Simple Approximate Form Research Articles

Novel constraints on the particle nature of dark matter from stellar streams

Tidal streams are highly sensitive to perturbations from passing dark matter (DM) subhalos and thus provide a means of measuring their abundance. In a recent paper, we analyzed the distribution of stars along the GD-1 stream with a combination of data from the Gaia satellite and the Pan-STARRS survey, and we demonstrated that the population of DM subhalos predicted by the cold dark matter (CDM) paradigm are necessary and sufficient to explain the perturbations observed in the linear density of stars. In this paper, we use the measurements of the subhalo mass function (SHMF) from the GD-1 data combined with a similar analysis of the Pal 5 stream to provide novel constraints on alternative DM scenarios that predict a suppression of the SHMF on scales smaller than the mass of dwarf galaxies, marginalizing over uncertainties in the slope and normalization of the unsuppressed SHMF and the susceptibility of DM subhalos in the inner Milky Way to tidal disruption. In particular, we derive a 95% lower limit on the mass of warm dark matter (WDM) thermal relics m WDM > 3.6 keV from streams alone that strengthens to m WDM > 6.2 keV when adding dwarf satellite counts. Similarly, we constrain the axion mass in ultra-light (“fuzzy”) dark matter (FDM) models to be m FDM > 1.4 × 10-21 eV from streams alone or m FDM > 2.2 × 10-21 eV when adding dwarf satellite counts. Because we make use of simple approximate forms of the streams' SHMF measurement, our analysis is easy to replicate with other alternative DM models that lead to a suppression of the SHMF.

Electron Broadening Operator Including Penetrating Collisions for Hydrogen

The expression of the electron broadening operator including the effect of penetrating collisions, i.e., for which the incoming electron enters the extent of bound-electron wave-functions, is rather complicated, even for hydrogen. It involves integrals of special functions, the evaluation of which deserves scrutiny. We present a simple approximate form of the electron collision operator for hydrogen including penetration effects, both in direct and interference terms. The new expression is accurate and easy to compute. In the Penetration Standard Theory, the collision operator is convergent whatever the value of the maximum impact parameter. However, when penetration theory is not valid anymore, it should be questioned. We discuss the problem of strong collisions when penetration effects are taken into account.

Modeling penetrating collisions in the standard line broadening impact theory for hydrogen

Spectral lines emitted by plasmas provide information about the thermodynamic conditions, the degree of randomness or the interactions prevailing in the medium. Collisions by plasma electrons penetrating the extent of bound-electron wavefunctions is important at high density, where short-range interactions become dominant. Such collisions are usually not taken into account properly in the standard lineshape theory, assuming long-range dipole approximation. The formalism of penetrating collisions for hydrogen relies on the introduction of a family of integrals calculated using a recursion relation. In this work, we show that such integrals can be expressed analytically, as a finite sum involving binomial coefficients and modified Bessel functions of the third kind. The explicit expression enabled us to obtain a simple approximate analytical form for the collision operator, making numerical implementation and physical interpretation easier. We also propose simple analytical forms of coefficients and integrals important for the modeling of penetrating collisions.

A New Chu Formula for Q

The Chu fundamental limitation on Q for electrically small antennas is well known and widely used. However, Chu did not include energy stored inside the hypothetical enveloping sphere. Exact calculation of total stored energy by Thal gives a new Q, which should better match well-designed electrically small antennas. The TM- and TE-mode Q formulas are derived, and a simple approximate form for the lowest-order TM mode is presented.

Testing New IndirectCPViolation

If new CP violating physics contributes to neutral meson mixing, but its contribution to CP violation in decay amplitudes is negligible, then there is a model independent relation between four (generally independent) observables related to the mixing: the mass splitting (x), the width splitting (y), the CP violation in mixing (1-|q/p|), and the CP violation in the interference of decays with and without mixing (phi). For the four neutral meson systems, this relation can be written in a simple approximate form: y tan phi approximately x(1-|q/p|). In the K system, all four observables have been measured and obey the relation to excellent accuracy. For the B(s) and D systems, new predictions are provided. The success or failure of these relations will probe the physics that is responsible for the CP violation.

A maximum likelihood method for particle momentum determination

We discuss a maximum likelihood method for determining a charged particle's momentum as it moves in a magnetic field. The formalism is presented in both rigorous and approximate forms. The rigorous form is valid when random processes include multiple scattering, energy loss and detector spatial resolution. When the measurement error is dominated by multiple scattering, it takes a particularly simple approximate form. The validity of both formalisms extends to include non-Gaussian multiple scattering distribution.

Unintegrated generalized parton distributions

We show how the generalized (or skewed) parton distributions of the proton, $\overline{H}(x,\ensuremath{\xi}{;k}_{t}^{2},{\ensuremath{\mu}}^{2}),$ unintegrated over the partonic transverse momenta, can be calculated from the known conventional parton distributions, $q(x,{\ensuremath{\mu}}^{2})$ and $g(x,{\ensuremath{\mu}}^{2}),$ for small values of the skewedness parameter $\ensuremath{\xi}.$ We demonstrate the procedure by numerically evaluating the skewed unintegrated gluon. We also provide a simple approximate phenomenological form of the distribution, which may be used to make more rapid predictions of observables.

Self-Preserving Theory for the Volume Distribution of Particles Undergoing Brownian Coagulation

In this paper, the self-preserving theory of coagulating aerosols is presented in a new way: the logarithmic volume (or mass) distribution of an aerosol undergoing coagulation stays invariant in shape at long times. This is shown for both the free molecular and continuum regime collision frequency functions as well as the constant collision frequency function. In addition, new simple approximate forms are presented for the self-preserving distributions, based on numerical solutions to the discrete coagulation equation.

SELF-PRESERVING THEORY FOR THE VOLUME DISTRIBUTION OF PARTICLES UNDERGOING BROWNIAN COAGULATION

In this paper, the self-preserving theory of coagulating aerosols is presented in a new way: the logarithmic volume (or mass) distribution of an aerosol undergoing coagulation stays invariant in shape at long times. This is shown for both the free molecular and continuum regime collision frequency functions as well as the constant collision frequency function. In addition, new simple approximate forms are presented for the self-preserving distributions, based on numerical solutions to the discrete coagulation equation.

Cumulative coordinates for approximations of high‐order atomic multipole moments in aluminosilicate and aluminophosphate sieves*

AbstractA scheme to obtain approximate analytical functions for the atomic distributed multipole moments of the crystallographically different atoms within aluminosilicate and aluminophosphate sieves is discussed. Respective atomic multipole moments are derived within the CRYSTAL95 ab initio periodic Hartree–Fock code with different basis sets, from minimal STO‐3G to 6‐21G*. In order to illustrate the possible applications, distributed analyses are carried out for various structural models from all‐siliceous zeolites and aluminophosphates with ratio Al/P=1 to hydrogen forms of aluminosilicates. Simple approximate forms based on charge and geometry coordinates are proposed for the high‐order moments of each atom, which are further required for the calculation of the electrostatic field within the structures. The possibility to use this analytical approach to evaluate the electrostatic field within embedded cluster models is also shortly discussed. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 70–85, 2001

Effective photon spectra for the photon colliders

The luminosity distribution in the effective $\gamma\gamma$ mass at a photon collider usually has two peaks which are well separated: a high energy peak with mean energy spread about 5–7% and a wide low energy peak. The low energy peak strongly depends on the details of the design and is unsuitable for the study of new physics phenomena. We find a simple approximate form for the spectra of colliding photons for $\gamma \gamma$ and $e\gamma $ colliders, whose convolution describes the high energy luminosity peak with a good accuracy in most of the essential preferable region of the parameters.

Modelling of charged particle decay in nitrogen afterglow

In this paper we extend the available experimental data for breakdown delay times as a function of the afterglow time to short afterglow times for a pulsed obstructed discharge in nitrogen. Measurements were performed at 6.6 mbar in research grade nitrogen by application of a high-vacuum procedure. The delay times are separated into formative and statistical time lags and their dependence on the afterglow time is discussed. A model is developed that is able to reproduce the breakdown delay time behaviour for short afterglow times which is based on ion density decay through the diffusion. A transition from the effective diffusion coefficient of to the free diffusion of ions is observed at times of the order of 20 ms. The model was applied both in a simple approximate analytical form and in a numerical form with accurate representation of the geometry of the tube. When applied in conjunction with our earlier model of the breakdown delay appropriate for longer afterglow times the theory can predict the behaviour of the breakdown delay times between 0.5 ms and several hours.

Beyond the instantaneous-exchange approximation for three-dimensional QED at finite temperature

Dynamical fermion mass generation is studied inN-flavour QED in three dimensions at finite temperature, using the Schwinger-Dyson (S-D) equation. Previous work (in the imaginary-time formalism) approximated the photon propagator Δμv by retaining only the μ=ν=0 part, at zero frequency (“instantaneous exchange”). Here we calculate the photon propagator to leading order in 1/N, in the real-time formalism. Simple approximate forms for the longitudinal (∏ L ) and transverse (∏ T ) photon self-energies are proposed. The S-D equation is solved in both the instantaneous and retarded cases. Numerical results for the temperature dependence of the dynamically generated mass (Σ(T)) are presented, for a range ofN values and for different approximate self-energies. The ratior=2Σ(T=0)/k B T c is generally found to be somewhat reduced when retardation is included, and to depend quite sensitively on the low-momentum behaviour of ∏ L and ∏ T . It is concluded that a valuer=6 is a reasonable estimate in this model.

Utilization of guided elastic waves for the characterization of circumferential cracks in hollow cylinders

The scattering of guided elastic waves from a circumferential crack in an isotropic hollow cylinder is investigated analytically using the S-parameter formalism. A general formula, analogous to that of [B. A. Auld, Acoustic Fields and Waves in Solids (Krieger, Malabar, FL, 1990), 2nd ed., Vol. II, p. 200], is derived for the scattered amplitude of an arbitrary mode due to the incidence of an arbitrary mode. The amplitudes are expressed as an integral of the inner product of the crack opening displacement caused by the incident mode and the traction field of the scattered mode. Using a simple approximate form of known elastostatic solutions for the crack opening displacements due to pure tension, a formula is derived for the scattering of guided wave modes of arbitrary circumferential order due to the incidence of axisymmetric wave modes with large wavelength and uniform stress distribution. These results are analyzed and conclusions drawn on the possibility of crack sizing using relatively simple scattering amplitude information.

Sound scattering by randomly rough elongated objects. I. Means of scattered field

By use of the recently published deformed cylinder formulation [T. K. Stanton, J. Acoust. Soc. Am. 86, 691–705 (1989)], the scattered field due to randomly rough elongated dense elastic objects is derived. The radius of the deformed cylinder is allowed to vary with respect to position along the lengthwise axis resulting in a one-dimensional roughness. Explicit expressions are derived describing both the mean and mean-square scattered field for the rough straight finite length cylinder for both ka ≪ 1 and ka ≫ 1 (k is the acoustic wavenumber and a is the radius), while only the mean is calculated for the prolate spheroid, uniformly bent finite cylinder, and infinitely long cylinder for ka ≫ 1. The modal-series-based solution is used in the ka ≪ 1 case and a recently published form of the Sommerfeld-Watson transformed solution is used in the ka ≫ 1 region where the coupling coefficients of the surface elastic wave components of the field are in simple approximate form [P. L. Marston, J. Acoust. Soc. Am. 83, 25–37 (1988)]. All ka ≫ 1 results in this work demonstrate attenuation of the “specular” (echo from front face) and Rayleigh surface elastic wave components of the scattered field due to the roughness. In addition, the attenuation of the Rayleigh wave depends on the number of times it has circumnavigated the object resulting in a loss of the higher “m” terms in excess to that due to radiation alone. The results are compared with the scattering by rough planar interfaces with distinct differences arising because of effects specific to volumetric scattering. [Work supported by ONR.]

Electron impact cross sections and spatial aspects of electron energy degradation in water vapor

Using a comprehensive set of electron cross sections, we have carried out a Monte Carlo study of the spatial and energetic aspects of electron degradation in the energy range from 30 eV to 1 MeV. The results are given in terms of a four-dimensional spatial yield spectral distribution. This is represented by a simple approximate analytic form which is convenient to use for calculating the spatial distribution of the initial species created due to electron energy degradation. We also investigate the sensitivity of these electron transport results to some of the input cross sections. We have specifically examined the influence of the assumed form of cross sections for vibrational states on the gross yields. We conclude that in transport calculations it is important to use approximate forms with reasonable asymptotic limits. Finally, we call attention to molecular processes which are important in microdosimetry but whose cross sections are particularly uncertain.

The kinetics of solubilization of nonpolar oils by nonionic surfactant solutions

A quantitative study of the rate of solubilization of nonpolar oils in aqueous surfactant solutions has been made using a new technique. The dissolution of a single microscopic oil droplet adhering to a thin fiber suspended in an aqueous surfactant solution is recorded photomicrographically and the solubilization rate (rate of volume change per unit interfacial area) is calculated from the shape of the droplet using previously published theory (B. J. Carroll, J. Colloid Interface Sci. 57, 488 (1976)), which in this case reduces to a simple approximate form. The new technique has been used to study solubilization kinetics in some well-characterized systems containing a high-purity nonionic surfactant. The effects of surfactant concentration, temperature, and oil type were studied. The rate of solubilization is proportional to the surfactant concentration above the CMC and depends on the oil polarity and molecular weight. The rate is strongly temperature dependent in the region of the nonionic cloud point: 15°K below the cloud point, the rate is extremely small relative to that at the cloud point. It is shown that in the case of highly insoluble oils the solubilization mechanism must involve the adsorption-desorption of micelles at the oil/water interface, rather than the diffusion of molecules of oil into the micelle via the aqueous phase. The experimental data are shown to be consistent with the slow stage of the process being the adsorption, not desorption, of the micelle at the interface. The magnitude of the measured rate constant is consistent with theory if dissociation of the micelle occurs prior to the adsorption step and an estimate of the relaxation time for demicellization is obtained. The temperature dependence of the rate in the cloud point region is discussed in terms of changes in the properties of the nonionic solution in this region.

Pitch angle dependence of the charge-exchange lifetime of ring current ions

Previous work has parameterized the pitch angle dependence of the charge-exchange lifetime τ of ring current ions in terms of γ, the power of the cosine of the mirror latitude λ m of the particles, such that τ(λ m) τ(0) ≌ cos γ λ m at given L. Using the atomic hydrogen density model of Johnson and Fish, previous authors have suggested values of γ = 5 or 6. We here evaluate γ as a function of λ m and L using the more recent Chamberlain density models, and show that γ = 3−4 is more appropriate over most of the pitch angle and L range. Consequently, ion distributions in the ring current decay phase are expected to become rather less anisotropic in pitch angle due to chargeexchange than previously believed. We have also investigated the use of several other simple approximate analytic forms for τ(λ m) τ(0) , one of which gives far better agreement with the numerical results than the cos γ λ m, variation, and should hence be used in future studies.

Temperature and polarization dependence of extended x-ray absorption fine-structure spectra

The temperature and polarization dependence of the extended x-ray absorption fine structure (EXAFS) is calculated within the framework of a simple approximate form for the theory. A discussion of the microscopic information contained in such dependences is given, and a specific application to the polarization-dependent EXAFS in zinc is made.

Atomic stopping-power problems encountered in measurements of nuclear γ-ray lifetimes by the Doppler-shift-attenuation method

The value of the nuclear lifetime of the 3.34-MeV level in 22Ne as determined by Broude et al. by the Doppler shift attenuation method exhibits variations depending on the atomic number Z 2 of the slowing down medium. We have re-evaluated the lifetime within the framework of the Lindhard-Winther stopping theory, assuming a simple approximate form for the density of electrons associated with an atom in a solid and an effective point charge for the projectile based on experimental stopping powers. Although there are still variations in the value of the lifetime the variations appear to be systematic through the region of the transition elements.