Accurate Analytical Formula Research Articles

Analytical opacity formulas for low Z plasmas

The accurate computation of radiative opacities is basic in the ICF target physics analysis, in which the radiation is an important feature to determine in detail. For this reason, accurate analytical formulas for giving mean opacities versus temperature and density of the plasma seem to be a useful tool. In this work we analyse some analytical expressions found in the literature for the opacity low Z plasmas in a wide range of temperature and densities. The validity of these formulas for computing the opacity under NLTE conditions is investigated using the new code ABAKO.

Inductance formulas adapted for direct use in Spice simulators

Accurate analytical formulas for the self and mutual inductance of rectangular wires, that only require basic arithmetic expressions, are presented. The main objective of these formulas is their use as user defined parameterised expressions in Spice and similar applications, offering a convenient way to simulate and optimise the inductive behaviour of on-chip metal wires.

Design of Spiral and Multiple Split-Ring Resonators for the Realization of Miniaturized Metamaterial Samples

We present the design of miniaturized resonant inclusions to be employed in the practical realization of metamaterial samples with anomalous values of the real part of the permeability. Such inclusions, in fact, can be employed in the design of both mu-negative (MNG) materials and artificial magnetodielectrics (with negative and high-positive values of the real part of the permeability, respectively). The inclusions here considered are the multiple split-ring resonators (MSRRs), that represent a straightforward extension of the commonly used split-ring resonators (SRRs), and the spiral resonators (SRs), that enable a greater miniaturization rate. Some physical insights on the resonance mechanism and on the inherent saturation of the resonant frequency when increasing the number of the rings of the MSRRs and the number of the turns of the SRs are given in the paper. New and accurate analytical design formulas, based on a quasi-static model, for both MSRRs and SRs are derived and tested through a proper comparison with the existing formulas and full-wave numerical results. Both MSRRs and SRs are shown to be useful to reduce the electrical dimensions of the resonant inclusions when synthesizing artificial metamaterials.

Ground-state energy of finite Fermi systems

We study the ground-state shell correction energy of a fermionic gas in a mean-field approximation. Considering the particular case of three-dimensional harmonic trapping potentials, we show the rich variety of different behaviors (erratic, regular, supershells) that appear when the number-theoretic properties of the frequency ratios are varied. For self-bound systems, where the shape of the trapping potential is determined by energy minimization, we obtain accurate analytic formulas for the deformation and the shell correction energy as a function of the particle number $N$. Special attention is devoted to the average of the shell correction energy. We explain why in self-bound systems it is a decreasing (and negative) function of $N$.

Phenomenological model to fit complex permittivity data of water from radio to optical frequencies

A general factorized form of the dielectric function together with a fractional model-based parameter estimation method is used to provide an accurate analytical formula for the complex refractive index in water for the frequency range 10(8)-10(16)Hz . The analytical formula is derived using a combination of a microscopic frequency-dependent rational function for adjusting zeros and poles of the dielectric dispersion together with the macroscopic statistical Fermi-Dirac distribution to provide a description of both the real and imaginary parts of the complex permittivity for water. The Fermi-Dirac distribution allows us to model the dramatic reduction in the imaginary part of the permittivity in the visible window of the water spectrum.

Analytical prediction of abnormal temporary overvoltages due to ground faults in MV networks

The paper deals with the temporary overvoltages that build up in radial MV distribution networks following the inception of a 1-phase-to-ground fault. It will be demonstrated that, for extended cable/overhead MV distribution networks with ungrounded neutral, in case of low resistance faults at critical locations along overhead lines, the neutral voltage can reach very large values, significantly higher than 1 p.u. (up to 2.5 p.u.). The attendant temporary overvoltages on healthy phases can be very large, much higher than √3 p.u. (up to 3.5 p.u.). Fault currents are also affected, reaching twice the value calculated with simplified methods, i.e. neglecting series impedances. Quick yet accurate analytical formulas for the prediction of maximum overvoltages, fault currents and critical fault distances are presented, together with their validation by detailed ATP-EMTP simulations. The results of a study on an existing Italian 15 kV–50 Hz extended mixed cable/overhead distribution network are finally presented and discussed.

Charm quark contribution toK+→π+νbar nuat next-to-next-to-leading order

We calculate the complete NNLO QCD corrections to the charm contribution of the rare decay K+ -> pi+ nu nu-bar. We encounter several new features, which were absent in lower orders. We discuss them in detail and present the results for the 2-loop matching conditions of the Wilson coefficients, the 3-loop anomalous dimensions, and the 2-loop matrix elements of the relevant operators that enter the NNLO renormalization group analysis of the Z-penguin and the electroweak box contribution. The inclusion of the NNLO QCD corrections leads to a significant reduction of the theoretical uncertainty from 9.8% down to 2.4% in the relevant parameter Pc, implying the leftover scale uncertainties in BR(K+ -> pi+ nu nu-bar) and in the determination of |V_td|, sin(2 beta), and gamma from the K -> pi nu nu system to be 1.3%, 1.0%, 0.006, and 1.2 degrees, respectively. For the charm quark MSbar mass mc=(1.30+-0.05) GeV and |V_us|= 0.2248 the NLO value Pc=0.37+-0.06 is modified to Pc=0.38+-0.04 at the NNLO level with the latter error fully dominated by the uncertainty in mc. We present tables for Pc as a function of mc and alphas(MZ) and a very accurate analytic formula that summarizes these two dependences as well as the dominant theoretical uncertainties. Adding the recently calculated long-distance contributions we find BR(K+ -> pi+ nu nu-bar)=(8.0+-1.1)*10^-11 with the present uncertainties in mc and the Cabibbo-Kobayashi-Maskawa elements being the dominant individual sources in the quoted error. We also emphasize that improved calculations of the long-distance contributions to K+ -> pi+ nu nu-bar and of the isospin breaking corrections in the evaluation of the weak current matrix elements from K+ -> pi0 e+ nu would be valuable in order to increase the potential of the two golden K -> pi nu nu decays in the search for new physics.

Analyses of elliptical coplanar coupled waveguides and coplanar coupled waveguides with finite ground width

In this paper, the quasi-TEM characteristic parameters of elliptical coplanar coupled waveguides and coupled coplanar waveguide with a finite ground width configuration are presented and analyzed. Computer-aided design (CAD)-oriented fast, simple, and accurate analytic formulas are derived by using conformal mapping techniques, which provide satisfactory accuracy at microwave frequencies and lead to closed-form analytical solutions suitable for CAD software packages. The results for the odd- and even-mode characteristic impedance, effective dielectric constant, and coupling coefficient have been computed by these formulas. Good agreement between the present results and published results is observed. For the planar case, simulations have also been undertaken with Sonnet electromagnetic-circuit solver software. The computed results agree well with those of the simulation ones.

A numerical–stochastic technique for underground cable system design

A numerical–stochastic technique for an optimisation of underground cable systems is based on a finite-element mesh generator carried out by means of the advancing front method, a nonlinear coupled electric-thermal model, a multiquadric ampacity approximations including successive zooming steps, and a stochastic optimisation method involving differential evolution. The proposed technique provides the optimisation of cable systems for which neither accurate analytical nor empirical ampacity formulas are available, thus resulting in more economical design.

Accurate Analytical Model of Equivalent Dielectric Constant for Rain Medium

In this paper, rain medium is considered as random statistically isotropy mixture, formed by distributing different shape and size raindrops following some raindrop-spectrum in atmosphere. The accurate analytical formula to calculate dielectric constant of rain medium is deduced, which take the influences of ⟨P⟩ and P on the field in rain region and raindrop inside and the effect of shape of raindrop on dielectric constant of raindrop into account. The condition of taking rain medium as continuous medium is discussed, and the dielectric constant of raindrop is found, if the condition is not satisfied. The important fact is that the interaction between targets in rain and rain medium can be expediently researched by the formula given in this paper. Attenuation of rain is computed using the conclusion in this paper, which is explanatorily compared with attenuation measured. The way applied in this paper is valid for other discrete medium, fog, dust storm, for example.

Optimal trap shape for a Bose gas with attractive interactions

Dilute Bose gas with attractive interactions is considered at zero temperature, when practically all atoms are in Bose-Einstein condensate. The problem is addressed aiming at answering the question: What is the optimal trap shape allowing for the condensation of the maximal number of atoms with negative scattering lengths? Simple and accurate analytical formulas are derived allowing for an easy analysis of the optimal trap shapes. These analytical formulas are the main result of the paper.

Coplanar propagation paths of 3D cracks in infinite bodies loaded in shear

Bower and Ortiz, recently followed by Lazarus, developed a powerful method, based on a theoretical work of Rice, for numerical simulation of planar propagation paths of mode 1 cracks in infinite isotropic elastic bodies. The efficiency of this method arose from the need for the sole 1D meshing of the crack front. This paper presents an extension of Rice’s theoretical work and the associated numerical scheme to mixed-mode (2 + 3) shear loadings. Propagation is supposed to be channeled along some weak planar layer and to remain therefore coplanar, as in the case of a geological fault for instance. The capabilities of the method are illustrated by computing the propagation paths of cracks with various initial contours (circular, elliptic, rectangular, heart-shaped) in both fatigue and brittle fracture. The crack quickly reaches a stable, almost elliptic shape in all cases. An approximate but accurate analytic formula for the ratio of the axes of this stable shape is derived.

The Rossiter‐McLaughlin Effect and Analytic Radial Velocity Curves for Transiting Extrasolar Planetary Systems

A transiting extrasolar planet sequentially blocks off the light coming from the different parts of the disk of the host star in a time-dependent manner. Because of the spin of the star, this produces an asymmetric distortion in the line profiles of the stellar spectrum, leading to an apparent anomaly in the the radial velocity curves, known as the Rossiter-McLaughlin effect. Here, we derive approximate but accurate analytic formulae for the anomaly in the radial velocity curves, taking into account the stellar limb darkening. The formulae are particularly useful in extracting information on the projected angle between the planetary orbit axis and the stellar spin axis, λ, and the projected stellar spin velocity, V sin Is. We create mock samples for the radial curves for the transiting extrasolar system HD 209458 and demonstrate that constraints on the spin parameters (V sin Is, λ) can be significantly improved by combining our analytic template formulae and the precision velocity curves from high-resolution spectroscopic observations with 8-10 m class telescopes. Thus, future observational exploration of transiting systems using the Rossiter-McLaughlin effect will be one of the most important probes for a better understanding of the origin of extrasolar planetary systems, especially the origin of their angular momentum.

An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate

In this paper, the unsteady boundary-layer flows caused by an impulsively stretching flat plate is solved by means of an analytic approach. Unlike perturbation techniques, this approach gives accurate analytic approximations uniformly valid for all dimensionless time. Besides, a simple but accurate analytic formula for the local skin friction is given, which agrees well with numerical results and thus is useful in the related industries. To the best of our knowledge, this type of analytic solutions has been never reported. Furthermore, the proposed analytic approach has general meaning and therefore may be applied in the similar way to other unsteady boundary-layer flows to get accurate analytic solutions valid for all time.

Ion‐Ion Correlation Effect on the Neutrino‐Nucleus Scattering in Supernova Cores

We calculate the ion-ion correlation effect on the neutrino-nucleus scattering in supernova cores, which is an important opacity source for the neutrinos and plays a vital role in the supernova explosion. In order to calculate the ion-ion correlation effect we use the results of the improved hypernetted-chain method calculations of the classical one-component plasma. As in the preceding studies on this effect, we find a dramatic decrease of the effective neutrino-nucleus scattering cross section for relatively low energy neutrinos with E < 20MeV. As a matter of fact, our calculation shows a much more dramatic reduction of the effective neutrino-nucleus scattering cross section for the low energy neutrinos with E < 10MeV than the results of Horowitz. Therefore, the ion-ion correlation effect will be more important than has hitherto been recognized. We present an accurate analytic fitting formula that summarizes our numerical results. This fitting formula will facilitate the application of the present results to the supernova explosion simulations.

Relativistic corrections to the Sunyaev-Zeldovich effect for extremely hot clusters of galaxies

We present an accurate numerical table for the relativistic corrections to the thermal Sunyaev-Zeldovich effect for clusters of galaxies. The numerical results for the relativistic corrections have been obtained by numerical integration of the collision term of the Boltzmann equation. The numerical table is provided for the ranges 0.002 ≤ θ e ≤ 0.100 and 0 ≤ X < 20, where θ e ≡ k B T e /m e c 2 , X ≡ hω/k B T 0 , T e is the electron temperature, ω is the angular frequency of the photon and T 0 is the temperature of the cosmic microwave background radiation. We also present an accurate analytic fitting formula that reproduces the numerical results with high precision.

Synthetic stellar populations: single stellar populations, stellar interior models and primordial protogalaxies

ABSTRACT We present a new set of stellar interior and synthesis models for predicting the inte-grated emission from stellar populations in star clusters and galaxies of arbitrary ageand metallicity. This work differs from existing spectral synthesis codes in a number ofimportant ways, namely (1) the incorporation of new stellar evolutionary tracks, withsufficient resolution in mass to sample rapid stages of stellar evolution; (2) a phys-ically consistent treatment of evolution in the HR diagram, including the approachto the main sequence and the effects of mass loss on the giant and horizontal-branchphases. Unlike several existing models, ours yield consistent ages when used to datea coeval stellar population from a wide range of spectral features and colour indexes.We use Hipparcos data to support the validity of our new evolutionary tracks. Werigorously discuss degeneracies in the age-metallicity plane and show that inclusion ofspectral features blueward of 4500 ˚A suffices to break any remaining degeneracy andthat with moderate S/N spectra (10 per 20˚A resolution element) age and metallicityare not degenerate. We also study sources of systematic errors in deriving the ageof a single stellar population and conclude that they are not larger than 10 − 15%.We illustrate the use of single stellar populations by predicting the colors of primor-dial proto-galaxies and show that one can first find them and then deduce the formof the IMF for the early generation of stars in the universe. Finally, we provide ac-curate analytic fitting formulas for ultra fast computation of colors of single stellarpopulations.Key words: galaxies: stellar populations; stars: stellar evolution

In-plane deformation of cantilever plates with applications to lateral force microscopy

The in-plane deformation of atomic force microscope (AFM) cantilevers under lateral loading is commonly assumed to have negligible effect in comparison to other deformation modes and ignored. In this article, we present a theoretical study of the behavior of cantilevers under lateral loading, and in so doing establish that in-plane deformation can strongly contribute to the total deformation, particularly for rectangular cantilevers of high aspect ratio (length/width). This has direct implications to lateral force microscopy, where the neglect of in-plane deformation can contribute to significant quantitative errors in force measurements and affect the interpretation of measurements. Consequently, criteria and approaches for minimizing the effects of in-plane deformation are presented, which will be of value to users and designers of AFM cantilevers. Accurate analytical formulas for the in-plane spring constants of both rectangular and V-shaped cantilevers are also presented.

On the nature of the X-ray emission from the accreting millisecond pulsar SAX J1808.4-3658

We study the pulse profiles of the accreting X-ray millisecond pulsar SAX J1808.4-3658 at different energies. The two main emission components, the blackbody and the Comptonized tail, which are clearly identified in the time-averaged spectrum, show strong variability with the first component lagging the second one. The observed variability can be explained if the emission is produced by Comptonization in a hot slab (radiative shock) of Thomson optical depth ∼0.3-1 at the neutron star surface. The blackbody radiation is strongly beamed along the normal to the slab (a 'knife'-like or 'pencil'-like emission pattern), while the Comptonized emission has a broader angular distribution peaking at about 50°-60° from the slab normal (a 'fan'-like pattern). We construct a detailed model of the X-ray production accounting for the Doppler boosting, relativistic aberration and gravitational light bending in the Schwarzschild space-time. We present also accurate analytical formulae for computations of the light curves from rapidly rotating neutron stars using the formalism recently developed by Beloborodov. Our model reproduces well the pulse profiles at different energies simultaneously, corresponding phase lags, as well as the time-averaged spectrum. We constrain the compact star mass to be bounded between 1.2 and 1.6 M O .. By fitting the observed profiles, we determine the radius of the compact object to be R ∼11 km if M = 1.6 M O ., while for M = 1.2 M O . the best-fitting radius is ∼6.5 km, indicating that the compact object in SAX J1808.4-3658 can be a strange star. We obtain a lower limit on the inclination of the system of 65°.

Enhancement of Resonant Thermonuclear Reaction Rates in Extremely Dense Stellar Plasmas

The enhancement factor of the resonant thermonuclear reaction rates is calculated for the extremely dense stellar plasmas in the liquid phase. In order to calculate the enhancement factor we use the screening potential which is deduced from the numerical experiment of the classical one-component plasma. It is found that the enhancement is tremendous for white dwarf densities if the ^{12}C + ^{12}C fusion cross sections show resonant behavior in the astrophysical energy range. We summarize our numerical results by accurate analytic fitting formulae.