Inverse Compton scattering in mildly relativistic plasma

We investigated the effect of inverse Compton scattering in mildly relativistic static and moving plasmas with low optical depth using Monte Carlo simulations, and calculated the Sunyaev-Zel'dovich effect in the cosmic background radiation. Our semi-analytic method is based on a separation of photon diffusion in frequency and real space. We use Monte Carlo simulation to derive the intensity and frequency of the scattered photons for a monochromatic incoming radiation. The outgoing spectrum is determined by integrating over the spectrum of the incoming radiation using the intensity to determine the correct weight. This method makes it possible to study the emerging radiation as a function of frequency and direction. As a first application we have studied the effects of finite optical depth and gas infall on the Sunyaev-Zel'dovich effect (not possible with the extended Kompaneets equation) and discuss the parameter range in which the Boltzmann equation and its expansions can be used. For high temperature clusters ($k_B T_e \gtrsim 15$ keV) relativistic corrections based on a fifth order expansion of the extended Kompaneets equation seriously underestimate the Sunyaev-Zel'dovich effect at high frequencies. The contribution from plasma infall is less important for reasonable velocities. We give a convenient analytical expression for the dependence of the cross-over frequency on temperature, optical depth, and gas infall speed. Optical depth effects are often more important than relativistic corrections, and should be taken into account for high-precision work, but are smaller than the typical kinematic effect from cluster radial velocities.Comment: LateX, 30 pages and 11 figures. Accepted for publication in the Astrophysical Journa

Determination the useful life of intangible assets with a negative depreciation

У статті проаналізовано вимоги до визначення терміну корисного використання специфічних нематеріальних активів, таких як товарні знаки та гудвіл, які мають властивість змінювати вартість протягом терміну економічного життя в обох напрямках та здатні таким чином змінювати знак накопиченого зносу. Розглянуто вимоги нормативних баз бухгалтерського обліку та незалежної оцінки, які є частково суперечливими, що створює нормативно-правову колізію між ними. Показано, що закономірність зміни ринкової вартості в часі, а, відповідно, і знак зносу досліджуваних активів залежить від проведення заходів щодо їх підтримки та розвитку, підкріплених відповідним інвестуванням грошових потоків витрат. Встановлено, що протягом періоду корисного використання товарного знака можливою є кількоразова зміна знаку зносу на окремих відрізках часу його економічного життя – залежно від того, чи у достатніх обсягах здійснюються заходи з його інформаційно-рекламної підтримки. Визначено та проаналізовано перелік факторів, які мають бути враховані при визначенні терміну корисного використання товарного знаку. Запропоновано, у найбільш загальному випадку, коли передбачається подальше ефективне використання товарного знаку, вважати його нематеріальним активом з невизначеним (необмеженим) строком корисного використання. Доведено, що нематеріальні активи цього класу недоцільно амортизувати в бухгалтерському обліку, оскільки їх накопичений знос за нормальних умов є від’ємним, а при оцінці їх вартість у загальному випадку нормальної прибуткової діяльності підприємства має визначатися із застосуванням моделі збільшення ринкової вартості в часі.The article analyzes the requirements for the useful lifetime determining of specific intangible assets, such as trademarks and goodwill, which are able to change value over the term of economic life in both directions and thus they can change the sign of accumulated depreciation. The requirements of the regulatory framework of accounting and independent evaluation, which are partially contradictory, that creates a normatively-legal collision between them, are considered. It is shown, that the pattern in market value changes with over time, and, accordingly, the depreciation sign of the investigated assets depends on the of measures implementation for their support and development, supported by appropriate investment of cash flow costs. It has been established, that during of a trademark useful livetime period, it is possible to repeatedly change of depreciation sign on certain periods of its economic life, depending on whether, the measures for its information and advertising support are carried out in sufficient quantities. The factors list, which should be taken into account when a trademark useful lifetime determining is identified and analyzed. Authors proposed, in the most general case, when further efficient use of the trademark is envisaged, to consider it an intangible asset with an indefinite (unlimited) useful lifetime. It is proved that intangible assets of this class are not advisable to depreciate in accounting, since their accumulated depreciation under normal conditions is negative, and when evaluating their value in the general case, the normal profitable activity of an enterprise should be determined with using of market value increasing model over time

The profile of a narrow line after single scattering by Maxwellian electrons: relativistic corrections to the kernel of the integral kinetic equation

The frequency distribution of photons in frequency that results from single Compton scattering of monochromatic radiation on thermal electrons is derived in the mildly relativistic limit. Algebraic expressions are given for (1) the photon redistribution function, K(nu,Omega -> nu',Omega'), and (2) the spectrum produced in the case of isotropic incident radiation, P(nu -> nu'). The former is a good approximation for electron temperatures kT_e < 25 keV and photon energies hnu < 50 keV, and the latter is applicable when hnu(hnu/m_ec^2) < kT_e < 25 keV, hnu < 50 keV. Both formulae can be used for describing the profiles of X-ray and low-frequency lines upon scattering in hot, optically thin plasmas, such as present in clusters of galaxies, in the coronae of accretion disks in X-ray binaries and AGNs, during supernova explosions, etc. Both formulae can also be employed as the kernels of the corresponding integral kinetic equations (direction-dependent and isotropic) in the general problem of Comptonization on thermal electrons. The K(nu,Omega -> nu',Omega') kernel, in particular, is applicable to the problem of induced Compton interaction of anisotropic low-frequency radiation of high brightness temperature with free electrons in the vicinity of powerful radiosources and masers. Fokker-Planck-type expansion (up to fourth order) of the integral kinetic equation with the P(nu -> nu') kernel derived here leads to a generalization of the Kompaneets equation. We further present (1) a simpler kernel that is necessary and sufficient to derive the Kompaneets equation and (2) an expression for the angular function for Compton scattering in a hot plasma, which includes temperature and photon energy corrections to the Rayleigh angular function.Comment: 29 pages, 17 figures, accepted for publication in ApJ, uses emulateapj.sty, corrects misprints in previous astro-ph versio

Method of substitution with group measurements as a metrological basis of comparative methodical approach application

У статті розглянуто метрологічні засади виконання економічних вимірювань із застосуванням порівняльного методичного підходу. Згідно з пропонованим підходом, за методом заміщення сукупність об’єктів порівняння трактується як групова міра, що з певною точністю відтворює значення одиничного показника вартості, властиве об’єктам сукупності. Окремі об’єкти порівняння, що складають цю сукупність, у такому випадку розглядаються як однозначні міри одиничного показника вартості. Показано, що за виконання умови однорідності сукупності за цим параметром, включно із об’єктом оцінки, він може бути з метою визначення його вартісних показників заміщений груповою мірою. Встановлено та проаналізовано основні співвідношення для визначення впливу систематичних та випадкових похибок у цьому випадку. Отримані аналітичні вирази для опису похибок методу заміщення з використанням групової міри дають можливість проаналізувати частку систематичної складової похибки вимірювання у загальній похибці визначення вартості об’єкта оцінки. Отримано та математично обґрунтовано можливості зниження впливу систематичної складової похибки вимірювання, привнесеної об’єктами порівняння. Сформульовано практичні рекомендації щодо збільшення точності та достовірності результатів оціночних робіт.The metrological principles of the economic measurements implementation using the comparative methodical approach are considered in the article. According to the proposed approach using the substitution method, the set of compared objects is treated as a group measurement standard, which with a certain accuracy reproduces the single indicator of value, which is appropriate to the selection objects. Every individual object of comparison representing this set in that case is treated as an unambiguous measurement of the single indicator of value. It is shown that in the case if for the set of compared objects, including evaluation object, homogeneity condition for this parameter is satisfied, it may be replaced by this group measurement in order to determine cost indicators of the evaluation object. The basic proportions to determine the influence of systematic and random errors in this case are formulated and analyzed. The obtained analytical equations describing the errors of the substitution method using the group measurement standard give an opportunity to analyze the share of the systematic component in the overall measurement error of estimating value of the evaluation object. The possibilities of reducing the influence of systematic component of measurement error, introduced by the compared objects, are obtained and mathematically proved. Practical recommendations for increasing accuracy and reliability of evaluation results are formulated

Compton Scattering by Static and Moving Media I. The Transfer Equation and Its Moments

Compton scattering of photons by nonrelativistic particles is thought to play an important role in forming the radiation spectrum of many astrophysical systems. Here we derive the time-dependent photon kinetic equation that describes spontaneous and induced Compton scattering as well as absorption and emission by static and moving media, the corresponding radiative transfer equation, and their zeroth and first moments, in both the system frame and in the frame comoving with the medium. We show that it is necessary to use the correct relativistic differential scattering cross section in order to obtain a photon kinetic equation that is correct to first order in epsilon/m_e, T_e/m_e, and V, where epsilon is the photon energy, T_e and m_e are the electron temperature and rest mass, and V is the electron bulk velocity in units of the speed of light. We also demonstrate that the terms in the radiative transfer equation that are second-order in V usually should be retained, because if the radiation energy density is sufficiently large compared to the radiation flux, the effects of bulk Comptonization described by the terms that are second-order in V are at least as important as the effects described by the terms that are first-order in V, even when V is small. Our equations are valid for systems of arbitrary optical depth and can therefore be used in both the free-streaming and the diffusion regimes. We demonstrate that Comptonization by the electron bulk motion occurs whether or not the radiation field is isotropic or the bulk flow converges and that it is more important than thermal Comptonization if V^2 > 3 T_e/m_e.Comment: 31 pages, accepted for publication in The Astrophysical Journa

Compton Scattering in Static and Moving Media. II. System-Frame Solutions for Spherically Symmetric Flows

I study the formation of Comptonization spectra in spherically symmetric, fast moving media in a flat spacetime. I analyze the mathematical character of the moments of the transfer equation in the system-frame and describe a numerical method that provides fast solutions of the time-independent radiative transfer problem that are accurate in both the diffusion and free-streaming regimes. I show that even if the flows are mildly relativistic (V~0.1, where V is the electron bulk velocity in units of the speed of light), terms that are second-order in V alter the emerging spectrum both quantitatively and qualitatively. In particular, terms that are second-order in V produce power-law spectral tails, which are the dominant feature at high energies, and therefore cannot be neglected. I further show that photons from a static source are upscattered by the bulk motion of the medium even if the velocity field does not converge. Finally, I discuss these results in the context of radial accretion onto and outflows from compact objects.Comment: 28 pages, 9 figures; minor changes, to appear in the Astrophysical Journa

Backward asymmetry of the Compton scattering by an isotropic distribution of relativistic electrons: astrophysical implications

The angular distribution of low-frequency radiation after single scattering by an isotropic distribution of relativistic electrons considerably differs from the Rayleigh angular function. In particular, the scattering by an ensemble of ultra-relativistic electrons obeys the law p=1-cos(alpha), where alpha is the scattering angle; hence photons are preferentially scattered backwards. We discuss some consequences of this fact for astrophysical problems. We show that a hot electron-scattering atmosphere is more reflective than a cold one: the fraction of incident photons which become reflected having suffered a single scattering event can be larger by up to 50 per cent in the former case. This should affect the photon exchange between cold accretion disks and hot coronae or ADAF flows in the vicinity of relativistic compact objects; as well as the rate of cooling (through multiple inverse-Compton scattering of seed photons supplied from outside) of optically thick clouds of relativistic electrons in compact radiosources. The forward-backward scattering asymmetry also causes spatial diffusion of photons to proceed slower in hot plasma than in cold one, which is important for the shapes of Comptonization spectra and the time delays between soft and hard radiations coming from variable X-ray sources.Comment: 20 pages, 3 figures, to appear in Astronomy Letters, added reference

Vertical Structure of the Outer Accretion Disk in Persistent Low-Mass X-Ray Binaries

We have investigated the influence of X-ray irradiation on the vertical structure of the outer accretion disk in low-mass X-ray binaries by performing a self-consistent calculation of the vertical structure and X-ray radiation transfer in the disk. Penetrating deep into the disk, the field of scattered X-ray photons with energy $E\gtrsim10$\,keV exerts a significant influence on the vertical structure of the accretion disk at a distance $R\gtrsim10^{10}$\,cm from the neutron star. At a distance $R\sim10^{11}$\,cm, where the total surface density in the disk reaches $\Sigma_0\sim20$\,g\,cm$^{-2}$, X-ray heating affects all layers of an optically thick disk. The X-ray heating effect is enhanced significantly in the presence of an extended atmospheric layer with a temperature $T_{atm}\sim(2\div3)\times10^6$\,K above the accretion disk. We have derived simple analytic formulas for the disk heating by scattered X-ray photons using an approximate solution of the transfer equation by the Sobolev method. This approximation has a $\gtrsim10$\,% accuracy in the range of X-ray photon energies $E<20$\,keV.Comment: 19 pages, 8 figures, published in Astronomy Letter

Three-dimensional Radiative Properties of Hot Accretion Flows onto the Galactic Centre Black Hole

By solving radiative transfer equations, we examine three-dimensional radiative properties of a magnetohydrodynamic accretion flow model confronting with the observed spectrum of Sgr A*, in the vicinity of supermassive black hole at the Galactic centre. As a result, we find that the core of radio emission is larger than the size of the event horizon shadow and its peak location is shifted from the gravitational centre. We also find that the self-absorbed synchrotron emissions by the superposition of thermal electrons within a few tens of the Schwartzschild radius can account for low-frequency spectra below the critical frequency $\nu_{c}\approx 10^{12}$ Hz. Above the critical frequency, the synchrotron self-Compton emission by thermal electrons can account for variable emissions in recent near-infrared observations. In contrast to the previous study by Ohsuga et al. (2005), we found that the X-ray spectra by Bremsstrahlung emission of thermal electrons for the different mass accretion rates can be consistent with both the flaring state and the quiescent state of Sgr A* observed by {\it Chandra}.Comment: 8 pages, 4 figures, submitted to MNRA

Effects of broadening and electron overheating in tunnel structures based on metallic clusters

We study the influence of energy levels broadening and electron subsystem overheating in island electrode (cluster) on current-voltage characteristics of three-electrode structure. A calculation scheme for broadening effect in one-dimensional case is suggested. Estimation of broadening is performed for electron levels in disc-like and spherical gold clusters. Within the two-temperature model of metallic cluster and by using a size dependence of the Debye frequency the effective electron temperature as a function of bias voltage is found approximately. We suggest that the effects of broadening and electron overheating are responsible for the strong smoothing of current-voltage curves, which is observed experimentally at low temperatures in structures based on clusters consisting of accountable number of atoms.Comment: 8 pages, 5 figure