Fast and Accurate Computation of Orbital Collision Probability for Short-Term Encounters

International audienceThis article provides a new method for computing the probability of collision between two spherical space objects involved in a short-term encounter under Gaussian-distributed uncertainty. In this model of conjunction, classical assumptions reduce the probability of collision to the integral of a two-dimensional Gaussian probability density function over a disk. The computational method presented here is based on an analytic expression for the integral, derived by use of Laplace transform and D-finite functions properties. The formula has the form of a product between an exponential term and a convergent power series with positive coefficients. Analytic bounds on the truncation error are also derived and are used to obtain a very accurate algorithm. Another contribution is the derivation of analytic bounds on the probability of collision itself, allowing for a very fast and - in most cases - very precise evaluation of the risk. The only other analytical method of the literature - based on an approximation - is shown to be a special case of the new formula. A numerical study illustrates the efficiency of the proposed algorithms on a broad variety of examples and favorably compares the approach to the other methods of the literature

Operator solutions for fractional Fokker-Planck equations

We obtain exact results for fractional equations of Fokker-Planck type using evolution operator method. We employ exact forms of one-sided Levy stable distributions to generate a set of self-reproducing solutions. Explicit cases are reported and studied for various fractional order of derivatives, different initial conditions, and for different versions of Fokker-Planck operators.Comment: 4 pages, 3 figure

Tur\'an type inequalities for Kr\"atzel functions

Complete monotonicity, Laguerre and Tur\'an type inequalities are established for the so-called Kr\"atzel function $Z_{\rho}^{\nu},$ defined by Z_{\rho}^{\nu}(u)=\int_0^{\infty}t^{\nu-1}e^{-t^{\rho}-\frac{u}{t}}\dt, where $u>0$ and $\rho,\nu\in\mathbb{R}.$ Moreover, we prove the complete monotonicity of a determinant function of which entries involve the Kr\"atzel function.Comment: 9 page

Upper critical field pecularities of superconducting YNi2B2C and LuNi2B2C

We present new upper critical field Hc2(T) data in a broad temperature region from 0.3K to Tc for LuNi2B2C and YNi2B2C single crystals with well characterized low impurity scattering rates. The absolute values for all T, in particular Hc2(0), and the sizeable positive curvature (PC) of Hc2(T) at high and intermediate T are explained quantitatively within an effective two-band model. The failure of the isotropic single band approach is discussed in detail. Supported by de Haas van Alphen data, the superconductivity reveals direct insight into details of the electronic structure. The observed maximal PC near Tc gives strong evidence for clean limit type II superconductors.Comment: 4 pages, 2 figures, Phys. Rev. Lett. accepte

Influence of oxygen ordering kinetics on Raman and optical response in YBa_2Cu_3O_{6.4}

Kinetics of the optical and Raman response in YBa_2Cu_3O_{6.4} were studied during room temperature annealing following heat treatment. The superconducting T_c, dc resistivity, and low-energy optical conductivity recover slowly, implying a long relaxation time for the carrier density. Short relaxation times are observed for the B_{1g} Raman scattering -- magnetic, continuum, and phonon -- and the charge transfer band. Monte Carlo simulations suggest that these two relaxation rates are related to two length scales corresponding to local oxygen ordering (fast) and long chain and twin formation (slow).Comment: REVTeX, 3 pages + 4 PostScript (compressed) figure

Infrared and optical properties of pure and cobalt-doped LuNi_2B_2C

We present optical conductivity data for Lu(Ni$_{1-x}$Co$_x$)$_2$B$_2$C over a wide range of frequencies and temperatures for x=0 and x=0.09. Both materials show evidence of being good Drude metals with the infrared data in reasonable agreement with dc resistivity measurements at low frequencies. An absorption threshold is seen at approximately 700 cm-1. In the cobalt-doped material we see a superconducting gap in the conductivity spectrum with an absorption onset at 24 +/- 2 cm-1 = 3.9$ +/- 0.4 k_BT_c suggestive of weak to moderately strong coupling. The pure material is in the clean limit and no gap can be seen. We discuss the data in terms of the electron-phonon interaction and find that it can be fit below 600 cm-1 with a plasma frequency of 3.3 eV and an electron-phonon coupling constant lambda_{tr}=0.33 using an alpha^{2}F(omega) spectrum fit to the resistivity.Comment: 10 pages with 10 embedded figures, submitted to PR

Anomalously large oxygen-ordering contribution to the thermal expansion of untwinned YBa2Cu3O6.95 single crystals: a glass-like transition near room temperature

We present high-resolution capacitance dilatometry studies from 5 - 500 K of untwinned YBa2Cu3Ox (Y123) single crystals for x ~ 6.95 and x = 7.0. Large contributions to the thermal expansivities due to O-ordering are found for x ~ 6.95, which disappear below a kinetic glass-like transition near room temperature. The kinetics at this glass transition is governed by an energy barrier of 0.98 +- 0.07 eV, in very good agreement with other O-ordering studies. Using thermodynamic arguments, we show that O-ordering in the Y123 system is particularly sensitive to uniaxial pressure (stress) along the chain axis and that the lack of well-ordered chains in Nd123 and La123 is most likely a consequence of a chemical-pressure effect.Comment: 4 pages, 3 figures, submitted to PR

Inter- and Intragranular Effects in Superconducting Compacted Platinum Powders

Compacted platinum powders exhibit a sharp onset of diamagnetic screening at $T \simeq 1.9$ mK in zero magnetic field in all samples investigated. This sharp onset is interpreted in terms of the intragranular transition into the superconducting state. At lower temperatures, the magnetic ac susceptibility strongly depends on the ac field amplitude and reflects the small intergranular critical current density $j_{c}$. This critical current density shows a strong dependence on the packing fraction f of the granular samples. Surprisingly, $j_{c}$ increases significantly with decreasing f ($j_{c}(B=0, T=0) \simeq 0.07$ A/cm$^{2}$ for f = 0.67 and $j_{c}(B=0, T=0) \simeq 0.8$ A/cm$^{2}$ for f = 0.50). The temperature dependence of $j_{c}$ shows strong positive curvature over a wide temperature range for both samples. The phase diagrams of inter- and intragranular superconductivity for different samples indicate that the granular structure might play the key role for an understanding of the origin of superconductivity in the platinum compacts.Comment: 11 pages including 9 figures. To appear in Phys. Rev. B in Nov. 0

NUMERIČKO INTEGRIRANJE KOD IZRAČUNA VOLUMENA NEPRAVILNIH ANTIKLINALA

The volume of geological structures is often calculated by using the definite integral. Though in some cases the integral can be solved analytically, in practice we usually approximate its value by numerical integration techniques. The application of definite integral in volume calculation is illustrated by two examples. The volume of Mount Fuji, the world-known “conic” geomorphological structure, is calculated by analytical integration. Two basic numerical integration methods, that is, the trapezoidal and Simpson’s rule are applied to subsurface hydrocarbon reservoir volume calculation, where irregular anticline is approximated by a frustum of a right circular cone.Pri izračunavanju volumena geoloških struktura često se koristi određeni integral. Iako se u nekim slučajevima integral može riješiti analitički, u praksi se njegova vrijednost obično procjenjuje koristeći tehnike numeričke integracije. Primjena određenog integrala u izračunavanju volumena ilustrirana je dvama primjerima. Volumen planine Fuji, koja je svjetski poznati geomorfološki primjer “stožaste” strukture, izračunat je analitičkom integracijom. Dvije temeljne metode numeričkog integriranja, tj. trapezno i Simpsonovo pravilo, primijenjene su na izračun volumena ležišta ugljikovodika, gdje je struktura nepravilne antiklinale aproksimirana pravilnim krnjim stošcem