Power-law Velocity Distribution Research Articles

Polarization from a Radially Stratified GRB Outflow

While the dominant radiation mechanism of gamma-ray bursts (GRBs) remains a question of debate, synchrotron emission is one of the foremost candidates to describe the multi-wavelength afterglow observations. As such, it is expected that GRBs should present some degree of polarization across their evolution—presenting a feasible means of probing these bursts’ energetic and angular properties. Although obtaining polarization data is difficult due to the inherent complexities regarding GRB observations, advances are being made, and theoretical modeling of synchrotron polarization is now more relevant than ever. In this manuscript, we present the polarization for a fiduciary model, where the synchrotron FS emission evolving in the radiative–adiabatic regime is described by a radially stratified off-axis outflow. This is parameterized with a power-law velocity distribution and decelerated in a constant-density and wind-like external environment. We apply this theoretical polarization model for two select GRBs, presenting upper limits in their polarization—GRB 170817A, a known off-axis GRB with radio polarization upper limits, and GRB 190014C, an on-axis case, where the burst was seen from within the half-opening angle of the jet, with observed optical polarization—in an attempt to constrain their magnetic field geometry in the emitting region.

On the impact of runaway stars on dwarf galaxies with resolved interstellar medium

ABSTRACT ‘Runaway stars’ might play a role in driving galactic outflows and enriching the circumgalactic medium with metals. To study this effect, we carry out high-resolution dwarf galaxy simulations that include velocity ‘kicks’ to massive stars above eigth solar masses. We consider two scenarios, one that adopts a power law velocity distribution for kick velocities, resulting in more stars with high-velocity kicks, and a more moderate scenario with a Maxwellian velocity distribution. We explicitly resolve the multiphase interstellar medium (ISM) and include non-equilibrium cooling and chemistry. We sample individual massive stars from an IMF and follow their radiation input and SN feedback (core-collapse) channel at the end of their lifetime. In the simulations with runaway stars, we add additional (natal) velocity kicks that mimic two- and three-body interactions that cannot be fully resolved in our simulations. We find that including runaway or ‘walkaway’ star scenarios impacts mass, metal, momentum, and energy outflows as well as the corresponding loading factors. The effect on the mass loading factor is small, but we find an increase in the metal loading by a factor of 1.5 to 2. The momentum loading increases by a factor of 1.5–2. The energy loading increases by roughly a factor of 5 when runaway stars are included. Additionally, we find that the overall level of star formation is increased in the models that include runaway stars. We conclude that the inclusion of runaway stars could have an impact on the global star formation and subsequent outflow properties of dwarf galaxies.

GRB Afterglow of the Sub-relativistic Materials with Energy Injection

Sub-relativistic materials launched during the merger of binary compact objects and the core collapse of massive stars acquire velocity structures when expanding in a stratified environment. The remnant (either a spinning magnetized neutron star (NS) or a central black hole) from the compact object or core collapse could additionally inject energy into the afterglow via spin-down luminosity or/and by accreting fallback material, producing a refreshed shock, modifying the dynamics, and leading to rich radiation signatures at distinct timescales and energy bands with contrasting intensities. We derive the synchrotron light curves evolving in a stratified environment when a power-law velocity distribution parameterizes the energy of the shock, and the remnant continuously injects energy into the blast wave. As the most relevant case, we describe the latest multiwavelength afterglow observations (≳900 days) of the GW170817/GRB 170817A event via a synchrotron afterglow model with energy injection of a sub-relativistic material. The features of the remnant and the synchrotron emission of the sub-relativistic material are consistent with a spinning magnetized NS and the faster blue kilonova afterglow, respectively. Using the multiband observations of some short bursts with evidence of kilonovae, we provide constraints on the expected afterglow emission.

Wave Excitation by Power-law-Distributed Energetic Electrons with Pitch-angle Anisotropy in the Solar Corona

Abstract Radio waves from the Sun are emitted, as a rule, due to energized electrons. Observations infer that the related energized electrons follow (negative) power-law velocity distributions above a break velocity U b . They might also distribute anisotropically in the pitch-angle space. To understand radio wave generation better, we study the consequences of anisotropic power-law-distributed energetic electrons in current-free collisionless coronal plasmas utilizing 2.5-dimensional particle-in-cell simulations. We assume that the velocity distribution f u of the energized electrons follows a plateau (∂f u /∂u = 0) and a power-law distribution with spectral index α for velocities below and above U b , respectively. In the pitch-angle space, these energized electrons are spread around a center μ c = 0.5. We found that the energetic plateau-power-law electrons can more efficiently generate coherent waves if the anisotropy of their pitch-angle distribution is sufficiently strong, i.e., a small pitch-angle spread μ s . The break velocity U b affects the excitation dominance between the electrostatic and electromagnetic waves: for larger U b electrostatic waves are mainly excited, while intermediate values of U b are required for an excitation dominated by electromagnetic waves. The spectral index α controls the growth rate, efficiency, saturation, and anisotropy of the excited electromagnetic waves as well as the energy partition in different wave modes. These excited electromagnetic waves are predominantly right-handed polarized, in X- and Z-modes, as observed, e.g., in solar radio spikes. Additionally about 90% of the kinetic energy loss of the energetic electrons is dissipated, heating the ambient thermal electrons. This may contribute to the coronal heating.

Description of Atypical Bursts Seen Slightly Off-axis

Abstract The detection of gravitational waves together with their electromagnetic counterpart, in the gamma-ray burst GRB 170817A, marked a new era of multi-messenger astronomy. Several theoretical models have been proposed to explain the atypical behavior of this event. Recently, it was shown th0at the multiwavelength afterglow of GRB 170817A was consistent with a synchrotron forward-shock model when the outflow was viewed off-axis, decelerated in a uniform medium and parameterized through a power-law velocity distribution. Motivated by the upper limits on the very high-energy emission, and the stratified medium in the close vicinity of a binary neutron star merger proposed to explain the gamma-ray flux in the short GRB 150101B, we extend the mechanism proposed to explain GRB 170817A to a more general scenario deriving the synchrotron self-Compton and synchrotron forward-shock model when the off-axis outflow is decelerated in a uniform and stratified circumburst density. As particular cases, we show that the delayed and long-lasting afterglow emission observed in GRB 080503, GRB 140903A, GRB 150101B, and GRB 160821B could be interpreted by a scenario similar to the one used to describe GRB 170817A. In addition, we show that the proposed scenario agrees with the Major Atmospheric Gamma-ray Imaging Cherenkov telescope, Fermi-Large Area Telescope, and High Energy Stereoscopic System upper limits on gamma-ray emission from GRB 160821B and GRB 170817A.

Light Curves of a Shock-breakout Material and a Relativistic Off-axis Jet from a Binary Neutron Star System

Abstract Binary neutron star mergers are believed to eject significant masses with a diverse range of velocities. Once these ejected materials begin to be decelerated by a homogeneous medium, relativistic electrons are mainly cooled down by synchrotron radiation, generating a multiwavelength long-lived afterglow. Analytic and numerical methods illustrate that the outermost matter, the merger shock-breakout material, can be parametrized by power-law velocity distributions . Considering that the shock-breakout material is moving on-axis toward the observer and the relativistic jet off-axis, we compute the light curves during the relativistic and the lateral expansion phase. As a particular case, we successfully describe the X-ray, optical, and radio light curves alongside the spectral energy distribution from the recently discovered gravitational-wave transient GW170817, when the merger shock-breakout material moves with mildly relativistic velocities and achieves the near-Newtonian phase and the jet moves with relativistic velocities. Future electromagnetic counterpart observations of this binary system could be able to evaluate different properties of these light curves.

The Short GRB 170817A: Modeling the Off-axis Emission and Implications on the Ejecta Magnetization

Abstract The short GRB 170817A, detected by the Fermi Gamma-ray Burst Monitor, orbiting satellites and ground-based telescopes, was the electromagnetic counterpart of a gravitational-wave transient (GW170817) from a binary neutron star merger. After this merger, the γ-ray light curve exhibited a faint peak at ∼1.7 s and the X-ray, optical, and radio light curves displayed extended emission that increased in brightness up to ∼160 days. In this paper, we show that the X-ray, optical and radio fluxes are consistent with the synchrotron forward-shock model viewed off-axis when the matter in the outflow is parameterized through a power-law velocity distribution. We discuss the origin of the γ-ray peak in terms of internal and external shocks. We show that the γ-ray flux might be consistent with a synchrotron self-Compton reverse-shock model observed at high latitudes. Comparing the best-fit values obtained after describing the γ-ray, X-ray, optical, and radio fluxes with our model, we find that the afterglow and γ-ray emission occurred in different regions and also find evidence to propose that the progenitor environment was entrained with magnetic fields; therefore we argue for the presence of magnetic field amplification in the binary neutron star merger.

Entropy of plasmas described with regularized κ distributions

In classical thermodynamics the entropy is an extensive quantity, i.e.\ the sum of the entropies of two subsystems in equilibrium with each other is equal to the entropy of the full system consisting of the two subsystems. The extensitivity of entropy has been questioned in the context of a theoretical foundation for the so-called $\kappa$-distributions, which describe plasma constituents with power-law velocity distributions. We demonstrate here, by employing the recently introduced {\it regularized $\kappa$-distributions}, that entropy can be defined as an extensive quantity even for such power-law-like distributions that truncate exponentially.

Constraints on the ejecta of the GW170817 neutron star merger from its electromagnetic emission

We present a simple analytic model, that captures the key features of the emission of radiation from material ejected by the merger of neutron stars (NS), and construct the multi-band and bolometric luminosity light curves of the transient associated with GW170817, AT\,2017gfo, using all available data. The UV to IR emission is shown to be consistent with a single $\approx0.05$\,M$_\odot$ component ejecta, with a power-law velocity distribution between $\approx 0.1\,c$ and $>0.3\,c$, a low opacity, {$\kappa<1$\,cm$^2$\,g$^{-1}$}, and a radioactive energy release rate consistent with an initial $Y_{\rm e}<0.4$. The late time spectra require an opacity of $\kappa_\nu\approx0.1$\,cm$^2$\,g$^{-1}$ at 1 to $2\mu$m. If this opacity is provided entirely by Lanthanides, their implied mass fraction is $X_{\rm Ln}\approx10^{-3}$, approximately 30 times below the value required to account for the solar abundance. The inferred value of $X_{\rm Ln}$ is uncertain due to uncertainties in the estimates of IR opacities of heavy elements, which also do not allow the exclusion of a significant contribution to the opacity by other elements (the existence of a slower ejecta rich in Lanthanides, that does not contribute significantly to the luminosity, can also not be ruled out). The existence of a relatively massive, $\approx 0.05$\,M$_\odot$, ejecta with high velocity and low opacity is in tension with the results of numerical simulations of NS mergers.

Weakly turbulent plasma processes in the presence of inverse power-law velocity tail population

Observations show that plasma particles in the solar wind frequently display power-law velocity distributions, which can be isotropic or anisotropic. Particularly, the velocity distribution functions of solar wind electrons are frequently modeled as a combination of a background Maxwellian distribution and a non-thermal distribution which is known as the “halo” distribution. For fast solar wind conditions, highly anisotropic field-aligned electrons, denominated as the “strahl” distribution, are also present. Motivated by these observations, the present paper considers a tenuous plasma with Maxwellian ions, and electrons described by a summation of an isotropic Maxwellian distribution and an isotropic Kappa distribution. The formalism of weak turbulence theory is utilized in order to discuss the spectra of electrostatic waves that must be present in such a plasma, satisfying the conditions of quasi-equilibrium between the processes of spontaneous fluctuations and of induced emission. The kappa index and relative density of the Kappa electron distribution are varied. By taking into account the effects due to electromagnetic waves into the weak turbulence formalism, we investigate the electromagnetic spectra that satisfy the conditions of “turbulent equilibrium,” and also the time evolution of the wave spectra and of the electron distribution, which occurs in the case of the presence of an electron beam in the electron distribution.

Nonlinear Radiative Effects on MHD Flow Past a Nonlinearly Stretching Surface Embedded in a Porous Medium

An analysis has been carried out to investigate the steady, laminar, two dimensional hydromagnetic flows with heat transfer of an incompressible, viscous and electrically conducting fluid over a surface stretching with a power-law velocity distribution and embedded in a porous medium in the presence of a variable magnetic field. The radiative heat flux term is taken to be nonlinear using Rosseland diffusion approximation. Governing nonlinear partial differential equations are transformed to nonlinear ordinary differential equations by utilizing suitable similarity transformation. Then the resulting nonlinear ordinary differential equations are solved numerically using Fourth-Order Runge-Kutta based shooting method along with Nachtsheim-Swigert iteration scheme for satisfaction of asymptotic boundary conditions and the numerical results for velocity and temperature distribution are obtained for different values of radiation parameter, permeability, velocity exponent parameter, surface temperature parameter, magnetic interaction parameter and Prandtl number. The dimensionless rates of heat transfer and skin friction coefficient are also obtained for different physical parameters and are presented graphically.

Bio-inspired computational heuristics to study the boundary layer flow of the Falkner-Scan system with mass transfer and wall stretching

In the present study, a bio-inspired computational intelligence technique is developed for finding the solutions of the celebrated Falkner-Skan equation arising in fluid mechanics problems using feed-forward Artificial Neural Networks (ANNs), Genetic Algorithms (GA), the Active-Set (AS) method and their combination namely a GA-AS approach. The differential equations based ANNs modeling of the Falkner-Skan system is constructed by defining an unsupervised error function. The training of the design parameters of ANNs is carried out with the help of viable global search through GAs and fine tuning of the results is achieved with an efficient local search using the AS method. The proposed scheme is applied to a number of scenarios for the Falkner-Skan system based on boundary layer flow over a moving wall with mass transfer in the presence of a free stream with power-law velocity distributions. The dynamics of the system are investigated for different cases of mass transfer and wall stretching. The proposed results are compared with analytical and numerical solutions to verify the correctness of the approach. The accuracy and convergence of the proposed solver are validated through sufficient large numbers of independent runs in terms of different performance indices based on mean absolute error, Thail’s inequality coefficient and Nash-Suitcliff efficiency. These solutions greatly enrich possible approaches for stochastic numerical solution of the celebrated Falkner-Skan system.

The doubloon models of dark haloes and galaxies

A family of spherical halo models with flat circular velocity curves is presented. This includes models in which the rotation curve has a finite central value but declines outwards (like the Jaffe model). It includes models in which the rotation curve is rising in the inner parts, but flattens asymptotically (like the Binney model). The family encompasses models with both finite and singular (cuspy) density profiles. The self-consistent distribution function depending on binding energy $E$ and angular momentum $L$ is derived and the kinematical properties of the models discussed. These really describe the properties of the total matter (both luminous and dark). For comparison with observations, it is better to consider tracer populations of stars. These can be used to represent elliptical galaxies or the spheroidal components of spiral galaxies. Accordingly, we study the properties of tracers with power-law or Einasto profiles moving in the doubloon potential. Under the assumption of spherical alignment, we provide a simple way to solve the Jeans equations for the velocity dispersions. This choice of alignment is supported by observations on the stellar halo of the Milky Way. Power-law tracers have prolate spheroidal velocity ellipsoids everywhere. However, this is not the case for Einasto tracers, for which the velocity ellipsoids change from prolate to oblate spheroidal near the pole. Asymptotic forms of the velocity distributions close to the escape speed are also derived, with an eye to application to the high velocity stars in the Milky Way. Power-law tracers have power-law or Maxwellian velocity distributions tails, whereas Einasto tracers have super-exponential cut-offs.

Fundamental Emission of Type III Bursts Produced in Non-Maxwellian Coronal Plasmas with Kappa-Distributed Background Particles

Detailed simulations based on quasi-linear theory are presented for fundamental ( $f_{\rm p}$ ) emission of type III bursts produced in non-Maxwellian, suprathermal, background coronal plasma by injection of energetic electrons during flares with a power-law or Maxwellian velocity distribution, where $f_{\rm p}$ is the electron plasma frequency. The background plasma is assumed to have a kappa (κ) distribution, as inferred from solar wind data and proposed by theories for the corona and solar wind. The predicted type III beam speeds, Langmuir wave levels, and the drift rate and flux of $f_{\rm p}$ emission are strongly sensitive to the presence of suprathermal background electrons in the corona. The simulations show the following results. i) Fast beams with speeds $v_{\rm b}>0.5c$ are produced for coronal background electrons with small κ (κ≲5) by injected electrons with power-law spectra. ii) Moderately fast beams with $v_{\rm b} \approx0.3\,\mbox{--}\,0.5c$ are generated in coronal plasma with κ≲8 by injections of power-law or Maxwellian electrons. iii) Slow beams with $v_{\rm b}<0.3c$ are produced for coronal background electrons with large κ (κ>8), including the asymptotic limit κ→∞ where the electrons are Maxwellian, for both power-law and Maxwellian injections. The observation of fast type III beams (with $v_{\rm b}>0.5c$ ) thus suggests that these beams are produced in coronal regions where the background electron distribution has small κ by injected electrons with power-law spectra, at least when such beams are observed. The simulations, from the viewpoint of type III bursts, thus support: i) the presence, at least sometimes, of suprathermal background electrons in the corona and the associated mechanisms for coronal heating and solar wind acceleration; ii) power-law spectra for injected energetic electrons, consistent with observations of such electrons in situ and of X-ray emission.

Derivation of 2D Power-Law Velocity Distribution Using Entropy Theory

The one-dimensional (1D) power law velocity distribution, commonly used for computing velocities in open channel flow, has been derived empirically. However, a multitude of problems, such as scour around bridge piers, cutoffs and diversions, pollutant dispersion, and so on, require the velocity distribution in two dimensions. This paper employs the Shannon entropy theory for deriving the power law velocity distribution in two-dimensions (2D). The development encompasses the rectangular domain, but can be extended to any arbitrary domain, including a trapezoidal domain. The derived methodology requires only a few parameters and the good agreement is confirmed by comparing the velocity values calculated using the proposed methodology with values derived from both the 1D power law model and a logarithmic velocity distribution available in the literature.

Dual solutions for flow and radiative heat transfer of a micropolar fluid over stretching/shrinking sheet

A boundary layer analysis is presented for the flow and radiative heat transfer of an incompressible micropolar fluid over stretching/shrinking sheet with power-law surface velocity and temperature distributions. Dual solutions are analytically obtained firstly by homotopy analysis method (HAM). It is found that dual solutions not only exist for the shrinking flow as reported in the previous literatures, but also exist for the stretching flow. The special case of the first branch (K=0, classical Newtonian fluid) is compared with the existing numerical results of stretching flow in good agreement. Our results show that both solutions are physically meaningful (two solutions are closely related to each other), unlike the results previously reported that only one solution is acceptable. Moreover, the effects of the material parameter K, the radiative Prandtl number Prn, the velocity exponent parameter m and the temperature exponent parameter λ on the flow and heat transfer characteristics are analyzed in detail.

Derivation of Power Law and Logarithmic Velocity Distributions Using the Shannon Entropy

Employing the Shannon entropy, this note derives the well-known power law and the Prandt–von Karman universal (or logarithmic) velocity distribution equations for open channel flow. The Shannon entropy yields probability distributions underlying these velocity equations. With the use of this entropy, one obtains an expression for the power law exponent in physically measurable quantities (surface flow velocity and average logarithmic velocity) and an expression for shear flow depth in flow depth, surface flow velocity and shear velocity, thus obviating the need for fitting.

Analytical solution of hydromagnetic flow with Hall effect over a surface stretching with a power-law velocity

The nonlinear magnetohydrodynamic (MHD) flow problem with Hall current caused by stretching surface having power law velocity distribution is solved by employing homotopy analysis method (HAM). Perturbation solution of stream function, the expression of skin friction coefficient and graphical results in absence of Hall current (Chiam, Int J Eng Sci 33 (1995), 429) are recovered as the limiting cases. It is found that unlike the solution obtained by Chiam (1995), the present results are valid for weak and large magnetic parameters. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 937–959, 2011

An exact analytical solution of the Falkner-Skan equation with mass transfer and wall stretching

In this paper, an exact analytical solution of the famous Falkner-Skan equation is obtained. The solution involves the boundary layer flow over a moving wall with mass transfer in presence of a free stream with a power-law velocity distribution. Multiple solution branches are observed. The effects of mass transfer and wall stretching are analyzed. Interesting velocity profiles including velocity overshoot and reversal flows are observed in the presence of both mass transfer and wall stretching. These solutions greatly enrich the analytical solution for the celebrated Falkner-Skan equation and the understanding of this important and interesting equation.

Power Law Velocity Profile in the Turbulent Boundary Layer on Transitional Rough Surfaces

A new approach to scaling of transitional wall roughness in turbulent flow is introduced by a new nondimensional roughness scale ϕ. This scale gives rise to an inner viscous length scale ϕν∕uτ, inner wall transitional variable, roughness friction Reynolds number, and roughness Reynolds number. The velocity distribution, just above the roughness level, turns out to be a universal relationship for all kinds of roughness (transitional, fully smooth, and fully rough surfaces), but depends implicitly on roughness scale. The open turbulent boundary layer equations, without any closure model, have been analyzed in the inner wall and outer wake layers, and matching by the Izakson-Millikan-Kolmogorov hypothesis leads to an open functional equation. An alternate open functional equation is obtained from the ratio of two successive derivatives of the basic functional equation of Izakson and Millikan, which admits two functional solutions: the power law velocity profile and the log law velocity profile. The envelope of the skin friction power law gives the log law, as well as the power law index and prefactor as the functions of roughness friction Reynolds number or skin friction coefficient as appropriate. All the results for power law and log law velocity and skin friction distributions, as well as power law constants are explicitly independent of the transitional wall roughness. The universality of these relations is supported very well by extensive experimental data from transitional rough walls for various different types of roughnesses. On the other hand, there are no universal scalings in traditional variables, and different expressions are needed for various types of roughness, such as inflectional roughness, monotonic roughness, and others. To the lowest order, the outer layer flow is governed by the nonlinear turbulent wake equations that match with the power law theory as well as log law theory, in the overlap region. These outer equations are in equilibrium for constant value of m, the pressure gradient parameter, and under constant eddy viscosity closure model, the analytical and numerical solutions are presented.