less sensitive to imperfections in the source mass. We discuss the design of these experiments and estimate their performance relative to currently established experimental limits on inverse-square law violation.

Phase-plane methods for central orbits

The elementary theory of central orbits is developed in terms of a convenient phase plane. The results of this analysis give considerable insight into the qualitative aspects of the problem; this is particularly true of the associated stability theory. The inverse-square law of force is considered in some detail, though other laws of force are also discussed. Phase-plane diagrams are given for some representative cases.

Winding strings, Kaluza-Klein monopoles and runge-lenz vectors

We study the interaction of a class of closed solitonic string states (“winders”) which wind around the Kaluza-Klein circle with a Kaluza-Klein monopole. We find that winders are attracted towards monopoles with an inverse-square law force and even when moving relativistically they pursue exactly elliptical parabolic or hyperbolic orbits. The simplicity of the motion is due to the existence of an extra conserved Runge-Lenz vector in addition to the conserved angular momentum. The Poisson algebra of these conserved vectors closes on so (3) ⊕ s R 3 or so (3), so (3) ⊕ s R or so (3) ⊕ so (2, 1). We also find a class of instanton solutions of the euclidean equations in the field of two or more monopoles. The possibility that monopoles catalyse the decay or annihilation of winders is discussed.

The inverse-square law of force and its spatial energy distribution

Using generalised mathematical considerations to inverse-square law of force is shown to imply specific spatial energy distributions relative to the interacting bodies. The retardation effects associated with energy redeployment when the bodies are in motion are examined. It is found that, as applied to the gravitational interaction between sun and planet and provided there is no discontinuity in the spatial energy distribution, retardation will give a law of motion conforming with Einstein's law of gravitation. A necessary condition is that the energy in transit in the field system is ineffective in determining force for a retardation period equal to the time required for a photon to travel from one body to the field and then return from the field to the other body. The implication is that gravitation could be a quantum interaction which assures causality and balance of action and reaction by this dual photon exchange interaction.

The Kepler orbit from initial conditions via the Lenz vector

The orientation and parameters of the orbit of a satellite under an inverse square law of force are obtained by use of the eccentricity vector, equivalent to the Lenz vector, from a typical set of initial conditions.

Variation of the galactic force law in the region of the Sun

From the radial velocity data for O-B5 stars within 3 kpc of the Sun the variation of the semi-axesΣ 1 andΣ 2 of the velocity ellipsoid in the solar region is estimated. It is found that both the semi-axes as well as their ratioΣ 2/Σ 1 decrease away from the galactic centre. The value ofΣ 2/Σ 1 changes from 1.17 atR=9 kpc to 0.48 atR=11 kpc passing through a value of 0.86 at the solar distance ofR 0=10 kpc. These results are consistent with the usually assumed inverse square law of force in the outer regions of the galaxy.